https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Apple321&feedformat=atomAoPS Wiki - User contributions [en]2021-01-16T22:00:31ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2020_AIME_I_Problems/Problem_13&diff=1238892020 AIME I Problems/Problem 132020-06-05T18:09:01Z<p>Apple321: /* THIS QUESTION IS VERY BAD */</p>
<hr />
<div><br />
== Problem ==<br />
Point <math>D</math> lies on side <math>\overline{BC}</math> of <math>\triangle ABC</math> so that <math>\overline{AD}</math> bisects <math>\angle BAC.</math> The perpendicular bisector of <math>\overline{AD}</math> intersects the bisectors of <math>\angle ABC</math> and <math>\angle ACB</math> in points <math>E</math> and <math>F,</math> respectively. Given that <math>AB=4,BC=5,</math> and <math>CA=6,</math> the area of <math>\triangle AEF</math> can be written as <math>\tfrac{m\sqrt{n}}p,</math> where <math>m</math> and <math>p</math> are relatively prime positive integers, and <math>n</math> is a positive integer not divisible by the square of any prime. Find <math>m+n+p.</math><br />
<br />
<br />
<asy><br />
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */<br />
import graph; size(18cm); <br />
real labelscalefactor = 0.5; /* changes label-to-point distance */<br />
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ <br />
pen dotstyle = black; /* point style */ <br />
real xmin = -10.645016481888238, xmax = 5.4445786933235505, ymin = 0.7766255516825293, ymax = 9.897545413994122; /* image dimensions */<br />
pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); <br />
<br />
draw((-6.837129089839387,8.163360372429347)--(-6.8268938290378,5.895596632024835)--(-4.33118398380513,6.851781504978754)--cycle, linewidth(2) + rvwvcq); <br />
draw((-6.837129089839387,8.163360372429347)--(-8.31920210577661,4.188003838050227)--(-3.319253031309944,4.210570466954303)--cycle, linewidth(2) + rvwvcq); <br />
/* draw figures */<br />
draw((-6.837129089839387,8.163360372429347)--(-7.3192122908832715,4.192517163831042), linewidth(2) + wrwrwr); <br />
draw((-7.3192122908832715,4.192517163831042)--(-2.319263216416622,4.2150837927351175), linewidth(2) + wrwrwr); <br />
draw((-2.319263216416622,4.2150837927351175)--(-6.837129089839387,8.163360372429347), linewidth(2) + wrwrwr); <br />
draw((xmin, -2.6100704119306224*xmin-9.68202796751058)--(xmax, -2.6100704119306224*xmax-9.68202796751058), linewidth(2) + wrwrwr); /* line */<br />
draw((xmin, 0.3831314264278095*xmin + 8.511194202815297)--(xmax, 0.3831314264278095*xmax + 8.511194202815297), linewidth(2) + wrwrwr); /* line */<br />
draw(circle((-6.8268938290378,5.895596632024835), 2.267786838055365), linewidth(2) + wrwrwr); <br />
draw(circle((-4.33118398380513,6.851781504978754), 2.828427124746193), linewidth(2) + wrwrwr); <br />
draw((xmin, 0.004513371749987873*xmin + 4.225551489816879)--(xmax, 0.004513371749987873*xmax + 4.225551489816879), linewidth(2) + wrwrwr); /* line */<br />
draw((-7.3192122908832715,4.192517163831042)--(-4.33118398380513,6.851781504978754), linewidth(2) + wrwrwr); <br />
draw((-6.8268938290378,5.895596632024835)--(-2.319263216416622,4.2150837927351175), linewidth(2) + wrwrwr); <br />
draw((-6.837129089839387,8.163360372429347)--(-8.31920210577661,4.188003838050227), linewidth(2) + wrwrwr); <br />
draw((xmin, 0.004513371749987873*xmin + 8.19421887771445)--(xmax, 0.004513371749987873*xmax + 8.19421887771445), linewidth(2) + wrwrwr); /* line */<br />
draw((-3.837159645159393,8.176900349771794)--(-8.31920210577661,4.188003838050227), linewidth(2) + wrwrwr); <br />
draw((-3.837159645159393,8.176900349771794)--(-5.3192326610966125,4.2015438153926725), linewidth(2) + wrwrwr); <br />
draw((-6.837129089839387,8.163360372429347)--(-6.8268938290378,5.895596632024835), linewidth(2) + rvwvcq); <br />
draw((-6.8268938290378,5.895596632024835)--(-4.33118398380513,6.851781504978754), linewidth(2) + rvwvcq); <br />
draw((-4.33118398380513,6.851781504978754)--(-6.837129089839387,8.163360372429347), linewidth(2) + rvwvcq); <br />
draw((-6.837129089839387,8.163360372429347)--(-8.31920210577661,4.188003838050227), linewidth(2) + rvwvcq); <br />
draw((-8.31920210577661,4.188003838050227)--(-3.319253031309944,4.210570466954303), linewidth(2) + rvwvcq); <br />
draw((-3.319253031309944,4.210570466954303)--(-6.837129089839387,8.163360372429347), linewidth(2) + rvwvcq); <br />
/* dots and labels */<br />
dot((-6.837129089839387,8.163360372429347),dotstyle); <br />
label("$A$", (-6.8002301023571095,8.267690318323321), NE * labelscalefactor); <br />
dot((-7.3192122908832715,4.192517163831042),dotstyle); <br />
label("$B$", (-7.2808283997985,4.29753046989445), NE * labelscalefactor); <br />
dot((-2.319263216416622,4.2150837927351175),linewidth(4pt) + dotstyle); <br />
label("$C$", (-2.276337432963145,4.29753046989445), NE * labelscalefactor); <br />
dot((-5.3192326610966125,4.2015438153926725),linewidth(4pt) + dotstyle); <br />
label("$D$", (-5.274852897434433,4.287082680819637), NE * labelscalefactor); <br />
dot((-6.8268938290378,5.895596632024835),linewidth(4pt) + dotstyle); <br />
label("$F$", (-6.789782313282296,5.979624510939313), NE * labelscalefactor); <br />
dot((-4.33118398380513,6.851781504978754),linewidth(4pt) + dotstyle); <br />
label("$E$", (-4.292760724402025,6.93037331674728), NE * labelscalefactor); <br />
dot((-8.31920210577661,4.188003838050227),linewidth(4pt) + dotstyle); <br />
label("$G$", (-8.273368361905721,4.276634891744824), NE * labelscalefactor); <br />
dot((-3.319253031309944,4.210570466954303),linewidth(4pt) + dotstyle); <br />
label("$H$", (-3.2793251841451787,4.29753046989445), NE * labelscalefactor); <br />
dot((-3.837159645159393,8.176900349771794),linewidth(4pt) + dotstyle); <br />
label("$I$", (-3.7912668488110084,8.257242529248508), NE * labelscalefactor); <br />
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); <br />
/* end of picture */<br />
</asy><br />
<br />
== Solution 1 ==<br />
<br />
Points are defined as shown. It is pretty easy to show that <math>\triangle AFE \sim \triangle AGH</math> by spiral similarity at <math>A</math> by some short angle chasing. Now, note that <math>AD</math> is the altitude of <math>\triangle AFE</math>, as the altitude of <math>AGH</math>. We need to compare these altitudes in order to compare their areas. Note that Stewart's theorem implies that <math>AD/2 = \frac{\sqrt{18}}{2}</math>, the altitude of <math>\triangle AFE</math>. Similarly, the altitude of <math>\triangle AGH</math> is the altitude of <math>\triangle ABC</math>, or <math>\frac{12}{\sqrt{7}}</math>. However, it's not too hard to see that <math>GB = HC = 1</math>, and therefore <math>[AGH] = [ABC]</math>. From here, we get that the area of <math>\triangle ABC</math> is <math>\frac{15\sqrt{7}}{14} \implies \boxed{036}</math>, by similarity. ~awang11<br />
<br />
==Solution 2(coord bash + basic geometry)==<br />
Let <math>\overline{BC}</math> lie on the x-axis and <math>B</math> be the origin. <math>C</math> is <math>(5,0)</math>. Use Heron's formula to compute the area of triangle <math>ABC</math>. We have <math>s=\frac{15}{2}</math>. and <math>[ABC]=\sqrt{\frac{15 \cdot 7 \cdot 5 \cdot 3}{2^4}}=\frac{15\sqrt{7}}{4}</math>. We now find the altitude, which is <math>\frac{\frac{15\sqrt{7}}{2}}{5}=\frac{3\sqrt{7}}{2}</math>, which is the y-coordinate of <math>A</math>. We now find the x-coordinate of <math>A</math>, which satisfies <math>x^2 + (\frac{3\sqrt{7}}{2})^{2}=16</math>, which gives <math>x=\frac{1}{2}</math> since the triangle is acute. Now using the Angle Bisector Theorem, we have <math>\frac{4}{6}=\frac{BD}{CD}</math> and <math>BD+CD=5</math> to get <math>BD=2</math>. The coordinates of D are <math>(2,0)</math>.<br />
Since we want the area of triangle <math>AEF</math>, we will find equations for perpendicular bisector of AD, and the other two angle bisectors. The perpendicular bisector is not too challenging: the midpoint of AD is <math>(\frac{5}{4}, \frac{3\sqrt{7}}{4})</math> and the slope of AD is <math>-\sqrt{7}</math>. The slope of the perpendicular bisector is <math>\frac{1}{\sqrt{7}}</math>. The equation is(in point slope form) <math>y-\frac{3\sqrt{7}}{4}=\frac{1}{\sqrt{7}}(x-\frac{5}{4})</math>.<br />
The slope of AB, or in trig words, the tangent of <math>\angle ABC</math> is <math>3\sqrt{7}</math>.<br />
Finding <math>\sin{\angle ABC}=\frac{\frac{3\sqrt{7}}{2}}{4}=\frac{3\sqrt{7}}{8}</math> and <math>\cos{\angle ABC}=\frac{\frac{1}{2}}{4}=\frac{1}{8}</math>. Plugging this in to half angle tangent, it gives <math>\frac{\frac{3\sqrt{7}}{8}}{1+\frac{1}{8}}=\frac{\sqrt{7}}{3}</math> as the slope of the angle bisector, since it passes through <math>B</math>, the equation is <math>y=\frac{\sqrt{7}}{3}x</math>.<br />
Similarly, the equation for the angle bisector of <math>C</math> will be <math>y=-\frac{1}{\sqrt{7}}(x-5)</math>.<br />
For <math>E</math> use the B-angle bisector and the perpendicular bisector of AD equations to intersect at <math>(3,\sqrt{7})</math>.<br />
For <math>F</math> use the C-angle bisector and the perpendicular bisector of AD equations to intersect at <math>(\frac{1}{2}, \frac{9}{2\sqrt{7}})</math>.<br />
The area of AEF is equal to <math>\frac{EF \cdot \frac{AD}{2}}{2}</math> since AD is the altitude of that triangle with EF as the base, with <math>\frac{AD}{2}</math> being the height. <math>EF=\frac{5\sqrt{2}}{\sqrt{7}}</math> and <math>AD=3\sqrt{2}</math>, so <math>[AEF]=\frac{15}{2\sqrt{7}}=\frac{15\sqrt{7}}{14}</math> which gives <math>\boxed{036}</math>. NEVER overlook coordinate bash in combination with beginner synthetic techniques.~vvluo<br />
<br />
==Solution 3 (Coordinate Bash + Trig)==<br />
<br />
<asy><br />
size(8cm); defaultpen(fontsize(10pt));<br />
<br />
pair A,B,C,I,D,M,T,Y,Z,EE,F;<br />
A=(0,3sqrt(7));<br />
B=(-1,0);<br />
C=(9,0);<br />
I=incenter(A,B,C);<br />
D=extension(A,I,B,C);<br />
M=(A+D)/2;<br />
<br />
draw(B--EE,gray+dashed);<br />
draw(C--F,gray+dashed);<br />
draw(A--B--C--A);<br />
draw(A--D);<br />
draw(B--(5,sqrt(28)));<br />
draw(M--(5,sqrt(28)));<br />
draw(C--(0,9sqrt(7)/7));<br />
draw(M--(0,9sqrt(7)/7));<br />
dot("$A$",A,NW);<br />
dot("$B$",B,SW);<br />
dot("$C$",C,SE);<br />
dot("$D$",D,S);<br />
dot("$E$",(5,sqrt(28)),N); <br />
dot("$M$",M,dir(70));<br />
dot("$F$",(0,9sqrt(7)/7),N);<br />
<br />
label("$2$",B--D,S);<br />
label("$3$",D--C,S);<br />
label("$6$",A--C,N);<br />
label("$4$",A--B,W);<br />
</asy><br />
<br />
Let <math>B=(0,0)</math> and <math>BC</math> be the line <math>y=0</math>.<br />
We compute that <math>\cos{\angle{ABC}}=\frac{1}{8}</math>, so <math>\tan{\angle{ABC}}=3\sqrt{7}</math>.<br />
Thus, <math>A</math> lies on the line <math>y=3x\sqrt{7}</math>. The length of <math>AB</math> at a point <math>x</math> is <math>8x</math>, so <math>x=\frac{1}{2}</math>.<br />
<br />
We now have the coordinates <math>A=\left(\frac{1}{2},\frac{3\sqrt{7}}{2}\right)</math>, <math>B=(0,0)</math> and <math>C=(5,0)</math>.<br />
We also have <math>D=(2,0)</math> by the angle-bisector theorem and <math>M=\left(\frac{5}{4},\frac{3\sqrt{7}}{4}\right)</math> by taking the midpoint.<br />
We have that because <math>\cos{\angle{ABC}}=\frac{1}{8}</math>, <math>\cos{\frac{\angle{ABC}}{2}}=\frac{3}{4}</math> by half angle formula.<br />
<br />
We also compute <math>\cos{\angle{ACB}}=\frac{3}{4}</math>, so <math>\cos{\frac{\angle{ACB}}{2}}=\frac{\sqrt{14}}{4}</math>.<br />
<br />
Now, <math>AD</math> has slope <math>-\frac{\frac{3\sqrt{7}}{2}}{2-\frac{1}{2}}=-\sqrt{7}</math>, so it's perpendicular bisector has slope <math>\frac{\sqrt{7}}{7}</math> and goes through <math>\left(\frac{5}{4},\frac{3\sqrt{7}}{4}\right)</math>.<br />
<br />
We find that this line has equation <math>y=\frac{\sqrt{7}}{7}x+\frac{4\sqrt{7}}{7}</math>.<br />
<br />
As <math>\cos{\angle{CBI}}=\frac{3}{4}</math>, we have that line <math>BI</math> has form <math>y=\frac{\sqrt{7}}{3}x</math>.<br />
Solving for the intersection point of these two lines, we get <math>x=3</math> and thus <math>E=\left(3, \sqrt{7}\right)</math><br />
<br />
We also have that because <math>\cos{\angle{ICB}}=\frac{\sqrt{14}}{4}</math>, <math>CI</math> has form <math>y=-\frac{x\sqrt{7}}{7}+\frac{5\sqrt{7}}{7}</math>.<br />
<br />
Intersecting the line <math>CI</math> and the perpendicular bisector of <math>AD</math> yields <math>-\frac{x\sqrt{7}}{7}+\frac{5\sqrt{7}}{7}=\frac{x\sqrt{7}}{7}+\frac{4\sqrt{7}}{7}</math>.<br />
<br />
Solving this, we get <math>x=\frac{1}{2}</math> and so <math>F=\left(\frac{1}{2},\frac{9\sqrt{7}}{14}\right)</math>.<br />
<br />
We now compute <math>EF=\sqrt{\left(\frac{5}{2}\right)^2+\left(\frac{5\sqrt{7}}{14}\right)^2}=\frac{5\sqrt{14}}{7}</math>.<br />
We also have <math>MA=\sqrt{\left(\frac{3}{4}\right)^2+\left(\frac{3\sqrt{7}}{4}\right)^2}=\frac{3\sqrt{2}}{2}</math>.<br />
<br />
As <math>{MA}\perp{EF}</math>, we have <math>[\triangle{AEF}]=\frac{1}{2}\left(\frac{3\sqrt{2}}{2}\times\frac{5\sqrt{14}}{7}\right)=\frac{15\sqrt{7}}{14}</math>.<br />
<br />
<br />
The desired answer is <math>15+7+14=\boxed{036}</math> ~Imayormaynotknowcalculus<br />
<br />
==Solution 4 (Barycentric Coordinates)==<br />
<br />
<asy><br />
size(8cm); defaultpen(fontsize(10pt));<br />
<br />
pair A,B,C,I,D,M,T,Y,Z,EE,F;<br />
A=(0,3sqrt(7));<br />
B=(-1,0);<br />
C=(9,0);<br />
I=incenter(A,B,C);<br />
D=extension(A,I,B,C);<br />
M=(A+D)/2;<br />
<br />
draw(B--EE,gray+dashed);<br />
draw(C--F,gray+dashed);<br />
draw(A--B--C--A);<br />
draw(A--D);<br />
draw(B--(5,sqrt(28)));<br />
draw(M--(5,sqrt(28)));<br />
draw(C--(0,9sqrt(7)/7));<br />
draw(M--(0,9sqrt(7)/7));<br />
dot("$A$",A,NW);<br />
dot("$B$",B,SW);<br />
dot("$C$",C,SE);<br />
dot("$D$",D,S);<br />
dot("$E$",(5,sqrt(28)),N); <br />
dot("$M$",M,dir(70));<br />
dot("$F$",(0,9sqrt(7)/7),N);<br />
<br />
label("$2$",B--D,S);<br />
label("$3$",D--C,S);<br />
label("$6$",A--C,N);<br />
label("$4$",A--B,W);<br />
</asy><br />
<br />
As usual, we will use homogenized barycentric coordinates.<br />
<br />
We have that <math>AD</math> will have form <math>3z=2y</math>. Similarly, <math>CF</math> has form <math>5y=6x</math> and <math>BE</math> has form <math>5z=4x</math>.<br />
Since <math>A=(1,0,0)</math> and <math>D=\left(0,\frac{3}{5},\frac{2}{5}\right)</math>, we also have <math>M=\left(\frac{1}{2},\frac{3}{10},\frac{1}{5}\right)</math>.<br />
It remains to determine the equation of the line formed by the perpendicular bisector of <math>AD</math>.<br />
<br />
This can be found using EFFT. Let a point <math>T</math> on <math>EF</math> have coordinates <math>(x, y, z)</math>.<br />
We then have that the displacement vector <math>\overrightarrow{AD}=\left(-1, \frac{3}{5}, \frac{2}{5}\right)</math> and that the displacement vector <math>\overrightarrow{TM}</math> has form <math>\left(x-\frac{1}{2},y-\frac{3}{10},z-\frac{1}{5}\right)</math>.<br />
Now, by EFFT, we have <math>5^2\left(\frac{3}{5}\times\left(z-\frac{1}{5}\right)+\frac{2}{5}\times\left(y-\frac{3}{10}\right)\right)+6^2\left(-1\times\left(z-\frac{1}{5}\right)+\frac{2}{5}\times\left(x-\frac{1}{2}\right)\right)+4^2\left(-1\times\left(y-\frac{3}{10}\right)+\frac{3}{5}\times\left(x-\frac{1}{2}\right)\right)=0</math>.<br />
This equates to <math>8x-2y-7z=2</math>.<br />
<br />
Now, intersecting this with <math>BE</math>, we have <math>5z=4x</math>, <math>8x-2y-7z=2</math>, and <math>x+y+z=1</math>.<br />
This yields <math>x=\frac{2}{3}</math>, <math>y=-\frac{1}{5}</math>, and <math>z=\frac{8}{15}</math>, or <math>E=\left(\frac{2}{3},-\frac{1}{5},\frac{8}{15}\right)</math>.<br />
<br />
Similarly, intersecting this with <math>CF</math>, we have <math>5y=6x</math>, <math>8x-2y-7z=2</math>, and <math>x+y+z=1</math>.<br />
Solving this, we obtain <math>x=\frac{3}{7}</math>, <math>y=\frac{18}{35}</math>, and <math>z=\frac{2}{35}</math>, or <math>F=\left(\frac{3}{7},\frac{18}{35},\frac{2}{35}\right)</math>.<br />
<br />
We finish by invoking the Barycentric Distance Formula twice; our first displacement vector being <math>\overrightarrow{FE}=\left(\frac{5}{21},-\frac{5}{7},\frac{10}{21}\right)</math>.<br />
We then have <math>FE^2=-25\left(-\frac{5}{7}\cdot\frac{10}{21}\right)-36\left(\frac{5}{21}\cdot\frac{10}{21}\right)-16\left(\frac{5}{21}\cdot-\frac{5}{7}\right)=\frac{50}{7}</math>, thus <math>FE=\frac{5\sqrt{14}}{7}</math>.<br />
<br />
Our second displacement vector is <math>\overrightarrow{AM}=\left(-\frac{1}{2},\frac{3}{10},\frac{1}{5}\right)</math>.<br />
As a result, <math>AM^2=-25\left(\frac{3}{10}\cdot\frac{1}{5}\right)-36\left(-\frac{1}{2}\cdot\frac{1}{5}\right)-16\left(-\frac{1}{2}\cdot\frac{3}{10}\right)=\frac{9}{2}</math>, so <math>AM=\frac{3\sqrt{2}}{2}</math>.<br />
<br />
As <math>{AM}\perp{EF}</math>, the desired area is <math>\frac{\frac{5\sqrt{14}}{7}\times\frac{3\sqrt{2}}{2}}{2}={\frac{15\sqrt{7}}{14}}\implies{m+n+p=\boxed{036}}</math>. ~Imayormaynotknowcalculus<br />
<br />
<br />
<b>Remark</b>: The area of <math>\triangle{AEF}</math> can also be computed using the <i>Barycentric Area Formula</i>, although it may increase the risk of computational errors; there are also many different ways to proceed once the coordinates are determined.<br />
<br />
==Solution 5 (geometry+trig)==<br />
<br />
<asy><br />
size(8cm); defaultpen(fontsize(10pt));<br />
<br />
pair A,B,C,I,D,M,T,Y,Z,EE,F;<br />
A=(0,3sqrt(7));<br />
B=(-1,0);<br />
C=(9,0);<br />
I=incenter(A,B,C);<br />
D=extension(A,I,B,C);<br />
M=(A+D)/2;<br />
<br />
draw(B--EE,gray+dashed);<br />
draw(C--F,gray+dashed);<br />
draw(A--B--C--A);<br />
draw(A--D);<br />
draw(A--(5,sqrt(28)));<br />
draw(A--(0,9sqrt(7)/7));<br />
draw(D--(0,9sqrt(7)/7));<br />
draw(D--(5,sqrt(28)));<br />
draw(B--(5,sqrt(28)));<br />
draw(M--(5,sqrt(28)));<br />
draw(C--(0,9sqrt(7)/7));<br />
draw(M--(0,9sqrt(7)/7));<br />
dot("$A$",A,NW);<br />
dot("$B$",B,SW);<br />
dot("$C$",C,SE);<br />
dot("$D$",D,S);<br />
dot("$E$",(5,sqrt(28)),N); <br />
dot("$M$",M,dir(70));<br />
dot("$F$",(0,9sqrt(7)/7),N);<br />
<br />
label("$2$",B--D,S);<br />
label("$3$",D--C,S);<br />
label("$6$",A--C,N);<br />
label("$4$",A--B,W);<br />
</asy><br />
<br />
To get the area of <math>\triangle AEF</math>, we try to find <math>AM</math> and <math>\angle EAF</math>.<br />
<br />
Since <math>AD</math> is the angle bisector, we can get that <math>BD=2</math> and <math>CD=3</math>. By applying Stewart's Theorem, we can get that <math>AD=3\sqrt{2}</math>. Therefore <math>AM=\frac{3\sqrt{2}}{2}</math>.<br />
<br />
Since <math>EF</math> is the perpendicular bisector of <math>AD</math>, we know that <math>AE = DE</math>. Since <math>BE</math> is the angle bisector of <math>\angle BAC</math>,<br />
we know that <math>\angle ABE = \angle DBE</math>. By applying the Law of Sines to <math>\triangle ABE</math> and <math>\triangle DBE</math>, we know that<br />
<math>\sin \angle BAE = \sin \angle BDE </math>. Since <math>BD</math> is not equal to <math>AB</math> and therefore these two triangles are not congruent, we know that <math>\angle BAE</math> and <math>\angle BDE</math> are supplementary. Then we know that <math>\angle ABD</math> and <math>\angle AED</math> are also supplementary. Given that <math>AE=DE</math>, we can get that <math>\angle DAE</math> is half of <math>\angle ABC</math>. Similarly, we have <math>\angle DAF</math> is half of <math>\angle ACB</math>.<br />
<br />
By applying the Law of Cosines, we get <math>\cos \angle ABC = \frac{1}{8}</math>, and then <math>\sin \angle ABC = \frac{3\sqrt{7}}{8}</math>. Similarly, we can get <math>\cos \angle ACB = \frac{3}{4}</math> and <math>\sin \angle ACB = \frac{\sqrt{7}}{4}</math>. Based on some trig identities, we can compute that <math>\tan \angle DAE = \frac{\sin \angle ABC}{1 + \cos \angle ABC} = \frac{\sqrt{7}}{3}</math>, and <math>\tan \angle DAF = \frac{\sqrt{7}}{7}</math>.<br />
<br />
Finally, the area of <math>\triangle AEF</math> equals <math>\frac{1}{2}AM^2(\tan \angle DAE + \tan \angle DAF)=\frac{15\sqrt{7}}{14}</math>. Therefore, the final answer is <math>15+7+14=\boxed{036}</math>. ~xamydad<br />
<br />
<b>Remark</b>: I didn't figure out how to add segments <math>AF</math>, <math>AE</math>, <math>DF</math> and <math>DE</math>. Can someone please help add these segments? <br />
<br />
(Added :) ~Math_Genius_164)<br />
<br />
== Solution 6 ==<br />
[[Image:question13.png|frame|none|###px|]]<br />
First and foremost <math>\big[\triangle{AEF}\big]=\big[\triangle{DEF}\big]</math> as <math>EF</math> is the perpendicular bisector of <math>AD</math>. Now note that quadrilateral <math>ABDF</math> is cyclic, because <math>\angle{ABF}=\angle{FBD}</math> and <math>FA=FD</math>. Similarly quadrilateral <math>AEDC</math> is cyclic, <cmath>\implies \angle{EDA}=\dfrac{C}{2}, \quad \angle{FDA}=\dfrac{B}{2}</cmath><br />
Let <math>A'</math>,<math>B'</math>, <math>C'</math> be the <math>A</math>,<math>B</math>, and <math>C</math> excenters of <math>\triangle{ABC}</math> respectively. Then it follows that <math>\triangle{DEF} \sim \triangle{A'C'B'}</math>. By angle bisector theorem we have <math>BD=2 \implies \dfrac{ID}{IA}=\dfrac{BD}{BA}=\dfrac{1}{2}</math>. Now let the feet of the perpendiculars from <math>I</math> and <math>A'</math> to <math>BC</math> be <math>X</math> and <math>Y</math> resptively. Then by tangents we have <cmath>BX=s-AC=\dfrac{3}{2} \implies XD=2-\dfrac{3}{2}=\dfrac{1}{2}</cmath> <cmath>CY=s-AC \implies YD=3-\dfrac{3}{2}=\dfrac{3}{2} \implies \dfrac{ID}{DA'}=\dfrac{XD}{YD}=\dfrac{1}{3} \implies \big[\triangle{DEF}\big]=\dfrac{1}{16}\big[\triangle{A'C'B'}\big]</cmath> From the previous ratios, <math>AI:ID:DA'=2:1:3 \implies AD=DA' \implies \big[\triangle{ABC}\big]=\big[\triangle{A'BC}\big]</math> Similarly we can find that <math>\big[\triangle{B'AC}\big]=2\big[\triangle{ABC}\big]</math> and <math>\big[\triangle{C'AB}\big]=\dfrac{4}{7}\big[\triangle{ABC}\big]</math> and thus <cmath>\big[\triangle{A'B'C'}\big]=\bigg(1+1+2+\dfrac{4}{7}\bigg)\big[\triangle{ABC}\big]=\dfrac{32}{7}\big[\triangle{ABC}\big] \implies \big[\triangle{DEF}\big]=\dfrac{2}{7}\big[\triangle{ABC}\big]=\dfrac{15\sqrt{7}}{14} \implies m+n+p = \boxed{036}</cmath><br />
-tkhalid<br />
==See Also==<br />
<br />
{{AIME box|year=2020|n=I|num-b=12|num-a=14}}<br />
{{MAA Notice}}</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=2020_AIME_I_Problems/Problem_13&diff=1238842020 AIME I Problems/Problem 132020-06-05T17:59:19Z<p>Apple321: /* Problem */</p>
<hr />
<div><br />
== THIS QUESTION IS VERY BAD ==<br />
Point <math>D</math> lies on side <math>\overline{BC}</math> of <math>\triangle ABC</math> so that <math>\overline{AD}</math> bisects <math>\angle BAC.</math> The perpendicular bisector of <math>\overline{AD}</math> intersects the bisectors of <math>\angle ABC</math> and <math>\angle ACB</math> in points <math>E</math> and <math>F,</math> respectively. Given that <math>AB=4,BC=5,</math> and <math>CA=6,</math> the area of <math>\triangle AEF</math> can be written as <math>\tfrac{m\sqrt{n}}p,</math> where <math>m</math> and <math>p</math> are relatively prime positive integers, and <math>n</math> is a positive integer not divisible by the square of any prime. Find <math>m+n+p.</math><br />
<br />
<br />
<asy><br />
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */<br />
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pen dotstyle = black; /* point style */ <br />
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<br />
draw((-6.837129089839387,8.163360372429347)--(-6.8268938290378,5.895596632024835)--(-4.33118398380513,6.851781504978754)--cycle, linewidth(2) + rvwvcq); <br />
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draw((-6.837129089839387,8.163360372429347)--(-8.31920210577661,4.188003838050227), linewidth(2) + rvwvcq); <br />
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draw((-3.319253031309944,4.210570466954303)--(-6.837129089839387,8.163360372429347), linewidth(2) + rvwvcq); <br />
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dot((-7.3192122908832715,4.192517163831042),dotstyle); <br />
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label("$F$", (-6.789782313282296,5.979624510939313), NE * labelscalefactor); <br />
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label("$E$", (-4.292760724402025,6.93037331674728), NE * labelscalefactor); <br />
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clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); <br />
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</asy><br />
<br />
== Solution 1 ==<br />
<br />
Points are defined as shown. It is pretty easy to show that <math>\triangle AFE \sim \triangle AGH</math> by spiral similarity at <math>A</math> by some short angle chasing. Now, note that <math>AD</math> is the altitude of <math>\triangle AFE</math>, as the altitude of <math>AGH</math>. We need to compare these altitudes in order to compare their areas. Note that Stewart's theorem implies that <math>AD/2 = \frac{\sqrt{18}}{2}</math>, the altitude of <math>\triangle AFE</math>. Similarly, the altitude of <math>\triangle AGH</math> is the altitude of <math>\triangle ABC</math>, or <math>\frac{12}{\sqrt{7}}</math>. However, it's not too hard to see that <math>GB = HC = 1</math>, and therefore <math>[AGH] = [ABC]</math>. From here, we get that the area of <math>\triangle ABC</math> is <math>\frac{15\sqrt{7}}{14} \implies \boxed{036}</math>, by similarity. ~awang11<br />
<br />
==Solution 2(coord bash + basic geometry)==<br />
Let <math>\overline{BC}</math> lie on the x-axis and <math>B</math> be the origin. <math>C</math> is <math>(5,0)</math>. Use Heron's formula to compute the area of triangle <math>ABC</math>. We have <math>s=\frac{15}{2}</math>. and <math>[ABC]=\sqrt{\frac{15 \cdot 7 \cdot 5 \cdot 3}{2^4}}=\frac{15\sqrt{7}}{4}</math>. We now find the altitude, which is <math>\frac{\frac{15\sqrt{7}}{2}}{5}=\frac{3\sqrt{7}}{2}</math>, which is the y-coordinate of <math>A</math>. We now find the x-coordinate of <math>A</math>, which satisfies <math>x^2 + (\frac{3\sqrt{7}}{2})^{2}=16</math>, which gives <math>x=\frac{1}{2}</math> since the triangle is acute. Now using the Angle Bisector Theorem, we have <math>\frac{4}{6}=\frac{BD}{CD}</math> and <math>BD+CD=5</math> to get <math>BD=2</math>. The coordinates of D are <math>(2,0)</math>.<br />
Since we want the area of triangle <math>AEF</math>, we will find equations for perpendicular bisector of AD, and the other two angle bisectors. The perpendicular bisector is not too challenging: the midpoint of AD is <math>(\frac{5}{4}, \frac{3\sqrt{7}}{4})</math> and the slope of AD is <math>-\sqrt{7}</math>. The slope of the perpendicular bisector is <math>\frac{1}{\sqrt{7}}</math>. The equation is(in point slope form) <math>y-\frac{3\sqrt{7}}{4}=\frac{1}{\sqrt{7}}(x-\frac{5}{4})</math>.<br />
The slope of AB, or in trig words, the tangent of <math>\angle ABC</math> is <math>3\sqrt{7}</math>.<br />
Finding <math>\sin{\angle ABC}=\frac{\frac{3\sqrt{7}}{2}}{4}=\frac{3\sqrt{7}}{8}</math> and <math>\cos{\angle ABC}=\frac{\frac{1}{2}}{4}=\frac{1}{8}</math>. Plugging this in to half angle tangent, it gives <math>\frac{\frac{3\sqrt{7}}{8}}{1+\frac{1}{8}}=\frac{\sqrt{7}}{3}</math> as the slope of the angle bisector, since it passes through <math>B</math>, the equation is <math>y=\frac{\sqrt{7}}{3}x</math>.<br />
Similarly, the equation for the angle bisector of <math>C</math> will be <math>y=-\frac{1}{\sqrt{7}}(x-5)</math>.<br />
For <math>E</math> use the B-angle bisector and the perpendicular bisector of AD equations to intersect at <math>(3,\sqrt{7})</math>.<br />
For <math>F</math> use the C-angle bisector and the perpendicular bisector of AD equations to intersect at <math>(\frac{1}{2}, \frac{9}{2\sqrt{7}})</math>.<br />
The area of AEF is equal to <math>\frac{EF \cdot \frac{AD}{2}}{2}</math> since AD is the altitude of that triangle with EF as the base, with <math>\frac{AD}{2}</math> being the height. <math>EF=\frac{5\sqrt{2}}{\sqrt{7}}</math> and <math>AD=3\sqrt{2}</math>, so <math>[AEF]=\frac{15}{2\sqrt{7}}=\frac{15\sqrt{7}}{14}</math> which gives <math>\boxed{036}</math>. NEVER overlook coordinate bash in combination with beginner synthetic techniques.~vvluo<br />
<br />
==Solution 3 (Coordinate Bash + Trig)==<br />
<br />
<asy><br />
size(8cm); defaultpen(fontsize(10pt));<br />
<br />
pair A,B,C,I,D,M,T,Y,Z,EE,F;<br />
A=(0,3sqrt(7));<br />
B=(-1,0);<br />
C=(9,0);<br />
I=incenter(A,B,C);<br />
D=extension(A,I,B,C);<br />
M=(A+D)/2;<br />
<br />
draw(B--EE,gray+dashed);<br />
draw(C--F,gray+dashed);<br />
draw(A--B--C--A);<br />
draw(A--D);<br />
draw(B--(5,sqrt(28)));<br />
draw(M--(5,sqrt(28)));<br />
draw(C--(0,9sqrt(7)/7));<br />
draw(M--(0,9sqrt(7)/7));<br />
dot("$A$",A,NW);<br />
dot("$B$",B,SW);<br />
dot("$C$",C,SE);<br />
dot("$D$",D,S);<br />
dot("$E$",(5,sqrt(28)),N); <br />
dot("$M$",M,dir(70));<br />
dot("$F$",(0,9sqrt(7)/7),N);<br />
<br />
label("$2$",B--D,S);<br />
label("$3$",D--C,S);<br />
label("$6$",A--C,N);<br />
label("$4$",A--B,W);<br />
</asy><br />
<br />
Let <math>B=(0,0)</math> and <math>BC</math> be the line <math>y=0</math>.<br />
We compute that <math>\cos{\angle{ABC}}=\frac{1}{8}</math>, so <math>\tan{\angle{ABC}}=3\sqrt{7}</math>.<br />
Thus, <math>A</math> lies on the line <math>y=3x\sqrt{7}</math>. The length of <math>AB</math> at a point <math>x</math> is <math>8x</math>, so <math>x=\frac{1}{2}</math>.<br />
<br />
We now have the coordinates <math>A=\left(\frac{1}{2},\frac{3\sqrt{7}}{2}\right)</math>, <math>B=(0,0)</math> and <math>C=(5,0)</math>.<br />
We also have <math>D=(2,0)</math> by the angle-bisector theorem and <math>M=\left(\frac{5}{4},\frac{3\sqrt{7}}{4}\right)</math> by taking the midpoint.<br />
We have that because <math>\cos{\angle{ABC}}=\frac{1}{8}</math>, <math>\cos{\frac{\angle{ABC}}{2}}=\frac{3}{4}</math> by half angle formula.<br />
<br />
We also compute <math>\cos{\angle{ACB}}=\frac{3}{4}</math>, so <math>\cos{\frac{\angle{ACB}}{2}}=\frac{\sqrt{14}}{4}</math>.<br />
<br />
Now, <math>AD</math> has slope <math>-\frac{\frac{3\sqrt{7}}{2}}{2-\frac{1}{2}}=-\sqrt{7}</math>, so it's perpendicular bisector has slope <math>\frac{\sqrt{7}}{7}</math> and goes through <math>\left(\frac{5}{4},\frac{3\sqrt{7}}{4}\right)</math>.<br />
<br />
We find that this line has equation <math>y=\frac{\sqrt{7}}{7}x+\frac{4\sqrt{7}}{7}</math>.<br />
<br />
As <math>\cos{\angle{CBI}}=\frac{3}{4}</math>, we have that line <math>BI</math> has form <math>y=\frac{\sqrt{7}}{3}x</math>.<br />
Solving for the intersection point of these two lines, we get <math>x=3</math> and thus <math>E=\left(3, \sqrt{7}\right)</math><br />
<br />
We also have that because <math>\cos{\angle{ICB}}=\frac{\sqrt{14}}{4}</math>, <math>CI</math> has form <math>y=-\frac{x\sqrt{7}}{7}+\frac{5\sqrt{7}}{7}</math>.<br />
<br />
Intersecting the line <math>CI</math> and the perpendicular bisector of <math>AD</math> yields <math>-\frac{x\sqrt{7}}{7}+\frac{5\sqrt{7}}{7}=\frac{x\sqrt{7}}{7}+\frac{4\sqrt{7}}{7}</math>.<br />
<br />
Solving this, we get <math>x=\frac{1}{2}</math> and so <math>F=\left(\frac{1}{2},\frac{9\sqrt{7}}{14}\right)</math>.<br />
<br />
We now compute <math>EF=\sqrt{\left(\frac{5}{2}\right)^2+\left(\frac{5\sqrt{7}}{14}\right)^2}=\frac{5\sqrt{14}}{7}</math>.<br />
We also have <math>MA=\sqrt{\left(\frac{3}{4}\right)^2+\left(\frac{3\sqrt{7}}{4}\right)^2}=\frac{3\sqrt{2}}{2}</math>.<br />
<br />
As <math>{MA}\perp{EF}</math>, we have <math>[\triangle{AEF}]=\frac{1}{2}\left(\frac{3\sqrt{2}}{2}\times\frac{5\sqrt{14}}{7}\right)=\frac{15\sqrt{7}}{14}</math>.<br />
<br />
<br />
The desired answer is <math>15+7+14=\boxed{036}</math> ~Imayormaynotknowcalculus<br />
<br />
==Solution 4 (Barycentric Coordinates)==<br />
<br />
<asy><br />
size(8cm); defaultpen(fontsize(10pt));<br />
<br />
pair A,B,C,I,D,M,T,Y,Z,EE,F;<br />
A=(0,3sqrt(7));<br />
B=(-1,0);<br />
C=(9,0);<br />
I=incenter(A,B,C);<br />
D=extension(A,I,B,C);<br />
M=(A+D)/2;<br />
<br />
draw(B--EE,gray+dashed);<br />
draw(C--F,gray+dashed);<br />
draw(A--B--C--A);<br />
draw(A--D);<br />
draw(B--(5,sqrt(28)));<br />
draw(M--(5,sqrt(28)));<br />
draw(C--(0,9sqrt(7)/7));<br />
draw(M--(0,9sqrt(7)/7));<br />
dot("$A$",A,NW);<br />
dot("$B$",B,SW);<br />
dot("$C$",C,SE);<br />
dot("$D$",D,S);<br />
dot("$E$",(5,sqrt(28)),N); <br />
dot("$M$",M,dir(70));<br />
dot("$F$",(0,9sqrt(7)/7),N);<br />
<br />
label("$2$",B--D,S);<br />
label("$3$",D--C,S);<br />
label("$6$",A--C,N);<br />
label("$4$",A--B,W);<br />
</asy><br />
<br />
As usual, we will use homogenized barycentric coordinates.<br />
<br />
We have that <math>AD</math> will have form <math>3z=2y</math>. Similarly, <math>CF</math> has form <math>5y=6x</math> and <math>BE</math> has form <math>5z=4x</math>.<br />
Since <math>A=(1,0,0)</math> and <math>D=\left(0,\frac{3}{5},\frac{2}{5}\right)</math>, we also have <math>M=\left(\frac{1}{2},\frac{3}{10},\frac{1}{5}\right)</math>.<br />
It remains to determine the equation of the line formed by the perpendicular bisector of <math>AD</math>.<br />
<br />
This can be found using EFFT. Let a point <math>T</math> on <math>EF</math> have coordinates <math>(x, y, z)</math>.<br />
We then have that the displacement vector <math>\overrightarrow{AD}=\left(-1, \frac{3}{5}, \frac{2}{5}\right)</math> and that the displacement vector <math>\overrightarrow{TM}</math> has form <math>\left(x-\frac{1}{2},y-\frac{3}{10},z-\frac{1}{5}\right)</math>.<br />
Now, by EFFT, we have <math>5^2\left(\frac{3}{5}\times\left(z-\frac{1}{5}\right)+\frac{2}{5}\times\left(y-\frac{3}{10}\right)\right)+6^2\left(-1\times\left(z-\frac{1}{5}\right)+\frac{2}{5}\times\left(x-\frac{1}{2}\right)\right)+4^2\left(-1\times\left(y-\frac{3}{10}\right)+\frac{3}{5}\times\left(x-\frac{1}{2}\right)\right)=0</math>.<br />
This equates to <math>8x-2y-7z=2</math>.<br />
<br />
Now, intersecting this with <math>BE</math>, we have <math>5z=4x</math>, <math>8x-2y-7z=2</math>, and <math>x+y+z=1</math>.<br />
This yields <math>x=\frac{2}{3}</math>, <math>y=-\frac{1}{5}</math>, and <math>z=\frac{8}{15}</math>, or <math>E=\left(\frac{2}{3},-\frac{1}{5},\frac{8}{15}\right)</math>.<br />
<br />
Similarly, intersecting this with <math>CF</math>, we have <math>5y=6x</math>, <math>8x-2y-7z=2</math>, and <math>x+y+z=1</math>.<br />
Solving this, we obtain <math>x=\frac{3}{7}</math>, <math>y=\frac{18}{35}</math>, and <math>z=\frac{2}{35}</math>, or <math>F=\left(\frac{3}{7},\frac{18}{35},\frac{2}{35}\right)</math>.<br />
<br />
We finish by invoking the Barycentric Distance Formula twice; our first displacement vector being <math>\overrightarrow{FE}=\left(\frac{5}{21},-\frac{5}{7},\frac{10}{21}\right)</math>.<br />
We then have <math>FE^2=-25\left(-\frac{5}{7}\cdot\frac{10}{21}\right)-36\left(\frac{5}{21}\cdot\frac{10}{21}\right)-16\left(\frac{5}{21}\cdot-\frac{5}{7}\right)=\frac{50}{7}</math>, thus <math>FE=\frac{5\sqrt{14}}{7}</math>.<br />
<br />
Our second displacement vector is <math>\overrightarrow{AM}=\left(-\frac{1}{2},\frac{3}{10},\frac{1}{5}\right)</math>.<br />
As a result, <math>AM^2=-25\left(\frac{3}{10}\cdot\frac{1}{5}\right)-36\left(-\frac{1}{2}\cdot\frac{1}{5}\right)-16\left(-\frac{1}{2}\cdot\frac{3}{10}\right)=\frac{9}{2}</math>, so <math>AM=\frac{3\sqrt{2}}{2}</math>.<br />
<br />
As <math>{AM}\perp{EF}</math>, the desired area is <math>\frac{\frac{5\sqrt{14}}{7}\times\frac{3\sqrt{2}}{2}}{2}={\frac{15\sqrt{7}}{14}}\implies{m+n+p=\boxed{036}}</math>. ~Imayormaynotknowcalculus<br />
<br />
<br />
<b>Remark</b>: The area of <math>\triangle{AEF}</math> can also be computed using the <i>Barycentric Area Formula</i>, although it may increase the risk of computational errors; there are also many different ways to proceed once the coordinates are determined.<br />
<br />
==Solution 5 (geometry+trig)==<br />
<br />
<asy><br />
size(8cm); defaultpen(fontsize(10pt));<br />
<br />
pair A,B,C,I,D,M,T,Y,Z,EE,F;<br />
A=(0,3sqrt(7));<br />
B=(-1,0);<br />
C=(9,0);<br />
I=incenter(A,B,C);<br />
D=extension(A,I,B,C);<br />
M=(A+D)/2;<br />
<br />
draw(B--EE,gray+dashed);<br />
draw(C--F,gray+dashed);<br />
draw(A--B--C--A);<br />
draw(A--D);<br />
draw(A--(5,sqrt(28)));<br />
draw(A--(0,9sqrt(7)/7));<br />
draw(D--(0,9sqrt(7)/7));<br />
draw(D--(5,sqrt(28)));<br />
draw(B--(5,sqrt(28)));<br />
draw(M--(5,sqrt(28)));<br />
draw(C--(0,9sqrt(7)/7));<br />
draw(M--(0,9sqrt(7)/7));<br />
dot("$A$",A,NW);<br />
dot("$B$",B,SW);<br />
dot("$C$",C,SE);<br />
dot("$D$",D,S);<br />
dot("$E$",(5,sqrt(28)),N); <br />
dot("$M$",M,dir(70));<br />
dot("$F$",(0,9sqrt(7)/7),N);<br />
<br />
label("$2$",B--D,S);<br />
label("$3$",D--C,S);<br />
label("$6$",A--C,N);<br />
label("$4$",A--B,W);<br />
</asy><br />
<br />
To get the area of <math>\triangle AEF</math>, we try to find <math>AM</math> and <math>\angle EAF</math>.<br />
<br />
Since <math>AD</math> is the angle bisector, we can get that <math>BD=2</math> and <math>CD=3</math>. By applying Stewart's Theorem, we can get that <math>AD=3\sqrt{2}</math>. Therefore <math>AM=\frac{3\sqrt{2}}{2}</math>.<br />
<br />
Since <math>EF</math> is the perpendicular bisector of <math>AD</math>, we know that <math>AE = DE</math>. Since <math>BE</math> is the angle bisector of <math>\angle BAC</math>,<br />
we know that <math>\angle ABE = \angle DBE</math>. By applying the Law of Sines to <math>\triangle ABE</math> and <math>\triangle DBE</math>, we know that<br />
<math>\sin \angle BAE = \sin \angle BDE </math>. Since <math>BD</math> is not equal to <math>AB</math> and therefore these two triangles are not congruent, we know that <math>\angle BAE</math> and <math>\angle BDE</math> are supplementary. Then we know that <math>\angle ABD</math> and <math>\angle AED</math> are also supplementary. Given that <math>AE=DE</math>, we can get that <math>\angle DAE</math> is half of <math>\angle ABC</math>. Similarly, we have <math>\angle DAF</math> is half of <math>\angle ACB</math>.<br />
<br />
By applying the Law of Cosines, we get <math>\cos \angle ABC = \frac{1}{8}</math>, and then <math>\sin \angle ABC = \frac{3\sqrt{7}}{8}</math>. Similarly, we can get <math>\cos \angle ACB = \frac{3}{4}</math> and <math>\sin \angle ACB = \frac{\sqrt{7}}{4}</math>. Based on some trig identities, we can compute that <math>\tan \angle DAE = \frac{\sin \angle ABC}{1 + \cos \angle ABC} = \frac{\sqrt{7}}{3}</math>, and <math>\tan \angle DAF = \frac{\sqrt{7}}{7}</math>.<br />
<br />
Finally, the area of <math>\triangle AEF</math> equals <math>\frac{1}{2}AM^2(\tan \angle DAE + \tan \angle DAF)=\frac{15\sqrt{7}}{14}</math>. Therefore, the final answer is <math>15+7+14=\boxed{036}</math>. ~xamydad<br />
<br />
<b>Remark</b>: I didn't figure out how to add segments <math>AF</math>, <math>AE</math>, <math>DF</math> and <math>DE</math>. Can someone please help add these segments? <br />
<br />
(Added :) ~Math_Genius_164)<br />
<br />
== Solution 6 ==<br />
[[Image:question13.png|frame|none|###px|]]<br />
First and foremost <math>\big[\triangle{AEF}\big]=\big[\triangle{DEF}\big]</math> as <math>EF</math> is the perpendicular bisector of <math>AD</math>. Now note that quadrilateral <math>ABDF</math> is cyclic, because <math>\angle{ABF}=\angle{FBD}</math> and <math>FA=FD</math>. Similarly quadrilateral <math>AEDC</math> is cyclic, <cmath>\implies \angle{EDA}=\dfrac{C}{2}, \quad \angle{FDA}=\dfrac{B}{2}</cmath><br />
Let <math>A'</math>,<math>B'</math>, <math>C'</math> be the <math>A</math>,<math>B</math>, and <math>C</math> excenters of <math>\triangle{ABC}</math> respectively. Then it follows that <math>\triangle{DEF} \sim \triangle{A'C'B'}</math>. By angle bisector theorem we have <math>BD=2 \implies \dfrac{ID}{IA}=\dfrac{BD}{BA}=\dfrac{1}{2}</math>. Now let the feet of the perpendiculars from <math>I</math> and <math>A'</math> to <math>BC</math> be <math>X</math> and <math>Y</math> resptively. Then by tangents we have <cmath>BX=s-AC=\dfrac{3}{2} \implies XD=2-\dfrac{3}{2}=\dfrac{1}{2}</cmath> <cmath>CY=s-AC \implies YD=3-\dfrac{3}{2}=\dfrac{3}{2} \implies \dfrac{ID}{DA'}=\dfrac{XD}{YD}=\dfrac{1}{3} \implies \big[\triangle{DEF}\big]=\dfrac{1}{16}\big[\triangle{A'C'B'}\big]</cmath> From the previous ratios, <math>AI:ID:DA'=2:1:3 \implies AD=DA' \implies \big[\triangle{ABC}\big]=\big[\triangle{A'BC}\big]</math> Similarly we can find that <math>\big[\triangle{B'AC}\big]=2\big[\triangle{ABC}\big]</math> and <math>\big[\triangle{C'AB}\big]=\dfrac{4}{7}\big[\triangle{ABC}\big]</math> and thus <cmath>\big[\triangle{A'B'C'}\big]=\bigg(1+1+2+\dfrac{4}{7}\bigg)\big[\triangle{ABC}\big]=\dfrac{32}{7}\big[\triangle{ABC}\big] \implies \big[\triangle{DEF}\big]=\dfrac{2}{7}\big[\triangle{ABC}\big]=\dfrac{15\sqrt{7}}{14} \implies m+n+p = \boxed{036}</cmath><br />
-tkhalid<br />
==See Also==<br />
<br />
{{AIME box|year=2020|n=I|num-b=12|num-a=14}}<br />
{{MAA Notice}}</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=MATHCOUNTS&diff=122729MATHCOUNTS2020-05-21T15:00:16Z<p>Apple321: /* MATHCOUNTS Curriculum */</p>
<hr />
<div>Many AoPS Community members and online school students have been participants at National MATHCOUNTS, including many Nationals Countdown Round participants in the past decade. '''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] program that has served millions of middle school students since 1984. Sponsored by the [http://cna.com CNA Foundation], [http://www.nspe.org/ National Society of Professional Engineers], the [http://www.nctm.org/ National Council of Teachers of Mathematics], and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.<br />
<br />
{{Contest Info|name=MATHCOUNTS|region=USA|type=Free Response|difficulty=0.5 - 2.5|breakdown=<u>Countdown</u>: 0.5 (School/Chapter), 1 (State/National)<br><u>Sprint</u>: 1-1.5 (School/Chapter), 2-2.5 (State/National)<br><u>Target:</u> 1.5 (School), 2 (Chapter), 2-2.5 (State/National)}}<br />
<br />
== METHCOUNTS Resources ==<br />
=== Books ===<br />
Art of Problem Solving's [http://artofproblemsolving.com/store/list/aops-curriculum Introductory subject textbooks] are ideal for students preparing for MATHCOUNTS, as are [http://artofproblemsolving.com/store/item/aops-vol1 AoPS Volume 1] and [http://artofproblemsolving.com/store/item/competition-math Competition Math for Middle School]<br />
<br />
=== AoPS Classes ===<br />
Art of Problem Solving hosts a [http://artofproblemsolving.com/school/course/mathcounts-basics Basic] and an [http://artofproblemsolving.com/school/course/mathcounts-advanced Advanced] MATHCOUNTS course. The AoPS Introduction-level subject courses also include a great deal of MATHCOUNTS preparation. Many AoPS instructors are former National MATHCOUNTS Mathletes.<br />
<br />
=== METHCOUNTS Online ===<br />
* [http://www.mathcounts.org Official MATHCOUNTS Homepage]<br />
* Art of Problem Solving hosts a large [http://artofproblemsolving.com/community/c3_middle_school_math Middle School Math Forum] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23209 MATHCOUNTS Coaches Forum].<br />
* The AoPS MATHCOUNTS Trainer is available on the [http://artofproblemsolving.com/mathcounts_trainer AoPS website], as well as on the [https://itunes.apple.com/us/app/mathcounts-trainer-math-contest/id1023961880?ls=1&mt=8 iPhone and iPad].<br />
* The free [http://www.artofproblemsolving.com/alcumus AoPS Alcumus learning system] includes thousands of MATHCOUNTS problems.<br />
* [http://artofproblemsolving.com/ftw/ftw.php For the Win!] gives students free Countdown Round-like practice against other AoPS students.<br />
* AoPS founder Richard Rusczyk has created dozens of [http://artofproblemsolving.com/videos/mathcounts MATHCOUNTS Mini video lessons].<br />
* [http://mathcounts.saab.org/ Elias Saab's MATHCOUNTS Preparation Homepage]<br />
* [http://www.unidata.ucar.edu/staff/russ/mathcounts/diaz.html The MATHCOUNTS Bible According to Mr. Diaz]<br />
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MATHCOUNTS Program] by [[Jeff Boyd]], who coached the 2005, 2007, and 2008 National Champion [[Texas MathCounts]] team.<br />
*[http://mathweb.scranton.edu/monks/courses/ProblemSolving/MathCountsPlaybookBW.pdf Coach Monk's MathCounts Playbook]<br />
* MathCounts Minis make hard problems easy<br />
<br />
== METHCOUNTS Curriculum ==<br />
MATHCOUNTS curriculum includes [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.<br />
<br />
Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.<br />
<br />
== Past State Team Winners ==<br />
* 1984: Virginia<br />
* 1985: Florida<br />
* 1986: California<br />
* 1987: New York<br />
* 1988: New York<br />
* 1989: North Carolina<br />
* 1990: Ohio<br />
* 1991: Alabama<br />
* 1992: California<br />
* 1993: Kansas<br />
* 1994: Pennsylvania<br />
* 1995: Indiana<br />
* 1996: Wisconsin<br />
* 1997: Massachusetts<br />
* 1998: Wisconsin<br />
* 1999: Massachusetts<br />
* 2000: California<br />
* 2001: Virginia<br />
* 2002: California<br />
* 2003: California<br />
* 2004: Illinois<br />
* 2005: Texas<br />
* 2006: Virginia<br />
* 2007: Texas<br />
* 2008: Texas<br />
* 2009: Texas<br />
* 2010: California<br />
* 2011: California<br />
* 2012: Massachusetts<br />
* 2013: Massachusetts<br />
* 2014: California<br />
* 2015: Indiana<br />
* 2016: Texas<br />
* 2017: Texas<br />
* 2018: Texas<br />
* 2019: Massachusetts<br />
* 2020: Florida<br />
<br />
== MATHCOUNTS Competition Structure ==<br />
<br />
=== Sprint Round ===<br />
<br />
30 problems are given all at once. Students have 40 minutes to complete the Sprint Round. This round is very fast-paced and requires speed and accuracy as well. The earlier problems are usually the easiest problems in the competition, and the later problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.<br />
<br />
=== Target Round ===<br />
8 problems given 2 at a time. Students have 6 minutes to complete each set of two problems. Students may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target Round. Usually comprised of one "confidence booster" and one hard problem.<br />
<br />
=== Team Round ===<br />
<br />
10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).<br />
<br />
=== Countdown Round ===<br />
Low scoring individuals compete head-to-head until a loser is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion. Calculators are not allowed, but scratch paper will be provided.<br />
<br />
<br />
====Chapter 1 the competitions====<br />
<br />
In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:<br />
<br />
*The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.<br />
<br />
*The winner of the first round goes up against the 8th place finisher.<br />
<br />
*The winner of the second round goes up against the 7th place finisher.<br />
<br />
This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three questions correct wins (as opposed to the best-out-of-three rule).<br />
<br />
If the countdown round is unofficial, any format may be used. Single-elimination bracket-style tournaments are common.<br />
<br />
====National Competition====<br />
<br />
At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first place written finisher to become champion, equalizing the field. But even then, a 12th place written competitor will have less of a chance to become champion than the top 4, because the top 4 get a bye. Until the semi-finals, the scoring is best out of five advances.<br />
<br />
At the first round and the second round, the person to correctly answer the most out of 5 questions wins. However, at the semifinals, the rules slightly change&mdash;the first person to correctly answer four questions wins.<br />
<br />
=== HELLOOOOOOOOOOOOOO ===<br />
In some states, (most notably Florida) there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "Should I jump in a 3 ft by 3 ft by 4 ft hole?" The answer is YES bc I am dumb.<br />
<br />
=== Masters Round ===<br />
Top students give in-depth explanations to challenging problems. This round is optional at the state level competition and is mandatory at the national competition (up to 2011). At nationals the top two on the written and countdown participate. <br />
In 2012, it was replaced by the Reel Math Challenge (now called the Math Video Challenge).<br />
<br />
=== Scoring and Ranking ===<br />
An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of <math>30 + 2(8) = 46</math> points.<br />
<br />
A team's score is the individual scores of its members divided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score. Note that when there are less than four members the score will become less.<br />
<br />
==Tiebreakers==<br />
If two or more students tie for a ranking with the same individual score, the people ranked in order is decided using the following algorithm(going to the next step if inconclusive)<br />
(1) Sprint score<br />
(2) Sum of the problem numbers correct(Unconfirmed)<br />
(3) Last problem done correctly(Unconfirmed), and checking down the list to previous problems if still tied<br />
(4) Flip a coin(Unconfirmed)<br />
<br />
In state contests, advancements to nationals in the case of a tie is often decided using a Tiebreaker Round(tied students are invited into a room and participate in a secret format of a contest to determine who advances to the National Contest)<br />
<br />
== MATHCOUNTS Competition Levels ==<br />
=== School Competition ===<br />
Students vie for the chance to make their school teams. Problems at this level are generally the easiest and most basic in curriculum.<br />
<br />
=== Chapter Competition ===<br />
Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.<br />
<br />
=== MINECRAFT Competition ===<br />
The top 4 GAMERS in each SCHOOL form the MINECRAFT team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. MINEMINEMINE<br />
<br />
=== BIG BRAINS ONLY Competition ===<br />
==== National Fortnite<br />
Sites ====<br />
For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.<br />
<br />
* The 2020 competition was canceled due to the COVID-19 pandemic.<br />
* The 2019 competition was held in Orlando, Florida.<br />
* The 2018 competition was held in Washington, D.C.<br />
* The 2017 competition was held in Orlando, Florida.<br />
* The 2016 competition was held in Washington, D.C.<br />
* The 2015 competition was held in Boston, Massachusetts.<br />
* The 2014 competition was held in Orlando, Florida.<br />
* The 2013 competition was held in Washington, D.C.<br />
* The 2012 competition was held in Orlando, Florida.<br />
* The 2011 competition was held in Washington, D.C.<br />
* The 2009 and 2010 competitions were held in Orlando, Florida.<br />
* The 2008 competition was held in Denver, Colorado.<br />
* The 2007 competition was held in Fort Worth, Texas.<br />
* The 2006 competition was held in Arlington, Virginia.<br />
* The 2005 competition was held in Detroit, Michigan.<br />
* The 2004 competition was held in Washington, D.C.<br />
* The 2002 and 2003 competitions were held in Chicago, Illinois.<br />
<br />
== LOLOLOLOLOLOLOLOLOLOL ==<br />
<br />
Give the following competitions a try and take a look at the [[List of United States high school mathematics competitions]].<br />
* [[American Mathematics Competitions]]<br />
* [[American Regions Math League]]<br />
* [[Mandelbrot Competition]]<br />
* [[Mu Alpha Theta]]<br />
* [[BLAH BLAH BLAH]]<br />
<br />
[[Category:Mathematics competitions]]<br />
<br />
== Please don't See also... ==<br />
* [[List of national MATHCOUNTS teams]]<br />
* [[Mathematics competition resources]]<br />
* [[Math contest books]]<br />
* [[Math books]]<br />
* [[List of United States middle school mathematics competitions]]<br />
* [[List of United States high school mathematics competitions]]<br />
* [http://www.mathcounts.org/webarticles/anmviewer.asp?a=921&z=71 2006 MATHCOUNTS Countdown Video]<br />
<br />
[[Category:Introductory mathematics competitions]]</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=MATHCOUNTS&diff=122728MATHCOUNTS2020-05-21T15:00:04Z<p>Apple321: /* MATHCOUNTS Online */</p>
<hr />
<div>Many AoPS Community members and online school students have been participants at National MATHCOUNTS, including many Nationals Countdown Round participants in the past decade. '''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] program that has served millions of middle school students since 1984. Sponsored by the [http://cna.com CNA Foundation], [http://www.nspe.org/ National Society of Professional Engineers], the [http://www.nctm.org/ National Council of Teachers of Mathematics], and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.<br />
<br />
{{Contest Info|name=MATHCOUNTS|region=USA|type=Free Response|difficulty=0.5 - 2.5|breakdown=<u>Countdown</u>: 0.5 (School/Chapter), 1 (State/National)<br><u>Sprint</u>: 1-1.5 (School/Chapter), 2-2.5 (State/National)<br><u>Target:</u> 1.5 (School), 2 (Chapter), 2-2.5 (State/National)}}<br />
<br />
== METHCOUNTS Resources ==<br />
=== Books ===<br />
Art of Problem Solving's [http://artofproblemsolving.com/store/list/aops-curriculum Introductory subject textbooks] are ideal for students preparing for MATHCOUNTS, as are [http://artofproblemsolving.com/store/item/aops-vol1 AoPS Volume 1] and [http://artofproblemsolving.com/store/item/competition-math Competition Math for Middle School]<br />
<br />
=== AoPS Classes ===<br />
Art of Problem Solving hosts a [http://artofproblemsolving.com/school/course/mathcounts-basics Basic] and an [http://artofproblemsolving.com/school/course/mathcounts-advanced Advanced] MATHCOUNTS course. The AoPS Introduction-level subject courses also include a great deal of MATHCOUNTS preparation. Many AoPS instructors are former National MATHCOUNTS Mathletes.<br />
<br />
=== METHCOUNTS Online ===<br />
* [http://www.mathcounts.org Official MATHCOUNTS Homepage]<br />
* Art of Problem Solving hosts a large [http://artofproblemsolving.com/community/c3_middle_school_math Middle School Math Forum] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23209 MATHCOUNTS Coaches Forum].<br />
* The AoPS MATHCOUNTS Trainer is available on the [http://artofproblemsolving.com/mathcounts_trainer AoPS website], as well as on the [https://itunes.apple.com/us/app/mathcounts-trainer-math-contest/id1023961880?ls=1&mt=8 iPhone and iPad].<br />
* The free [http://www.artofproblemsolving.com/alcumus AoPS Alcumus learning system] includes thousands of MATHCOUNTS problems.<br />
* [http://artofproblemsolving.com/ftw/ftw.php For the Win!] gives students free Countdown Round-like practice against other AoPS students.<br />
* AoPS founder Richard Rusczyk has created dozens of [http://artofproblemsolving.com/videos/mathcounts MATHCOUNTS Mini video lessons].<br />
* [http://mathcounts.saab.org/ Elias Saab's MATHCOUNTS Preparation Homepage]<br />
* [http://www.unidata.ucar.edu/staff/russ/mathcounts/diaz.html The MATHCOUNTS Bible According to Mr. Diaz]<br />
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MATHCOUNTS Program] by [[Jeff Boyd]], who coached the 2005, 2007, and 2008 National Champion [[Texas MathCounts]] team.<br />
*[http://mathweb.scranton.edu/monks/courses/ProblemSolving/MathCountsPlaybookBW.pdf Coach Monk's MathCounts Playbook]<br />
* MathCounts Minis make hard problems easy<br />
<br />
== MATHCOUNTS Curriculum ==<br />
MATHCOUNTS curriculum includes [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.<br />
<br />
Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.<br />
<br />
== Past State Team Winners ==<br />
* 1984: Virginia<br />
* 1985: Florida<br />
* 1986: California<br />
* 1987: New York<br />
* 1988: New York<br />
* 1989: North Carolina<br />
* 1990: Ohio<br />
* 1991: Alabama<br />
* 1992: California<br />
* 1993: Kansas<br />
* 1994: Pennsylvania<br />
* 1995: Indiana<br />
* 1996: Wisconsin<br />
* 1997: Massachusetts<br />
* 1998: Wisconsin<br />
* 1999: Massachusetts<br />
* 2000: California<br />
* 2001: Virginia<br />
* 2002: California<br />
* 2003: California<br />
* 2004: Illinois<br />
* 2005: Texas<br />
* 2006: Virginia<br />
* 2007: Texas<br />
* 2008: Texas<br />
* 2009: Texas<br />
* 2010: California<br />
* 2011: California<br />
* 2012: Massachusetts<br />
* 2013: Massachusetts<br />
* 2014: California<br />
* 2015: Indiana<br />
* 2016: Texas<br />
* 2017: Texas<br />
* 2018: Texas<br />
* 2019: Massachusetts<br />
* 2020: Florida<br />
<br />
== MATHCOUNTS Competition Structure ==<br />
<br />
=== Sprint Round ===<br />
<br />
30 problems are given all at once. Students have 40 minutes to complete the Sprint Round. This round is very fast-paced and requires speed and accuracy as well. The earlier problems are usually the easiest problems in the competition, and the later problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.<br />
<br />
=== Target Round ===<br />
8 problems given 2 at a time. Students have 6 minutes to complete each set of two problems. Students may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target Round. Usually comprised of one "confidence booster" and one hard problem.<br />
<br />
=== Team Round ===<br />
<br />
10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).<br />
<br />
=== Countdown Round ===<br />
Low scoring individuals compete head-to-head until a loser is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion. Calculators are not allowed, but scratch paper will be provided.<br />
<br />
<br />
====Chapter 1 the competitions====<br />
<br />
In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:<br />
<br />
*The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.<br />
<br />
*The winner of the first round goes up against the 8th place finisher.<br />
<br />
*The winner of the second round goes up against the 7th place finisher.<br />
<br />
This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three questions correct wins (as opposed to the best-out-of-three rule).<br />
<br />
If the countdown round is unofficial, any format may be used. Single-elimination bracket-style tournaments are common.<br />
<br />
====National Competition====<br />
<br />
At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first place written finisher to become champion, equalizing the field. But even then, a 12th place written competitor will have less of a chance to become champion than the top 4, because the top 4 get a bye. Until the semi-finals, the scoring is best out of five advances.<br />
<br />
At the first round and the second round, the person to correctly answer the most out of 5 questions wins. However, at the semifinals, the rules slightly change&mdash;the first person to correctly answer four questions wins.<br />
<br />
=== HELLOOOOOOOOOOOOOO ===<br />
In some states, (most notably Florida) there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "Should I jump in a 3 ft by 3 ft by 4 ft hole?" The answer is YES bc I am dumb.<br />
<br />
=== Masters Round ===<br />
Top students give in-depth explanations to challenging problems. This round is optional at the state level competition and is mandatory at the national competition (up to 2011). At nationals the top two on the written and countdown participate. <br />
In 2012, it was replaced by the Reel Math Challenge (now called the Math Video Challenge).<br />
<br />
=== Scoring and Ranking ===<br />
An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of <math>30 + 2(8) = 46</math> points.<br />
<br />
A team's score is the individual scores of its members divided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score. Note that when there are less than four members the score will become less.<br />
<br />
==Tiebreakers==<br />
If two or more students tie for a ranking with the same individual score, the people ranked in order is decided using the following algorithm(going to the next step if inconclusive)<br />
(1) Sprint score<br />
(2) Sum of the problem numbers correct(Unconfirmed)<br />
(3) Last problem done correctly(Unconfirmed), and checking down the list to previous problems if still tied<br />
(4) Flip a coin(Unconfirmed)<br />
<br />
In state contests, advancements to nationals in the case of a tie is often decided using a Tiebreaker Round(tied students are invited into a room and participate in a secret format of a contest to determine who advances to the National Contest)<br />
<br />
== MATHCOUNTS Competition Levels ==<br />
=== School Competition ===<br />
Students vie for the chance to make their school teams. Problems at this level are generally the easiest and most basic in curriculum.<br />
<br />
=== Chapter Competition ===<br />
Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.<br />
<br />
=== MINECRAFT Competition ===<br />
The top 4 GAMERS in each SCHOOL form the MINECRAFT team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. MINEMINEMINE<br />
<br />
=== BIG BRAINS ONLY Competition ===<br />
==== National Fortnite<br />
Sites ====<br />
For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.<br />
<br />
* The 2020 competition was canceled due to the COVID-19 pandemic.<br />
* The 2019 competition was held in Orlando, Florida.<br />
* The 2018 competition was held in Washington, D.C.<br />
* The 2017 competition was held in Orlando, Florida.<br />
* The 2016 competition was held in Washington, D.C.<br />
* The 2015 competition was held in Boston, Massachusetts.<br />
* The 2014 competition was held in Orlando, Florida.<br />
* The 2013 competition was held in Washington, D.C.<br />
* The 2012 competition was held in Orlando, Florida.<br />
* The 2011 competition was held in Washington, D.C.<br />
* The 2009 and 2010 competitions were held in Orlando, Florida.<br />
* The 2008 competition was held in Denver, Colorado.<br />
* The 2007 competition was held in Fort Worth, Texas.<br />
* The 2006 competition was held in Arlington, Virginia.<br />
* The 2005 competition was held in Detroit, Michigan.<br />
* The 2004 competition was held in Washington, D.C.<br />
* The 2002 and 2003 competitions were held in Chicago, Illinois.<br />
<br />
== LOLOLOLOLOLOLOLOLOLOL ==<br />
<br />
Give the following competitions a try and take a look at the [[List of United States high school mathematics competitions]].<br />
* [[American Mathematics Competitions]]<br />
* [[American Regions Math League]]<br />
* [[Mandelbrot Competition]]<br />
* [[Mu Alpha Theta]]<br />
* [[BLAH BLAH BLAH]]<br />
<br />
[[Category:Mathematics competitions]]<br />
<br />
== Please don't See also... ==<br />
* [[List of national MATHCOUNTS teams]]<br />
* [[Mathematics competition resources]]<br />
* [[Math contest books]]<br />
* [[Math books]]<br />
* [[List of United States middle school mathematics competitions]]<br />
* [[List of United States high school mathematics competitions]]<br />
* [http://www.mathcounts.org/webarticles/anmviewer.asp?a=921&z=71 2006 MATHCOUNTS Countdown Video]<br />
<br />
[[Category:Introductory mathematics competitions]]</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=MATHCOUNTS&diff=122727MATHCOUNTS2020-05-21T14:59:19Z<p>Apple321: /* MATHCOUNTS Resources */</p>
<hr />
<div>Many AoPS Community members and online school students have been participants at National MATHCOUNTS, including many Nationals Countdown Round participants in the past decade. '''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] program that has served millions of middle school students since 1984. Sponsored by the [http://cna.com CNA Foundation], [http://www.nspe.org/ National Society of Professional Engineers], the [http://www.nctm.org/ National Council of Teachers of Mathematics], and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.<br />
<br />
{{Contest Info|name=MATHCOUNTS|region=USA|type=Free Response|difficulty=0.5 - 2.5|breakdown=<u>Countdown</u>: 0.5 (School/Chapter), 1 (State/National)<br><u>Sprint</u>: 1-1.5 (School/Chapter), 2-2.5 (State/National)<br><u>Target:</u> 1.5 (School), 2 (Chapter), 2-2.5 (State/National)}}<br />
<br />
== METHCOUNTS Resources ==<br />
=== Books ===<br />
Art of Problem Solving's [http://artofproblemsolving.com/store/list/aops-curriculum Introductory subject textbooks] are ideal for students preparing for MATHCOUNTS, as are [http://artofproblemsolving.com/store/item/aops-vol1 AoPS Volume 1] and [http://artofproblemsolving.com/store/item/competition-math Competition Math for Middle School]<br />
<br />
=== AoPS Classes ===<br />
Art of Problem Solving hosts a [http://artofproblemsolving.com/school/course/mathcounts-basics Basic] and an [http://artofproblemsolving.com/school/course/mathcounts-advanced Advanced] MATHCOUNTS course. The AoPS Introduction-level subject courses also include a great deal of MATHCOUNTS preparation. Many AoPS instructors are former National MATHCOUNTS Mathletes.<br />
<br />
=== MATHCOUNTS Online ===<br />
* [http://www.mathcounts.org Official MATHCOUNTS Homepage]<br />
* Art of Problem Solving hosts a large [http://artofproblemsolving.com/community/c3_middle_school_math Middle School Math Forum] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23209 MATHCOUNTS Coaches Forum].<br />
* The AoPS MATHCOUNTS Trainer is available on the [http://artofproblemsolving.com/mathcounts_trainer AoPS website], as well as on the [https://itunes.apple.com/us/app/mathcounts-trainer-math-contest/id1023961880?ls=1&mt=8 iPhone and iPad].<br />
* The free [http://www.artofproblemsolving.com/alcumus AoPS Alcumus learning system] includes thousands of MATHCOUNTS problems.<br />
* [http://artofproblemsolving.com/ftw/ftw.php For the Win!] gives students free Countdown Round-like practice against other AoPS students.<br />
* AoPS founder Richard Rusczyk has created dozens of [http://artofproblemsolving.com/videos/mathcounts MATHCOUNTS Mini video lessons].<br />
* [http://mathcounts.saab.org/ Elias Saab's MATHCOUNTS Preparation Homepage]<br />
* [http://www.unidata.ucar.edu/staff/russ/mathcounts/diaz.html The MATHCOUNTS Bible According to Mr. Diaz]<br />
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MATHCOUNTS Program] by [[Jeff Boyd]], who coached the 2005, 2007, and 2008 National Champion [[Texas MathCounts]] team.<br />
*[http://mathweb.scranton.edu/monks/courses/ProblemSolving/MathCountsPlaybookBW.pdf Coach Monk's MathCounts Playbook]<br />
* MathCounts Minis make hard problems easy<br />
<br />
== MATHCOUNTS Curriculum ==<br />
MATHCOUNTS curriculum includes [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.<br />
<br />
Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.<br />
<br />
== Past State Team Winners ==<br />
* 1984: Virginia<br />
* 1985: Florida<br />
* 1986: California<br />
* 1987: New York<br />
* 1988: New York<br />
* 1989: North Carolina<br />
* 1990: Ohio<br />
* 1991: Alabama<br />
* 1992: California<br />
* 1993: Kansas<br />
* 1994: Pennsylvania<br />
* 1995: Indiana<br />
* 1996: Wisconsin<br />
* 1997: Massachusetts<br />
* 1998: Wisconsin<br />
* 1999: Massachusetts<br />
* 2000: California<br />
* 2001: Virginia<br />
* 2002: California<br />
* 2003: California<br />
* 2004: Illinois<br />
* 2005: Texas<br />
* 2006: Virginia<br />
* 2007: Texas<br />
* 2008: Texas<br />
* 2009: Texas<br />
* 2010: California<br />
* 2011: California<br />
* 2012: Massachusetts<br />
* 2013: Massachusetts<br />
* 2014: California<br />
* 2015: Indiana<br />
* 2016: Texas<br />
* 2017: Texas<br />
* 2018: Texas<br />
* 2019: Massachusetts<br />
* 2020: Florida<br />
<br />
== MATHCOUNTS Competition Structure ==<br />
<br />
=== Sprint Round ===<br />
<br />
30 problems are given all at once. Students have 40 minutes to complete the Sprint Round. This round is very fast-paced and requires speed and accuracy as well. The earlier problems are usually the easiest problems in the competition, and the later problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.<br />
<br />
=== Target Round ===<br />
8 problems given 2 at a time. Students have 6 minutes to complete each set of two problems. Students may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target Round. Usually comprised of one "confidence booster" and one hard problem.<br />
<br />
=== Team Round ===<br />
<br />
10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).<br />
<br />
=== Countdown Round ===<br />
Low scoring individuals compete head-to-head until a loser is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion. Calculators are not allowed, but scratch paper will be provided.<br />
<br />
<br />
====Chapter 1 the competitions====<br />
<br />
In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:<br />
<br />
*The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.<br />
<br />
*The winner of the first round goes up against the 8th place finisher.<br />
<br />
*The winner of the second round goes up against the 7th place finisher.<br />
<br />
This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three questions correct wins (as opposed to the best-out-of-three rule).<br />
<br />
If the countdown round is unofficial, any format may be used. Single-elimination bracket-style tournaments are common.<br />
<br />
====National Competition====<br />
<br />
At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first place written finisher to become champion, equalizing the field. But even then, a 12th place written competitor will have less of a chance to become champion than the top 4, because the top 4 get a bye. Until the semi-finals, the scoring is best out of five advances.<br />
<br />
At the first round and the second round, the person to correctly answer the most out of 5 questions wins. However, at the semifinals, the rules slightly change&mdash;the first person to correctly answer four questions wins.<br />
<br />
=== HELLOOOOOOOOOOOOOO ===<br />
In some states, (most notably Florida) there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "Should I jump in a 3 ft by 3 ft by 4 ft hole?" The answer is YES bc I am dumb.<br />
<br />
=== Masters Round ===<br />
Top students give in-depth explanations to challenging problems. This round is optional at the state level competition and is mandatory at the national competition (up to 2011). At nationals the top two on the written and countdown participate. <br />
In 2012, it was replaced by the Reel Math Challenge (now called the Math Video Challenge).<br />
<br />
=== Scoring and Ranking ===<br />
An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of <math>30 + 2(8) = 46</math> points.<br />
<br />
A team's score is the individual scores of its members divided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score. Note that when there are less than four members the score will become less.<br />
<br />
==Tiebreakers==<br />
If two or more students tie for a ranking with the same individual score, the people ranked in order is decided using the following algorithm(going to the next step if inconclusive)<br />
(1) Sprint score<br />
(2) Sum of the problem numbers correct(Unconfirmed)<br />
(3) Last problem done correctly(Unconfirmed), and checking down the list to previous problems if still tied<br />
(4) Flip a coin(Unconfirmed)<br />
<br />
In state contests, advancements to nationals in the case of a tie is often decided using a Tiebreaker Round(tied students are invited into a room and participate in a secret format of a contest to determine who advances to the National Contest)<br />
<br />
== MATHCOUNTS Competition Levels ==<br />
=== School Competition ===<br />
Students vie for the chance to make their school teams. Problems at this level are generally the easiest and most basic in curriculum.<br />
<br />
=== Chapter Competition ===<br />
Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.<br />
<br />
=== MINECRAFT Competition ===<br />
The top 4 GAMERS in each SCHOOL form the MINECRAFT team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. MINEMINEMINE<br />
<br />
=== BIG BRAINS ONLY Competition ===<br />
==== National Fortnite<br />
Sites ====<br />
For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.<br />
<br />
* The 2020 competition was canceled due to the COVID-19 pandemic.<br />
* The 2019 competition was held in Orlando, Florida.<br />
* The 2018 competition was held in Washington, D.C.<br />
* The 2017 competition was held in Orlando, Florida.<br />
* The 2016 competition was held in Washington, D.C.<br />
* The 2015 competition was held in Boston, Massachusetts.<br />
* The 2014 competition was held in Orlando, Florida.<br />
* The 2013 competition was held in Washington, D.C.<br />
* The 2012 competition was held in Orlando, Florida.<br />
* The 2011 competition was held in Washington, D.C.<br />
* The 2009 and 2010 competitions were held in Orlando, Florida.<br />
* The 2008 competition was held in Denver, Colorado.<br />
* The 2007 competition was held in Fort Worth, Texas.<br />
* The 2006 competition was held in Arlington, Virginia.<br />
* The 2005 competition was held in Detroit, Michigan.<br />
* The 2004 competition was held in Washington, D.C.<br />
* The 2002 and 2003 competitions were held in Chicago, Illinois.<br />
<br />
== LOLOLOLOLOLOLOLOLOLOL ==<br />
<br />
Give the following competitions a try and take a look at the [[List of United States high school mathematics competitions]].<br />
* [[American Mathematics Competitions]]<br />
* [[American Regions Math League]]<br />
* [[Mandelbrot Competition]]<br />
* [[Mu Alpha Theta]]<br />
* [[BLAH BLAH BLAH]]<br />
<br />
[[Category:Mathematics competitions]]<br />
<br />
== Please don't See also... ==<br />
* [[List of national MATHCOUNTS teams]]<br />
* [[Mathematics competition resources]]<br />
* [[Math contest books]]<br />
* [[Math books]]<br />
* [[List of United States middle school mathematics competitions]]<br />
* [[List of United States high school mathematics competitions]]<br />
* [http://www.mathcounts.org/webarticles/anmviewer.asp?a=921&z=71 2006 MATHCOUNTS Countdown Video]<br />
<br />
[[Category:Introductory mathematics competitions]]</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=MATHCOUNTS&diff=122716MATHCOUNTS2020-05-21T14:34:59Z<p>Apple321: /* See also... */</p>
<hr />
<div>Many AoPS Community members and online school students have been participants at National MATHCOUNTS, including many Nationals Countdown Round participants in the past decade. '''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] program that has served millions of middle school students since 1984. Sponsored by the [http://cna.com CNA Foundation], [http://www.nspe.org/ National Society of Professional Engineers], the [http://www.nctm.org/ National Council of Teachers of Mathematics], and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.<br />
<br />
{{Contest Info|name=MATHCOUNTS|region=USA|type=Free Response|difficulty=0.5 - 2.5|breakdown=<u>Countdown</u>: 0.5 (School/Chapter), 1 (State/National)<br><u>Sprint</u>: 1-1.5 (School/Chapter), 2-2.5 (State/National)<br><u>Target:</u> 1.5 (School), 2 (Chapter), 2-2.5 (State/National)}}<br />
<br />
== MATHCOUNTS Resources ==<br />
=== STOOOOOOOOOPID Books ===<br />
Art of Problem Solving's [http://artofproblemsolving.com/store/list/aops-curriculum Introductory subject textbooks] are ideal for students preparing for MATHCOUNTS, as are [http://artofproblemsolving.com/store/item/aops-vol1 AoPS Volume 1] and [http://artofproblemsolving.com/store/item/competition-math Competition Math for Middle School]<br />
<br />
=== Fortnite Classes ===<br />
Art of Problem Solving hosts a [http://artofproblemsolving.com/school/course/mathcounts-basics Basic] and an [http://artofproblemsolving.com/school/course/mathcounts-advanced Advanced] MATHCOUNTS course. The AoPS Introduction-level subject courses also include a great deal of MATHCOUNTS preparation. Many AoPS instructors are former National MATHCOUNTS Mathletes.<br />
<br />
=== MATHCOUNTS Online ===<br />
* [http://www.mathcounts.org Official MATHCOUNTS Homepage]<br />
* Art of Problem Solving hosts a large [http://artofproblemsolving.com/community/c3_middle_school_math Middle School Math Forum] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23209 MATHCOUNTS Coaches Forum].<br />
* The AoPS MATHCOUNTS Trainer is available on the [http://artofproblemsolving.com/mathcounts_trainer AoPS website], as well as on the [https://itunes.apple.com/us/app/mathcounts-trainer-math-contest/id1023961880?ls=1&mt=8 iPhone and iPad].<br />
* The free [http://www.artofproblemsolving.com/alcumus AoPS Alcumus learning system] includes thousands of MATHCOUNTS problems.<br />
* [http://artofproblemsolving.com/ftw/ftw.php For the Win!] gives students free Countdown Round-like practice against other AoPS students.<br />
* AoPS founder Richard Rusczyk has created dozens of [http://artofproblemsolving.com/videos/mathcounts MATHCOUNTS Mini video lessons].<br />
* [http://mathcounts.saab.org/ Elias Saab's MATHCOUNTS Preparation Homepage]<br />
* [http://www.unidata.ucar.edu/staff/russ/mathcounts/diaz.html The MATHCOUNTS Bible According to Mr. Diaz]<br />
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MATHCOUNTS Program] by [[Jeff Boyd]], who coached the 2005, 2007, and 2008 National Champion [[Texas MathCounts]] team.<br />
*[http://mathweb.scranton.edu/monks/courses/ProblemSolving/MathCountsPlaybookBW.pdf Coach Monk's MathCounts Playbook]<br />
* MathCounts Minis make hard problems easy<br />
<br />
== MATHCOUNTS Curriculum ==<br />
MATHCOUNTS curriculum includes [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.<br />
<br />
Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.<br />
<br />
== Past State Team Winners ==<br />
* 1984: Virginia<br />
* 1985: Florida<br />
* 1986: California<br />
* 1987: New York<br />
* 1988: New York<br />
* 1989: North Carolina<br />
* 1990: Ohio<br />
* 1991: Alabama<br />
* 1992: California<br />
* 1993: Kansas<br />
* 1994: Pennsylvania<br />
* 1995: Indiana<br />
* 1996: Wisconsin<br />
* 1997: Massachusetts<br />
* 1998: Wisconsin<br />
* 1999: Massachusetts<br />
* 2000: California<br />
* 2001: Virginia<br />
* 2002: California<br />
* 2003: California<br />
* 2004: Illinois<br />
* 2005: Texas<br />
* 2006: Virginia<br />
* 2007: Texas<br />
* 2008: Texas<br />
* 2009: Texas<br />
* 2010: California<br />
* 2011: California<br />
* 2012: Massachusetts<br />
* 2013: Massachusetts<br />
* 2014: California<br />
* 2015: Indiana<br />
* 2016: Texas<br />
* 2017: Texas<br />
* 2018: Texas<br />
* 2019: Massachusetts<br />
<br />
== MATHCOUNTS Competition Structure ==<br />
<br />
=== Sprint Round ===<br />
<br />
30 problems are given all at once. Students have 40 minutes to complete the Sprint Round. This round is very fast-paced and requires speed and accuracy as well. The earlier problems are usually the easiest problems in the competition, and the later problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.<br />
<br />
=== Target Round ===<br />
8 problems given 2 at a time. Students have 6 minutes to complete each set of two problems. Students may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target Round. Usually comprised of one "confidence booster" and one hard problem.<br />
<br />
=== Team Round ===<br />
<br />
10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).<br />
<br />
=== Countdown Round ===<br />
High scoring individuals compete head-to-head until a champion is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion. Calculators are not allowed, but scratch paper will be provided.<br />
<br />
<br />
====Chapter 1 the competitions====<br />
<br />
In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:<br />
<br />
*The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.<br />
<br />
*The winner of the first round goes up against the 8th place finisher.<br />
<br />
*The winner of the second round goes up against the 7th place finisher.<br />
<br />
This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three questions correct wins (as opposed to the best-out-of-three rule).<br />
<br />
If the countdown round is unofficial, any format may be used. Single-elimination bracket-style tournaments are common.<br />
<br />
====National Competition====<br />
<br />
At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first place written finisher to become champion, equalizing the field. But even then, a 12th place written competitor will have less of a chance to become champion than the top 4, because the top 4 get a bye. Until the semi-finals, the scoring is best out of five advances.<br />
<br />
At the first round and the second round, the person to correctly answer the most out of 5 questions wins. However, at the semifinals, the rules slightly change&mdash;the first person to correctly answer four questions wins.<br />
<br />
=== HELLOOOOOOOOOOOOOO ===<br />
In some states, (most notably Florida) there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "How much dirt is in a 3 ft by 3 ft by 4 ft hole?" The answer was 0 because there is no dirt in a hole.<br />
<br />
=== Masters Round ===<br />
Top students give in-depth explanations to challenging problems. This round is optional at the state level competition and is mandatory at the national competition (up to 2011). At nationals the top two on the written and countdown participate. <br />
In 2012, it was replaced by the Reel Math Challenge (now called the Math Video Challenge).<br />
<br />
=== Scoring and Ranking ===<br />
An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of <math>30 + 2(8) = 46</math> points.<br />
<br />
A team's score is the individual scores of its members divided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score. Note that when there are less than four members the score will become less.<br />
<br />
==Tiebreakers==<br />
If two or more students tie for a ranking with the same individual score, the people ranked in order is decided using the following algorithm(going to the next step if inconclusive)<br />
(1) Sprint score<br />
(2) Sum of the problem numbers correct(Unconfirmed)<br />
(3) Last problem done correctly(Unconfirmed), and checking down the list to previous problems if still tied<br />
(4) Flip a coin(Unconfirmed)<br />
<br />
In state contests, advancements to nationals in the case of a tie is often decided using a Tiebreaker Round(tied students are invited into a room and participate in a secret format of a contest to determine who advances to the National Contest)<br />
<br />
== MATHCOUNTS Competition Levels ==<br />
=== School Competition ===<br />
Students vie for the chance to make their school teams. Problems at this level are generally the easiest and most basic in curriculum.<br />
<br />
=== Chapter Competition ===<br />
Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.<br />
CHAPTER IS EZ AF!!!<br />
<br />
<br />
<br />
HELLLLLLLO<br />
I CAN EDIT THIS 4 SOME REASON<br />
<br />
=== DOG eating Competition ===<br />
The top 4 students in each state form the state team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. JIJIJIJIJII<br />
<br />
=== BIG BRAINS ONLY Competition ===<br />
==== National Fortnite<br />
Sites ====<br />
For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.<br />
<br />
* The 2020 competition was canceled due to the COVID-19 pandemic.<br />
* The 2019 competition was held in Orlando, Florida.<br />
* The 2018 competition was held in Washington, D.C.<br />
* The 2017 competition was held in Orlando, Florida.<br />
* The 2016 competition was held in Washington, D.C.<br />
* The 2015 competition was held in Boston, Massachusetts.<br />
* The 2014 competition was held in Orlando, Florida.<br />
* The 2013 competition was held in Washington, D.C.<br />
* The 2012 competition was held in Orlando, Florida.<br />
* The 2011 competition was held in Washington, D.C.<br />
* The 2009 and 2010 competitions were held in Orlando, Florida.<br />
* The 2008 competition was held in Denver, Colorado.<br />
* The 2007 competition was held in Fort Worth, Texas.<br />
* The 2006 competition was held in Arlington, Virginia.<br />
* The 2005 competition was held in Detroit, Michigan.<br />
* The 2004 competition was held in Washington, D.C.<br />
* The 2002 and 2003 competitions were held in Chicago, Illinois.<br />
<br />
== LOLOLOLOLOLOLOLOLOLOL ==<br />
<br />
Give the following competitions a try and take a look at the [[List of United States high school mathematics competitions]].<br />
* [[American Mathematics Competitions]]<br />
* [[American Regions Math League]]<br />
* [[Mandelbrot Competition]]<br />
* [[Mu Alpha Theta]]<br />
* [[BLAH BLAH BLAH]]<br />
<br />
[[Category:Mathematics competitions]]<br />
<br />
== Please don't See also... ==<br />
* [[List of national MATHCOUNTS teams]]<br />
* [[Mathematics competition resources]]<br />
* [[Math contest books]]<br />
* [[Math books]]<br />
* [[List of United States middle school mathematics competitions]]<br />
* [[List of United States high school mathematics competitions]]<br />
* [http://www.mathcounts.org/webarticles/anmviewer.asp?a=921&z=71 2006 MATHCOUNTS Countdown Video]<br />
<br />
[[Category:Introductory mathematics competitions]]</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=MATHCOUNTS&diff=122715MATHCOUNTS2020-05-21T14:34:19Z<p>Apple321: /* jiji eating Competition */</p>
<hr />
<div>Many AoPS Community members and online school students have been participants at National MATHCOUNTS, including many Nationals Countdown Round participants in the past decade. '''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] program that has served millions of middle school students since 1984. Sponsored by the [http://cna.com CNA Foundation], [http://www.nspe.org/ National Society of Professional Engineers], the [http://www.nctm.org/ National Council of Teachers of Mathematics], and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.<br />
<br />
{{Contest Info|name=MATHCOUNTS|region=USA|type=Free Response|difficulty=0.5 - 2.5|breakdown=<u>Countdown</u>: 0.5 (School/Chapter), 1 (State/National)<br><u>Sprint</u>: 1-1.5 (School/Chapter), 2-2.5 (State/National)<br><u>Target:</u> 1.5 (School), 2 (Chapter), 2-2.5 (State/National)}}<br />
<br />
== MATHCOUNTS Resources ==<br />
=== STOOOOOOOOOPID Books ===<br />
Art of Problem Solving's [http://artofproblemsolving.com/store/list/aops-curriculum Introductory subject textbooks] are ideal for students preparing for MATHCOUNTS, as are [http://artofproblemsolving.com/store/item/aops-vol1 AoPS Volume 1] and [http://artofproblemsolving.com/store/item/competition-math Competition Math for Middle School]<br />
<br />
=== Fortnite Classes ===<br />
Art of Problem Solving hosts a [http://artofproblemsolving.com/school/course/mathcounts-basics Basic] and an [http://artofproblemsolving.com/school/course/mathcounts-advanced Advanced] MATHCOUNTS course. The AoPS Introduction-level subject courses also include a great deal of MATHCOUNTS preparation. Many AoPS instructors are former National MATHCOUNTS Mathletes.<br />
<br />
=== MATHCOUNTS Online ===<br />
* [http://www.mathcounts.org Official MATHCOUNTS Homepage]<br />
* Art of Problem Solving hosts a large [http://artofproblemsolving.com/community/c3_middle_school_math Middle School Math Forum] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23209 MATHCOUNTS Coaches Forum].<br />
* The AoPS MATHCOUNTS Trainer is available on the [http://artofproblemsolving.com/mathcounts_trainer AoPS website], as well as on the [https://itunes.apple.com/us/app/mathcounts-trainer-math-contest/id1023961880?ls=1&mt=8 iPhone and iPad].<br />
* The free [http://www.artofproblemsolving.com/alcumus AoPS Alcumus learning system] includes thousands of MATHCOUNTS problems.<br />
* [http://artofproblemsolving.com/ftw/ftw.php For the Win!] gives students free Countdown Round-like practice against other AoPS students.<br />
* AoPS founder Richard Rusczyk has created dozens of [http://artofproblemsolving.com/videos/mathcounts MATHCOUNTS Mini video lessons].<br />
* [http://mathcounts.saab.org/ Elias Saab's MATHCOUNTS Preparation Homepage]<br />
* [http://www.unidata.ucar.edu/staff/russ/mathcounts/diaz.html The MATHCOUNTS Bible According to Mr. Diaz]<br />
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MATHCOUNTS Program] by [[Jeff Boyd]], who coached the 2005, 2007, and 2008 National Champion [[Texas MathCounts]] team.<br />
*[http://mathweb.scranton.edu/monks/courses/ProblemSolving/MathCountsPlaybookBW.pdf Coach Monk's MathCounts Playbook]<br />
* MathCounts Minis make hard problems easy<br />
<br />
== MATHCOUNTS Curriculum ==<br />
MATHCOUNTS curriculum includes [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.<br />
<br />
Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.<br />
<br />
== Past State Team Winners ==<br />
* 1984: Virginia<br />
* 1985: Florida<br />
* 1986: California<br />
* 1987: New York<br />
* 1988: New York<br />
* 1989: North Carolina<br />
* 1990: Ohio<br />
* 1991: Alabama<br />
* 1992: California<br />
* 1993: Kansas<br />
* 1994: Pennsylvania<br />
* 1995: Indiana<br />
* 1996: Wisconsin<br />
* 1997: Massachusetts<br />
* 1998: Wisconsin<br />
* 1999: Massachusetts<br />
* 2000: California<br />
* 2001: Virginia<br />
* 2002: California<br />
* 2003: California<br />
* 2004: Illinois<br />
* 2005: Texas<br />
* 2006: Virginia<br />
* 2007: Texas<br />
* 2008: Texas<br />
* 2009: Texas<br />
* 2010: California<br />
* 2011: California<br />
* 2012: Massachusetts<br />
* 2013: Massachusetts<br />
* 2014: California<br />
* 2015: Indiana<br />
* 2016: Texas<br />
* 2017: Texas<br />
* 2018: Texas<br />
* 2019: Massachusetts<br />
<br />
== MATHCOUNTS Competition Structure ==<br />
<br />
=== Sprint Round ===<br />
<br />
30 problems are given all at once. Students have 40 minutes to complete the Sprint Round. This round is very fast-paced and requires speed and accuracy as well. The earlier problems are usually the easiest problems in the competition, and the later problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.<br />
<br />
=== Target Round ===<br />
8 problems given 2 at a time. Students have 6 minutes to complete each set of two problems. Students may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target Round. Usually comprised of one "confidence booster" and one hard problem.<br />
<br />
=== Team Round ===<br />
<br />
10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).<br />
<br />
=== Countdown Round ===<br />
High scoring individuals compete head-to-head until a champion is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion. Calculators are not allowed, but scratch paper will be provided.<br />
<br />
<br />
====Chapter 1 the competitions====<br />
<br />
In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:<br />
<br />
*The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.<br />
<br />
*The winner of the first round goes up against the 8th place finisher.<br />
<br />
*The winner of the second round goes up against the 7th place finisher.<br />
<br />
This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three questions correct wins (as opposed to the best-out-of-three rule).<br />
<br />
If the countdown round is unofficial, any format may be used. Single-elimination bracket-style tournaments are common.<br />
<br />
====National Competition====<br />
<br />
At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first place written finisher to become champion, equalizing the field. But even then, a 12th place written competitor will have less of a chance to become champion than the top 4, because the top 4 get a bye. Until the semi-finals, the scoring is best out of five advances.<br />
<br />
At the first round and the second round, the person to correctly answer the most out of 5 questions wins. However, at the semifinals, the rules slightly change&mdash;the first person to correctly answer four questions wins.<br />
<br />
=== HELLOOOOOOOOOOOOOO ===<br />
In some states, (most notably Florida) there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "How much dirt is in a 3 ft by 3 ft by 4 ft hole?" The answer was 0 because there is no dirt in a hole.<br />
<br />
=== Masters Round ===<br />
Top students give in-depth explanations to challenging problems. This round is optional at the state level competition and is mandatory at the national competition (up to 2011). At nationals the top two on the written and countdown participate. <br />
In 2012, it was replaced by the Reel Math Challenge (now called the Math Video Challenge).<br />
<br />
=== Scoring and Ranking ===<br />
An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of <math>30 + 2(8) = 46</math> points.<br />
<br />
A team's score is the individual scores of its members divided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score. Note that when there are less than four members the score will become less.<br />
<br />
==Tiebreakers==<br />
If two or more students tie for a ranking with the same individual score, the people ranked in order is decided using the following algorithm(going to the next step if inconclusive)<br />
(1) Sprint score<br />
(2) Sum of the problem numbers correct(Unconfirmed)<br />
(3) Last problem done correctly(Unconfirmed), and checking down the list to previous problems if still tied<br />
(4) Flip a coin(Unconfirmed)<br />
<br />
In state contests, advancements to nationals in the case of a tie is often decided using a Tiebreaker Round(tied students are invited into a room and participate in a secret format of a contest to determine who advances to the National Contest)<br />
<br />
== MATHCOUNTS Competition Levels ==<br />
=== School Competition ===<br />
Students vie for the chance to make their school teams. Problems at this level are generally the easiest and most basic in curriculum.<br />
<br />
=== Chapter Competition ===<br />
Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.<br />
CHAPTER IS EZ AF!!!<br />
<br />
<br />
<br />
HELLLLLLLO<br />
I CAN EDIT THIS 4 SOME REASON<br />
<br />
=== DOG eating Competition ===<br />
The top 4 students in each state form the state team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. JIJIJIJIJII<br />
<br />
=== BIG BRAINS ONLY Competition ===<br />
==== National Fortnite<br />
Sites ====<br />
For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.<br />
<br />
* The 2020 competition was canceled due to the COVID-19 pandemic.<br />
* The 2019 competition was held in Orlando, Florida.<br />
* The 2018 competition was held in Washington, D.C.<br />
* The 2017 competition was held in Orlando, Florida.<br />
* The 2016 competition was held in Washington, D.C.<br />
* The 2015 competition was held in Boston, Massachusetts.<br />
* The 2014 competition was held in Orlando, Florida.<br />
* The 2013 competition was held in Washington, D.C.<br />
* The 2012 competition was held in Orlando, Florida.<br />
* The 2011 competition was held in Washington, D.C.<br />
* The 2009 and 2010 competitions were held in Orlando, Florida.<br />
* The 2008 competition was held in Denver, Colorado.<br />
* The 2007 competition was held in Fort Worth, Texas.<br />
* The 2006 competition was held in Arlington, Virginia.<br />
* The 2005 competition was held in Detroit, Michigan.<br />
* The 2004 competition was held in Washington, D.C.<br />
* The 2002 and 2003 competitions were held in Chicago, Illinois.<br />
<br />
== LOLOLOLOLOLOLOLOLOLOL ==<br />
<br />
Give the following competitions a try and take a look at the [[List of United States high school mathematics competitions]].<br />
* [[American Mathematics Competitions]]<br />
* [[American Regions Math League]]<br />
* [[Mandelbrot Competition]]<br />
* [[Mu Alpha Theta]]<br />
* [[BLAH BLAH BLAH]]<br />
<br />
[[Category:Mathematics competitions]]<br />
<br />
== See also... ==<br />
* [[List of national MATHCOUNTS teams]]<br />
* [[Mathematics competition resources]]<br />
* [[Math contest books]]<br />
* [[Math books]]<br />
* [[List of United States middle school mathematics competitions]]<br />
* [[List of United States high school mathematics competitions]]<br />
* [http://www.mathcounts.org/webarticles/anmviewer.asp?a=921&z=71 2006 MATHCOUNTS Countdown Video]<br />
<br />
[[Category:Introductory mathematics competitions]]</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=MATHCOUNTS&diff=122714MATHCOUNTS2020-05-21T14:33:52Z<p>Apple321: /* National Competition */</p>
<hr />
<div>Many AoPS Community members and online school students have been participants at National MATHCOUNTS, including many Nationals Countdown Round participants in the past decade. '''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] program that has served millions of middle school students since 1984. Sponsored by the [http://cna.com CNA Foundation], [http://www.nspe.org/ National Society of Professional Engineers], the [http://www.nctm.org/ National Council of Teachers of Mathematics], and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.<br />
<br />
{{Contest Info|name=MATHCOUNTS|region=USA|type=Free Response|difficulty=0.5 - 2.5|breakdown=<u>Countdown</u>: 0.5 (School/Chapter), 1 (State/National)<br><u>Sprint</u>: 1-1.5 (School/Chapter), 2-2.5 (State/National)<br><u>Target:</u> 1.5 (School), 2 (Chapter), 2-2.5 (State/National)}}<br />
<br />
== MATHCOUNTS Resources ==<br />
=== STOOOOOOOOOPID Books ===<br />
Art of Problem Solving's [http://artofproblemsolving.com/store/list/aops-curriculum Introductory subject textbooks] are ideal for students preparing for MATHCOUNTS, as are [http://artofproblemsolving.com/store/item/aops-vol1 AoPS Volume 1] and [http://artofproblemsolving.com/store/item/competition-math Competition Math for Middle School]<br />
<br />
=== Fortnite Classes ===<br />
Art of Problem Solving hosts a [http://artofproblemsolving.com/school/course/mathcounts-basics Basic] and an [http://artofproblemsolving.com/school/course/mathcounts-advanced Advanced] MATHCOUNTS course. The AoPS Introduction-level subject courses also include a great deal of MATHCOUNTS preparation. Many AoPS instructors are former National MATHCOUNTS Mathletes.<br />
<br />
=== MATHCOUNTS Online ===<br />
* [http://www.mathcounts.org Official MATHCOUNTS Homepage]<br />
* Art of Problem Solving hosts a large [http://artofproblemsolving.com/community/c3_middle_school_math Middle School Math Forum] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23209 MATHCOUNTS Coaches Forum].<br />
* The AoPS MATHCOUNTS Trainer is available on the [http://artofproblemsolving.com/mathcounts_trainer AoPS website], as well as on the [https://itunes.apple.com/us/app/mathcounts-trainer-math-contest/id1023961880?ls=1&mt=8 iPhone and iPad].<br />
* The free [http://www.artofproblemsolving.com/alcumus AoPS Alcumus learning system] includes thousands of MATHCOUNTS problems.<br />
* [http://artofproblemsolving.com/ftw/ftw.php For the Win!] gives students free Countdown Round-like practice against other AoPS students.<br />
* AoPS founder Richard Rusczyk has created dozens of [http://artofproblemsolving.com/videos/mathcounts MATHCOUNTS Mini video lessons].<br />
* [http://mathcounts.saab.org/ Elias Saab's MATHCOUNTS Preparation Homepage]<br />
* [http://www.unidata.ucar.edu/staff/russ/mathcounts/diaz.html The MATHCOUNTS Bible According to Mr. Diaz]<br />
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MATHCOUNTS Program] by [[Jeff Boyd]], who coached the 2005, 2007, and 2008 National Champion [[Texas MathCounts]] team.<br />
*[http://mathweb.scranton.edu/monks/courses/ProblemSolving/MathCountsPlaybookBW.pdf Coach Monk's MathCounts Playbook]<br />
* MathCounts Minis make hard problems easy<br />
<br />
== MATHCOUNTS Curriculum ==<br />
MATHCOUNTS curriculum includes [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.<br />
<br />
Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.<br />
<br />
== Past State Team Winners ==<br />
* 1984: Virginia<br />
* 1985: Florida<br />
* 1986: California<br />
* 1987: New York<br />
* 1988: New York<br />
* 1989: North Carolina<br />
* 1990: Ohio<br />
* 1991: Alabama<br />
* 1992: California<br />
* 1993: Kansas<br />
* 1994: Pennsylvania<br />
* 1995: Indiana<br />
* 1996: Wisconsin<br />
* 1997: Massachusetts<br />
* 1998: Wisconsin<br />
* 1999: Massachusetts<br />
* 2000: California<br />
* 2001: Virginia<br />
* 2002: California<br />
* 2003: California<br />
* 2004: Illinois<br />
* 2005: Texas<br />
* 2006: Virginia<br />
* 2007: Texas<br />
* 2008: Texas<br />
* 2009: Texas<br />
* 2010: California<br />
* 2011: California<br />
* 2012: Massachusetts<br />
* 2013: Massachusetts<br />
* 2014: California<br />
* 2015: Indiana<br />
* 2016: Texas<br />
* 2017: Texas<br />
* 2018: Texas<br />
* 2019: Massachusetts<br />
<br />
== MATHCOUNTS Competition Structure ==<br />
<br />
=== Sprint Round ===<br />
<br />
30 problems are given all at once. Students have 40 minutes to complete the Sprint Round. This round is very fast-paced and requires speed and accuracy as well. The earlier problems are usually the easiest problems in the competition, and the later problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.<br />
<br />
=== Target Round ===<br />
8 problems given 2 at a time. Students have 6 minutes to complete each set of two problems. Students may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target Round. Usually comprised of one "confidence booster" and one hard problem.<br />
<br />
=== Team Round ===<br />
<br />
10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).<br />
<br />
=== Countdown Round ===<br />
High scoring individuals compete head-to-head until a champion is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion. Calculators are not allowed, but scratch paper will be provided.<br />
<br />
<br />
====Chapter 1 the competitions====<br />
<br />
In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:<br />
<br />
*The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.<br />
<br />
*The winner of the first round goes up against the 8th place finisher.<br />
<br />
*The winner of the second round goes up against the 7th place finisher.<br />
<br />
This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three questions correct wins (as opposed to the best-out-of-three rule).<br />
<br />
If the countdown round is unofficial, any format may be used. Single-elimination bracket-style tournaments are common.<br />
<br />
====National Competition====<br />
<br />
At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first place written finisher to become champion, equalizing the field. But even then, a 12th place written competitor will have less of a chance to become champion than the top 4, because the top 4 get a bye. Until the semi-finals, the scoring is best out of five advances.<br />
<br />
At the first round and the second round, the person to correctly answer the most out of 5 questions wins. However, at the semifinals, the rules slightly change&mdash;the first person to correctly answer four questions wins.<br />
<br />
=== HELLOOOOOOOOOOOOOO ===<br />
In some states, (most notably Florida) there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "How much dirt is in a 3 ft by 3 ft by 4 ft hole?" The answer was 0 because there is no dirt in a hole.<br />
<br />
=== Masters Round ===<br />
Top students give in-depth explanations to challenging problems. This round is optional at the state level competition and is mandatory at the national competition (up to 2011). At nationals the top two on the written and countdown participate. <br />
In 2012, it was replaced by the Reel Math Challenge (now called the Math Video Challenge).<br />
<br />
=== Scoring and Ranking ===<br />
An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of <math>30 + 2(8) = 46</math> points.<br />
<br />
A team's score is the individual scores of its members divided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score. Note that when there are less than four members the score will become less.<br />
<br />
==Tiebreakers==<br />
If two or more students tie for a ranking with the same individual score, the people ranked in order is decided using the following algorithm(going to the next step if inconclusive)<br />
(1) Sprint score<br />
(2) Sum of the problem numbers correct(Unconfirmed)<br />
(3) Last problem done correctly(Unconfirmed), and checking down the list to previous problems if still tied<br />
(4) Flip a coin(Unconfirmed)<br />
<br />
In state contests, advancements to nationals in the case of a tie is often decided using a Tiebreaker Round(tied students are invited into a room and participate in a secret format of a contest to determine who advances to the National Contest)<br />
<br />
== MATHCOUNTS Competition Levels ==<br />
=== School Competition ===<br />
Students vie for the chance to make their school teams. Problems at this level are generally the easiest and most basic in curriculum.<br />
<br />
=== Chapter Competition ===<br />
Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.<br />
CHAPTER IS EZ AF!!!<br />
<br />
<br />
<br />
HELLLLLLLO<br />
I CAN EDIT THIS 4 SOME REASON<br />
<br />
=== jiji eating Competition ===<br />
The top 4 students in each state form the state team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. JIJIJIJIJII<br />
<br />
=== BIG BRAINS ONLY Competition ===<br />
==== National Fortnite<br />
Sites ====<br />
For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.<br />
<br />
* The 2020 competition was canceled due to the COVID-19 pandemic.<br />
* The 2019 competition was held in Orlando, Florida.<br />
* The 2018 competition was held in Washington, D.C.<br />
* The 2017 competition was held in Orlando, Florida.<br />
* The 2016 competition was held in Washington, D.C.<br />
* The 2015 competition was held in Boston, Massachusetts.<br />
* The 2014 competition was held in Orlando, Florida.<br />
* The 2013 competition was held in Washington, D.C.<br />
* The 2012 competition was held in Orlando, Florida.<br />
* The 2011 competition was held in Washington, D.C.<br />
* The 2009 and 2010 competitions were held in Orlando, Florida.<br />
* The 2008 competition was held in Denver, Colorado.<br />
* The 2007 competition was held in Fort Worth, Texas.<br />
* The 2006 competition was held in Arlington, Virginia.<br />
* The 2005 competition was held in Detroit, Michigan.<br />
* The 2004 competition was held in Washington, D.C.<br />
* The 2002 and 2003 competitions were held in Chicago, Illinois.<br />
<br />
== LOLOLOLOLOLOLOLOLOLOL ==<br />
<br />
Give the following competitions a try and take a look at the [[List of United States high school mathematics competitions]].<br />
* [[American Mathematics Competitions]]<br />
* [[American Regions Math League]]<br />
* [[Mandelbrot Competition]]<br />
* [[Mu Alpha Theta]]<br />
* [[BLAH BLAH BLAH]]<br />
<br />
[[Category:Mathematics competitions]]<br />
<br />
== See also... ==<br />
* [[List of national MATHCOUNTS teams]]<br />
* [[Mathematics competition resources]]<br />
* [[Math contest books]]<br />
* [[Math books]]<br />
* [[List of United States middle school mathematics competitions]]<br />
* [[List of United States high school mathematics competitions]]<br />
* [http://www.mathcounts.org/webarticles/anmviewer.asp?a=921&z=71 2006 MATHCOUNTS Countdown Video]<br />
<br />
[[Category:Introductory mathematics competitions]]</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=MATHCOUNTS&diff=122713MATHCOUNTS2020-05-21T14:33:12Z<p>Apple321: /* Chapter Competition */</p>
<hr />
<div>Many AoPS Community members and online school students have been participants at National MATHCOUNTS, including many Nationals Countdown Round participants in the past decade. '''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] program that has served millions of middle school students since 1984. Sponsored by the [http://cna.com CNA Foundation], [http://www.nspe.org/ National Society of Professional Engineers], the [http://www.nctm.org/ National Council of Teachers of Mathematics], and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.<br />
<br />
{{Contest Info|name=MATHCOUNTS|region=USA|type=Free Response|difficulty=0.5 - 2.5|breakdown=<u>Countdown</u>: 0.5 (School/Chapter), 1 (State/National)<br><u>Sprint</u>: 1-1.5 (School/Chapter), 2-2.5 (State/National)<br><u>Target:</u> 1.5 (School), 2 (Chapter), 2-2.5 (State/National)}}<br />
<br />
== MATHCOUNTS Resources ==<br />
=== STOOOOOOOOOPID Books ===<br />
Art of Problem Solving's [http://artofproblemsolving.com/store/list/aops-curriculum Introductory subject textbooks] are ideal for students preparing for MATHCOUNTS, as are [http://artofproblemsolving.com/store/item/aops-vol1 AoPS Volume 1] and [http://artofproblemsolving.com/store/item/competition-math Competition Math for Middle School]<br />
<br />
=== Fortnite Classes ===<br />
Art of Problem Solving hosts a [http://artofproblemsolving.com/school/course/mathcounts-basics Basic] and an [http://artofproblemsolving.com/school/course/mathcounts-advanced Advanced] MATHCOUNTS course. The AoPS Introduction-level subject courses also include a great deal of MATHCOUNTS preparation. Many AoPS instructors are former National MATHCOUNTS Mathletes.<br />
<br />
=== MATHCOUNTS Online ===<br />
* [http://www.mathcounts.org Official MATHCOUNTS Homepage]<br />
* Art of Problem Solving hosts a large [http://artofproblemsolving.com/community/c3_middle_school_math Middle School Math Forum] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23209 MATHCOUNTS Coaches Forum].<br />
* The AoPS MATHCOUNTS Trainer is available on the [http://artofproblemsolving.com/mathcounts_trainer AoPS website], as well as on the [https://itunes.apple.com/us/app/mathcounts-trainer-math-contest/id1023961880?ls=1&mt=8 iPhone and iPad].<br />
* The free [http://www.artofproblemsolving.com/alcumus AoPS Alcumus learning system] includes thousands of MATHCOUNTS problems.<br />
* [http://artofproblemsolving.com/ftw/ftw.php For the Win!] gives students free Countdown Round-like practice against other AoPS students.<br />
* AoPS founder Richard Rusczyk has created dozens of [http://artofproblemsolving.com/videos/mathcounts MATHCOUNTS Mini video lessons].<br />
* [http://mathcounts.saab.org/ Elias Saab's MATHCOUNTS Preparation Homepage]<br />
* [http://www.unidata.ucar.edu/staff/russ/mathcounts/diaz.html The MATHCOUNTS Bible According to Mr. Diaz]<br />
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MATHCOUNTS Program] by [[Jeff Boyd]], who coached the 2005, 2007, and 2008 National Champion [[Texas MathCounts]] team.<br />
*[http://mathweb.scranton.edu/monks/courses/ProblemSolving/MathCountsPlaybookBW.pdf Coach Monk's MathCounts Playbook]<br />
* MathCounts Minis make hard problems easy<br />
<br />
== MATHCOUNTS Curriculum ==<br />
MATHCOUNTS curriculum includes [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.<br />
<br />
Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.<br />
<br />
== Past State Team Winners ==<br />
* 1984: Virginia<br />
* 1985: Florida<br />
* 1986: California<br />
* 1987: New York<br />
* 1988: New York<br />
* 1989: North Carolina<br />
* 1990: Ohio<br />
* 1991: Alabama<br />
* 1992: California<br />
* 1993: Kansas<br />
* 1994: Pennsylvania<br />
* 1995: Indiana<br />
* 1996: Wisconsin<br />
* 1997: Massachusetts<br />
* 1998: Wisconsin<br />
* 1999: Massachusetts<br />
* 2000: California<br />
* 2001: Virginia<br />
* 2002: California<br />
* 2003: California<br />
* 2004: Illinois<br />
* 2005: Texas<br />
* 2006: Virginia<br />
* 2007: Texas<br />
* 2008: Texas<br />
* 2009: Texas<br />
* 2010: California<br />
* 2011: California<br />
* 2012: Massachusetts<br />
* 2013: Massachusetts<br />
* 2014: California<br />
* 2015: Indiana<br />
* 2016: Texas<br />
* 2017: Texas<br />
* 2018: Texas<br />
* 2019: Massachusetts<br />
<br />
== MATHCOUNTS Competition Structure ==<br />
<br />
=== Sprint Round ===<br />
<br />
30 problems are given all at once. Students have 40 minutes to complete the Sprint Round. This round is very fast-paced and requires speed and accuracy as well. The earlier problems are usually the easiest problems in the competition, and the later problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.<br />
<br />
=== Target Round ===<br />
8 problems given 2 at a time. Students have 6 minutes to complete each set of two problems. Students may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target Round. Usually comprised of one "confidence booster" and one hard problem.<br />
<br />
=== Team Round ===<br />
<br />
10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).<br />
<br />
=== Countdown Round ===<br />
High scoring individuals compete head-to-head until a champion is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion. Calculators are not allowed, but scratch paper will be provided.<br />
<br />
<br />
====Chapter 1 the competitions====<br />
<br />
In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:<br />
<br />
*The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.<br />
<br />
*The winner of the first round goes up against the 8th place finisher.<br />
<br />
*The winner of the second round goes up against the 7th place finisher.<br />
<br />
This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three questions correct wins (as opposed to the best-out-of-three rule).<br />
<br />
If the countdown round is unofficial, any format may be used. Single-elimination bracket-style tournaments are common.<br />
<br />
====National Competition====<br />
<br />
At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first place written finisher to become champion, equalizing the field. But even then, a 12th place written competitor will have less of a chance to become champion than the top 4, because the top 4 get a bye. Until the semi-finals, the scoring is best out of five advances.<br />
<br />
At the first round and the second round, the person to correctly answer the most out of 5 questions wins. However, at the semifinals, the rules slightly change&mdash;the first person to correctly answer four questions wins.<br />
<br />
=== HELLOOOOOOOOOOOOOO ===<br />
In some states, (most notably Florida) there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "How much dirt is in a 3 ft by 3 ft by 4 ft hole?" The answer was 0 because there is no dirt in a hole.<br />
<br />
=== Masters Round ===<br />
Top students give in-depth explanations to challenging problems. This round is optional at the state level competition and is mandatory at the national competition (up to 2011). At nationals the top two on the written and countdown participate. <br />
In 2012, it was replaced by the Reel Math Challenge (now called the Math Video Challenge).<br />
<br />
=== Scoring and Ranking ===<br />
An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of <math>30 + 2(8) = 46</math> points.<br />
<br />
A team's score is the individual scores of its members divided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score. Note that when there are less than four members the score will become less.<br />
<br />
==Tiebreakers==<br />
If two or more students tie for a ranking with the same individual score, the people ranked in order is decided using the following algorithm(going to the next step if inconclusive)<br />
(1) Sprint score<br />
(2) Sum of the problem numbers correct(Unconfirmed)<br />
(3) Last problem done correctly(Unconfirmed), and checking down the list to previous problems if still tied<br />
(4) Flip a coin(Unconfirmed)<br />
<br />
In state contests, advancements to nationals in the case of a tie is often decided using a Tiebreaker Round(tied students are invited into a room and participate in a secret format of a contest to determine who advances to the National Contest)<br />
<br />
== MATHCOUNTS Competition Levels ==<br />
=== School Competition ===<br />
Students vie for the chance to make their school teams. Problems at this level are generally the easiest and most basic in curriculum.<br />
<br />
=== Chapter Competition ===<br />
Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.<br />
CHAPTER IS EZ AF!!!<br />
<br />
<br />
<br />
HELLLLLLLO<br />
I CAN EDIT THIS 4 SOME REASON<br />
<br />
=== jiji eating Competition ===<br />
The top 4 students in each state form the state team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. JIJIJIJIJII<br />
<br />
=== National Competition ===<br />
==== National Fortnite<br />
Sites ====<br />
For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.<br />
<br />
* The 2020 competition was canceled due to the COVID-19 pandemic.<br />
* The 2019 competition was held in Orlando, Florida.<br />
* The 2018 competition was held in Washington, D.C.<br />
* The 2017 competition was held in Orlando, Florida.<br />
* The 2016 competition was held in Washington, D.C.<br />
* The 2015 competition was held in Boston, Massachusetts.<br />
* The 2014 competition was held in Orlando, Florida.<br />
* The 2013 competition was held in Washington, D.C.<br />
* The 2012 competition was held in Orlando, Florida.<br />
* The 2011 competition was held in Washington, D.C.<br />
* The 2009 and 2010 competitions were held in Orlando, Florida.<br />
* The 2008 competition was held in Denver, Colorado.<br />
* The 2007 competition was held in Fort Worth, Texas.<br />
* The 2006 competition was held in Arlington, Virginia.<br />
* The 2005 competition was held in Detroit, Michigan.<br />
* The 2004 competition was held in Washington, D.C.<br />
* The 2002 and 2003 competitions were held in Chicago, Illinois.<br />
<br />
== LOLOLOLOLOLOLOLOLOLOL ==<br />
<br />
Give the following competitions a try and take a look at the [[List of United States high school mathematics competitions]].<br />
* [[American Mathematics Competitions]]<br />
* [[American Regions Math League]]<br />
* [[Mandelbrot Competition]]<br />
* [[Mu Alpha Theta]]<br />
* [[BLAH BLAH BLAH]]<br />
<br />
[[Category:Mathematics competitions]]<br />
<br />
== See also... ==<br />
* [[List of national MATHCOUNTS teams]]<br />
* [[Mathematics competition resources]]<br />
* [[Math contest books]]<br />
* [[Math books]]<br />
* [[List of United States middle school mathematics competitions]]<br />
* [[List of United States high school mathematics competitions]]<br />
* [http://www.mathcounts.org/webarticles/anmviewer.asp?a=921&z=71 2006 MATHCOUNTS Countdown Video]<br />
<br />
[[Category:Introductory mathematics competitions]]</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=MATHCOUNTS&diff=122711MATHCOUNTS2020-05-21T14:32:41Z<p>Apple321: /* State Competition */</p>
<hr />
<div>Many AoPS Community members and online school students have been participants at National MATHCOUNTS, including many Nationals Countdown Round participants in the past decade. '''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] program that has served millions of middle school students since 1984. Sponsored by the [http://cna.com CNA Foundation], [http://www.nspe.org/ National Society of Professional Engineers], the [http://www.nctm.org/ National Council of Teachers of Mathematics], and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.<br />
<br />
{{Contest Info|name=MATHCOUNTS|region=USA|type=Free Response|difficulty=0.5 - 2.5|breakdown=<u>Countdown</u>: 0.5 (School/Chapter), 1 (State/National)<br><u>Sprint</u>: 1-1.5 (School/Chapter), 2-2.5 (State/National)<br><u>Target:</u> 1.5 (School), 2 (Chapter), 2-2.5 (State/National)}}<br />
<br />
== MATHCOUNTS Resources ==<br />
=== STOOOOOOOOOPID Books ===<br />
Art of Problem Solving's [http://artofproblemsolving.com/store/list/aops-curriculum Introductory subject textbooks] are ideal for students preparing for MATHCOUNTS, as are [http://artofproblemsolving.com/store/item/aops-vol1 AoPS Volume 1] and [http://artofproblemsolving.com/store/item/competition-math Competition Math for Middle School]<br />
<br />
=== Fortnite Classes ===<br />
Art of Problem Solving hosts a [http://artofproblemsolving.com/school/course/mathcounts-basics Basic] and an [http://artofproblemsolving.com/school/course/mathcounts-advanced Advanced] MATHCOUNTS course. The AoPS Introduction-level subject courses also include a great deal of MATHCOUNTS preparation. Many AoPS instructors are former National MATHCOUNTS Mathletes.<br />
<br />
=== MATHCOUNTS Online ===<br />
* [http://www.mathcounts.org Official MATHCOUNTS Homepage]<br />
* Art of Problem Solving hosts a large [http://artofproblemsolving.com/community/c3_middle_school_math Middle School Math Forum] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23209 MATHCOUNTS Coaches Forum].<br />
* The AoPS MATHCOUNTS Trainer is available on the [http://artofproblemsolving.com/mathcounts_trainer AoPS website], as well as on the [https://itunes.apple.com/us/app/mathcounts-trainer-math-contest/id1023961880?ls=1&mt=8 iPhone and iPad].<br />
* The free [http://www.artofproblemsolving.com/alcumus AoPS Alcumus learning system] includes thousands of MATHCOUNTS problems.<br />
* [http://artofproblemsolving.com/ftw/ftw.php For the Win!] gives students free Countdown Round-like practice against other AoPS students.<br />
* AoPS founder Richard Rusczyk has created dozens of [http://artofproblemsolving.com/videos/mathcounts MATHCOUNTS Mini video lessons].<br />
* [http://mathcounts.saab.org/ Elias Saab's MATHCOUNTS Preparation Homepage]<br />
* [http://www.unidata.ucar.edu/staff/russ/mathcounts/diaz.html The MATHCOUNTS Bible According to Mr. Diaz]<br />
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MATHCOUNTS Program] by [[Jeff Boyd]], who coached the 2005, 2007, and 2008 National Champion [[Texas MathCounts]] team.<br />
*[http://mathweb.scranton.edu/monks/courses/ProblemSolving/MathCountsPlaybookBW.pdf Coach Monk's MathCounts Playbook]<br />
* MathCounts Minis make hard problems easy<br />
<br />
== MATHCOUNTS Curriculum ==<br />
MATHCOUNTS curriculum includes [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.<br />
<br />
Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.<br />
<br />
== Past State Team Winners ==<br />
* 1984: Virginia<br />
* 1985: Florida<br />
* 1986: California<br />
* 1987: New York<br />
* 1988: New York<br />
* 1989: North Carolina<br />
* 1990: Ohio<br />
* 1991: Alabama<br />
* 1992: California<br />
* 1993: Kansas<br />
* 1994: Pennsylvania<br />
* 1995: Indiana<br />
* 1996: Wisconsin<br />
* 1997: Massachusetts<br />
* 1998: Wisconsin<br />
* 1999: Massachusetts<br />
* 2000: California<br />
* 2001: Virginia<br />
* 2002: California<br />
* 2003: California<br />
* 2004: Illinois<br />
* 2005: Texas<br />
* 2006: Virginia<br />
* 2007: Texas<br />
* 2008: Texas<br />
* 2009: Texas<br />
* 2010: California<br />
* 2011: California<br />
* 2012: Massachusetts<br />
* 2013: Massachusetts<br />
* 2014: California<br />
* 2015: Indiana<br />
* 2016: Texas<br />
* 2017: Texas<br />
* 2018: Texas<br />
* 2019: Massachusetts<br />
<br />
== MATHCOUNTS Competition Structure ==<br />
<br />
=== Sprint Round ===<br />
<br />
30 problems are given all at once. Students have 40 minutes to complete the Sprint Round. This round is very fast-paced and requires speed and accuracy as well. The earlier problems are usually the easiest problems in the competition, and the later problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.<br />
<br />
=== Target Round ===<br />
8 problems given 2 at a time. Students have 6 minutes to complete each set of two problems. Students may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target Round. Usually comprised of one "confidence booster" and one hard problem.<br />
<br />
=== Team Round ===<br />
<br />
10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).<br />
<br />
=== Countdown Round ===<br />
High scoring individuals compete head-to-head until a champion is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion. Calculators are not allowed, but scratch paper will be provided.<br />
<br />
<br />
====Chapter and State Competitions====<br />
<br />
In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:<br />
<br />
*The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.<br />
<br />
*The winner of the first round goes up against the 8th place finisher.<br />
<br />
*The winner of the second round goes up against the 7th place finisher.<br />
<br />
This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three questions correct wins (as opposed to the best-out-of-three rule).<br />
<br />
If the countdown round is unofficial, any format may be used. Single-elimination bracket-style tournaments are common.<br />
<br />
====National Competition====<br />
<br />
At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first place written finisher to become champion, equalizing the field. But even then, a 12th place written competitor will have less of a chance to become champion than the top 4, because the top 4 get a bye. Until the semi-finals, the scoring is best out of five advances.<br />
<br />
At the first round and the second round, the person to correctly answer the most out of 5 questions wins. However, at the semifinals, the rules slightly change&mdash;the first person to correctly answer four questions wins.<br />
<br />
=== HELLOOOOOOOOOOOOOO ===<br />
In some states, (most notably Florida) there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "How much dirt is in a 3 ft by 3 ft by 4 ft hole?" The answer was 0 because there is no dirt in a hole.<br />
<br />
=== Masters Round ===<br />
Top students give in-depth explanations to challenging problems. This round is optional at the state level competition and is mandatory at the national competition (up to 2011). At nationals the top two on the written and countdown participate. <br />
In 2012, it was replaced by the Reel Math Challenge (now called the Math Video Challenge).<br />
<br />
=== Scoring and Ranking ===<br />
An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of <math>30 + 2(8) = 46</math> points.<br />
<br />
A team's score is the individual scores of its members divided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score. Note that when there are less than four members the score will become less.<br />
<br />
==Tiebreakers==<br />
If two or more students tie for a ranking with the same individual score, the people ranked in order is decided using the following algorithm(going to the next step if inconclusive)<br />
(1) Sprint score<br />
(2) Sum of the problem numbers correct(Unconfirmed)<br />
(3) Last problem done correctly(Unconfirmed), and checking down the list to previous problems if still tied<br />
(4) Flip a coin(Unconfirmed)<br />
<br />
In state contests, advancements to nationals in the case of a tie is often decided using a Tiebreaker Round(tied students are invited into a room and participate in a secret format of a contest to determine who advances to the National Contest)<br />
<br />
== MATHCOUNTS Competition Levels ==<br />
=== School Competition ===<br />
Students vie for the chance to make their school teams. Problems at this level are generally the easiest and most basic in curriculum.<br />
<br />
=== Chapter Competition ===<br />
Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.<br />
<br />
<br />
<br />
HELLLLLLLO<br />
I CAN EDIT THIS 4 SOME REASON<br />
<br />
=== jiji eating Competition ===<br />
The top 4 students in each state form the state team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. JIJIJIJIJII<br />
<br />
=== National Competition ===<br />
==== National Fortnite<br />
Sites ====<br />
For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.<br />
<br />
* The 2020 competition was canceled due to the COVID-19 pandemic.<br />
* The 2019 competition was held in Orlando, Florida.<br />
* The 2018 competition was held in Washington, D.C.<br />
* The 2017 competition was held in Orlando, Florida.<br />
* The 2016 competition was held in Washington, D.C.<br />
* The 2015 competition was held in Boston, Massachusetts.<br />
* The 2014 competition was held in Orlando, Florida.<br />
* The 2013 competition was held in Washington, D.C.<br />
* The 2012 competition was held in Orlando, Florida.<br />
* The 2011 competition was held in Washington, D.C.<br />
* The 2009 and 2010 competitions were held in Orlando, Florida.<br />
* The 2008 competition was held in Denver, Colorado.<br />
* The 2007 competition was held in Fort Worth, Texas.<br />
* The 2006 competition was held in Arlington, Virginia.<br />
* The 2005 competition was held in Detroit, Michigan.<br />
* The 2004 competition was held in Washington, D.C.<br />
* The 2002 and 2003 competitions were held in Chicago, Illinois.<br />
<br />
== LOLOLOLOLOLOLOLOLOLOL ==<br />
<br />
Give the following competitions a try and take a look at the [[List of United States high school mathematics competitions]].<br />
* [[American Mathematics Competitions]]<br />
* [[American Regions Math League]]<br />
* [[Mandelbrot Competition]]<br />
* [[Mu Alpha Theta]]<br />
* [[BLAH BLAH BLAH]]<br />
<br />
[[Category:Mathematics competitions]]<br />
<br />
== See also... ==<br />
* [[List of national MATHCOUNTS teams]]<br />
* [[Mathematics competition resources]]<br />
* [[Math contest books]]<br />
* [[Math books]]<br />
* [[List of United States middle school mathematics competitions]]<br />
* [[List of United States high school mathematics competitions]]<br />
* [http://www.mathcounts.org/webarticles/anmviewer.asp?a=921&z=71 2006 MATHCOUNTS Countdown Video]<br />
<br />
[[Category:Introductory mathematics competitions]]</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=MATHCOUNTS&diff=122710MATHCOUNTS2020-05-21T14:32:24Z<p>Apple321: /* National Competition */</p>
<hr />
<div>Many AoPS Community members and online school students have been participants at National MATHCOUNTS, including many Nationals Countdown Round participants in the past decade. '''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] program that has served millions of middle school students since 1984. Sponsored by the [http://cna.com CNA Foundation], [http://www.nspe.org/ National Society of Professional Engineers], the [http://www.nctm.org/ National Council of Teachers of Mathematics], and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.<br />
<br />
{{Contest Info|name=MATHCOUNTS|region=USA|type=Free Response|difficulty=0.5 - 2.5|breakdown=<u>Countdown</u>: 0.5 (School/Chapter), 1 (State/National)<br><u>Sprint</u>: 1-1.5 (School/Chapter), 2-2.5 (State/National)<br><u>Target:</u> 1.5 (School), 2 (Chapter), 2-2.5 (State/National)}}<br />
<br />
== MATHCOUNTS Resources ==<br />
=== STOOOOOOOOOPID Books ===<br />
Art of Problem Solving's [http://artofproblemsolving.com/store/list/aops-curriculum Introductory subject textbooks] are ideal for students preparing for MATHCOUNTS, as are [http://artofproblemsolving.com/store/item/aops-vol1 AoPS Volume 1] and [http://artofproblemsolving.com/store/item/competition-math Competition Math for Middle School]<br />
<br />
=== Fortnite Classes ===<br />
Art of Problem Solving hosts a [http://artofproblemsolving.com/school/course/mathcounts-basics Basic] and an [http://artofproblemsolving.com/school/course/mathcounts-advanced Advanced] MATHCOUNTS course. The AoPS Introduction-level subject courses also include a great deal of MATHCOUNTS preparation. Many AoPS instructors are former National MATHCOUNTS Mathletes.<br />
<br />
=== MATHCOUNTS Online ===<br />
* [http://www.mathcounts.org Official MATHCOUNTS Homepage]<br />
* Art of Problem Solving hosts a large [http://artofproblemsolving.com/community/c3_middle_school_math Middle School Math Forum] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23209 MATHCOUNTS Coaches Forum].<br />
* The AoPS MATHCOUNTS Trainer is available on the [http://artofproblemsolving.com/mathcounts_trainer AoPS website], as well as on the [https://itunes.apple.com/us/app/mathcounts-trainer-math-contest/id1023961880?ls=1&mt=8 iPhone and iPad].<br />
* The free [http://www.artofproblemsolving.com/alcumus AoPS Alcumus learning system] includes thousands of MATHCOUNTS problems.<br />
* [http://artofproblemsolving.com/ftw/ftw.php For the Win!] gives students free Countdown Round-like practice against other AoPS students.<br />
* AoPS founder Richard Rusczyk has created dozens of [http://artofproblemsolving.com/videos/mathcounts MATHCOUNTS Mini video lessons].<br />
* [http://mathcounts.saab.org/ Elias Saab's MATHCOUNTS Preparation Homepage]<br />
* [http://www.unidata.ucar.edu/staff/russ/mathcounts/diaz.html The MATHCOUNTS Bible According to Mr. Diaz]<br />
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MATHCOUNTS Program] by [[Jeff Boyd]], who coached the 2005, 2007, and 2008 National Champion [[Texas MathCounts]] team.<br />
*[http://mathweb.scranton.edu/monks/courses/ProblemSolving/MathCountsPlaybookBW.pdf Coach Monk's MathCounts Playbook]<br />
* MathCounts Minis make hard problems easy<br />
<br />
== MATHCOUNTS Curriculum ==<br />
MATHCOUNTS curriculum includes [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.<br />
<br />
Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.<br />
<br />
== Past State Team Winners ==<br />
* 1984: Virginia<br />
* 1985: Florida<br />
* 1986: California<br />
* 1987: New York<br />
* 1988: New York<br />
* 1989: North Carolina<br />
* 1990: Ohio<br />
* 1991: Alabama<br />
* 1992: California<br />
* 1993: Kansas<br />
* 1994: Pennsylvania<br />
* 1995: Indiana<br />
* 1996: Wisconsin<br />
* 1997: Massachusetts<br />
* 1998: Wisconsin<br />
* 1999: Massachusetts<br />
* 2000: California<br />
* 2001: Virginia<br />
* 2002: California<br />
* 2003: California<br />
* 2004: Illinois<br />
* 2005: Texas<br />
* 2006: Virginia<br />
* 2007: Texas<br />
* 2008: Texas<br />
* 2009: Texas<br />
* 2010: California<br />
* 2011: California<br />
* 2012: Massachusetts<br />
* 2013: Massachusetts<br />
* 2014: California<br />
* 2015: Indiana<br />
* 2016: Texas<br />
* 2017: Texas<br />
* 2018: Texas<br />
* 2019: Massachusetts<br />
<br />
== MATHCOUNTS Competition Structure ==<br />
<br />
=== Sprint Round ===<br />
<br />
30 problems are given all at once. Students have 40 minutes to complete the Sprint Round. This round is very fast-paced and requires speed and accuracy as well. The earlier problems are usually the easiest problems in the competition, and the later problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.<br />
<br />
=== Target Round ===<br />
8 problems given 2 at a time. Students have 6 minutes to complete each set of two problems. Students may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target Round. Usually comprised of one "confidence booster" and one hard problem.<br />
<br />
=== Team Round ===<br />
<br />
10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).<br />
<br />
=== Countdown Round ===<br />
High scoring individuals compete head-to-head until a champion is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion. Calculators are not allowed, but scratch paper will be provided.<br />
<br />
<br />
====Chapter and State Competitions====<br />
<br />
In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:<br />
<br />
*The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.<br />
<br />
*The winner of the first round goes up against the 8th place finisher.<br />
<br />
*The winner of the second round goes up against the 7th place finisher.<br />
<br />
This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three questions correct wins (as opposed to the best-out-of-three rule).<br />
<br />
If the countdown round is unofficial, any format may be used. Single-elimination bracket-style tournaments are common.<br />
<br />
====National Competition====<br />
<br />
At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first place written finisher to become champion, equalizing the field. But even then, a 12th place written competitor will have less of a chance to become champion than the top 4, because the top 4 get a bye. Until the semi-finals, the scoring is best out of five advances.<br />
<br />
At the first round and the second round, the person to correctly answer the most out of 5 questions wins. However, at the semifinals, the rules slightly change&mdash;the first person to correctly answer four questions wins.<br />
<br />
=== HELLOOOOOOOOOOOOOO ===<br />
In some states, (most notably Florida) there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "How much dirt is in a 3 ft by 3 ft by 4 ft hole?" The answer was 0 because there is no dirt in a hole.<br />
<br />
=== Masters Round ===<br />
Top students give in-depth explanations to challenging problems. This round is optional at the state level competition and is mandatory at the national competition (up to 2011). At nationals the top two on the written and countdown participate. <br />
In 2012, it was replaced by the Reel Math Challenge (now called the Math Video Challenge).<br />
<br />
=== Scoring and Ranking ===<br />
An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of <math>30 + 2(8) = 46</math> points.<br />
<br />
A team's score is the individual scores of its members divided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score. Note that when there are less than four members the score will become less.<br />
<br />
==Tiebreakers==<br />
If two or more students tie for a ranking with the same individual score, the people ranked in order is decided using the following algorithm(going to the next step if inconclusive)<br />
(1) Sprint score<br />
(2) Sum of the problem numbers correct(Unconfirmed)<br />
(3) Last problem done correctly(Unconfirmed), and checking down the list to previous problems if still tied<br />
(4) Flip a coin(Unconfirmed)<br />
<br />
In state contests, advancements to nationals in the case of a tie is often decided using a Tiebreaker Round(tied students are invited into a room and participate in a secret format of a contest to determine who advances to the National Contest)<br />
<br />
== MATHCOUNTS Competition Levels ==<br />
=== School Competition ===<br />
Students vie for the chance to make their school teams. Problems at this level are generally the easiest and most basic in curriculum.<br />
<br />
=== Chapter Competition ===<br />
Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.<br />
<br />
<br />
<br />
HELLLLLLLO<br />
I CAN EDIT THIS 4 SOME REASON<br />
<br />
=== State Competition ===<br />
The top 4 students in each state form the state team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. JIJIJIJIJII<br />
<br />
=== National Competition ===<br />
==== National Fortnite<br />
Sites ====<br />
For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.<br />
<br />
* The 2020 competition was canceled due to the COVID-19 pandemic.<br />
* The 2019 competition was held in Orlando, Florida.<br />
* The 2018 competition was held in Washington, D.C.<br />
* The 2017 competition was held in Orlando, Florida.<br />
* The 2016 competition was held in Washington, D.C.<br />
* The 2015 competition was held in Boston, Massachusetts.<br />
* The 2014 competition was held in Orlando, Florida.<br />
* The 2013 competition was held in Washington, D.C.<br />
* The 2012 competition was held in Orlando, Florida.<br />
* The 2011 competition was held in Washington, D.C.<br />
* The 2009 and 2010 competitions were held in Orlando, Florida.<br />
* The 2008 competition was held in Denver, Colorado.<br />
* The 2007 competition was held in Fort Worth, Texas.<br />
* The 2006 competition was held in Arlington, Virginia.<br />
* The 2005 competition was held in Detroit, Michigan.<br />
* The 2004 competition was held in Washington, D.C.<br />
* The 2002 and 2003 competitions were held in Chicago, Illinois.<br />
<br />
== LOLOLOLOLOLOLOLOLOLOL ==<br />
<br />
Give the following competitions a try and take a look at the [[List of United States high school mathematics competitions]].<br />
* [[American Mathematics Competitions]]<br />
* [[American Regions Math League]]<br />
* [[Mandelbrot Competition]]<br />
* [[Mu Alpha Theta]]<br />
* [[BLAH BLAH BLAH]]<br />
<br />
[[Category:Mathematics competitions]]<br />
<br />
== See also... ==<br />
* [[List of national MATHCOUNTS teams]]<br />
* [[Mathematics competition resources]]<br />
* [[Math contest books]]<br />
* [[Math books]]<br />
* [[List of United States middle school mathematics competitions]]<br />
* [[List of United States high school mathematics competitions]]<br />
* [http://www.mathcounts.org/webarticles/anmviewer.asp?a=921&z=71 2006 MATHCOUNTS Countdown Video]<br />
<br />
[[Category:Introductory mathematics competitions]]</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=MATHCOUNTS&diff=122707MATHCOUNTS2020-05-21T14:31:33Z<p>Apple321: /* MATHCOUNTS Books */</p>
<hr />
<div>Many AoPS Community members and online school students have been participants at National MATHCOUNTS, including many Nationals Countdown Round participants in the past decade. '''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] program that has served millions of middle school students since 1984. Sponsored by the [http://cna.com CNA Foundation], [http://www.nspe.org/ National Society of Professional Engineers], the [http://www.nctm.org/ National Council of Teachers of Mathematics], and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.<br />
<br />
{{Contest Info|name=MATHCOUNTS|region=USA|type=Free Response|difficulty=0.5 - 2.5|breakdown=<u>Countdown</u>: 0.5 (School/Chapter), 1 (State/National)<br><u>Sprint</u>: 1-1.5 (School/Chapter), 2-2.5 (State/National)<br><u>Target:</u> 1.5 (School), 2 (Chapter), 2-2.5 (State/National)}}<br />
<br />
== MATHCOUNTS Resources ==<br />
=== STOOOOOOOOOPID Books ===<br />
Art of Problem Solving's [http://artofproblemsolving.com/store/list/aops-curriculum Introductory subject textbooks] are ideal for students preparing for MATHCOUNTS, as are [http://artofproblemsolving.com/store/item/aops-vol1 AoPS Volume 1] and [http://artofproblemsolving.com/store/item/competition-math Competition Math for Middle School]<br />
<br />
=== Fortnite Classes ===<br />
Art of Problem Solving hosts a [http://artofproblemsolving.com/school/course/mathcounts-basics Basic] and an [http://artofproblemsolving.com/school/course/mathcounts-advanced Advanced] MATHCOUNTS course. The AoPS Introduction-level subject courses also include a great deal of MATHCOUNTS preparation. Many AoPS instructors are former National MATHCOUNTS Mathletes.<br />
<br />
=== MATHCOUNTS Online ===<br />
* [http://www.mathcounts.org Official MATHCOUNTS Homepage]<br />
* Art of Problem Solving hosts a large [http://artofproblemsolving.com/community/c3_middle_school_math Middle School Math Forum] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23209 MATHCOUNTS Coaches Forum].<br />
* The AoPS MATHCOUNTS Trainer is available on the [http://artofproblemsolving.com/mathcounts_trainer AoPS website], as well as on the [https://itunes.apple.com/us/app/mathcounts-trainer-math-contest/id1023961880?ls=1&mt=8 iPhone and iPad].<br />
* The free [http://www.artofproblemsolving.com/alcumus AoPS Alcumus learning system] includes thousands of MATHCOUNTS problems.<br />
* [http://artofproblemsolving.com/ftw/ftw.php For the Win!] gives students free Countdown Round-like practice against other AoPS students.<br />
* AoPS founder Richard Rusczyk has created dozens of [http://artofproblemsolving.com/videos/mathcounts MATHCOUNTS Mini video lessons].<br />
* [http://mathcounts.saab.org/ Elias Saab's MATHCOUNTS Preparation Homepage]<br />
* [http://www.unidata.ucar.edu/staff/russ/mathcounts/diaz.html The MATHCOUNTS Bible According to Mr. Diaz]<br />
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MATHCOUNTS Program] by [[Jeff Boyd]], who coached the 2005, 2007, and 2008 National Champion [[Texas MathCounts]] team.<br />
*[http://mathweb.scranton.edu/monks/courses/ProblemSolving/MathCountsPlaybookBW.pdf Coach Monk's MathCounts Playbook]<br />
* MathCounts Minis make hard problems easy<br />
<br />
== MATHCOUNTS Curriculum ==<br />
MATHCOUNTS curriculum includes [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.<br />
<br />
Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.<br />
<br />
== Past State Team Winners ==<br />
* 1984: Virginia<br />
* 1985: Florida<br />
* 1986: California<br />
* 1987: New York<br />
* 1988: New York<br />
* 1989: North Carolina<br />
* 1990: Ohio<br />
* 1991: Alabama<br />
* 1992: California<br />
* 1993: Kansas<br />
* 1994: Pennsylvania<br />
* 1995: Indiana<br />
* 1996: Wisconsin<br />
* 1997: Massachusetts<br />
* 1998: Wisconsin<br />
* 1999: Massachusetts<br />
* 2000: California<br />
* 2001: Virginia<br />
* 2002: California<br />
* 2003: California<br />
* 2004: Illinois<br />
* 2005: Texas<br />
* 2006: Virginia<br />
* 2007: Texas<br />
* 2008: Texas<br />
* 2009: Texas<br />
* 2010: California<br />
* 2011: California<br />
* 2012: Massachusetts<br />
* 2013: Massachusetts<br />
* 2014: California<br />
* 2015: Indiana<br />
* 2016: Texas<br />
* 2017: Texas<br />
* 2018: Texas<br />
* 2019: Massachusetts<br />
<br />
== MATHCOUNTS Competition Structure ==<br />
<br />
=== Sprint Round ===<br />
<br />
30 problems are given all at once. Students have 40 minutes to complete the Sprint Round. This round is very fast-paced and requires speed and accuracy as well. The earlier problems are usually the easiest problems in the competition, and the later problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.<br />
<br />
=== Target Round ===<br />
8 problems given 2 at a time. Students have 6 minutes to complete each set of two problems. Students may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target Round. Usually comprised of one "confidence booster" and one hard problem.<br />
<br />
=== Team Round ===<br />
<br />
10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).<br />
<br />
=== Countdown Round ===<br />
High scoring individuals compete head-to-head until a champion is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion. Calculators are not allowed, but scratch paper will be provided.<br />
<br />
<br />
====Chapter and State Competitions====<br />
<br />
In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:<br />
<br />
*The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.<br />
<br />
*The winner of the first round goes up against the 8th place finisher.<br />
<br />
*The winner of the second round goes up against the 7th place finisher.<br />
<br />
This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three questions correct wins (as opposed to the best-out-of-three rule).<br />
<br />
If the countdown round is unofficial, any format may be used. Single-elimination bracket-style tournaments are common.<br />
<br />
====National Competition====<br />
<br />
At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first place written finisher to become champion, equalizing the field. But even then, a 12th place written competitor will have less of a chance to become champion than the top 4, because the top 4 get a bye. Until the semi-finals, the scoring is best out of five advances.<br />
<br />
At the first round and the second round, the person to correctly answer the most out of 5 questions wins. However, at the semifinals, the rules slightly change&mdash;the first person to correctly answer four questions wins.<br />
<br />
=== HELLOOOOOOOOOOOOOO ===<br />
In some states, (most notably Florida) there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "How much dirt is in a 3 ft by 3 ft by 4 ft hole?" The answer was 0 because there is no dirt in a hole.<br />
<br />
=== Masters Round ===<br />
Top students give in-depth explanations to challenging problems. This round is optional at the state level competition and is mandatory at the national competition (up to 2011). At nationals the top two on the written and countdown participate. <br />
In 2012, it was replaced by the Reel Math Challenge (now called the Math Video Challenge).<br />
<br />
=== Scoring and Ranking ===<br />
An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of <math>30 + 2(8) = 46</math> points.<br />
<br />
A team's score is the individual scores of its members divided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score. Note that when there are less than four members the score will become less.<br />
<br />
==Tiebreakers==<br />
If two or more students tie for a ranking with the same individual score, the people ranked in order is decided using the following algorithm(going to the next step if inconclusive)<br />
(1) Sprint score<br />
(2) Sum of the problem numbers correct(Unconfirmed)<br />
(3) Last problem done correctly(Unconfirmed), and checking down the list to previous problems if still tied<br />
(4) Flip a coin(Unconfirmed)<br />
<br />
In state contests, advancements to nationals in the case of a tie is often decided using a Tiebreaker Round(tied students are invited into a room and participate in a secret format of a contest to determine who advances to the National Contest)<br />
<br />
== MATHCOUNTS Competition Levels ==<br />
=== School Competition ===<br />
Students vie for the chance to make their school teams. Problems at this level are generally the easiest and most basic in curriculum.<br />
<br />
=== Chapter Competition ===<br />
Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.<br />
<br />
<br />
<br />
HELLLLLLLO<br />
I CAN EDIT THIS 4 SOME REASON<br />
<br />
=== State Competition ===<br />
The top 4 students in each state form the state team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. JIJIJIJIJII<br />
<br />
=== National Competition ===<br />
==== National Fornite<br />
Sites ====<br />
For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.<br />
<br />
* The 2020 competition was canceled due to the COVID-19 pandemic.<br />
* The 2019 competition was held in Orlando, Florida.<br />
* The 2018 competition was held in Washington, D.C.<br />
* The 2017 competition was held in Orlando, Florida.<br />
* The 2016 competition was held in Washington, D.C.<br />
* The 2015 competition was held in Boston, Massachusetts.<br />
* The 2014 competition was held in Orlando, Florida.<br />
* The 2013 competition was held in Washington, D.C.<br />
* The 2012 competition was held in Orlando, Florida.<br />
* The 2011 competition was held in Washington, D.C.<br />
* The 2009 and 2010 competitions were held in Orlando, Florida.<br />
* The 2008 competition was held in Denver, Colorado.<br />
* The 2007 competition was held in Fort Worth, Texas.<br />
* The 2006 competition was held in Arlington, Virginia.<br />
* The 2005 competition was held in Detroit, Michigan.<br />
* The 2004 competition was held in Washington, D.C.<br />
* The 2002 and 2003 competitions were held in Chicago, Illinois.<br />
<br />
== What comes after MATHCOUNTS? ==<br />
<br />
Give the following competitions a try and take a look at the [[List of United States high school mathematics competitions]].<br />
* [[American Mathematics Competitions]]<br />
* [[American Regions Math League]]<br />
* [[Mandelbrot Competition]]<br />
* [[Mu Alpha Theta]]<br />
<br />
[[Category:Mathematics competitions]]<br />
<br />
== See also... ==<br />
* [[List of national MATHCOUNTS teams]]<br />
* [[Mathematics competition resources]]<br />
* [[Math contest books]]<br />
* [[Math books]]<br />
* [[List of United States middle school mathematics competitions]]<br />
* [[List of United States high school mathematics competitions]]<br />
* [http://www.mathcounts.org/webarticles/anmviewer.asp?a=921&z=71 2006 MATHCOUNTS Countdown Video]<br />
<br />
[[Category:Introductory mathematics competitions]]</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=MATHCOUNTS&diff=122706MATHCOUNTS2020-05-21T14:30:58Z<p>Apple321: /* National Competition Sites */</p>
<hr />
<div>Many AoPS Community members and online school students have been participants at National MATHCOUNTS, including many Nationals Countdown Round participants in the past decade. '''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] program that has served millions of middle school students since 1984. Sponsored by the [http://cna.com CNA Foundation], [http://www.nspe.org/ National Society of Professional Engineers], the [http://www.nctm.org/ National Council of Teachers of Mathematics], and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.<br />
<br />
{{Contest Info|name=MATHCOUNTS|region=USA|type=Free Response|difficulty=0.5 - 2.5|breakdown=<u>Countdown</u>: 0.5 (School/Chapter), 1 (State/National)<br><u>Sprint</u>: 1-1.5 (School/Chapter), 2-2.5 (State/National)<br><u>Target:</u> 1.5 (School), 2 (Chapter), 2-2.5 (State/National)}}<br />
<br />
== MATHCOUNTS Resources ==<br />
=== MATHCOUNTS Books ===<br />
Art of Problem Solving's [http://artofproblemsolving.com/store/list/aops-curriculum Introductory subject textbooks] are ideal for students preparing for MATHCOUNTS, as are [http://artofproblemsolving.com/store/item/aops-vol1 AoPS Volume 1] and [http://artofproblemsolving.com/store/item/competition-math Competition Math for Middle School]<br />
<br />
=== Fortnite Classes ===<br />
Art of Problem Solving hosts a [http://artofproblemsolving.com/school/course/mathcounts-basics Basic] and an [http://artofproblemsolving.com/school/course/mathcounts-advanced Advanced] MATHCOUNTS course. The AoPS Introduction-level subject courses also include a great deal of MATHCOUNTS preparation. Many AoPS instructors are former National MATHCOUNTS Mathletes.<br />
<br />
=== MATHCOUNTS Online ===<br />
* [http://www.mathcounts.org Official MATHCOUNTS Homepage]<br />
* Art of Problem Solving hosts a large [http://artofproblemsolving.com/community/c3_middle_school_math Middle School Math Forum] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23209 MATHCOUNTS Coaches Forum].<br />
* The AoPS MATHCOUNTS Trainer is available on the [http://artofproblemsolving.com/mathcounts_trainer AoPS website], as well as on the [https://itunes.apple.com/us/app/mathcounts-trainer-math-contest/id1023961880?ls=1&mt=8 iPhone and iPad].<br />
* The free [http://www.artofproblemsolving.com/alcumus AoPS Alcumus learning system] includes thousands of MATHCOUNTS problems.<br />
* [http://artofproblemsolving.com/ftw/ftw.php For the Win!] gives students free Countdown Round-like practice against other AoPS students.<br />
* AoPS founder Richard Rusczyk has created dozens of [http://artofproblemsolving.com/videos/mathcounts MATHCOUNTS Mini video lessons].<br />
* [http://mathcounts.saab.org/ Elias Saab's MATHCOUNTS Preparation Homepage]<br />
* [http://www.unidata.ucar.edu/staff/russ/mathcounts/diaz.html The MATHCOUNTS Bible According to Mr. Diaz]<br />
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MATHCOUNTS Program] by [[Jeff Boyd]], who coached the 2005, 2007, and 2008 National Champion [[Texas MathCounts]] team.<br />
*[http://mathweb.scranton.edu/monks/courses/ProblemSolving/MathCountsPlaybookBW.pdf Coach Monk's MathCounts Playbook]<br />
* MathCounts Minis make hard problems easy<br />
<br />
== MATHCOUNTS Curriculum ==<br />
MATHCOUNTS curriculum includes [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.<br />
<br />
Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.<br />
<br />
== Past State Team Winners ==<br />
* 1984: Virginia<br />
* 1985: Florida<br />
* 1986: California<br />
* 1987: New York<br />
* 1988: New York<br />
* 1989: North Carolina<br />
* 1990: Ohio<br />
* 1991: Alabama<br />
* 1992: California<br />
* 1993: Kansas<br />
* 1994: Pennsylvania<br />
* 1995: Indiana<br />
* 1996: Wisconsin<br />
* 1997: Massachusetts<br />
* 1998: Wisconsin<br />
* 1999: Massachusetts<br />
* 2000: California<br />
* 2001: Virginia<br />
* 2002: California<br />
* 2003: California<br />
* 2004: Illinois<br />
* 2005: Texas<br />
* 2006: Virginia<br />
* 2007: Texas<br />
* 2008: Texas<br />
* 2009: Texas<br />
* 2010: California<br />
* 2011: California<br />
* 2012: Massachusetts<br />
* 2013: Massachusetts<br />
* 2014: California<br />
* 2015: Indiana<br />
* 2016: Texas<br />
* 2017: Texas<br />
* 2018: Texas<br />
* 2019: Massachusetts<br />
<br />
== MATHCOUNTS Competition Structure ==<br />
<br />
=== Sprint Round ===<br />
<br />
30 problems are given all at once. Students have 40 minutes to complete the Sprint Round. This round is very fast-paced and requires speed and accuracy as well. The earlier problems are usually the easiest problems in the competition, and the later problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.<br />
<br />
=== Target Round ===<br />
8 problems given 2 at a time. Students have 6 minutes to complete each set of two problems. Students may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target Round. Usually comprised of one "confidence booster" and one hard problem.<br />
<br />
=== Team Round ===<br />
<br />
10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).<br />
<br />
=== Countdown Round ===<br />
High scoring individuals compete head-to-head until a champion is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion. Calculators are not allowed, but scratch paper will be provided.<br />
<br />
<br />
====Chapter and State Competitions====<br />
<br />
In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:<br />
<br />
*The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.<br />
<br />
*The winner of the first round goes up against the 8th place finisher.<br />
<br />
*The winner of the second round goes up against the 7th place finisher.<br />
<br />
This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three questions correct wins (as opposed to the best-out-of-three rule).<br />
<br />
If the countdown round is unofficial, any format may be used. Single-elimination bracket-style tournaments are common.<br />
<br />
====National Competition====<br />
<br />
At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first place written finisher to become champion, equalizing the field. But even then, a 12th place written competitor will have less of a chance to become champion than the top 4, because the top 4 get a bye. Until the semi-finals, the scoring is best out of five advances.<br />
<br />
At the first round and the second round, the person to correctly answer the most out of 5 questions wins. However, at the semifinals, the rules slightly change&mdash;the first person to correctly answer four questions wins.<br />
<br />
=== HELLOOOOOOOOOOOOOO ===<br />
In some states, (most notably Florida) there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "How much dirt is in a 3 ft by 3 ft by 4 ft hole?" The answer was 0 because there is no dirt in a hole.<br />
<br />
=== Masters Round ===<br />
Top students give in-depth explanations to challenging problems. This round is optional at the state level competition and is mandatory at the national competition (up to 2011). At nationals the top two on the written and countdown participate. <br />
In 2012, it was replaced by the Reel Math Challenge (now called the Math Video Challenge).<br />
<br />
=== Scoring and Ranking ===<br />
An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of <math>30 + 2(8) = 46</math> points.<br />
<br />
A team's score is the individual scores of its members divided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score. Note that when there are less than four members the score will become less.<br />
<br />
==Tiebreakers==<br />
If two or more students tie for a ranking with the same individual score, the people ranked in order is decided using the following algorithm(going to the next step if inconclusive)<br />
(1) Sprint score<br />
(2) Sum of the problem numbers correct(Unconfirmed)<br />
(3) Last problem done correctly(Unconfirmed), and checking down the list to previous problems if still tied<br />
(4) Flip a coin(Unconfirmed)<br />
<br />
In state contests, advancements to nationals in the case of a tie is often decided using a Tiebreaker Round(tied students are invited into a room and participate in a secret format of a contest to determine who advances to the National Contest)<br />
<br />
== MATHCOUNTS Competition Levels ==<br />
=== School Competition ===<br />
Students vie for the chance to make their school teams. Problems at this level are generally the easiest and most basic in curriculum.<br />
<br />
=== Chapter Competition ===<br />
Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.<br />
<br />
<br />
<br />
HELLLLLLLO<br />
I CAN EDIT THIS 4 SOME REASON<br />
<br />
=== State Competition ===<br />
The top 4 students in each state form the state team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. JIJIJIJIJII<br />
<br />
=== National Competition ===<br />
==== National Fornite<br />
Sites ====<br />
For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.<br />
<br />
* The 2020 competition was canceled due to the COVID-19 pandemic.<br />
* The 2019 competition was held in Orlando, Florida.<br />
* The 2018 competition was held in Washington, D.C.<br />
* The 2017 competition was held in Orlando, Florida.<br />
* The 2016 competition was held in Washington, D.C.<br />
* The 2015 competition was held in Boston, Massachusetts.<br />
* The 2014 competition was held in Orlando, Florida.<br />
* The 2013 competition was held in Washington, D.C.<br />
* The 2012 competition was held in Orlando, Florida.<br />
* The 2011 competition was held in Washington, D.C.<br />
* The 2009 and 2010 competitions were held in Orlando, Florida.<br />
* The 2008 competition was held in Denver, Colorado.<br />
* The 2007 competition was held in Fort Worth, Texas.<br />
* The 2006 competition was held in Arlington, Virginia.<br />
* The 2005 competition was held in Detroit, Michigan.<br />
* The 2004 competition was held in Washington, D.C.<br />
* The 2002 and 2003 competitions were held in Chicago, Illinois.<br />
<br />
== What comes after MATHCOUNTS? ==<br />
<br />
Give the following competitions a try and take a look at the [[List of United States high school mathematics competitions]].<br />
* [[American Mathematics Competitions]]<br />
* [[American Regions Math League]]<br />
* [[Mandelbrot Competition]]<br />
* [[Mu Alpha Theta]]<br />
<br />
[[Category:Mathematics competitions]]<br />
<br />
== See also... ==<br />
* [[List of national MATHCOUNTS teams]]<br />
* [[Mathematics competition resources]]<br />
* [[Math contest books]]<br />
* [[Math books]]<br />
* [[List of United States middle school mathematics competitions]]<br />
* [[List of United States high school mathematics competitions]]<br />
* [http://www.mathcounts.org/webarticles/anmviewer.asp?a=921&z=71 2006 MATHCOUNTS Countdown Video]<br />
<br />
[[Category:Introductory mathematics competitions]]</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=2019_AIME_Problems/Problem_2&diff=1021102019 AIME Problems/Problem 22019-02-14T14:34:44Z<p>Apple321: Created page with "ommkdmskmdkasdfsfdsff"</p>
<hr />
<div>ommkdmskmdkasdfsfdsff</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_10B&diff=1021092019 AMC 10B2019-02-14T14:34:20Z<p>Apple321: </p>
<hr />
<div>'''2019 AIME''' problems and solutions. The test was held on February 13, 2019. The problems in 2019 AMC 10B are not available. They will be available within 24 hours. Do not post 2019 AIME problems or answers here within 24 hours because some students have not completed the competition.<br />
JL08<br />
*[[2019 AIME Problems]]<br />
*[[2019 AIME Answer Key]]<br />
**[[2019 AIME Problems/Problem 1|poooppppooooooyayayayadoodooeasysooeasy]]<br />
**[[2019 AIME Problems/Problem 2|Problem 2]]<br />
**[[2019 AIME Problems/Problem 3|Problem 3]]<br />
**[[2019 AIME Problems/Problem 4|Problem 4]]<br />
**[[2019 AIME Problems/Problem 5|Problem 5]]<br />
**[[2019 AIME Problems/Problem 6|Problem 6]]<br />
**[[2019 AIME Problems/Problem 7|Problem 7]]<br />
**[[2019 AIME Problems/Problem 8|Problem 8]]<br />
**[[2019 AIME Problems/Problem 9|Problem 9]]<br />
**[[2019 AIME Problems/Problem 10|Problem 10]]<br />
**[[2019 AIME Problems/Problem 11|Problem 11]]<br />
**[[2019 AIME Problems/Problem 12|Problem 12]]<br />
**[[2019 AIME Problems/Problem 13|Problem 13]]<br />
**[[2019 AIME Problems/Problem 14|Problem 14]]<br />
**[[2019 AIME Problems/Problem 15|Problem 15]]<br />
**[[2019 AIME Problems/Problem 16|Problem 16]]<br />
**[[2019 AIME Problems/Problem 17|Problem 17]]<br />
**[[2019 AIME Problems/Problem 18|Problem 18]]<br />
**[[2019 AMC 10B Problems/Problem 19|Problem 19]]<br />
**[[2019 AMC 10B Problems/Problem 20|Problem 20]]<br />
**[[2019 AMC 10B Problems/Problem 21|Problem 21]]<br />
**[[2019 AMC 10B Problems/Problem 22|Problem 22]]<br />
**[[2019 AMC 10B Problems/Problem 23|Problem 23]]<br />
**[[2019 AMC 10B Problems/Problem 24|Problem 24]]<br />
**[[2019 AMC 10B Problems/Problem 25|Problem 25]]<br />
==See also==<br />
{{AMC10 box|year=2019|ab=B|before=[[2019 AMC 10A]]|after=[[2020 AMC 10A]]}}<br />
* [[AMC 10]]<br />
* [[AMC 10 Problems and Solutions]]<br />
* [[Mathematics competitions]]<br />
* [[Mathematics competition resources]]<br />
{{MAA Notice}}</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_10B_Problems/Problem_25&diff=1021082019 AMC 10B Problems/Problem 252019-02-14T14:30:44Z<p>Apple321: </p>
<hr />
<div>dontsub2pewds if u think this test was gg2ez</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_10B_Problems/Problem_9&diff=1021072019 AMC 10B Problems/Problem 92019-02-14T14:30:24Z<p>Apple321: </p>
<hr />
<div>nosub2pewds</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_10B_Problems/Problem_1&diff=1021062019 AMC 10B Problems/Problem 12019-02-14T14:29:54Z<p>Apple321: 3498</p>
<hr />
<div>Its soooooooooo ez lololol</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_10B&diff=1021052019 AMC 10B2019-02-14T14:28:57Z<p>Apple321: </p>
<hr />
<div>'''2019 AMC 10B''' problems and solutions. The test was held on February 13, 2019. The problems in 2019 AMC 10B are not available. They will be available within 24 hours. Do not post 2019 AMC 10B problems or answers here within 24 hours because some students have not completed the competition.<br />
<br />
*[[2019 AMC 10B Problems]]<br />
*[[2019 AMC 10B Answer Key]]<br />
**[[2019 AMC 10B Problems/Problem 1|poooppppooooooyayayayadoodooeasysooeasy]]<br />
**[[2019 AMC 10B Problems/Problem 2|Problem 2]]<br />
**[[2019 AMC 10B Problems/Problem 3|Problem 3]]<br />
**[[2019 AMC 10B Problems/Problem 4|Problem 4]]<br />
**[[2019 AMC 10B Problems/Problem 5|Problem 5]]<br />
**[[2019 AMC 10B Problems/Problem 6|Problem 6]]<br />
**[[2019 AMC 10B Problems/Problem 7|Problem 7]]<br />
**[[2019 AMC 10B Problems/Problem 8|Problem 8]]<br />
**[[2019 AMC 10B Problems/Problem 9|Problem 9]]<br />
**[[2019 AMC 10B Problems/Problem 10|Problem 10]]<br />
**[[2019 AMC 10B Problems/Problem 11|Problem 11]]<br />
**[[2019 AMC 10B Problems/Problem 12|Problem 12]]<br />
**[[2019 AMC 10B Problems/Problem 13|Problem 13]]<br />
**[[2019 AMC 10B Problems/Problem 14|Problem 14]]<br />
**[[2019 AMC 10B Problems/Problem 15|Problem 15]]<br />
**[[2019 AMC 10B Problems/Problem 16|Problem 16]]<br />
**[[2019 AMC 10B Problems/Problem 17|Problem 17]]<br />
**[[2019 AMC 10B Problems/Problem 18|Problem 18]]<br />
**[[2019 AMC 10B Problems/Problem 19|Problem 19]]<br />
**[[2019 AMC 10B Problems/Problem 20|Problem 20]]<br />
**[[2019 AMC 10B Problems/Problem 21|Problem 21]]<br />
**[[2019 AMC 10B Problems/Problem 22|Problem 22]]<br />
**[[2019 AMC 10B Problems/Problem 23|Problem 23]]<br />
**[[2019 AMC 10B Problems/Problem 24|Problem 24]]<br />
**[[2019 AMC 10B Problems/Problem 25|Problem 25]]<br />
==See also==<br />
{{AMC10 box|year=2019|ab=B|before=[[2019 AMC 10A]]|after=[[2020 AMC 10A]]}}<br />
* [[AMC 10]]<br />
* [[AMC 10 Problems and Solutions]]<br />
* [[Mathematics competitions]]<br />
* [[Mathematics competition resources]]<br />
{{MAA Notice}}</div>Apple321https://artofproblemsolving.com/wiki/index.php?title=File:Depression.PNG&diff=102104File:Depression.PNG2019-02-14T14:23:22Z<p>Apple321: </p>
<hr />
<div></div>Apple321