Search results

Create the page "J- -" on this wiki! See also the search results found.

Hölder's Inequality
...bda_a = \lambda_b = 1/2</math>, this is the elementary form of the [[Cauchy-Schwarz Inequality]]. ...he left-hand side of the desired inequality but may contribute to the right-hand side.

4 KB (762 words) - 00:11, 18 June 2023
Mobius function
...>b</math>. Via a combinatorial argument using the [[Principle of Inclusion-Exclusion]] (PIE), it can be seen that the number of integers less than or e ...u{p_i}+\sum_{i,j}\mu(p_ip_j)+\ldots=1-k+\binom{k}{2}-\binom{k}{3}-\ldots=(1-1)^k=0^k=0</math>.</center>

5 KB (910 words) - 02:58, 1 March 2022
2012 IMO Problems/Problem 1
Let <math>R</math> be the radius of the A-excircle. Because <math>CM = CL</math>, we have <math>CML</math> isosceles a ...w of Sines on triangle <math>BFM</math> and <math>ABC</math> and the double-angle formula for sine, we have

7 KB (1,189 words) - 01:22, 19 November 2023
Energy

636 bytes (99 words) - 21:44, 12 April 2021
2002 AMC 8 Problems
A birdbath is designed to overflow so that it will be self-cleaning. Water flows in at the rate of 20 milliliters per minute and drains Right isosceles triangles are constructed on the sides of a 3-4-5 right triangle, as shown. A capital letter represents the area of each tri

15 KB (2,102 words) - 09:58, 5 May 2024
1994 USAMO Problems/Problem 4
...timal equation is <math>a_n = \sqrt{n} - \sqrt{n-1} = 1/(\sqrt{n} + \sqrt{n-1})</math>. Note that this is strictly greater than <math>1/(2\sqrt{n})</mat {{USAMO box|year=1994|num-b=3|num-a=5}}

1 KB (210 words) - 13:30, 4 July 2013
2011 AIME I Problems/Problem 11
...ed by <math>d = \text{ord}_{125}(2).</math> To begin with, we'll use a well-known property of the order to get a bound on <math>d.</math> <cmath>2^b\left(2^{a-b} - 1\right) \equiv 0 \mod 1000</cmath>

11 KB (1,668 words) - 22:10, 24 February 2023
2011 AIME I Problems/Problem 7
...h> to be anything we want (including 0), all we care about is that <math>(m-1)\text{ }|\text{ } 2010</math> which happens <math>\boxed{016}</math> times ...<math>(m-1)Q(m) = 2010</math>. Thus, if <math>P(m) = 0</math>, then <math>m-1 | 2010</math> .

10 KB (1,673 words) - 19:27, 8 April 2024
2011 AIME II Problems/Problem 8
<center>[[File:2011_AIME_II_-8.png‎]]</center> <cmath>(z^6-2^{18})(z^6+2^{18})=0</cmath>

5 KB (805 words) - 18:46, 27 January 2024
2011 AIME II Problems
The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Fi

8 KB (1,301 words) - 08:43, 11 October 2020
2011 AIME II Problems/Problem 14
Each POSITION in the 30-position permutation is uniquely defined by an ordered triple <math>(i, j, k ...or the number 2, there are (2-1) choices for i, (3-1) choices for j, and (5-1) choices for k. Thus, there are 1*2*4=8 possible placements for the number

10 KB (1,581 words) - 22:09, 27 August 2023
2011 AIME II Problems/Problem 9
Let <math>x_1, x_2, ... , x_6</math> be non-negative real numbers such that <math>x_1 +x_2 +x_3 +x_4 +x_5 +x_6 =1</math> ...n is the product of three non-negative terms whose sum is equal to 1. By AM-GM this product is at most <math>\frac1{27}</math>. Since we have added at l

3 KB (466 words) - 15:06, 16 January 2023
1998 USAMO Problems
...> (<math>1\leq i\leq 999</math>) so that for all <math>i</math>, <math>|a_i-b_i|</math> equals <math>1</math> or <math>6</math>. Prove that the sum <cmath> |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| </cmath>

3 KB (486 words) - 06:11, 24 November 2020
1999 AMC 8 Problems
...is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost w Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cookies? (Some eggs and some cookies may

17 KB (2,394 words) - 19:51, 8 May 2023
KGS math club/solution 11 5
...<math>\frac{1}{y}, \frac{2}{y}, \ldots, \frac{y}{y}=1</math>, i.e. <math>y-x</math> needs to divide <math>y</math> perfectly. This condition is also en ...now that for every pair <math>i</math> and <math>j</math> we know <math>s_j-s_i</math> divides <math>s_j</math> exactly an integer number of times. We w

4 KB (712 words) - 10:51, 11 May 2011
2011 USAJMO Problems/Problem 3
..._1), l(P_2), l(P_3)</math> are <cmath>y=2a_1x-a_1^2, y=2a_2x-a_2^2, y=2a_3x-a_3^2,</cmath> respectively. It is easy to deduce that the three points of i &= \frac{\Im(1+6a_ni-12a_n^2-8a_n^3i)}{\Re(1+6a_ni-12a_n^2-8a_n^3i)}\\

15 KB (2,593 words) - 13:37, 29 January 2021
2011 USAMO Problems/Problem 6
...ach intersection of three of the 11 sets. Since each pair of sets is in 9 3-way intersections—one with each of the 9 remaining sets—any two ...he number of <math>b_k>0</math>, where <math>1\le k\le225</math>. By the QM-AM inequality, we know <math>\frac{\sum_{k=1}^{225}b_k^2}n\ge\Bigg(\frac{\su

7 KB (1,209 words) - 12:50, 25 August 2023
2011 AMC 10B Problems/Problem 16
...}} = \frac{1}{2+2\sqrt{2}} \times \frac{2-2\sqrt{2}}{2-2\sqrt{2}} = \frac{2-2\sqrt{2}}{-4} = \boxed{\textbf{(A)} \frac{\sqrt{2}-1}{2}}</cmath> ...he denominator reduces to <math>-4</math> and the fraction is <math>\frac{2-2\sqrt{2}}{-4}=\frac{\sqrt{2}-1}{2}</math> which is <math>\boxed{\textbf{(A)

4 KB (606 words) - 06:14, 25 June 2022
2001 AMC 8 Problems/Problem 11
Using the diagram above, the figure can be divided along the x-axis into two familiar regions that do not overlap: a right triangle and a r {{AMC8 box|year=2001|num-b=10|num-a=12}}

2 KB (383 words) - 19:36, 24 December 2023
2002 AMC 10B Problems/Problem 23
...ore, <math>a_m = a_{m-1} + m, a_{m-1}=a_{m-2}+(m-1), a_{m-2} = a_{m-3} + (m-2)</math>, and so on until <math>a_2 = a_1 + 2</math>. ...Sides of all of these equations gives <math>a_m + a_{m-1} + a_{m-2} + a_{m-3} + \cdots + a_2</math>.

5 KB (781 words) - 06:38, 23 October 2023
Olympiad books
Here is a list of '''Olympiad Books''' that have Olympiad-level problems used to train students for future [[mathematics]] competition ...ric/dp/1794193928/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=1556093238&sr=1-1-catcorr Advanced Olympiad Inequalities: Algebraic & Geometric Olympiad Inequ

17 KB (2,261 words) - 00:30, 22 April 2024
2007 AMC 10B Problems/Problem 16
{{AMC12 box|year=2007|ab=B|num-b=11|num-a=13}} {{AMC10 box|year=2007|ab=B|num-b=15|num-a=17}}

1 KB (236 words) - 17:11, 6 July 2023
Block walking

9 KB (1,228 words) - 21:47, 26 February 2022
AoPS Wiki:Problem of the Day/June 11, 2011

879 bytes (145 words) - 23:17, 7 June 2011
AoPS Wiki talk:Problem of the Day/June 11, 2011

1 KB (215 words) - 22:55, 10 June 2011
Mock Amc 12
{| class="wikitable" style="text-align:center;width:100%" | [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9572 1-5]

18 KB (2,206 words) - 19:41, 24 December 2020
AoPS Wiki:FAQ
:This likely to be because of zoom-in area. Please make sure your zoom level is at 100% or is reset for correct ...sers can go [https://www.windowscentral.com/how-manage-time-servers-windows-10 here.] You can also check [http://www.time.gov the US official time] whic

25 KB (4,184 words) - 21:16, 13 May 2024
2003 AMC 8 Problems
Blake and Jenny each took four 100-point tests. Blake averaged 78 on the four tests. Jenny scored 10 points hig When a fair six-sided die is tossed on a table top, the bottom face cannot be seen. What is

16 KB (2,236 words) - 12:02, 19 February 2024
1997 USAMO Problems
...\text{ if }x_{k-1}=0\\ \left\{\frac{p_{k}}{x_{k-1}}\right\}&\text{ if }x_{k-1}\ne0\end{cases} </math>

3 KB (507 words) - 14:31, 12 April 2023
1997 USAMO Problems/Problem 6
<cmath>a_n\ge a_{n-1}+a_1\ge (a_{n-2}+a_1)+a_1\ge \cdots \ge na_1.</cmath> <cmath>a_k\le a_{k-1}+(a_1+1)\le (a_{k-2}+(a_1+1))+(a_1+1)\le\cdots\le ka_1+(k-1).</cmath>

6 KB (1,107 words) - 14:12, 12 April 2023
2011 IMO Problems
Let <math>f : \mathbb{R} \rightarrow \mathbb{R}</math> be a real-valued function defined on the set of real numbers that satisfies ...a balance and <math>n</math> weights of weight <math>2^0, 2^1,\ldots, 2^{n-1}</math> . We are to place each of the <math>n</math> weights on the balanc

4 KB (682 words) - 00:12, 18 February 2021
2011 IMO Problems/Problem 1
.../math> isn't a good couple. We have in total 6 couples. So <math>n_A \leq 6-2=4</math>. ...So we have <math>2a+x | 2a+2b</math> and <math>a+b-x|2a+2b \Rightarrow a+b-x|2x</math>.

9 KB (1,718 words) - 23:08, 26 June 2014
1987 USAMO Problems
...olutions to <math>(m^2+n)(m + n^2)= (m - n)^3</math>, where m and n are non-zero integers. ...ctors of <math>\Delta ABC</math> form a right-angled triangle. If the right-angle is at <math>X</math>, where <math>AX</math> is the bisector of <math>\

2 KB (356 words) - 18:42, 18 July 2016
2002 AMC 8 Problems/Problem 15
{{AMC8 box|year=2002|num-b=14|num-a=16}}

1 KB (197 words) - 04:47, 25 November 2019
1999 AMC 8 Problems/Problem 25
...ake an underestimate by using the current diagram. All triangles are right-isosceles triangles. {{AMC8 box|year=1999|num-b=24|after=Last Question}}

4 KB (655 words) - 05:21, 12 March 2024
1997 AJHSME Problems/Problem 12
{{AJHSME box|year=1997|num-b=11|num-a=13}}

1 KB (188 words) - 12:37, 2 May 2021
1979 USAMO Problems/Problem 5
...do not contain <math>k, m</math>. Because the latter subset has <math>(n-t-2) < (n+1)-t - 1</math> elements, we may use infinite descent to contradict {{USAMO box|year=1979|num-b=4|after=Last Question}}

2 KB (410 words) - 21:04, 11 November 2015
2003 AMC 8 Problems/Problem 18
{{AMC8 box|year=2003|num-b=17|num-a=19}}

2 KB (250 words) - 22:17, 5 January 2024
1984 AHSME Problems/Problem 30
Now we multiply <math>S</math> by <math>1-w</math>: <cmath>S(1-w)=\sum_{i=1}^{9}\sum_{j=i}^{9} w^j(1-w)</cmath>

3 KB (447 words) - 21:21, 17 July 2020
1985 AHSME Problems
...ame as <math>a(b-c)</math> in ordinary algebraic notation. If <math>a\div b-c+d</math> is evaluated in such a language, the result in ordinary algebraic ...\qquad \mathrm{(D) \ } \frac{a}{b-c+d} \qquad \mathrm{(E) \ }\frac{a}{b-c-d} </math>

17 KB (2,488 words) - 03:26, 20 March 2024
1985 AHSME Problems/Problem 9
label(string(16*i+15-2*j), (2*j,-2*i-1)); label(string(15-2*i), (2*(i-1),-2*1.35));

1 KB (213 words) - 21:08, 19 March 2024
1985 AHSME Problems/Problem 13
{{AHSME box|year=1985|num-b=12|num-a=14}}

2 KB (287 words) - 21:05, 19 March 2024
2010 AMC 8 Problems
<math>\textbf{(B)}\ \text{non-square rhombus} </math> <math>\textbf{(C)}\ \text{non-square rectangle}</math>

18 KB (2,768 words) - 21:05, 9 January 2024
2011 AMC 8 Problems
...h> apples at a cost of <math> 50 </math> cents per apple. She paid with a 5-dollar bill. How much change did Margie receive? A fair 6-sided die is rolled twice. What is the probability that the first number tha

16 KB (2,371 words) - 17:34, 9 January 2024
2011 AMC 8 Problems/Problem 19
{{AMC8 box|year=2011|num-b=18|num-a=20}}

1 KB (158 words) - 15:06, 17 December 2023
1973 Canadian MO Problems
Find all real numbers that satisfy the equation <math>|x+3|-|x-1|=x+1</math>. (Note: <math>|a| = a</math> if <math>a\ge 0; |a|=-a</math> if Prove that <math>n+h(1)+h(2)+h(3)+\cdots+h(n-1) = nh(n)\qquad</math> for <math>n=2,3,4,\ldots</math>

4 KB (540 words) - 18:23, 8 October 2014
1973 Canadian MO Problems/Problem 7
{{CanadaMO box|year=1973||num-b=6||num-a=1}}

2 KB (351 words) - 18:56, 27 May 2022
Euler's Four-Square Identity
...)^2</cmath> <cmath>+(x_1y_3-x_3y_1+x_4y_2-x_2y_4)^2</cmath> <cmath>+(x_1y_4-x_4y_1 + x_2y_3 - x_3y_2)^2.</cmath> Then Euler's Four-Square Identity simply reads <math>|XY|^2 = |X|^2 |Y|^2</math>; i.e. the qua

1 KB (179 words) - 23:24, 5 November 2019
2011 AMC 8 Problems/Problem 7
{{AMC8 box|year=2011|num-b=6|num-a=8}}

4 KB (557 words) - 00:30, 4 January 2023
1971 Canadian MO Problems/Problem 9
{{Old CanadaMO box|num-b=8|num-a=10|year=1971}}

1 KB (204 words) - 18:09, 19 September 2012
1989 AHSME Problems
...th>n</math> between <math>1</math> and <math>100</math> does <math> x^{2}+x-n </math> factor into the product of two linear factors with integer coeffic ...e following is closest to the number of westbound vehicles present in a 100-mile section of highway?

15 KB (2,247 words) - 13:44, 19 February 2020
2012 AMC 10A Problems/Problem 25
...ints satisfying <math>|x-y|\geq1</math>, <math>|y-z|\geq1</math>, <math>|z-x|\geq1</math>. ...n\geq x \geq y \geq z \geq 0</math>, we have <math>x-y\geq1</math>, <math>y-z\geq1</math>.

13 KB (2,133 words) - 01:22, 6 February 2024
2012 AMC 12A Problems
a^{2}+3b^{2}+3c^{2}-3ab-2ac-2bc&=-1997 (0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}} & \text{if }k\text{ is odd,}\\

14 KB (2,197 words) - 13:34, 12 August 2020
2012 AMC 12A Problems/Problem 22
...ath>k</math> is <math>7</math>. The answer to this problem is then <math>27-7=20</math> ... <math>\framebox{C}</math>. {{AMC12 box|year=2012|ab=A|num-b=21|num-a=23}}

5 KB (940 words) - 17:13, 4 April 2020
2014 USAMO Problems
Let <math>a,b,c,d</math> be real numbers such that <math>b-d \ge 5</math> and all zeros <math>x_1, x_2, x_3,</math> and <math>x_4</math ...athbb{Z} \rightarrow \mathbb{Z}</math> such that <cmath>xf(2f(y)-x)+y^2f(2x-f(y))=\frac{(f(x))^2}{x}+f(yf(y))</cmath> for all <math>x, y \in \mathbb{Z}<

3 KB (496 words) - 02:14, 14 February 2024
2012 AMC 12A Problems/Problem 23
{{AMC12 box|year=2012|ab=A|num-b=22|num-a=24}}

4 KB (658 words) - 20:37, 19 January 2021
2011 IMO Shortlist Problems/A1

500 bytes (91 words) - 06:44, 28 October 2013
1989 AHSME Problems/Problem 13
{{AHSME box|year=1989|num-b=12|num-a=14}}

2 KB (242 words) - 07:51, 22 October 2014
2012 AMC 12B Problems
...abbits. How many more students than rabbits are there in all 4 of the third-grade classrooms? In order to estimate the value of <math>x-y</math> where <math>x</math> and <math>y</math> are real numbers with <math

20 KB (2,681 words) - 09:47, 29 June 2023
2012 AMC 12B Problems/Problem 19
[[File:2012_AMC-12B-19‎.jpg]] .... This will occur when the distance from <math>(x,0,0)</math> to <math>(1,1-x, 1)</math> is <math>(x)(\sqrt{2}</math>).

5 KB (815 words) - 21:59, 19 September 2023
2015 AIME I Problems
...ates a sequence according to the rule <math>a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |</math> for all <math>n\ge 4</math>. Find the number of such sequences f Let <math>f(x)</math> be a third-degree polynomial with real coefficients satisfying

10 KB (1,615 words) - 21:48, 13 January 2024
2012 AIME I Problems
...eaches the first of the hitching posts that are conveniently located at one-mile intervals along their route, he ties Sparky to the post and begins walk ...math>\mathcal{T}</math> be the set of all numbers of the form <math>\frac{x-256}{1000}</math>, where <math>x</math> is in <math>\mathcal{S}</math>. In o

10 KB (1,617 words) - 14:49, 2 June 2023
2012 AIME I Problems/Problem 7
{{AIME box|year=2012|n=I|num-b=6|num-a=8}}

6 KB (1,058 words) - 01:49, 25 November 2023
2007 iTest Problems/Problem 15
{{iTest box|year=2007|num-b=14|num-a=16}}

3 KB (539 words) - 21:01, 29 July 2018
Mock AIME II 2012 Problems/Problem 8
.... We can use the law of cosines to find <math>EC</math>, <math>EC^2=9^2+9^2-2(\cos 135)\cdot 9\cdot 9=2(81)+81\sqrt{2}=162+81\sqrt{2}</math>.

1 KB (184 words) - 03:11, 5 April 2012
Pascal (unit)

309 bytes (52 words) - 16:59, 28 February 2020
1979 USAMO Problems
Determine all non-negative integral solutions <math>(n_1,n_2,\dots , n_{14})</math> if any, ap

2 KB (321 words) - 13:04, 24 December 2015
2000 USAMO Problems/Problem 6
'''Lemma 1:''' If <math>r_1, r_2, \ldots , r_n</math> are non-negative reals and <math>x_1, x_2, \ldots x_n</math> are reals, then <cmath>\sum_{i}(r_{i}-r_{i-1})\biggl(\sum_{j=i}^{n}x_{j}\biggr)^{2}\, .</cmath>

3 KB (556 words) - 15:08, 15 July 2021
2000 USAMO Problems/Problem 3
<cmath>f(B,W,R) = \min(W+f(B-1,W,R),2R+f(B,W-1,R),3B+f(B,W,R-1)) </cmath> ...o <math>W+3R(B-1) < BW</math>, whence <math>3R < W</math>. But <math>W+3R(B-1) < 3RB</math>, so that <math>W < 3R</math>, a contradiction.

4 KB (800 words) - 21:24, 28 August 2019
2012 USAMO Problems/Problem 6
...=1}^{n} x_i^2\right)+2^{n-1}\left(\sum_{1\leq i<j\leq n} x_ix_j\right)=2^{n-2}.</cmath> Since <math>S_A^2\geq 0</math>, it follows that at most <math>2^{n-2}/\lambda^2</math> sets <math>A\subseteq N</math> have <math>|S_A|\geq \lam

6 KB (1,081 words) - 13:37, 21 June 2023
2015 IMO Problems/Problem 1
<math>\mathcal{S}</math> is centre-free if for any three points <math>A</math>, <math>B</math>, <math>C</ma <ol style="list-style-type: lower-latin;">

4 KB (773 words) - 08:14, 19 July 2016
1966 AHSME Problems/Problem 14
{{AHSME box|year=1966|num-b=13|num-a=15}}

3 KB (434 words) - 23:00, 12 December 2019
2006 SMT/Advanced Topics Problems
Ten teams of five runners each compete in a cross-country race. A runner finishing in <math> n\text{th} </math> place contribu Evaluate: <math> \sum_{n=1}^{\infty}\arctan\left(\frac{1}{n^2-n+1}\right) </math>

4 KB (596 words) - 20:09, 27 May 2012
2006 SMT/Team Problems
...th>, for terms <math> n\ge3 </math>, <math> S_n=\sum_{i=1}^{n-1}i\cdot S_{n-i} </math>. For example, if the first two elements are <math> 2 </math> and ...igin tangent to the graph of the upper part of the hyperbola <math> y^2=x^2-x+1 </math> in the first quadrant. This ray makes an angle of <math> \theta

6 KB (910 words) - 17:32, 27 May 2012
2006 SMT/Advanced Topics Problems/Problem 3

434 bytes (58 words) - 20:20, 27 May 2012
2010 AMC 8 Problems/Problem 3
...$10. 17 is <math>\frac{17}{10}\cdot100=170\%</math> of 10, and is <math>170-100=70\%</math> more than 10. {{AMC8 box|year=2010|num-b=2|num-a=4}}

5 KB (847 words) - 19:59, 5 February 2023
2012 AMC 12B Problems/Problem 16
...But this overcounts the case 2 with 36 ways. So total ways are <math>168-36=132</math>. {{AMC10 box|year=2012|ab=B|num-b=23|num-a=25}}

8 KB (1,462 words) - 11:33, 21 April 2024
2005 IMO Shortlist Problems/N2
...a_1,a_2,\ldots a_{n}</math> are all in the closed interval <math>[a_k,a_k+n-1]</math>, and hence <math>a_1,a_2,\ldots a_n</math> is a permutation of <ma

2 KB (333 words) - 10:43, 1 September 2012
1998 USAMO Problems/Problem 6
C = (sqrt(2)/2-1,sqrt(2)/2); J = (1-sqrt(2)/2, sqrt(2)/2);

5 KB (871 words) - 18:59, 10 May 2023
1989 AHSME Problems/Problem 30
{{AHSME box|year=1989|num-b=29|after=Last Question}}

2 KB (293 words) - 17:13, 24 August 2020
1989 AHSME Problems/Problem 5
...the frame can be dissected into <math>20\cdot10</math> L shapes, the right-hand border, and the bottom border: p = (i+0.75,j)--(i,j)--(i,j-0.75);

2 KB (295 words) - 07:43, 22 October 2014
2012 IMO Problems
Find all positive integers n for which there exist non-negative integers <math>a_1</math>, <math>a_2</math>, <math>\ldots</math> ,

5 KB (834 words) - 11:00, 24 February 2021
2020 Mock Combo AMC 10
...ths from the origin <math>(0, 0)</math> to a point on the line <math>y=2020-2x</math> such that each step is from <math>(x, y)</math> to either <math>(x <cmath>r_1 = 1</cmath><cmath>r_n = nr_{n-1} + n</cmath>Let <math>S</math> be the sum of all <math>n</math> such that

15 KB (2,452 words) - 03:03, 4 July 2020
2012 AMC 8 Problems
Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally Isabella must take four 100-point tests in her math class. Her goal is to achieve an average grade of 95

13 KB (1,835 words) - 08:51, 8 March 2024
2012 AMC 8 Problems/Problem 5
https://youtu.be/m4g-Nmot-c8 ~savannahsolver {{AMC8 box|year=2012|num-b=4|num-a=6}}

2 KB (208 words) - 20:16, 3 January 2024
2004 AMC 8 Problems/Problem 23
...he lives at corner <math>J</math>. Which graph could represent her straight-line distance from home? {{AMC8 box|year=2004|num-b=22|num-a=24}}

2 KB (308 words) - 17:26, 25 July 2021
2013 AMC 10A Problems/Problem 25
...ed to subtract <math>\dbinom{4}{2} -1 = 5</math> from this count, <math>70-5 = 65</math>. Note that diagonals like <math>\overline{AD}</math>, <math>\ ..., not double-counted -which the calculation by Lcz accounts for- but triple-counted. Thus, taking away one for each of these points is more than enough

6 KB (1,054 words) - 09:21, 28 December 2021
2013 AMC 10B Problems/Problem 22
{{AMC10 box|year=2013|ab=B|num-b=21|num-a=23}}

4 KB (746 words) - 17:29, 30 September 2023
User:Wuwang2002/Sandbox

1 KB (73 words) - 16:04, 8 January 2021
2013 AMC 12B Problems/Problem 20
<math>\textbf{(A)} \ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qqua ...points to be parallel, so that <math>\frac{g^2-f^2}{g-f} = \frac{j^2-h^2}{j-h}</math>, or <math>g+f = j+h</math>.

2 KB (375 words) - 00:54, 28 September 2021
2013 AMC 10B Problems
...ir parents is 33. What is the average age of all of these parents and fifth-graders? ...y attempted 50% more two-point shots than three-point shots. How many three-point shots did they attempt?

12 KB (1,926 words) - 21:54, 6 October 2022
1982 USAMO Problems/Problem 5
{{USAMO box|year=1982|num-b=4|after=Last Question}}

3 KB (524 words) - 16:08, 11 July 2018
1991 AHSME Problems
...>, <math>b</math>, and <math>c</math> we define <math>f(a,b,c)=\frac{c+a}{c-b}</math>, then <math>f(1,-2,-3)</math> is ...\qquad \textbf{(C) }3-\pi \qquad \textbf{(D) }3+\pi \qquad \textbf{(E) }\pi-3 </math>

16 KB (2,451 words) - 04:27, 6 September 2021
2013 AIME II Problems/Problem 8
A=expi(pi); B=expi(pi-a); C=expi(a); D=expi(0); E=expi(-a); F=expi(pi+a); ...means <cmath>\sin{\theta}=\frac{BY}{r}=\frac{11}{r}</cmath> <cmath>\sin{(90-2\theta)}=\frac{BX}{r}=\frac{10}{r}</cmath>

8 KB (1,357 words) - 09:23, 11 March 2024
2013 USAMO Problems/Problem 2
...ending at <math>A</math>, without repeating a move. Prove that <math>a_{n-1}+a_n=2^n</math> for all <math>n\geq 4</math>. label("$\dots$",(8-1,1));

14 KB (2,076 words) - 20:29, 10 July 2023
1989 AHSME Problems/Problem 27
...)^3\cdot\left(\frac{1}{1+k}\right)^2\ =\ k^5\cdot(1+k)\cdot\left(\frac{1}{1-k^2}\right)^3</math> ...k^6+...)</math>, the last part having general term <math>\tbinom{j}{2}k^{2j-4}</math>

2 KB (285 words) - 12:39, 5 July 2021
2014 AIME II Problems/Problem 15
...ath>x_{n+2^{k-1}} = P_k \cdot x_{n}</math> for all <math>0 \leq n \leq 2^{k-1} - 1.</math> .../math> is now the least <math>n</math> such that <cmath>P_{k+1} = p(x_{2^{n-1}}), </cmath> in which <math>P_a = p(x_n)</math> where <math>a > k</math> i

4 KB (770 words) - 17:44, 11 October 2023
1986 USAMO Problems/Problem 3
...smallest integer <math>n</math>, greater than one, for which the root-mean-square of the first <math>n</math> positive integers is an integer? <math>\mathbf{Note.}</math> The root-mean-square of <math>n</math> numbers <math>a_1, a_2, \cdots, a_n</math> is defin

3 KB (554 words) - 13:19, 19 July 2023
User talk:Bobthesmartypants/Sandbox
...repeating decimal would be <math>\dfrac{1}{n}=\dfrac{a_1a_2\cdots a_x}{10^x-1}</math>. Taking the reciprocal of both sides you get <math>n=\dfrac{10^x-1}{a_1a_2\cdots a_x}</math>.

17 KB (2,269 words) - 21:25, 6 November 2014
2013 IMO Problems/Problem 1
<math>1+\frac{2^k-1}{n}=(1+\frac{1}{m_1})(1+\frac{1}{m_2})...(1+\frac{1}{m_k})</math>. '''Base case:''' If <math>k = 1</math> then <math>1 +\frac{2^1-1}{n} = 1 + \frac{1}{n}</math>, so the claim is true for all positive intege

4 KB (654 words) - 01:30, 19 November 2023
1971 IMO Problems/Problem 5
...l)</math> iff either <math>v_i=v_k,|w_j-w_l|=1</math> or <math>w_j=w_l,|v_i-v_k|=1</math>. clearly if the condition holds then the adjacency is satisfie ...>. since we can form the graph <math>K_2</math>, we can form <math>Q_n=Q_{n-1}\times K_2</math>(the unit cube) and that solves the problem.

4 KB (749 words) - 14:09, 29 January 2021
2013 AMC 8 Problems
A sign at the fish market says, "50% off, today only: half-pound packages for just \$3 per package." What is the regular price for a fu What is the value of <math>4 \cdot (-1+2-3+4-5+6-7+\cdots+1000)</math>?

15 KB (2,162 words) - 20:05, 8 May 2023
2013 AMC 8 Problems/Problem 18
Isabella uses one-foot cubical blocks to build a rectangular fort that is <math>12</math> feet ...</math>. Therefore, the number of blocks contained in the fort is <math>600-320=\boxed{\textbf{(B)}\ 280}</math>

3 KB (412 words) - 22:30, 18 December 2023
2013 AMC 8 Problems/Problem 21
https://youtu.be/OOdK-nOzaII?t=2679 ==Video Solution for Problems 21-25 By Rohan Dhillon==

2 KB (263 words) - 19:21, 19 December 2023
2013 AMC 8 Problems/Problem 22
for (int i=0; i<3; ++i){for (int j=0; j>-2; --j){if ((i-j)<3){add(corner,(50i,50j));}}} ==Video Solution for Problems 21-25==

2 KB (256 words) - 19:06, 22 January 2024
2013 AMC 8 Problems/Problem 24
==Video Solution for Problems 21-25= Shaded area = 12-8=4 

6 KB (1,076 words) - 22:15, 20 December 2023
1958 AHSME Problems/Problem 49
<cmath>\sum\limits_{k=0}^{10} \binom{10}{k} (a+b)^k c^{10-k}</cmath> {{AHSME 50p box|year=1958|num-b=48|num-a=50}}

1 KB (191 words) - 00:59, 1 January 2024
1981 AHSME Problems
If <math> \dfrac{x}{x-1}=\dfrac{y^2+2y-1}{y^2+2y-2} </math>, then <math>x</math> equals ...tbf{(C)}\ y^2+2y+2 \qquad \\ \textbf{(D)}\ y^2+2y+1\qquad\textbf{(E)}\ -y^2-2y+1 </math>

17 KB (2,633 words) - 15:44, 16 September 2023
1968 AHSME Problems
...lue of <math>x</math> such that <math>64^{x-1}</math> divided by <math>4^{x-1}</math> equals <math>256^{2x}</math> is: ...hrough the point <math>(0,4)</math> is perpendicular to the line <math>x-3y-7=0</math>. Its equation is:

16 KB (2,571 words) - 14:13, 20 February 2020
2014 AMC 10A Problems/Problem 13
Since <math>\angle C=360</math>, <math>\angle GCH=360-90-90-60=120</math>. ...itude from <math>C</math> to <math>GH</math> allows to create a <math>30-60-90</math> triangle since <math>\triangle GCH</math> is isosceles. This means

7 KB (1,040 words) - 02:07, 8 October 2022
2014 AMC 10A Problems/Problem 14
draw((-3,0)--(9,0),linewidth(1),Arrows); //x-axis draw((0,-13)--(0,13),linewidth(1),Arrows); //y-axis

6 KB (1,001 words) - 13:07, 25 July 2022
2014 AMC 10A Problems/Problem 16
...math>, <math>IJ = 2x</math>. So each perpendicular is length <math>\dfrac{1-2x}{2}</math>. So taking our numbers and plugging them into <math> [ABCD] = {{AMC10 box|year=2014|ab=A|num-b=15|num-a=17}}

9 KB (1,411 words) - 19:51, 25 July 2023
2013 Mock AIME I Problems/Problem 14
...{2011}+16x^{2010}+\cdots + 4052169x + 4056196 = \sum_{j=1}^{2014}j^2x^{2014-j}.</math> If <math>a_1, a_2, \cdots a_{2013}</math> are its roots, then com

783 bytes (105 words) - 19:04, 15 December 2020
1998 PMWC Problems/Problem I2
for(int j = n-i; j > 0; --j){ We want <math>n^2-(n-1)^2=11</math> or <math>2n-1=11</math> for <math>n=6</math>. Therefore the two square numbers are <math

1 KB (231 words) - 15:06, 23 July 2014
2014 USAJMO Problems/Problem 2
Let <math>\triangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60^{\circ}</math>, and let < i1=2*O-H;

7 KB (1,197 words) - 19:40, 28 February 2023
2014 USAMO Problems/Problem 6

2 KB (361 words) - 11:55, 25 June 2020
2007 iTest Problems/Problem 12
This problem is phrased oddly, leading to the non-standard result of <math>\frac{3}{2}</math> instead of the standard <math>\f {{iTest box|year=2007|num-b=11|num-a=13}}

3 KB (500 words) - 04:44, 10 June 2018
2006 IMO Problems/Problem 1
...ath>\triangle ABC\, </math> at point <math>J</math>. Then, by the Incenter-Excenter Lemma, <math>JB=JC=JI=JP</math>. {{IMO box|year=2006|before=First Problem|num-a=2}}

1 KB (247 words) - 01:02, 19 November 2023
2014 AMC 12B Problems
...l road, the next <math> 20 </math> miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip? Doug constructs a square window using <math> 8 </math> equal-size panes of glass, as shown. The ratio of the height to width for each pan

13 KB (2,066 words) - 14:08, 1 November 2022
2014 AMC 10B Problems
...vel road, the next <math>20</math> miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Peter's trip? Camden constructs a square window using <math> 8 </math> equal-size panes of glass, as shown. The ratio of the height to width for each pan

13 KB (2,011 words) - 21:54, 8 November 2022
2014 AMC 10B Problems/Problem 18
{{AMC10 box|year=2014|ab=B|num-b=17|num-a=19}}

2 KB (369 words) - 17:44, 30 January 2021
2014 AMC 12B Problems/Problem 11
{{AMC12 box|year=2014|ab=B|num-b=10|num-a=12}}

2 KB (277 words) - 10:13, 3 March 2015
2014 AMC 12B Problems/Problem 22
...is on pad <math>N</math>, <math>0<N<10</math>, it will jump to pad <math>N-1</math> with probability <math>\frac{N}{10}</math> and to pad <math>N+1</ma ...es b=\frac{10a-\emptyset}{9},c=\frac{5b-a}{4},d=\frac{10c-3b}{7},e=\frac{5d-2c}{3}=1/2</cmath>

7 KB (1,082 words) - 22:35, 3 April 2024
2014 AMC 12B Problems/Problem 21
...(2-sqrt(3),1), G=(1,sqrt(3)/2), H=(2.5-sqrt(3),1), J=(.5,0), K=(2-sqrt(3),1-sqrt(3)/2); ...so we must have <math>\triangle HGJ\cong\triangle JLH</math> by Hypotenuse-Leg congruence. From this congruence we have <math>LJ=HG=BE</math>.

8 KB (1,307 words) - 02:07, 1 January 2023
2014 AIME I Problems
Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west direction. Jon rides east at <math>2 ...<math>2014</math>, respectively, and each graph has two positive integer x-intercepts. Find <math>h</math>.

9 KB (1,472 words) - 13:59, 30 November 2021
2014 AIME I Problems/Problem 6
...<math>2014</math>, respectively, and each graph has two positive integer x-intercepts. Find <math>h</math>. ...respectively. Notice that because the two parabolas have to have positive x-intercepts, <math>h\ge32</math>.

7 KB (1,158 words) - 20:50, 8 December 2021
2014 AIME I Problems/Problem 13
...(A,D,O,dir(90+theta)); J=extension(B,C,O,dir(-90+theta)); H=(A+I)/2; F=H+(J-I); R=midpoint(H--F); S=midpoint(C--D); T=(R.x, A.y); draw(A--B--C--D--cycle Because these areas are in the ratio <math>411:405=(408+3):(408-3)</math>, it follows that <cmath>\frac{\frac 1{400}\sin 2\theta}{\frac 3{10

9 KB (1,404 words) - 21:07, 13 October 2023
2014 AIME I Problems/Problem 15
...rc</math>. But <math>DF=FE</math>, so <math>\triangle DEF</math> is a 45-45-90 triangle. Letting <math>DG=3x</math>, we have that <math>EG=4x</math>, <m ...while the segment <math>CJ</math> is <math>2x\sqrt{2}</math> since its 3-4-5 again. Now adding all those segments together we can find that <math>AC=5=

10 KB (1,643 words) - 22:30, 28 January 2024
1993 AHSME Problems/Problem 29
{{AHSME box|year=1993|num-b=28|num-a=30}}

2 KB (322 words) - 15:07, 6 July 2022
1991 AHSME Problems/Problem 9
{{AHSME box|year=1991|num-b=8|num-a=10}}

1,012 bytes (158 words) - 13:29, 14 March 2023
1988 AHSME Problems
<math>{a \choose k} = \frac{a(a-1)(a-2)\cdots(a-(k-1))}{k(k-1)(k-2)\cdots(2)(1)}</math> ...e <math>x_{0}</math>, consider the sequence defined by <math>x_{n} = f(x_{n-1})</math> for all <math>n \ge 1</math>.

17 KB (2,535 words) - 13:45, 19 February 2020
2015 AMC 10B Problems/Problem 22
<math>\textbf{(A) } 3 \qquad\textbf{(B) } 12-4\sqrt5 \qquad\textbf{(C) } \dfrac{5+2\sqrt5}{3} \qquad\textbf{(D) } 1+\sqrt <cmath>FG^2+FG-1 = 0</cmath>

7 KB (1,146 words) - 14:09, 13 April 2024
1982 AHSME Problems
When the polynomial <math>x^3-2</math> is divided by the polynomial <math>x^2-2</math>, the remainder is \textbf{(C)} \ -2x-2 \qquad

17 KB (2,500 words) - 19:05, 11 September 2023
2007 iTest Problems/Problem 10
{{iTest box|year=2007|num-b=9|num-a=11}}

2 KB (308 words) - 06:13, 14 August 2021
2007 iTest Problems/Problem 11
{{iTest box|year=2007|num-b=10|num-a=12}}

931 bytes (130 words) - 04:31, 10 June 2018
2007 iTest Problems/Problem 13
{{iTest box|year=2007|num-b=12|num-a=14}}

1 KB (212 words) - 14:35, 4 August 2018
2007 iTest Problems/Problem 14
...p_s^{k_s}</math> then <math>\phi(n)=p_1^{k_1-1}\cdots p_s^{k_s-1}\cdot (p_1-1)\cdots (p_s-1)</math>. Hence every prime factor of <math>n</math> is contained in <math>

2 KB (265 words) - 21:23, 4 February 2023
2007 iTest Problems/Problem 19
{{iTest box|year=2007|num-b=18|num-a=20}}

3 KB (397 words) - 01:18, 29 October 2023
2007 iTest Problems/Problem 18
<cmath>(x-a)^2 (x-b)</cmath> <cmath>(x^2 - 2ax + a^2)(x-b)</cmath>

2 KB (393 words) - 17:01, 10 June 2018
2007 iTest Problems/Problem 17
{{iTest box|year=2007|num-b=16|num-a=18}}

2 KB (266 words) - 21:30, 4 February 2023
2007 iTest Problems/Problem 16
{{iTest box|year=2007|num-b=15|num-a=17}}

3 KB (420 words) - 20:28, 17 June 2018
2007 iTest Problems/Problem 21
<cmath>1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.</cmath> {{iTest box|year=2007|num-b=20|num-a=22}}

2 KB (247 words) - 03:34, 14 June 2018
2007 iTest Problems/Problem 22
<cmath> |x+y|=2007 \\ |x-y|=c</cmath> <math>2007|x-y|=c</math> must be one line; otherwise, the system would have four solution

929 bytes (137 words) - 22:05, 10 January 2019
2007 iTest Problems/Problem 25
We know that <math>\binom{2007}{i} = \binom{2007}{2007-i}</math>, so ...7} (i \cdot \binom{2007}{i}) + \sum_{j=1}^{2007} (j \cdot \binom{2007}{2007-j})</cmath>

2 KB (270 words) - 14:35, 29 July 2018
2007 iTest Problems/Problem 47
<cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\ Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}</cmath>

2 KB (415 words) - 11:46, 18 April 2024
2007 iTest Problems/Problem 58
...red the answer of a previous problem. When the problem is rewritten, the T-value is substituted.'' ...3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor</cmath>

2 KB (307 words) - 17:29, 22 July 2018
2013 Canadian MO Problems
<math>(x+1)P(x-1)-(x-1)P(x)</math> Let <math>G</math> be the centroid of a right-angled triangle <math>ABC</math> with <math>\angle BCA = 90^\circ</math>. Le

2 KB (381 words) - 13:45, 8 October 2014
2007 iTest Problems/Problem 23
Find the product of the non-real roots of the equation <cmath>(x^2-3x)^2+5(x^2-3x)+6=0</cmath>

1 KB (202 words) - 16:48, 24 November 2018
2007 iTest Problems/Problem 24
{{iTest box|year=2007|num-b=23|num-a=25}}

2 KB (340 words) - 19:49, 30 June 2018
Mock AIME 5 2005-2006 Problems/Problem 15
...math> and positive integer <math>n < 1003</math> such that <math>a_m = a_{m-n} = a_{m+n}</math>, where all subscripts are taken <math>\pmod{2006}</math> {{Mock AIME box|year=2005-2006|n=5|source=76847|num-b=14|num-a=15}}

955 bytes (157 words) - 21:20, 8 October 2014
1956 AHSME Problems
\textbf{(B)}\ \frac{1}{x-1}\qquad \textbf{(C)}\ \frac{-1}{x-1}\qquad

20 KB (3,039 words) - 22:44, 12 February 2021
2014 UMO Problems
Completely describe the set of all right triangles with positive integer-valued legs such that when four draw((10-sqrt(3)/2,-1/2)--(10,1.5-1/2),arrow=Arrow());

6 KB (982 words) - 15:11, 22 October 2018
2014 UMO Problems/Problem 6
draw((10-sqrt(3)/2,-1/2)--(10,1.5-1/2),arrow=Arrow()); draw((12.5-sqrt(3)/2,-1/2)--(12.5+sqrt(3)/2,-1/2),arrow=Arrow());

3 KB (628 words) - 19:22, 20 May 2021
Asymptote: Crash Course
Suppose you are floating in a three-dimensional universe that extends infinitely in each direction. A bright sou

3 KB (410 words) - 20:22, 15 December 2020
2006 UNCO Math Contest II Problems
For Colorado Students Grades 7-12. A=dir(90-(j-1)*(360/14));

5 KB (725 words) - 22:15, 7 November 2014
2006 UNCO Math Contest II Problems/Problem 3
A=dir(90-(j-1)*(360/14)); {{UNCO Math Contest box|n=II|year=2006|num-b=2|num-a=4}}

713 bytes (103 words) - 19:47, 4 December 2016
2006 UNCO Math Contest II Problems/Problem 11
draw((j,max(0,j-1))--(j,4),black); {{UNCO Math Contest box|n=II|year=2006|num-b=10|after=Last Question}}

863 bytes (128 words) - 02:35, 13 January 2019
1988 AHSME Problems/Problem 9
{{AHSME box|year=1988|num-b=8|num-a=10}}

2 KB (367 words) - 08:24, 12 July 2022
1988 AHSME Problems/Problem 11
{{AHSME box|year=1988|num-b=10|num-a=12}}

2 KB (306 words) - 18:25, 21 November 2017
2014 UNM-PNM Statewide High School Mathematics Contest II Problems
[[UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 1|Solution [[2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 1|Solution

5 KB (774 words) - 06:07, 29 January 2019
2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7
{{UNM-PNM Math Contest box|year=2014|n=II|num-b=6|num-a=8}}

700 bytes (112 words) - 13:23, 20 December 2018
2014 AMC 8 Problems
...wer is <math>H</math> and Terry's answer is <math>T</math>, what is <math>H-T</math>? ...Paula 35 cents and has a pocket full of 5-cent coins, 10-cent coins, and 25-cent coins that he can use to pay her. What is the difference between the la

13 KB (1,957 words) - 12:08, 13 January 2024
2014 AMC 8 Problems/Problem 15
circum[i]=dir(120-30*i); https://youtu.be/3QHH9xV-QDw

2 KB (280 words) - 11:10, 2 July 2023
2015 AMC 10A Problems
What is the value of <math>(2^0-1+5^2+0)^{-1}\times5?</math> ...wn in the figure. How many toothpicks does she need to add to complete a 5-step staircase?

12 KB (1,887 words) - 18:52, 14 May 2024
2015 AMC 12A Problems
What is the value of <math>(2^0-1+5^2-0)^{-1}\times5?</math> A league with 12 teams holds a round-robin tournament, with each team playing every other team exactly once. Game

13 KB (2,117 words) - 12:33, 24 August 2023
2015 AMC 10A Problems/Problem 3
for(int j=0;j<=3-i;j=j+1) ...have <math>2[2(5)+4+3+2+1]=40</math> toothpicks. So our answer is <math>40-18=22</math> or <math>D</math>.

2 KB (280 words) - 23:07, 26 June 2023
2015 AMC 12A Problems/Problem 25
...two circles in the pair. Layer <math>L_k</math> consists of the <math>2^{k-1}</math> circles constructed in this way. Let <math>S=\bigcup_{j=0}^{6}L_j< fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75));

8 KB (1,340 words) - 14:24, 23 November 2023
Mock AIME 3 2006-2007 Problems/Problem 3

769 bytes (139 words) - 23:38, 15 February 2015
2015 AMC 10B Problems
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at ...imum value of the sum of the roots of the equation <math>(x-a)(x-b)+(x-b)(x-c)=0</math>?

14 KB (2,247 words) - 11:38, 26 March 2024
2015 AMC 12B Problems
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at David, Hikmet, Jack, Marta, Rand, and Todd were in a 12-person race with 6 other people. Rand finished 6 places ahead of Hikmet. Mar

13 KB (2,064 words) - 13:39, 1 October 2022
2015 AMC 12B Problems/Problem 20
f(i-1,1) & \text{ if } i \ge 1 \text{ and } j = 0 \text{, and} \\ f(i-1, f(i,j-1)) & \text{ if } i \ge 1 \text{ and } 1 \le j \le 4.

5 KB (745 words) - 20:35, 9 December 2020
2015 AMC 12B Problems/Problem 21
<cmath>5s+5a-2s-2b=190</cmath> <cmath>3s = 190+2b-5a</cmath>

6 KB (1,007 words) - 13:54, 12 October 2022
2015 AMC 12B Problems/Problem 25
...ude of <math>P_{2015}</math> on the complex plane. This is an [[arithmetico-geometric series]]. (1-x)S &=1+x+x^2+\cdots+x^{2014}-2015x^{2015} \\

13 KB (2,117 words) - 11:30, 8 April 2023
2015 AMC 10B Problems/Problem 21
...least <math>3 \cdot 19=57</math> steps high. We then start using guess-and-check: {{AMC10 box|year=2015|ab=B|num-b=20|num-a=22}}

6 KB (983 words) - 01:51, 16 June 2022
2015 AIME I Problems/Problem 7
...ta JFC</math> are similar to <math>\Delta EDC</math> since they are <math>1-2-\sqrt{5}</math> triangles. Thus, we can rewrite <math>BC</math> in terms o ...m \triangle JFC</math> (and similarly the rest of the triangles are <math>1-2-\sqrt5</math> triangles). We let the sidelength of <math>FGHJ</math> be <m

6 KB (1,105 words) - 21:02, 9 November 2023
Hawaii AMC

356 bytes (44 words) - 15:47, 7 August 2017
2015 AIME II Problems/Problem 15
..._C) = \left(\frac{24}{15}, \frac{-48}{15}\right)</math>. Using the point-to-line distance formula and the condition <math>n = 4p</math>, we have <cmath> ...g a perpendicular from <math>P</math> to <math>Q</math> creates a <math>3-4-5</math> right triangle, from which <math>BC = 4</math> and, if <math>\alpha

31 KB (5,086 words) - 19:15, 20 December 2023
2016 AIME I Problems
...55,-0.85),G=B-(0,1.1),H=F-(0,0.6),I=E-(0,0.6),J=D-(0,1.1),K=C-(0,1.4),L=C+K-A; ...that for every positive integer <math>k</math>, the subsequence <math>a_{2k-1}</math>, <math>a_{2k}</math>, <math>a_{2k+1}</math> is geometric and the s

8 KB (1,360 words) - 12:19, 29 January 2022
2015 USAMO Problems/Problem 4
How many different non-equivalent ways can Steve pile the stones on the grid? ...\forall i</math>. By stars and bars, the number of ways is <math>\binom{n+m-1}{m}^{2}</math>.

4 KB (770 words) - 20:09, 10 March 2016
2015 USAMO Problems
How many different non-equivalent ways can Steve pile the stones on the grid? ...math> is at most <math>\frac{n(n+1)}{2}\lambda</math>. (A multiset is a set-like collection of elements in which order is ignored, but repetition of ele

4 KB (629 words) - 13:49, 22 November 2023
2015 USAJMO Problems
How many different non-equivalent ways can Steve pile the stones on the grid?

3 KB (565 words) - 16:42, 5 August 2023
2015 USAJMO Problems/Problem 1
...11}, m_{111})</math> for all the eight elements. Since the sum of the eight-group is <math>0</math>, <math>m_{111}</math> must also be <math>0</math>. T ..., namely the mean. Then before that, if we could always choose <math>M\ge N-2^k</math> members to form pairs, each yielding the average of the total gro

8 KB (1,405 words) - 20:13, 26 July 2022
Aops font
For example, to type the symbol for the re-sort button |, you would type: :<code> <nowiki>text here!</nowiki></code>

6 KB (924 words) - 21:31, 22 February 2024
1993 IMO Problems/Problem 1
...<math>f\left(x\right)</math> cannot be expressed as the product of two non-constant polynomials with integer coefficients. ...gives that <math>f\left(x\right)=\sum_{i=0}^{n}\left(\sum_{j=0}^{i}b_jc_{i-j}\right)x^i</math>.

3 KB (536 words) - 11:26, 21 November 2023
2015 AMC 8 Problems/Problem 24
A baseball league consists of two four-team divisions. Each team plays every other team in its division <math>N</ma ...tart by solving this Diophantine equation. In other words, <math>N=\frac{76-4M}{3}</math>.

4 KB (669 words) - 20:41, 17 January 2024
2015 AMC 8 Problems
...\text{mean} \qquad \textbf{(D) } \text{mode} \qquad \textbf{(E) } \text{mid-range}</math> ...econd and third must be two different letters among the <math>21</math> non-vowels, and the fourth must be a digit (<math>0</math> through <math>9</math

16 KB (2,322 words) - 14:04, 2 February 2024
2015 AMC 8 Problems/Problem 21
draw((-8,0)--(-4,-6*sqrt(2))--(-4-6*sqrt(2),-4-6*sqrt(2))--(-8-6*sqrt(2),-4)--cycle); label("$H$",(-4-6*sqrt(2),-4-6*sqrt(2)),S);

3 KB (471 words) - 20:36, 17 January 2024
2015 AMC 8 Problems/Problem 18
...th term <math>25</math>, so we have the common difference is <math>\frac{25-1}4=6</math>. This means we can fill in the first row of the table: ...a fifth term of <math>81</math>, so the common difference is <math>\frac{81-17}4=16</math>. We can fill in the fifth row of the table as shown:

5 KB (641 words) - 10:28, 13 January 2024
2016 AMC 12A Problems/Problem 25
...t squares needs to be at least <math>100</math>. Since <math>100\le (x+1)^2-x^2=2x+1</math>, this first happens at <math>x\ge \lfloor 99/2\rfloor = 50</ ...btracted from <math>f(k)</math>. In other words, <math>f(k) = 25\cdot 10^{k-2}+1\cdot 10^{k/2}</math>.

8 KB (1,224 words) - 18:28, 16 November 2022
2016 AMC 10B Problems/Problem 25
...k \{ x \} \rfloor</math> can equal integers from <math>0</math> to <math>k-1</math>. ...equal to any of the fractions <math>\frac{1}{k}, \frac{2}{k} \dots \frac{k-1}{k}</math>.

12 KB (1,951 words) - 18:23, 4 April 2023
2016 AMC 10B Problems/Problem 15
with the numbers <math>1-8</math> around the outsides and <math>9</math> in the middle. We see that t {{AMC10 box|year=2016|ab=B|num-b=14|num-a=16}}

2 KB (378 words) - 21:13, 18 June 2022
2016 AMC 10B Problems/Problem 14
...le 5</math>. Then to include all other corners, we need <math>1\le y\le 3(x-1)</math>. ...le x\le 5</math>. To include all other corners, we need <math>2\le y\le 3(x-2)</math>.

3 KB (497 words) - 08:50, 9 March 2024
2016 AIME I Problems/Problem 14
https://youtu.be/diLCwN-358s {{AIME box|year=2016|n=I|num-b=13|num-a=15}}

6 KB (967 words) - 21:08, 22 November 2022
2016 AIME I Problems/Problem 3
...55,-0.85),G=B-(0,1.1),H=F-(0,0.6),I=E-(0,0.6),J=D-(0,1.1),K=C-(0,1.4),L=C+K-A; {{AIME box|year=2016|n=I|num-b=2|num-a=4}}

4 KB (636 words) - 13:00, 22 December 2020
2016 AIME II Problems/Problem 7
...th>IB=b</math>, we have <cmath>IJ^{2}=a^{2}+b^{2} \geq 1008</cmath> by [[AM-GM inequality]]. Also, since <math>EFGH||ABCD</math>, the angles that each s ...quare whose sides its vertices touch), so the desired answer is <cmath>1848-1008=\boxed{840}</cmath>

2 KB (316 words) - 00:10, 2 September 2021
2016 AIME II Problems/Problem 9
...c_n=a_n+b_n</math>. There is an integer <math>k</math> such that <math>c_{k-1}=100</math> and <math>c_{k+1}=1000</math>. Find <math>c_k</math>. ...h>a_k=r^{k-1}</math> and <math>b_k=(k-1)d</math>. First, <math>b_{k-1}<c_{k-1}=100</math> implies <math>d<100</math>, so <math>b_{k+1}<300</math>.

6 KB (983 words) - 01:18, 2 February 2023
2016 AIME II Problems/Problem 12
<cmath>f(n,C_i)=\sum_{j\ne i} f(n-1,C_j),</cmath> i.e., <math>f(n,C_i)</math> is equal to the number of colorings of <math>n-1</math> sections that end in any color other than <math>C_i</math>. Using t

20 KB (3,328 words) - 21:13, 5 April 2024
2016 AIME II Problems/Problem 13
...however we like, so by PIE we have <math>54 \cdot 4-18 \cdot 6 + 3! \cdot 4-2!=130</math> ways to get a score of <math>5</math>. ...e number of ways to get a sum of <math>6</math> is <math>6!-120-216-222-130-1=31</math>.

9 KB (1,564 words) - 16:53, 1 December 2021
Gmaas
...lem. EDIT: That theorem was proved by GMAAS. EDIT: GMAAS's Theorem has real-world applications: because GMAAS knows the answer to any math problem, you ...respected. If you want to respect him the most, you should NOT have a high-quality sculpture of Gmaas.

69 KB (11,805 words) - 20:49, 18 December 2019
2006 iTest Problems
...a right angle to the line <math>2x + y = 3</math>. Find the x-intercept of the line. Questions 19 and 20 are Sudoku-related questions. Sudoku is a puzzle game that has one and only

31 KB (4,811 words) - 00:02, 4 November 2023
2015 USAJMO Problems/Problem 6
How many different non-equivalent ways can Steve pile the stones on the grid? ...\forall i</math>. By stars and bars, the number of ways is <math>\binom{n+m-1}{m}^{2}</math>.

4 KB (769 words) - 20:24, 16 March 2019
1982 AHSME Problems/Problem 26
If <math>k=8j+5</math>, then <math>(8j+5)\equiv 64j^2+80j-1\equiv 16j-1\equiv 24+c \implies 16j\equiv 25+c</math>, which clearly can only have the {{AHSME box|year=1982|num-b=25|num-a=27}}

2 KB (403 words) - 15:50, 17 June 2021
2016 USAMO Problems
<cmath>\left(k^2\right)!\cdot\prod_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}</cmath> ...math> and <math>y</math>, <cmath>(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.</cmath>

4 KB (608 words) - 13:49, 22 November 2023
2016 USAMO Problems/Problem 2
<cmath>\left(k^2\right)!\cdot\prod_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}</cmath> ...{\frac{k^{2}}{m}}\right\rfloor-\frac{\sum_{i=0}^{k-1} r(i,m)-\sum_{i=k}^{2k-1} r(i,m)}{m} \geq \sum_{m=p^{i}, i\in \mathbb{N}} \left\lceil -\frac{k^{2}}

4 KB (732 words) - 20:59, 8 April 2021
2016 USAMO Problems/Problem 5
...lane with <math>A</math> at the origin and <math>C</math> on the positive x-axis. Assume without loss of generality that C is acute. ...ta</math> and <math>\gamma</math> as the angles made between the positive x-axis and <math>\overrightarrow{MN}</math> and <math>\overrightarrow{QP}</mat

8 KB (1,449 words) - 00:09, 12 October 2023
2016 USAMO Problems/Problem 6
...bitrarily permutes the <math>k</math> chosen cards and turns them back face-down. Then, it is your turn again. ...hat you do not win on any earlier move, repeat this for <math>1\le i \le 2n-k+1.</math>

8 KB (1,516 words) - 10:11, 8 April 2023
1954 AHSME Problems/Problem 3
{{AHSME 50p box|year=1954|num-b=2|num-a=4}}

627 bytes (105 words) - 20:34, 17 February 2020
1990 IMO Problems/Problem 6
...en if we can find a sequence <math>\{b_n\}</math> such that <math>b_i=a_p^2-a_q^2, (p\not=q)</math> and <math>\sum_{i=1}^{995}b_ie^{i\theta}=0</math>, t Note that <math>2^2-1^2=3,4^2-3^2=7,\cdots,1990^2-1989^2=3979</math>, then <math>3,7,\cdots,3979</math> can be written as a su

3 KB (522 words) - 13:54, 30 January 2021
1985 USAMO Problems/Problem 1
{{USAMO box|year=1985|before=First Problem|num-a=2}}

2 KB (261 words) - 04:18, 23 October 2022
1985 USAMO Problems/Problem 5
Let <math>a_1,a_2,a_3,\cdots</math> be a non-decreasing sequence of positive integers. For <math>m\ge1</math>, define <ma ...ed cells in the first 19 columns of row <math>j</math> is equal to <math>20-b_j</math>.

2 KB (295 words) - 13:03, 18 July 2016
1987 USAMO Problems/Problem 5
{{USAMO box|year=1987|num-b=4|after=Last Question}}

662 bytes (128 words) - 18:44, 18 July 2016
2016 AMC 8 Problems/Problem 22
...t, we find the height of each triangle to calculate their area. The two non-colored isosceles triangles are similar, and are in a <math>3:1</math> ratio ...dot 4 + 1\cdot 0)-(\dfrac{3}{2}\cdot 0 + 1\cdot 3 + 4\cdot 0)}{2} = \frac{6-3}{2} = \frac{3}{2}</math>

8 KB (1,157 words) - 16:12, 18 January 2024
Mock AIME 3 2010 Problems
Let x be the integer two below one hundred, and let y be the number of two-digit primes. Find x − y. How many three-digit numbers (with no leading zeroes) with three distinct digits satisfy th

14 KB (2,904 words) - 18:24, 16 May 2017
2017 AMC 10A Problems/Problem 23
...l the ones in a line from <math>2300</math>, so we have <math>2300-120-16-4-12=\boxed{(\textbf{B})\ 2148}</math> ...cases where the chosen 3 points make a line. The answer would be <math>2300-152=\boxed{(\textbf{B})\ 2148}</math>

3 KB (456 words) - 23:08, 7 October 2023
2017 AMC 12A Problems/Problem 5
https://youtu.be/TCxjwO-I8kQ {{AMC10 box|year=2017|ab=A|num-b=7|num-a=9}}

3 KB (495 words) - 15:40, 9 June 2023
2017 AMC 12A Problems/Problem 4
<math>\frac{j-s}{j} \approx \frac{2 - 1.4}{2} = 0.3 = 30\%</math> {{AMC10 box|year=2017|ab=A|num-b=6|num-a=8}}

1 KB (220 words) - 15:41, 9 June 2023
2017 AMC 10A Problems/Problem 7
<math>\frac{j-s}{j} \approx \frac{2 - 1.4}{2}=0.3 \implies \boxed{\textbf{(A)}\ 30\%}</mat https://youtu.be/MUxbCp-2OYQ

1 KB (213 words) - 21:40, 1 November 2023
2017 AMC 10A Problems
...r garden. The beds are separated and also surrounded by <math>1</math>-foot-wide walkways, as shown on the diagram. What is the total area of the walkwa The region consisting of all points in three-dimensional space within <math>3</math> units of line segment <math>\overlin

15 KB (2,285 words) - 18:02, 28 October 2023
2017 AMC 12A Problems/Problem 25
...1+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}.</cmath> ...we draw these 6 complex numbers out on the complex plane, we get a crystal-looking thing. Note that the total number of ways to choose 12 complex numbe

18 KB (2,878 words) - 01:47, 16 December 2023
2017 AMC 12A Problems
...store sells single popsicles for \$1 each, 3-popsicle boxes for \$2, and 5-popsicle boxes for \$3. What is the greatest number of popsicles that Pablo ...th>f(n) = f(n-1) + 1</math> if <math>n</math> is even, and <math>f(n) = f(n-2) + 2</math> if <math>n</math> is odd and greater than <math>1</math>. What

15 KB (2,418 words) - 16:58, 7 November 2022
Mock AIME I 2015 Problems/Problem 11
.... Define a polynomial <math>P(x)</math> such that <math>P(x) = (x-a)(x-b)(x-c) = x^3 - (a+b+c)x^2 + (ab+ac+bc)x - abc</math>. Let <math>j = ab + ac + bc

3 KB (435 words) - 11:29, 29 October 2019
2017 AMC 10B Problems/Problem 21
{{AMC10 box|year=2017|ab=B|num-b=20|num-a=22}}

4 KB (601 words) - 00:34, 8 August 2023
Areteem Institute's Math Zoom Winter Boot Camp (Locations Vary each year including one in Irvine, CA
...ely practice sessions with challenging written and oral problems and a good-natured spirit of competition. Locations vary each year with one in Irvine, ...place Dec 19-23 and the Irvine Winter Boot Camp always taking place Dec 26-30. We will have updated information on the 2017 Winter Math Boot Camps in L

1 KB (166 words) - 17:04, 9 March 2017
2017 AIME II Problems
...ber of positive integers less than or equal to <math>2017</math> whose base-three representation contains no digit equal to <math>0</math>. ...</math> to <math>7</math> and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of e

7 KB (1,200 words) - 15:02, 8 September 2020
2017 AIME II Problems/Problem 7
...ath>(x+2)^2</math> is always non-negative, <math>kx</math> must also be non-negative; therefore this takes care of the <math>kx>0</math> condition as lo ...kx = (x+2)^2</math> (the converse isn't necessarily true!), or <math>x^2+(4-k)x+4=0</math>. Our original equation has exactly one solution if and only i

6 KB (983 words) - 23:03, 5 January 2024
2017 AIME II Problems/Problem 2
{{AIME box|year=2017|n=II|num-b=1|num-a=3}}

2 KB (358 words) - 18:57, 24 March 2017
2017 AIME II Problems/Problem 14
...3)</math>. If <math>\textrm{min} > 0</math> then the point <math>(a-1,b-1,c-1)</math> will also be on the line for example, 3 applies to the other end. So the answer is <math>192-24=\boxed{168}</math>.

6 KB (1,015 words) - 18:02, 31 December 2023
2017 USAJMO Problems/Problem 6
*Here is a counter-case. <math>BRR_{1}B_{1}RR</math> : j = 1, k = 2 => <math>BRRR</math> : j = ...e, by assumption, the total amount of arcs is <math>1+\text{max}(0,k-1)=1+k-1=k=\text{max}(j,k)</math>.

8 KB (1,465 words) - 15:30, 12 June 2020
2017 UNCO Math Contest II Problems
Twenty-fifth Annual UNC Math Contest Final Round January 21, 2017 Harold writes an integer; its right-most digit is 4. When Curious George

7 KB (1,192 words) - 15:14, 20 August 2020
2017 UNCO Math Contest II Problems/Problem 2
{{UNCO Math Contest box|year=2017|n=II|num-b=1|num-a=3}}

1 KB (195 words) - 18:22, 16 January 2023
Bill's Triangle
draw((0,8-j)--(j,8)); draw((8-j,0)--(8,j));

2 KB (307 words) - 20:25, 12 December 2017
2020 AMC 12A Problems
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (pos

14 KB (2,073 words) - 15:15, 21 October 2021
2020 AMC 12B Problems
<cmath>\frac{100^2-7^2}{70^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)}</cmath><math>\textbf{(A) } 1 \qquad \textbf{(B) } \frac{9951}{995 A three-quarter sector of a circle of radius <math>4</math> inches together with its

16 KB (2,417 words) - 01:03, 28 April 2022
2017 AMC 8 Problems/Problem 24
==Solution 1 (Principle of Inclusion-Exclusion)== We use Principle of Inclusion-Exclusion. There are <math>365</math> days in the year, and we subtract the

6 KB (907 words) - 21:11, 20 January 2024
The Pot Method
The Pot Method is a double-counting method used to evaluate series with binomial coefficients, especial The other way to count this expected value is case-by-case based on the number of blue marbles.

3 KB (414 words) - 19:10, 4 April 2019
MIE 2015
...ree sets <math>F</math>, <math>G</math> and <math>H</math>. The set <math>G-H</math> is equal to the set: (a) <math>(G\cup F)-(F-H)</math>

7 KB (1,127 words) - 18:23, 11 January 2018
2018 AMC 10A Problems/Problem 20
...y it in particular. Each of these sections has <math>10</math> distinct sub-squares, whether partially or in full. So since each can be colored either w .... Since we must subtract off <math>2</math> cases for the all-black and all-white cases, the answer is <math>2^{10}-2=\boxed{\textbf{(B) } 1022.}</math>

5 KB (827 words) - 21:31, 24 October 2023
2018 AMC 10A Problems/Problem 22
== Video Solution by OmegaLearn (Meta-Solving Technique) == {{AMC10 box|year=2018|ab=A|num-b=21|num-a=23}}

10 KB (1,599 words) - 04:51, 6 August 2023
2018 AMC 10A Problems/Problem 21
...math>a</math> for which the curves <math>x^2+y^2=a^2</math> and <math>y=x^2-a</math> in the real <math>xy</math>-plane intersect at exactly <math>3</mat Substituting <math>y=x^2-a</math> into <math>x^2+y^2=a^2</math>, we get

9 KB (1,502 words) - 23:31, 19 August 2023
2018 AMC 12A Problems/Problem 19
{{AMC12 box|year=2018|ab=A|num-b=18|num-a=20}}

7 KB (1,137 words) - 22:27, 18 May 2024
2018 AMC 12B Problems
...sqrt2 \qquad \textbf{(D) } \frac{1}{2}\sqrt{2} \qquad \textbf{(E) } \sqrt 3-1</math> ...ac{4}{7}</cmath>and <math>q</math> is as small as possible. What is <math>q-p</math>?

14 KB (2,118 words) - 15:36, 28 October 2021
2018 AMC 10B Problems/Problem 19
...>p^2q^2.</math> Since <math>c-1<100,</math> the only possibility is <math>c-1=2^2\cdot3^2=36,</math> from which <math>c=37.</math> We conclude that Joey ...</math>. Now, since <math>j-1=37</math>, by similar logic, <math>37=(1+k)(a-1)</math>, so <math>k=36</math> and Joey will be <math>38+36=74</math> and t

5 KB (888 words) - 05:20, 12 April 2024
2018 AMC 10B Problems/Problem 16
...ath> consecutive numbers must be a multiple of <math>k</math>, so <math>a^3-a</math> is both divisible by <math>2</math> and <math>3</math>. This provid {{AMC10 box|year=2018|ab=B|num-b=15|num-a=17}}

5 KB (814 words) - 19:27, 5 March 2024
2018 AMC 10B Problems
A three-dimensional rectangular box with dimensions <math>X</math>, <math>Y</math>, for(int j=0;j<=3-i;j=j+1)

15 KB (2,237 words) - 23:08, 15 November 2023
2018 AMC 12B Problems/Problem 9
<ol style="margin-left: 1.5em;"> More generally, the term <math>k+101</math> occurs <math>100-k</math> times for <math>k\in\{1,2,3,\ldots,99\}.</math></li>

5 KB (594 words) - 22:03, 27 May 2023
2018 AMC 10B Problems/Problem 8
for(int j=0;j<=3-i;j=j+1) This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircas

4 KB (609 words) - 19:49, 27 August 2023
2018 USAJMO Problems/Problem 5
...th>\{1, 2, ..., p\},</math> so they have a total of exactly <math>\frac{p(p-1)}{2}</math> edges. Therefore, there exists at least one graph <math>G_k</m {{USAJMO newbox|year=2018|num-b=4|num-a=6}}

2 KB (455 words) - 13:53, 21 April 2018
2018 USAMO Problems/Problem 4
...th>\{1, 2, ..., p\},</math> so they have a total of exactly <math>\frac{p(p-1)}{2}</math> edges. Therefore, there exists at least one graph <math>G_k</m The first equality is given by a well known theorem, which can be proven by C-S or Jensen's.

4 KB (794 words) - 21:46, 20 March 2023
2018 USAMO Problems/Problem 3
s+1. Therefore, the desired sum of k-th powers is equal is a complete reduced residue set modulo nq. The new sum of k-th powers is equal to

5 KB (939 words) - 22:00, 11 February 2024
2018 USAMO Problems/Problem 6
Now, consider what happens when the two rows are swapped (and the top-bottom pairs are reordered so that the top reads (1,2,3,...,n)). This will ...ds <math>b_n=n\cdot b_{n-1}</math>* which has the same parity as <math>b_{n-1}</math>, so we need only consider the parity of <math>b_n</math> for odd <

4 KB (700 words) - 04:36, 5 March 2023

Page

Toolbox

Search

Search results