https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Makorn&feedformat=atom AoPS Wiki - User contributions [en] 2021-01-18T01:10:52Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2010_AIME_I_Problems/Problem_15&diff=91229 2010 AIME I Problems/Problem 15 2018-02-15T04:14:13Z <p>Makorn: /* Solution 1 */</p> <hr /> <div>__TOC__<br /> == Problem ==<br /> In &lt;math&gt;\triangle{ABC}&lt;/math&gt; with &lt;math&gt;AB = 12&lt;/math&gt;, &lt;math&gt;BC = 13&lt;/math&gt;, and &lt;math&gt;AC = 15&lt;/math&gt;, let &lt;math&gt;M&lt;/math&gt; be a point on &lt;math&gt;\overline{AC}&lt;/math&gt; such that the [[incircle]]s of &lt;math&gt;\triangle{ABM}&lt;/math&gt; and &lt;math&gt;\triangle{BCM}&lt;/math&gt; have equal [[inradius|radii]]. Let &lt;math&gt;p&lt;/math&gt; and &lt;math&gt;q&lt;/math&gt; be positive [[relatively prime]] integers such that &lt;math&gt;\frac {AM}{CM} = \frac {p}{q}&lt;/math&gt;. Find &lt;math&gt;p + q&lt;/math&gt;.<br /> <br /> == Solution ==<br /> &lt;center&gt;&lt;asy&gt; /* from geogebra: see azjps, userscripts.org/scripts/show/72997 */<br /> import graph; defaultpen(linewidth(0.7)+fontsize(10)); size(200);<br /> <br /> /* segments and figures */<br /> draw((0,0)--(15,0));<br /> draw((15,0)--(6.66667,9.97775));<br /> draw((6.66667,9.97775)--(0,0));<br /> draw((7.33333,0)--(6.66667,9.97775));<br /> draw(circle((4.66667,2.49444),2.49444));<br /> draw(circle((9.66667,2.49444),2.49444));<br /> draw((4.66667,0)--(4.66667,2.49444));<br /> draw((9.66667,2.49444)--(9.66667,0));<br /> <br /> /* points and labels */<br /> label(&quot;r&quot;,(10.19662,1.92704),SE);<br /> label(&quot;r&quot;,(5.02391,1.8773),SE);<br /> dot((0,0));<br /> label(&quot;$A$&quot;,(-1.04408,-0.60958),NE);<br /> dot((15,0));<br /> label(&quot;$C$&quot;,(15.41907,-0.46037),NE);<br /> dot((6.66667,9.97775));<br /> label(&quot;$B$&quot;,(6.66525,10.23322),NE);<br /> label(&quot;$15$&quot;,(6.01866,-1.15669),NE);<br /> label(&quot;$13$&quot;,(11.44006,5.50815),NE);<br /> label(&quot;$12$&quot;,(2.28834,5.75684),NE);<br /> dot((7.33333,0));<br /> label(&quot;$M$&quot;,(7.56053,-0.908),NE);<br /> dot((4.66667,2.49444));<br /> label(&quot;$I_1$&quot;,(3.97942,2.92179),NE);<br /> dot((9.66667,2.49444));<br /> label(&quot;$I_2$&quot;,(10.04741,2.97153),NE);<br /> clip((-3.72991,-6.47862)--(-3.72991,17.44518)--(32.23039,17.44518)--(32.23039,-6.47862)--cycle);<br /> &lt;/asy&gt;&lt;/center&gt;<br /> === Solution 1 ===<br /> Let &lt;math&gt;AM = x&lt;/math&gt;, then &lt;math&gt;CM = 15 - x&lt;/math&gt;. Also let &lt;math&gt;BM = d&lt;/math&gt; Clearly, &lt;math&gt;\frac {[ABM]}{[CBM]} = \frac {x}{15 - x}&lt;/math&gt;. We can also express each area by the rs formula. Then &lt;math&gt;\frac {[ABM]}{[CBM]} = \frac {p(ABM)}{p(CBM)} = \frac {12 + d + x}{28 + d - x}&lt;/math&gt;. Equating and cross-multiplying yields &lt;math&gt;25x + 2dx = 15d + 180&lt;/math&gt; or &lt;math&gt;d = \frac {25x - 180}{15 - 2x}.&lt;/math&gt; Note that for &lt;math&gt;d&lt;/math&gt; to be positive, we must have &lt;math&gt;7.2 &lt; x &lt; 7.5&lt;/math&gt;.<br /> <br /> By [[Stewart's Theorem]], we have &lt;math&gt;12^2(15 - x) + 13^2x = d^215 + 15x(15 - x)&lt;/math&gt; or &lt;math&gt;432 = 3d^2 + 40x - 3x^2.&lt;/math&gt; Brute forcing by plugging in our previous result for &lt;math&gt;d&lt;/math&gt;, we have &lt;math&gt;432 = \frac {3(25x - 180)^2}{(15 - 2x)^2} + 40x - 3x^2.&lt;/math&gt; Clearing the fraction and gathering like terms, we get &lt;math&gt;0 = 12x^4 - 340x^3 + 2928x^2 - 7920x.&lt;/math&gt;<br /> <br /> ''Aside: Since &lt;math&gt;x&lt;/math&gt; must be rational in order for our answer to be in the desired form, we can use the [[Rational Root Theorem]] to reveal that &lt;math&gt;12x&lt;/math&gt; is an integer. The only such &lt;math&gt;x&lt;/math&gt; in the above-stated range is &lt;math&gt;\frac {22}3&lt;/math&gt;.''<br /> <br /> Legitimately solving that quartic, note that &lt;math&gt;x = 0&lt;/math&gt; and &lt;math&gt;x = 15&lt;/math&gt; should clearly be solutions, corresponding to the sides of the triangle and thus degenerate cevians. Factoring those out, we get &lt;math&gt;0 = 4x(x - 15)(3x^2 - 40x + 132) = x(x - 15)(x - 6)(3x - 22).&lt;/math&gt; The only solution in the desired range is thus &lt;math&gt;\frac {22}3&lt;/math&gt;. Then &lt;math&gt;CM = \frac {23}3&lt;/math&gt;, and our desired ratio &lt;math&gt;\frac {AM}{CM} = \frac {22}{23}&lt;/math&gt;, giving us an answer of &lt;math&gt;\boxed{045}&lt;/math&gt;.<br /> <br /> === Solution 2 ===<br /> Let &lt;math&gt;AM = 2x&lt;/math&gt; and &lt;math&gt;BM = 2y&lt;/math&gt; so &lt;math&gt;CM = 15 - 2x&lt;/math&gt;. Let the [[incenter]]s of &lt;math&gt;\triangle ABM&lt;/math&gt; and &lt;math&gt;\triangle BCM&lt;/math&gt; be &lt;math&gt;I_1&lt;/math&gt; and &lt;math&gt;I_2&lt;/math&gt; respectively, and their equal inradii be &lt;math&gt;r&lt;/math&gt;. From &lt;math&gt;r = \sqrt {(s - a)(s - b)(s - c)/s}&lt;/math&gt;, we find that<br /> <br /> &lt;cmath&gt;\begin{align*}r^2 &amp; = \frac {(x + y - 6)( - x + y + 6)(x - y + 6)}{x + y + 6} &amp; (1) \\<br /> &amp; = \frac {( - x + y + 1)(x + y - 1)( - x - y + 14)}{ - x + y + 14}. &amp; (2) \end{align*}&lt;/cmath&gt;<br /> <br /> Let the incircle of &lt;math&gt;\triangle ABM&lt;/math&gt; meet &lt;math&gt;AM&lt;/math&gt; at &lt;math&gt;P&lt;/math&gt; and the incircle of &lt;math&gt;\triangle BCM&lt;/math&gt; meet &lt;math&gt;CM&lt;/math&gt; at &lt;math&gt;Q&lt;/math&gt;. Then note that &lt;math&gt;I_1 P Q I_2&lt;/math&gt; is a rectangle. Also, &lt;math&gt;\angle I_1 M I_2&lt;/math&gt; is right because &lt;math&gt;MI_1&lt;/math&gt; and &lt;math&gt;MI_2&lt;/math&gt; are the angle bisectors of &lt;math&gt;\angle AMB&lt;/math&gt; and &lt;math&gt;\angle CMB&lt;/math&gt; respectively and &lt;math&gt;\angle AMB + \angle CMB = 180^\circ&lt;/math&gt;. By properties of [[tangent (geometry)|tangents]] to [[circle]]s &lt;math&gt;MP = (MA + MB - AB)/2 = x + y - 6&lt;/math&gt; and &lt;math&gt;MQ = (MB + MC - BC)/2 = - x + y + 1&lt;/math&gt;. Now notice that the altitude of &lt;math&gt;M&lt;/math&gt; to &lt;math&gt;I_1 I_2&lt;/math&gt; is of length &lt;math&gt;r&lt;/math&gt;, so by similar triangles we find that &lt;math&gt;r^2 = MP \cdot MQ = (x + y - 6)( - x + y + 1)&lt;/math&gt; (3). Equating (3) with (1) and (2) separately yields<br /> <br /> &lt;cmath&gt;\begin{align*}<br /> 2y^2 - 30 = 2xy + 5x - 7y \\<br /> 2y^2 - 70 = - 2xy - 5x + 7y, \end{align*}<br /> &lt;/cmath&gt;<br /> <br /> and adding these we have<br /> <br /> &lt;cmath&gt;<br /> 4y^2 - 100 = 0\implies y = 5\implies x = 11/3 \\<br /> \implies AM/MC = (22/3)/(15 - 22/3) = 22/23 \implies \boxed{045}.<br /> &lt;/cmath&gt;<br /> <br /> === Solution 3 ===<br /> Let the incircle of &lt;math&gt;ABM&lt;/math&gt; hit &lt;math&gt;AM&lt;/math&gt;, &lt;math&gt;AB&lt;/math&gt;, &lt;math&gt;BM&lt;/math&gt; at &lt;math&gt;X_{1},Y_{1},Z_{1}&lt;/math&gt;, and let the incircle of &lt;math&gt;CBM&lt;/math&gt; hit &lt;math&gt;MC&lt;/math&gt;, &lt;math&gt;BC&lt;/math&gt;, &lt;math&gt;BM&lt;/math&gt; at &lt;math&gt;X_{2},Y_{2},Z_{2}&lt;/math&gt;. Draw the incircle of &lt;math&gt;ABC&lt;/math&gt;, and let it be tangent to &lt;math&gt;AC&lt;/math&gt; at &lt;math&gt;X&lt;/math&gt;. Observe that we have a homothety centered at A sending the incircle of &lt;math&gt;ABM&lt;/math&gt; to that of &lt;math&gt;ABC&lt;/math&gt;, and one centered at &lt;math&gt;C&lt;/math&gt; taking the incircle of &lt;math&gt;BCM&lt;/math&gt; to that of &lt;math&gt;ABC&lt;/math&gt;. These have the same power, since they take incircles of the same size to the same circle. Also, the power of the homothety is &lt;math&gt;AX_{1}/AX=CX_{2}/CX&lt;/math&gt;.<br /> <br /> By standard computations, &lt;math&gt;AX=\dfrac{AB+AC-BC}{2}=7&lt;/math&gt; and &lt;math&gt;CX=\dfrac{BC+AC-AB}{2}=8&lt;/math&gt;. Now, let &lt;math&gt;AX_{1}=7x&lt;/math&gt; and &lt;math&gt;CX_{2}=8x&lt;/math&gt;. We will now go around and chase lengths. Observe that &lt;math&gt;BY_{1}=BA-AY_{1}=BA-AX_{1}=12-7x&lt;/math&gt;. Then, &lt;math&gt;BZ_{1}=12-7x&lt;/math&gt;. We also have &lt;math&gt;CY_{2}=CX_{2}=8x&lt;/math&gt;, so &lt;math&gt;BY_{2}=13-8x&lt;/math&gt; and &lt;math&gt;BZ_{2}=13-8x&lt;/math&gt;.<br /> <br /> Observe now that &lt;math&gt;X_{1}M+MX_{2}=AC-15x=15(1-x)&lt;/math&gt;. Also,&lt;math&gt;X_{1}M-MX_{2}=MZ_{1}-MZ_{2}=BZ_{2}-BZ_{1}=BY_{2}-BY_{1}=(1-x)&lt;/math&gt;. Solving, we get &lt;math&gt;X_{1}M=8-8x&lt;/math&gt; and &lt;math&gt;MX_{2}=7-7x&lt;/math&gt; (as a side note, note that &lt;math&gt;AX_{1}+MX_{2}=X_{1}M+X_{2}C&lt;/math&gt;, a result that I actually believe appears in Mandelbrot 1995-2003, or some book in that time-range).<br /> <br /> Now, we get &lt;math&gt;BM=BZ_{2}+Z_{2}M=BZ_{2}+MX_{2}=20-15x&lt;/math&gt;. To finish, we will compute area ratios. &lt;math&gt;\dfrac{[ABM]}{[CBM]}=\dfrac{AM}{MC}=\dfrac{8-x}{7+x}&lt;/math&gt;. Also, since their inradii are equal, we get &lt;math&gt;\dfrac{[ABM]}{[CBM]}=\dfrac{40-16x}{40-14x}&lt;/math&gt;. Equating and cross multiplying yields the quadratic &lt;math&gt;3x^{2}-8x+4=0&lt;/math&gt;, so &lt;math&gt;x=2/3,2&lt;/math&gt;. However, observe that &lt;math&gt;AX_{1}+CX_{2}=15x&lt;15&lt;/math&gt;, so we take &lt;math&gt;x=2/3&lt;/math&gt;. Our ratio is therefore &lt;math&gt;\dfrac{8-2/3}{7+2/3}=\dfrac{22}{23}&lt;/math&gt;, giving the answer &lt;math&gt;\boxed{045}&lt;/math&gt;.<br /> <br /> === Solution 4 ===<br /> Suppose the incircle of &lt;math&gt;ABM&lt;/math&gt; touches &lt;math&gt;AM&lt;/math&gt; at &lt;math&gt;X&lt;/math&gt;, and the incircle of &lt;math&gt;CBM&lt;/math&gt; touches &lt;math&gt;CM&lt;/math&gt; at &lt;math&gt;Y&lt;/math&gt;. Then<br /> <br /> &lt;cmath&gt;r = AX \tan(A/2) = CY \tan(C/2)&lt;/cmath&gt;<br /> <br /> We have &lt;math&gt;\cos A = \frac{12^2+15^2-13^2}{2\cdot 12\cdot 15} = \frac{200}{30\cdot 12}=\frac{5}{9}&lt;/math&gt;, &lt;math&gt;\tan(A/2) = \sqrt{\frac{1-\cos A}{1+\cos A}} = \sqrt{\frac{9-5}{9+5}} = \frac{2}{\sqrt{14}}&lt;/math&gt;<br /> <br /> &lt;math&gt;\cos C = \frac{13^2+15^2-12^2}{2\cdot 13\cdot 15} = \frac{250}{30\cdot 13} = \frac{25}{39}&lt;/math&gt;, &lt;math&gt;\tan(C/2) = \sqrt{\frac{39-25}{39+25}}=\frac{\sqrt{14}}{8}&lt;/math&gt;,<br /> <br /> Therefore &lt;math&gt;AX/CY = \tan(C/2)/\tan(A/2) = \frac{14}{2\cdot 8}= \frac{7}{8}.&lt;/math&gt;<br /> <br /> And since &lt;math&gt;AX=\frac{1}{2}(12+AM-BM)&lt;/math&gt;, &lt;math&gt;CY = \frac{1}{2}(13+CM-BM)&lt;/math&gt;, <br /> <br /> &lt;cmath&gt; \frac{12+AM-BM}{13+CM-BM} = \frac{7}{8}&lt;/cmath&gt;<br /> <br /> &lt;cmath&gt; 96+8AM-8BM = 91 +7CM-7BM&lt;/cmath&gt;<br /> <br /> &lt;cmath&gt;BM= 5 + 8AM-7CM = 5 + 15AM - 7(CM+AM) = 5+15(AM-7) \dots\dots (1)&lt;/cmath&gt;<br /> <br /> Now,<br /> <br /> &lt;math&gt;\frac{AM}{CM} = \frac{[ABM]}{[CBM]} = \frac{\frac{1}{2}(12+AM+BM)r}{\frac{1}{2}(13+CM+BM)r}=\frac{12+AM+BM}{13+CM+BM}= \frac{12+BM}{13+BM} = \frac{17+15(AM-7)}{18+15(AM-7)}&lt;/math&gt;<br /> <br /> &lt;cmath&gt;\frac{AM}{15} = \frac{17+15(AM-7)}{35+30(AM-7)} = \frac{15AM-88}{30AM-175}&lt;/cmath&gt;<br /> &lt;cmath&gt;6AM^2 - 35AM = 45AM-264&lt;/cmath&gt;<br /> &lt;cmath&gt;3AM^2 -40AM+132=0&lt;/cmath&gt;<br /> &lt;cmath&gt;(3AM-22)(AM-6)=0&lt;/cmath&gt;<br /> <br /> So &lt;math&gt;AM=22/3&lt;/math&gt; or &lt;math&gt;6&lt;/math&gt;. But from (1) we know that &lt;math&gt;5+15(AM-7)&gt;0&lt;/math&gt;, or &lt;math&gt;AM&gt;7-1/3&gt;6&lt;/math&gt;, so &lt;math&gt;AM=22/3&lt;/math&gt;, &lt;math&gt;CM=15-22/3=23/3&lt;/math&gt;, &lt;math&gt;AM/CM=22/23&lt;/math&gt;.<br /> <br /> '''Sidenote'''<br /> <br /> In the problem, &lt;math&gt;BM=10&lt;/math&gt; and the equal inradius of the two triangles happens to be &lt;math&gt; \frac {2\sqrt{14}}{3}&lt;/math&gt;.<br /> <br /> == See Also ==<br /> <br /> <br /> {{AIME box|year=2010|num-b=14|after=Last Problem|n=I}}<br /> <br /> [[Category:Intermediate Geometry Problems]]<br /> {{MAA Notice}}</div> Makorn https://artofproblemsolving.com/wiki/index.php?title=2018_AMC_10A_Problems/Problem_21&diff=91083 2018 AMC 10A Problems/Problem 21 2018-02-12T03:00:32Z <p>Makorn: /* Solution 6 (Even Simpler Cheating with Answer Choices) */</p> <hr /> <div>== Problem ==<br /> <br /> Which of the following describes the set of values of &lt;math&gt;a&lt;/math&gt; for which the curves &lt;math&gt;x^2+y^2=a^2&lt;/math&gt; and &lt;math&gt;y=x^2-a&lt;/math&gt; in the real &lt;math&gt;xy&lt;/math&gt;-plane intersect at exactly &lt;math&gt;3&lt;/math&gt; points?<br /> <br /> &lt;math&gt;<br /> \textbf{(A) }a=\frac14 \qquad<br /> \textbf{(B) }\frac14 &lt; a &lt; \frac12 \qquad<br /> \textbf{(C) }a&gt;\frac14 \qquad<br /> \textbf{(D) }a=\frac12 \qquad<br /> \textbf{(E) }a&gt;\frac12 \qquad<br /> &lt;/math&gt;<br /> <br /> == Solution 1 ==<br /> <br /> Substituting &lt;math&gt;y=x^2-a&lt;/math&gt; into &lt;math&gt;x^2+y^2=a^2&lt;/math&gt;, we get<br /> &lt;cmath&gt;<br /> x^2+(x^2-a)^2=a^2 \implies x^2+x^4-2ax^2=0 \implies x^2(x^2-(2a-1))=0<br /> &lt;/cmath&gt;<br /> Since this is a quartic, there are 4 total roots (counting multiplicity). We see that &lt;math&gt;x=0&lt;/math&gt; always at least one intersection at &lt;math&gt;(0,-a)&lt;/math&gt; (and is in fact a double root). <br /> <br /> The other two intersection points have &lt;math&gt;x&lt;/math&gt; coordinates &lt;math&gt;\sqrt{2a-1}&lt;/math&gt;. We must have &lt;math&gt;2a-1&gt; 0,&lt;/math&gt; otherwise we are in the case where the parabola lies entirely above the circle (tangent to it at the point &lt;math&gt;(0,a)&lt;/math&gt;). This only results in a single intersection point in the real coordinate plane. Thus, we see &lt;math&gt;a&gt;\frac{1}{2}&lt;/math&gt;.<br /> <br /> (projecteulerlover)<br /> <br /> == Solution 2 ==<br /> <br /> &lt;asy&gt;<br /> Label f; <br /> f.p=fontsize(6);<br /> xaxis(-2,2,Ticks(f, 0.2)); <br /> yaxis(-2,2,Ticks(f, 0.2)); <br /> real g(real x) <br /> { <br /> return x^2-1; <br /> } <br /> draw(graph(g, 1.7, -1.7));<br /> real h(real x) <br /> { <br /> return sqrt(1-x^2); <br /> } <br /> draw(graph(h, 1, -1));<br /> real j(real x) <br /> { <br /> return -sqrt(1-x^2); <br /> } <br /> draw(graph(j, 1, -1));<br /> &lt;/asy&gt;<br /> <br /> Looking at a graph, it is obvious that the two curves intersect at (0, -a). We also see that if the parabola go's 'in' the circle, than by going out of it (as it will) it will intersect five times, an impossibility. Thus we only look for cases where the parabola becomes externally tangent to the circle. We have &lt;math&gt;x^2 - a = -\sqrt(a^2 - x^2)&lt;/math&gt;. Squaring both sides and solving yields &lt;math&gt;x^4 - (2a - 1)x^2 = 0&lt;/math&gt;. Since x = 0 is already accounted for, we only need to find 1 solution for &lt;math&gt;x^2 = 2a - 1&lt;/math&gt;, where the right hand side portion is obviously increasing. Since a = 1/2 begets x = 0 (an overcount), we have &lt;math&gt;a &gt; 1/2 -&gt; E&lt;/math&gt; is the right answer.<br /> <br /> Solution by JohnHankock<br /> <br /> == Solution 3 ==<br /> <br /> This describes a unit parabola, with a circle centered at the axis of symmetry and tangent to the vertex. As the curvature of the unit parabola at the vertex is 2, the radius of the circle that matches it has a radius of &lt;math&gt;\frac{1}{2}&lt;/math&gt;. This circle is tangent to an infinitesimally close pair of points, one on each side. Therefore, it is tangent to only 1 point. When a larger circle is used, it is tangent to 3 points because the points on either side are now separated from the vertex. Therefore, &lt;math&gt;\boxed{a &gt; \frac{1}{2}}&lt;/math&gt; or &lt;math&gt;\boxed{E}&lt;/math&gt; is correct.<br /> <br /> &lt;math&gt;QED \blacksquare&lt;/math&gt;<br /> <br /> <br /> == Solution 4 ==<br /> <br /> Notice, the equations are of that of a circle of radius a centered at the origin and a parabola translated down by a units. They always intersect at the point &lt;math&gt;(0, a)&lt;/math&gt;, and they have symmetry across the y-axis, thus, for them to intersect at exactly 3 points, it suffices to find the y solution. <br /> <br /> First, rewrite the second equation to &lt;math&gt;y=x^2-a\implies x^2=y+a&lt;/math&gt;<br /> And substitute into the first equation: &lt;math&gt;y+a+y^2=a^2&lt;/math&gt; <br /> Since we're only interested in seeing the interval in which a can exist, we find the discriminant: &lt;math&gt;1-4a+4a^2&lt;/math&gt;. This value must not be less than 0 (It is the square root part of the quadratic formula). To find when it is 0, we find the roots: <br /> &lt;cmath&gt;4a^2-4a+1=0 \implies a=\frac{4\pm\sqrt{16-16}}{8}=\frac{1}{2}&lt;/cmath&gt;<br /> Since &lt;math&gt;\lim_{a\to \infty}(4a^2-4a+1)=\infty&lt;/math&gt;, our range is &lt;math&gt;\boxed{a&gt;\frac{1}{2}}&lt;/math&gt;<br /> <br /> Solution by ktong<br /> <br /> == Solution 5 (Cheating with Answer Choices) ==<br /> Simply plug in &lt;math&gt;a = 0, \frac{1}{2}, \frac{1}{4}, 1&lt;/math&gt; and solve the systems. (This shouldn't take too long.) And realized that only &lt;math&gt;a=1&lt;/math&gt; yields three real solutions for &lt;math&gt;x&lt;/math&gt;, so we are done and the answer is &lt;math&gt;\boxed{a&gt;\frac{1}{2}}&lt;/math&gt;<br /> <br /> ~ ccx09<br /> == See Also ==<br /> <br /> {{AMC10 box|year=2018|ab=A|num-b=20|num-a=22}}<br /> {{AMC12 box|year=2018|ab=A|num-b=15|num-a=17}}<br /> {{MAA Notice}}</div> Makorn https://artofproblemsolving.com/wiki/index.php?title=AMC_historical_results&diff=91060 AMC historical results 2018-02-11T22:56:38Z <p>Makorn: /* AMC 10A */</p> <hr /> <div>&lt;!-- Post AMC statistics and lists of high scorers here so that the AMC page doesn't get cluttered. --&gt;<br /> This is the '''AMC historical results''' page. This page should include results for the [[AIME]] as well. For [[USAMO]] results, see [[USAMO historical results]].<br /> <br /> ==2018==<br /> ===AMC 10A===<br /> *Average score: tba<br /> *AIME floor: tba<br /> *Distinguished Honor Roll floor: tba<br /> <br /> ===AMC 10B===<br /> This exam has not occurred yet.<br /> <br /> ===AMC 12A===<br /> *Average score: <br /> *AIME floor: <br /> *Distinguished Honor Roll floor:<br /> <br /> ===AMC 12B===<br /> This exam has not occurred yet.<br /> <br /> ==2017==<br /> ===AMC 10A===<br /> *Average score: 59.33<br /> *AIME floor: 110<br /> *DHR: 127.5<br /> <br /> ===AMC 10B===<br /> *Average score: 66.55<br /> *AIME floor: 120<br /> *DHR: 136.5<br /> <br /> ===AMC 12A===<br /> *Average score: 60.32<br /> *AIME floor: 92<br /> *DHR: 110<br /> <br /> ===AMC 12B===<br /> *Average score: 58.35<br /> *AIME floor: 100.5<br /> *DHR: 129<br /> <br /> ===AIME I===<br /> *Average score: 5.69<br /> *Median score: 6<br /> *USAMO cutoff: 225(AMC 12A), 235(AMC 12B)<br /> *USAJMO cutoff: 224.5(AMC 10A), 233(AMC 10B)<br /> <br /> ===AIME II===<br /> *Average score: 5.64<br /> *Median score: 6<br /> *USAMO cutoff: 221(AMC 12A), 230.5(AMC 12B)<br /> *USAJMO cutoff: 219(AMC 10A), 225(AMC 10B)<br /> <br /> ==2016==<br /> ===AMC 10A===<br /> *Average score: 65.3<br /> *AIME floor: 110<br /> *DHR: 120<br /> <br /> ===AMC 10B===<br /> *Average score: 65.4<br /> *AIME floor: 110<br /> *DHR: 124.5<br /> <br /> ===AMC 12A===<br /> *Average score: 59.06<br /> *AIME floor: 92<br /> *DHR: 110<br /> <br /> ===AMC 12B===<br /> *Average score: 67.96<br /> *AIME floor: 100<br /> *DHR: 127.5<br /> <br /> ===AIME I===<br /> *Average score: 5.83<br /> *Median score: 6<br /> *USAMO cutoff: 220<br /> *USAJMO cutoff: 210.5<br /> <br /> ===AIME II===<br /> *Average score: 4.33<br /> *Median score: 4<br /> *USAMO cutoff: 205<br /> *USAJMO cutoff: 200<br /> <br /> ==2015==<br /> ===AMC 10A===<br /> *Average score: 73.39<br /> *AIME floor: 106.5<br /> *DHR: 115.5<br /> <br /> ===AMC 10B===<br /> *Average score: 76.10<br /> *AIME floor: 120<br /> *DHR: 132<br /> <br /> ===AMC 12A===<br /> *Average score: 69.90<br /> *AIME floor: 99<br /> *DHR: 117<br /> <br /> ===AMC 12B===<br /> *Average score: 66.92<br /> *AIME floor: 100<br /> *DHR: 126<br /> <br /> ===AIME I===<br /> *Average score: 5.29<br /> *Median score: 5<br /> *USAMO cutoff: 219.0<br /> *USAJMO cutoff: 213.0<br /> <br /> ===AIME II===<br /> *Average score: 6.63<br /> *Median score: 6<br /> *USAMO cutoff: 229.0<br /> *USAJMO cutoff: 223.5<br /> <br /> ==2014==<br /> ===AMC 10A===<br /> *Average score: 63.83<br /> *AIME floor: 120<br /> <br /> ===AMC 10B===<br /> *Average score: 71.44<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 64.01<br /> *AIME floor: 93<br /> <br /> ===AMC 12B===<br /> *Average score: 68.11<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score: 4.88<br /> *Median score: 5<br /> *USAMO cutoff: 211.5<br /> *USAJMO cutoff: 211<br /> <br /> ===AIME II===<br /> *Average score: 5.49<br /> *Median score: 5<br /> *USAMO cutoff: 211.5<br /> *USAJMO cutoff: 211<br /> <br /> ==2013==<br /> ===AMC 10A===<br /> *Average score: 72.50<br /> *AIME floor: 108<br /> <br /> ===AMC 10B===<br /> *Average score: 72.62<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 65.06<br /> *AIME floor:88.5<br /> <br /> ===AMC 12B===<br /> *Average score: 64.21<br /> *AIME floor: 93<br /> <br /> ===AIME I===<br /> *Average score: 4.69<br /> *Median score: <br /> *USAMO cutoff: 209<br /> *USAJMO cutoff: 210.5<br /> <br /> ===AIME II===<br /> *Average score: 6.56<br /> *Median score: <br /> *USAMO cutoff: 209<br /> *USAJMO cutoff: 210.5<br /> <br /> ==2012==<br /> ===AMC 10A===<br /> *Average score: 72.51<br /> *AIME floor: 115.5<br /> <br /> ===AMC 10B===<br /> *Average score: 76.59<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 64.62<br /> *AIME floor: 94.5<br /> <br /> ===AMC 12B===<br /> *Average score: 70.08<br /> *AIME floor: 99<br /> <br /> ===AIME I===<br /> *Average score: 5.13<br /> *Median score: <br /> *USAMO cutoff: 204.5<br /> *USAJMO cutoff: 204<br /> <br /> ===AIME II===<br /> *Average score: 4.94<br /> *Median score: <br /> *USAMO cutoff: 204.5<br /> *USAJMO cutoff: 204<br /> <br /> ==2011==<br /> ===AMC 10A===<br /> *Average score: 67.61<br /> *AIME floor: 117<br /> <br /> ===AMC 10B===<br /> *Average score: 71.78<br /> *AIME floor: 117<br /> <br /> ===AMC 12A===<br /> *Average score: 66.77<br /> *AIME floor: 93<br /> <br /> ===AMC 12B===<br /> *Average score: 64.71<br /> *AIME floor: 97.5<br /> <br /> ===AIME I===<br /> *Average score: 5.47<br /> *Median score: <br /> *USAMO cutoff: 215.5<br /> *USAJMO cutoff: 196.5<br /> <br /> ===AIME II===<br /> *Average score: 2.23<br /> *Median score: <br /> *USAMO cutoff: 188<br /> *USAJMO cutoff: 179<br /> <br /> ==2010==<br /> ===AMC 10A===<br /> *Average score: 68.11<br /> *AIME floor: 115.5<br /> <br /> ===AMC 10B===<br /> *Average score: 68.57<br /> *AIME floor: 118.5<br /> <br /> ===AMC 12A===<br /> *Average score: 61.02<br /> *AIME floor: 88.5<br /> <br /> ===AMC 12B===<br /> *Average score: 59.58<br /> *AIME floor: 88.5<br /> <br /> ===AIME I===<br /> *Average score: 5.90<br /> *Median score: <br /> *USAMO cutoff: 208.5 (204.5 for non juniors and seniors)<br /> *USAJMO cutoff: 188.5<br /> <br /> ===AIME II===<br /> *Average score: 3.39<br /> *Median score: <br /> *USAMO cutoff: 208.5 (204.5 for non juniors and seniors)<br /> *USAJMO cutoff: 188.5<br /> <br /> ==2009==<br /> ===AMC 10A===<br /> *Average score: 67.41<br /> *AIME floor: 120<br /> <br /> ===AMC 10B===<br /> *Average score: 74.73<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 66.37<br /> *AIME floor: 97.5<br /> <br /> ===AMC 12B===<br /> *Average score: 71.88<br /> *AIME floor: 100 (Top 5% (1.00))<br /> <br /> ===AIME I===<br /> *Average score: 4.17<br /> *Median score: 4<br /> *USAMO floor: <br /> <br /> ===AIME II===<br /> *Average score: 3.27<br /> *Median score: 3<br /> *USAMO floor:<br /> <br /> ==2008==<br /> ===AMC 10A===<br /> *Average score: <br /> *AIME floor: <br /> <br /> ===AMC 10B===<br /> *Average score: <br /> *AIME floor: <br /> <br /> ===AMC 12A===<br /> *Average score: 65.6<br /> *AIME floor: 97.5<br /> <br /> ===AMC 12B===<br /> *Average score: 68.9<br /> *AIME floor: 97.5<br /> <br /> ===AIME I===<br /> *Average score: <br /> *Median score: <br /> *USAMO floor: <br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2007==<br /> <br /> ===AMC 10A===<br /> *Average score: 67.9<br /> *AIME floor: 117<br /> <br /> ===AMC 10B===<br /> *Average score: 61.5<br /> *AIME floor: 115.5<br /> <br /> ===AMC 12A===<br /> *Average score: 66.8<br /> *AIME floor: 97.5<br /> <br /> ===AMC 12B===<br /> *Average score: 73.1<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score: 5<br /> *Median score: 3<br /> *USAMO floor: 6<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2006==<br /> ===AMC 10A===<br /> *Average score: 79.0<br /> *AIME floor: 120<br /> <br /> ===AMC 10B===<br /> *Average score: 68.5<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 85.7<br /> *AIME floor: 100<br /> <br /> ===AMC 12B===<br /> *Average score: 85.5<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2005==<br /> ===AMC 10A===<br /> *Average score: 74.0<br /> *AIME floor: 120<br /> <br /> ===AMC 10B===<br /> *Average score: 79.0<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 78.7<br /> *AIME floor: 100<br /> <br /> ===AMC 12B===<br /> *Average score: 83.4<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2004==<br /> ===AMC 10A===<br /> *Average score: 69.1<br /> *AIME floor: 110<br /> <br /> ===AMC 10B===<br /> *Average score: 80.4<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 73.9<br /> *AIME floor: 100<br /> <br /> ===AMC 12B===<br /> *Average score: 84.5<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2003==<br /> ===AMC 10A===<br /> *Average score: 74.4<br /> *AIME floor: 119<br /> <br /> ===AMC 10B===<br /> *Average score: 79.6<br /> *AIME floor: 121<br /> <br /> ===AMC 12A===<br /> *Average score: 77.8<br /> *AIME floor: 100<br /> <br /> ===AMC 12B===<br /> *Average score: 76.6<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2002==<br /> ===AMC 10A===<br /> *Average score: 68.5<br /> *AIME floor: 115<br /> <br /> ===AMC 10B===<br /> *Average score: 74.9<br /> *AIME floor: 118<br /> <br /> ===AMC 12A===<br /> *Average score: 72.7<br /> *AIME floor: 100<br /> <br /> ===AMC 12B===<br /> *Average score: 80.8<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2001==<br /> ===AMC 10===<br /> *Average score: 67.8<br /> *AIME floor: 116<br /> <br /> ===AMC 12===<br /> *Average score: 56.6<br /> *AIME floor: 84<br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2000==<br /> ===AMC 10===<br /> *Average score: 64.2<br /> *AIME floor: <br /> <br /> ===AMC 12===<br /> *Average score: 64.9<br /> *AIME floor: <br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==1999==<br /> ===AHSME===<br /> *Average score: 68.8<br /> *AIME floor:<br /> <br /> ===AIME===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==1998==<br /> none<br /> <br /> ==1997==<br /> ==1996==<br /> ==1995==<br /> ==1994==<br /> ==1993==<br /> ==1992==<br /> ==1991==<br /> ==1990==<br /> ==1989==<br /> ==1988==<br /> ==1987==<br /> ==1986==<br /> ==1985==<br /> ==1984==<br /> ==1983==<br /> ==1982==<br /> ==1981==<br /> ==1980==<br /> ==1979==<br /> ==1978==<br /> ==1977==<br /> ==1976==<br /> ==1975==<br /> ==1974==<br /> ==1973==<br /> ==1972==<br /> ==1971==<br /> ==1970==<br /> ==1969==<br /> ==1968==<br /> ==1967==<br /> ==1966==<br /> ==1965==<br /> ==1964==<br /> ==1963==<br /> ==1962==<br /> ==1961==<br /> ==1960==<br /> <br /> ==1959==</div> Makorn https://artofproblemsolving.com/wiki/index.php?title=AMC_historical_results&diff=91059 AMC historical results 2018-02-11T22:51:03Z <p>Makorn: /* AMC 10A */</p> <hr /> <div>&lt;!-- Post AMC statistics and lists of high scorers here so that the AMC page doesn't get cluttered. --&gt;<br /> This is the '''AMC historical results''' page. This page should include results for the [[AIME]] as well. For [[USAMO]] results, see [[USAMO historical results]].<br /> <br /> ==2018==<br /> ===AMC 10A===<br /> *Average score: tba<br /> *AIME floor: tba<br /> *Distinguished Honor Roll floor: tba<br /> <br /> ===AMC 10B===<br /> This exam has not occurred yet.<br /> <br /> ===AMC 12A===<br /> *Average score: <br /> *AIME floor: <br /> *Distinguished Honor Roll floor:<br /> <br /> ===AMC 12B===<br /> This exam has not occurred yet.<br /> <br /> ==2017==<br /> ===AMC 10A===<br /> *Average score: 59.33<br /> *AIME floor: 112.5<br /> *DHR: 127.5<br /> <br /> ===AMC 10B===<br /> *Average score: 66.55<br /> *AIME floor: 120<br /> *DHR: 136.5<br /> <br /> ===AMC 12A===<br /> *Average score: 60.32<br /> *AIME floor: 92<br /> *DHR: 110<br /> <br /> ===AMC 12B===<br /> *Average score: 58.35<br /> *AIME floor: 100.5<br /> *DHR: 129<br /> <br /> ===AIME I===<br /> *Average score: 5.69<br /> *Median score: 6<br /> *USAMO cutoff: 225(AMC 12A), 235(AMC 12B)<br /> *USAJMO cutoff: 224.5(AMC 10A), 233(AMC 10B)<br /> <br /> ===AIME II===<br /> *Average score: 5.64<br /> *Median score: 6<br /> *USAMO cutoff: 221(AMC 12A), 230.5(AMC 12B)<br /> *USAJMO cutoff: 219(AMC 10A), 225(AMC 10B)<br /> <br /> ==2016==<br /> ===AMC 10A===<br /> *Average score: 65.3<br /> *AIME floor: 110<br /> *DHR: 120<br /> <br /> ===AMC 10B===<br /> *Average score: 65.4<br /> *AIME floor: 110<br /> *DHR: 124.5<br /> <br /> ===AMC 12A===<br /> *Average score: 59.06<br /> *AIME floor: 92<br /> *DHR: 110<br /> <br /> ===AMC 12B===<br /> *Average score: 67.96<br /> *AIME floor: 100<br /> *DHR: 127.5<br /> <br /> ===AIME I===<br /> *Average score: 5.83<br /> *Median score: 6<br /> *USAMO cutoff: 220<br /> *USAJMO cutoff: 210.5<br /> <br /> ===AIME II===<br /> *Average score: 4.33<br /> *Median score: 4<br /> *USAMO cutoff: 205<br /> *USAJMO cutoff: 200<br /> <br /> ==2015==<br /> ===AMC 10A===<br /> *Average score: 73.39<br /> *AIME floor: 106.5<br /> *DHR: 115.5<br /> <br /> ===AMC 10B===<br /> *Average score: 76.10<br /> *AIME floor: 120<br /> *DHR: 132<br /> <br /> ===AMC 12A===<br /> *Average score: 69.90<br /> *AIME floor: 99<br /> *DHR: 117<br /> <br /> ===AMC 12B===<br /> *Average score: 66.92<br /> *AIME floor: 100<br /> *DHR: 126<br /> <br /> ===AIME I===<br /> *Average score: 5.29<br /> *Median score: 5<br /> *USAMO cutoff: 219.0<br /> *USAJMO cutoff: 213.0<br /> <br /> ===AIME II===<br /> *Average score: 6.63<br /> *Median score: 6<br /> *USAMO cutoff: 229.0<br /> *USAJMO cutoff: 223.5<br /> <br /> ==2014==<br /> ===AMC 10A===<br /> *Average score: 63.83<br /> *AIME floor: 120<br /> <br /> ===AMC 10B===<br /> *Average score: 71.44<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 64.01<br /> *AIME floor: 93<br /> <br /> ===AMC 12B===<br /> *Average score: 68.11<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score: 4.88<br /> *Median score: 5<br /> *USAMO cutoff: 211.5<br /> *USAJMO cutoff: 211<br /> <br /> ===AIME II===<br /> *Average score: 5.49<br /> *Median score: 5<br /> *USAMO cutoff: 211.5<br /> *USAJMO cutoff: 211<br /> <br /> ==2013==<br /> ===AMC 10A===<br /> *Average score: 72.50<br /> *AIME floor: 108<br /> <br /> ===AMC 10B===<br /> *Average score: 72.62<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 65.06<br /> *AIME floor:88.5<br /> <br /> ===AMC 12B===<br /> *Average score: 64.21<br /> *AIME floor: 93<br /> <br /> ===AIME I===<br /> *Average score: 4.69<br /> *Median score: <br /> *USAMO cutoff: 209<br /> *USAJMO cutoff: 210.5<br /> <br /> ===AIME II===<br /> *Average score: 6.56<br /> *Median score: <br /> *USAMO cutoff: 209<br /> *USAJMO cutoff: 210.5<br /> <br /> ==2012==<br /> ===AMC 10A===<br /> *Average score: 72.51<br /> *AIME floor: 115.5<br /> <br /> ===AMC 10B===<br /> *Average score: 76.59<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 64.62<br /> *AIME floor: 94.5<br /> <br /> ===AMC 12B===<br /> *Average score: 70.08<br /> *AIME floor: 99<br /> <br /> ===AIME I===<br /> *Average score: 5.13<br /> *Median score: <br /> *USAMO cutoff: 204.5<br /> *USAJMO cutoff: 204<br /> <br /> ===AIME II===<br /> *Average score: 4.94<br /> *Median score: <br /> *USAMO cutoff: 204.5<br /> *USAJMO cutoff: 204<br /> <br /> ==2011==<br /> ===AMC 10A===<br /> *Average score: 67.61<br /> *AIME floor: 117<br /> <br /> ===AMC 10B===<br /> *Average score: 71.78<br /> *AIME floor: 117<br /> <br /> ===AMC 12A===<br /> *Average score: 66.77<br /> *AIME floor: 93<br /> <br /> ===AMC 12B===<br /> *Average score: 64.71<br /> *AIME floor: 97.5<br /> <br /> ===AIME I===<br /> *Average score: 5.47<br /> *Median score: <br /> *USAMO cutoff: 215.5<br /> *USAJMO cutoff: 196.5<br /> <br /> ===AIME II===<br /> *Average score: 2.23<br /> *Median score: <br /> *USAMO cutoff: 188<br /> *USAJMO cutoff: 179<br /> <br /> ==2010==<br /> ===AMC 10A===<br /> *Average score: 68.11<br /> *AIME floor: 115.5<br /> <br /> ===AMC 10B===<br /> *Average score: 68.57<br /> *AIME floor: 118.5<br /> <br /> ===AMC 12A===<br /> *Average score: 61.02<br /> *AIME floor: 88.5<br /> <br /> ===AMC 12B===<br /> *Average score: 59.58<br /> *AIME floor: 88.5<br /> <br /> ===AIME I===<br /> *Average score: 5.90<br /> *Median score: <br /> *USAMO cutoff: 208.5 (204.5 for non juniors and seniors)<br /> *USAJMO cutoff: 188.5<br /> <br /> ===AIME II===<br /> *Average score: 3.39<br /> *Median score: <br /> *USAMO cutoff: 208.5 (204.5 for non juniors and seniors)<br /> *USAJMO cutoff: 188.5<br /> <br /> ==2009==<br /> ===AMC 10A===<br /> *Average score: 67.41<br /> *AIME floor: 120<br /> <br /> ===AMC 10B===<br /> *Average score: 74.73<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 66.37<br /> *AIME floor: 97.5<br /> <br /> ===AMC 12B===<br /> *Average score: 71.88<br /> *AIME floor: 100 (Top 5% (1.00))<br /> <br /> ===AIME I===<br /> *Average score: 4.17<br /> *Median score: 4<br /> *USAMO floor: <br /> <br /> ===AIME II===<br /> *Average score: 3.27<br /> *Median score: 3<br /> *USAMO floor:<br /> <br /> ==2008==<br /> ===AMC 10A===<br /> *Average score: <br /> *AIME floor: <br /> <br /> ===AMC 10B===<br /> *Average score: <br /> *AIME floor: <br /> <br /> ===AMC 12A===<br /> *Average score: 65.6<br /> *AIME floor: 97.5<br /> <br /> ===AMC 12B===<br /> *Average score: 68.9<br /> *AIME floor: 97.5<br /> <br /> ===AIME I===<br /> *Average score: <br /> *Median score: <br /> *USAMO floor: <br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2007==<br /> <br /> ===AMC 10A===<br /> *Average score: 67.9<br /> *AIME floor: 117<br /> <br /> ===AMC 10B===<br /> *Average score: 61.5<br /> *AIME floor: 115.5<br /> <br /> ===AMC 12A===<br /> *Average score: 66.8<br /> *AIME floor: 97.5<br /> <br /> ===AMC 12B===<br /> *Average score: 73.1<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score: 5<br /> *Median score: 3<br /> *USAMO floor: 6<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2006==<br /> ===AMC 10A===<br /> *Average score: 79.0<br /> *AIME floor: 120<br /> <br /> ===AMC 10B===<br /> *Average score: 68.5<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 85.7<br /> *AIME floor: 100<br /> <br /> ===AMC 12B===<br /> *Average score: 85.5<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2005==<br /> ===AMC 10A===<br /> *Average score: 74.0<br /> *AIME floor: 120<br /> <br /> ===AMC 10B===<br /> *Average score: 79.0<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 78.7<br /> *AIME floor: 100<br /> <br /> ===AMC 12B===<br /> *Average score: 83.4<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2004==<br /> ===AMC 10A===<br /> *Average score: 69.1<br /> *AIME floor: 110<br /> <br /> ===AMC 10B===<br /> *Average score: 80.4<br /> *AIME floor: 120<br /> <br /> ===AMC 12A===<br /> *Average score: 73.9<br /> *AIME floor: 100<br /> <br /> ===AMC 12B===<br /> *Average score: 84.5<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2003==<br /> ===AMC 10A===<br /> *Average score: 74.4<br /> *AIME floor: 119<br /> <br /> ===AMC 10B===<br /> *Average score: 79.6<br /> *AIME floor: 121<br /> <br /> ===AMC 12A===<br /> *Average score: 77.8<br /> *AIME floor: 100<br /> <br /> ===AMC 12B===<br /> *Average score: 76.6<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2002==<br /> ===AMC 10A===<br /> *Average score: 68.5<br /> *AIME floor: 115<br /> <br /> ===AMC 10B===<br /> *Average score: 74.9<br /> *AIME floor: 118<br /> <br /> ===AMC 12A===<br /> *Average score: 72.7<br /> *AIME floor: 100<br /> <br /> ===AMC 12B===<br /> *Average score: 80.8<br /> *AIME floor: 100<br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2001==<br /> ===AMC 10===<br /> *Average score: 67.8<br /> *AIME floor: 116<br /> <br /> ===AMC 12===<br /> *Average score: 56.6<br /> *AIME floor: 84<br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==2000==<br /> ===AMC 10===<br /> *Average score: 64.2<br /> *AIME floor: <br /> <br /> ===AMC 12===<br /> *Average score: 64.9<br /> *AIME floor: <br /> <br /> ===AIME I===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ===AIME II===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==1999==<br /> ===AHSME===<br /> *Average score: 68.8<br /> *AIME floor:<br /> <br /> ===AIME===<br /> *Average score:<br /> *Median score:<br /> *USAMO floor:<br /> <br /> ==1998==<br /> none<br /> <br /> ==1997==<br /> ==1996==<br /> ==1995==<br /> ==1994==<br /> ==1993==<br /> ==1992==<br /> ==1991==<br /> ==1990==<br /> ==1989==<br /> ==1988==<br /> ==1987==<br /> ==1986==<br /> ==1985==<br /> ==1984==<br /> ==1983==<br /> ==1982==<br /> ==1981==<br /> ==1980==<br /> ==1979==<br /> ==1978==<br /> ==1977==<br /> ==1976==<br /> ==1975==<br /> ==1974==<br /> ==1973==<br /> ==1972==<br /> ==1971==<br /> ==1970==<br /> ==1969==<br /> ==1968==<br /> ==1967==<br /> ==1966==<br /> ==1965==<br /> ==1964==<br /> ==1963==<br /> ==1962==<br /> ==1961==<br /> ==1960==<br /> <br /> ==1959==</div> Makorn https://artofproblemsolving.com/wiki/index.php?title=Mock_MathCounts&diff=84688 Mock MathCounts 2017-03-13T04:02:37Z <p>Makorn: /* National */</p> <hr /> <div>A Mock MathCounts is a contest intended to mimic an actual MathCounts exam. A number of Mock MathCounts competitions have been hosted on the Art of Problem Solving message boards. Some are made by one community member, while others may be sourced from a group or the general community. Then they are administered to the AoPS community. Different users may have a different way of participating: some may require signups, while others do not.<br /> <br /> Mock MathCounts' are usually very popular in the months leading up to the actual MathCounts competition. There is no guarantee that community members will make Mock MathCounts' in any given year, but it's usually a good bet that someone will.<br /> <br /> == Tips for Writing a Mock MathCounts ==<br /> Here are some tips to write a good mock MathCounts:<br /> * Have multiple people work on the problems. Getting more people will get variety and prevents mocks from having too many problems from one subject. Having a group is also good so they can discuss which problems are good or need improvement, and fix errors. More than one person working on a mock can get the test done faster and often times better quality.<br /> * Get people to proofread it. Have someone to look over the problems, make sure they are appropriate in difficulty and order, and understandable by the general audience.<br /> * Get people to work on the problems. Sometimes the actual problem difficulty may vary from what it appears when the problem is actually worked on.<br /> * Don't be afraid to harshly criticize your problems. Usually criticism is what helps the mocks get better!<br /> * Once the test has been released, it is usually a good idea to keep the answers from being released at the same time. This can help prevent cheating, and also gives the problem solvers more time to make sure the problems are solved correctly.<br /> * Make sure the problems are original. The AoPS community does not want to waste time seeing a collection of boring, cliche problems that in principle are identical to an existing problem from a past MathCounts, with a few numbers modified.<br /> * If you are really out of problems, it is fine to get problems from past competitions. If you do so, choose with caution, beware of copyright, and choose from lesser known competitions. Don't choose a problem from a recent MathCounts competition; choose one from a trivial competition few people have heard of.<br /> * Add variety. Nobody wants to see a round with 15 counting problems or 20 algebra problems. Generally, algebra and geometry should have a larger emphasis than number theory and counting. The target round should have problems from all four subjects.<br /> <br /> Basically, getting many people to work on the mock is a great way to help improve the quality, though it is not always the case if everyone gets off task and &quot;helpers&quot; becomes &quot;parasites&quot;.<br /> <br /> [https://docs.google.com/document/d/1-Q2I48r6tRtKEo9SI5jP4NPFCzL61XIK-4CHcIrCfO0/edit?usp=sharing Mock Mathcounts Template]<br /> <br /> == Past Mock Mathcounts ==<br /> <br /> Listed below are several Mock Mathcounts' which have been hosted over AoPS in the past. <br /> <br /> Difficulty is rated on a scale from 0 to 5, where 1 is school level and 5 is national level. Note that this is not necessarily the same as the intended difficulty. If you take a contest, please contribute by rating it.<br /> <br /> Also note that &quot;author&quot; is the user who started the mock competition, not his or her helpers.<br /> <br /> === Unrated ===<br /> <br /> If you attempt one of these contests, please help out by rating it.<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align:center;width:100%&quot;<br /> |-<br /> |<br /> ! scope=&quot;col&quot; | '''Author'''<br /> ! scope=&quot;col&quot; | '''Year'''<br /> ! scope=&quot;col&quot; | '''Initial Discussion'''<br /> ! scope=&quot;col&quot; | '''Problems'''<br /> ! scope=&quot;col&quot; width=80 | '''Answers'''<br /> ! scope=&quot;col&quot; | '''Results/Discussion'''<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts'''<br /> | ragnarok23<br /> | 2006<br /> | [http://www.artofproblemsolving.com/community/c3h126233p715641 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c3h126233p715641 Problems]<br /> | [http://www.artofproblemsolving.com/community/c3h126233p715805 Answers]<br /> | [http://www.artofproblemsolving.com/community/c3h126233p716320 Results / Discussion]<br /> |-<br /> <br /> ! scope = &quot;row&quot; | '''Mock Mathcounts'''<br /> | #H34N1<br /> | 2008<br /> | [http://artofproblemsolving.com/community/c3h189486p1040797 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c3h189486p1040797 Sprint] [http://www.artofproblemsolving.com/community/c3h189847p1042672 Target]<br /> | n/a<br /> | n/a<br /> |-<br /> ! scope = &quot;row&quot; | '''My Mock Test'''<br /> | Gauss1181<br /> | 2008<br /> | [http://www.artofproblemsolving.com/community/q2h219024p1214709 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/q2h219024p1214709 Target]<br /> [http://www.artofproblemsolving.com/community/c5h219024p1234025 Sprint]<br /> | n/a<br /> | [http://www.artofproblemsolving.com/community/c5h245901p1351126 Results / Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts Competition'''<br /> | andersonw<br /> | 2008<br /> | [http://www.artofproblemsolving.com/community/c3h203522p1120153 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c3h203522p1120153 Sprint / Target p1]<br /> [http://www.artofproblemsolving.com/community/c3h203522p1120164 Target p2]<br /> | n/a<br /> | n/a<br /> |-<br /> ! scope = &quot;row&quot; | '''Mock Mathcounts Round'''<br /> | abacadaea<br /> | 2008<br /> | [http://www.artofproblemsolving.com/community/c3h205996p1133776 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c3h205996p1135277 Sprint]<br /> | n/a<br /> | n/a<br /> |-<br /> ! scope=&quot;row&quot; | '''My Mock Test'''<br /> | runpengFAILS<br /> | 2008<br /> | [http://www.artofproblemsolving.com/community/c3h207334p1140925 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c3h207334p1151054 Problems]<br /> |<br /> | [http://www.artofproblemsolving.com/community/c3h207334p1157638 Results / Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''42nd Mock Mathcounts'''<br /> | xpmath<br /> | 2009<br /> | [http://www.artofproblemsolving.com/community/c5h244364 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c5h244364p1350625 Sprint/Target] [http://www.artofproblemsolving.com/community/c5h244364p1359885 Team]<br /> | n/a<br /> | [http://www.artofproblemsolving.com/community/c5h244364p1365622 Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Waffle's Mock Mathcounts'''<br /> | Waffle<br /> | 2009<br /> | [http://artofproblemsolving.com/community/c3h248999p1365677 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c3h248999p1365677 Problems]<br /> | n/a<br /> | n/a<br /> |-<br /> ! scope=&quot;row&quot; | '''Izzy's Mock Mathcounts'''<br /> | isabella2296<br /> | 2009<br /> | [http://artofproblemsolving.com/community/c5h250456p1372363 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c5h250456p1373251 Problems]<br /> | n/a<br /> | [http://artofproblemsolving.com/community/c5h250456p1407978 Results / Discussion]<br /> |}<br /> <br /> === School ===<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align:center;width:100%&quot;<br /> |-<br /> |<br /> ! scope=&quot;col&quot; | '''Author'''<br /> ! scope=&quot;col&quot; | '''Year'''<br /> ! scope=&quot;col&quot; | '''Initial Discussion'''<br /> ! scope=&quot;col&quot; | '''Problems'''<br /> ! scope=&quot;col&quot; width=80 | '''Answers'''<br /> ! scope=&quot;col&quot; | '''Results/Discussion'''<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock MC School'''<br /> | PowerOfPi<br /> | 2009<br /> | [http://artofproblemsolving.com/community/q1h303085p1640160 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c5h310266 Problems]<br /> | [http://www.artofproblemsolving.com/community/c5h310266p1684004 Answers]<br /> | [http://www.artofproblemsolving.com/community/c5h310266p1684004 Results / Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock MATHCOUNTS School'''<br /> | KingSmasher3<br /> | 2009<br /> | [http://artofproblemsolving.com/community/c5h318058p1710148 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c5h318058p1716110 Problems]<br /> | n/a<br /> | [http://artofproblemsolving.com/community/c5h318058p1722860 Results / Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mathcounts School Round'''<br /> | Maybach<br /> | 2009<br /> | [http://artofproblemsolving.com/community/c5h314524p1694449 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c5h314524p1694449 Problems]<br /> | [http://artofproblemsolving.com/community/c5h314524p1708822 Answers]<br /> | [http://artofproblemsolving.com/community/c5h314524p1708822 Results / Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts School'''<br /> | Th3Numb3rThr33<br /> | 2014<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&amp;t=616764 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c3h616764p3680716 Problems]<br /> | [http://www.artofproblemsolving.com/community/c3h616764p3695696 Answers]<br /> | [http://www.artofproblemsolving.com/community/c3h616764p3695692 Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mental MathCounts'''<br /> | suli<br /> | 2015<br /> | [http://www.artofproblemsolving.com/community/c3h626046 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c3h626046 Problems]<br /> | [http://www.artofproblemsolving.com/community/c3h626046 Answers]<br /> | [http://www.artofproblemsolving.com/community/c3h626046 Results/Discussion]<br /> |}<br /> <br /> === Chapter ===<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align:center;width:100%&quot;<br /> |-<br /> |<br /> ! scope=&quot;col&quot; | '''Author'''<br /> ! scope=&quot;col&quot; | '''Year'''<br /> ! scope=&quot;col&quot; | '''Initial Discussion'''<br /> ! scope=&quot;col&quot; | '''Problems'''<br /> ! scope=&quot;col&quot; width=80 | '''Answers'''<br /> ! scope=&quot;col&quot; | '''Results/Discussion'''<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts A'''<br /> | #H34N1<br /> | 2006<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=112370 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=112370 Problems]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=112370 Answers]<br /> | n/a<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock MathCounts Chapter'''<br /> | AIME15<br /> | 2008<br /> | [http://artofproblemsolving.com/community/c5h215501p1192141 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c5h215501p1194450 Sprint] [http://artofproblemsolving.com/community/c5h215501p1194458 Target]<br /> | n/a<br /> | [http://artofproblemsolving.com/community/c5h215501p1197115 Results / Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock MC Chapter'''<br /> | PowerOfPi<br /> | 2009<br /> | [http://www.artofproblemsolving.com/community/c5h310266p1674315 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c5h310266p1687744 Problems]<br /> | [http://www.artofproblemsolving.com/community/c5h310266p1696480 Answers]<br /> | [http://www.artofproblemsolving.com/community/c5h310266p1696480 Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock MATHCOUNTS Chapter'''<br /> | KingSmasher3<br /> | 2009<br /> | [http://artofproblemsolving.com/community/c5h318058p1710148 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c5h318058p1728962 Problems]<br /> | n/a<br /> | n/a<br /> |-<br /> ! scope=&quot;row&quot; | '''Mathcounts Chapter Round'''<br /> | Maybach<br /> | 2009<br /> | [http://artofproblemsolving.com/community/c5h314524p1709157 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c5h314524p1709157 Problems]<br /> | [http://artofproblemsolving.com/community/c5h314524p1738213 Answers]<br /> | [http://artofproblemsolving.com/community/c5h314524p1738213 Results / Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts B'''<br /> | Mathc314<br /> | 2011<br /> | [http://artofproblemsolving.com/community/q1h408536p2283460 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c3h408536p2323201 Problems]<br /> | n/a<br /> | n/a<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts'''<br /> | iNomOnCountdown<br /> | 2014<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&amp;t=615892 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&amp;t=615892 Problems]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&amp;t=615892&amp;start=100 Answers]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3681419#p3681419 Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts Chapter'''<br /> | RadioActive<br /> | 2015<br /> | [http://artofproblemsolving.com/community/c3h1095344_mock_mathcounts Initial Discussion]<br /> | [http://artofproblemsolving.com/community/q1h1095344p4904988 Problems]<br /> | [http://www.artofproblemsolving.com/community/c3h1095344p4963396 Answers]<br /> | [http://www.artofproblemsolving.com/community/c3h1095344p4963396 Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts Chapter'''<br /> | uwotm8<br /> | 2015<br /> | [http://www.artofproblemsolving.com/community/c3h1065066p4624179 Initial Discussion ]<br /> | [https://docs.google.com/document/d/1GnHrzm-pUk9l27d7U5xhGc0iik9WLLOZEhF4eXNRMFo/edit?usp=sharing Problems]<br /> | [https://docs.google.com/document/d/1ZnIKzvOiygCae1v_qH0UrgYp9qmarbBUA4KPBVMehKo/edit?usp=sharing Answers]<br /> | [http://www.artofproblemsolving.com/community/c3h1065066p4624179 Results/Discussion]<br /> |}<br /> <br /> === State ===<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align:center;width:100%&quot;<br /> |-<br /> |<br /> ! scope=&quot;col&quot; | '''Author'''<br /> ! scope=&quot;col&quot; | '''Year'''<br /> ! scope=&quot;col&quot; | '''Initial Discussion'''<br /> ! scope=&quot;col&quot; | '''Problems'''<br /> ! scope=&quot;col&quot; width=80 | '''Answers'''<br /> ! scope=&quot;col&quot; | '''Results/Discussion'''<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts National 2006'''<br /> | biffanddoc<br /> | 2006<br /> | [http://www.artofproblemsolving.com/community/c3h80490p460687 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c3h80490p460687 Problems]<br /> | n/a<br /> | n/a<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock MATHCOUNTS State'''<br /> | KingSmasher3<br /> | 2009<br /> | [http://artofproblemsolving.com/community/c5h318058p1710148 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c5h318058p1816174 Problems]<br /> | n/a<br /> | n/a<br /> |-<br /> ! scope = &quot;row&quot; | '''Mathcounts State Round'''<br /> | Maybach<br /> | 2009<br /> | [http://artofproblemsolving.com/community/c5h320439p1721733 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c5h320439p1721901 Sprint] [http://artofproblemsolving.com/community/c5h320439p1721835 Target]<br /> | n/a<br /> | [http://artofproblemsolving.com/community/c5h320439p1761051 Results / Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts A'''<br /> | BOGTRO<br /> | 2011<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=532&amp;t=397897 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2235592#p2235592 Problems]<br /> | [http://www.artofproblemsolving.com/community/c5h397897p2252152 Answers]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=532&amp;t=397897&amp;start=120 Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts Chapter'''<br /> | Th3Numb3rThr33<br /> | 2015<br /> | [http://www.artofproblemsolving.com/community/c3h616764p3674322 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c3h616764p3747811 Problems]<br /> | [http://www.artofproblemsolving.com/community/c3h616764p4578506 Answers]<br /> | [http://www.artofproblemsolving.com/community/c3h616764p4578506 Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Chapter Competition'''<br /> | SuperMathWiz<br /> | 2015<br /> | [http://artofproblemsolving.com/community/c3h1167839p5589956 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c188248h1169330p5750047 Problems]<br /> | [http://www.artofproblemsolving.com/community/c188248h1169330p5750047 Answers]<br /> | [http://artofproblemsolving.com/community/c188248h1173496_leaderboard Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''July 4th Mock State Competition'''<br /> | ishankhare<br /> | 2016<br /> | [http://www.artofproblemsolving.com/community/c3h1237230_july_4th_mock_mathcounts_state_round_released Initial Discussion]<br /> | [https://docs.google.com/document/d/16WS0lX5Mka8fnDkkOyMyufyOj9dIhlN5rWDFsy5J5Lk/edit?usp=sharing Problems]<br /> | [https://docs.google.com/document/d/10_4a_tKn1zDk18kgyxdbhBQJUVfJbGvx3haVCyI4GAE/edit Answers] <br /> | n/a<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock MathCounts State'''<br /> | reun<br /> | 2016<br /> | [http://www.artofproblemsolving.com/community/c3h1232462p6238505 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c3h1232462p6238505 Problems]<br /> | n/a<br /> | n/a<br /> |}<br /> <br /> === National ===<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align:center;width:100%&quot;<br /> |-<br /> |<br /> ! scope=&quot;col&quot; | '''Author'''<br /> ! scope=&quot;col&quot; | '''Year'''<br /> ! scope=&quot;col&quot; | '''Initial Discussion'''<br /> ! scope=&quot;col&quot; | '''Problems'''<br /> ! scope=&quot;col&quot; width=80 | '''Answers'''<br /> ! scope=&quot;col&quot; | '''Results/Discussion'''<br /> |-<br /> ! scope=&quot;row&quot; | '''My Second Mock Test!'''<br /> | Yongyi781<br /> | 2008<br /> | [http://www.artofproblemsolving.com/community/c5h217443p1205050 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c5h217443p1205050 Problems]<br /> | [http://www.artofproblemsolving.com/community/c5h221270 Answers]<br /> | [http://www.artofproblemsolving.com/community/c5h221270 Results / Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts Test'''<br /> | sansuunoousama<br /> | 2009<br /> | [http://artofproblemsolving.com/community/c300h328753p1760385 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c300h328753p1760385 Problems]<br /> | n/a<br /> | n/a<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts Exam'''<br /> | luppleAOPS<br /> | 2011<br /> | [http://artofproblemsolving.com/community/q1h394236p2190992 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c3h394236p2217970 Problems]<br /> | n/a<br /> | n/a<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts National'''<br /> | forthegreatergood<br /> | 2013<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=532&amp;t=529084&amp;hilit=mock+mathcounts Initial Discussion]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=532&amp;t=529084&amp;hilit=mock+mathcounts Problems]<br /> | [http://www.artofproblemsolving.com/community/c5h529084p3475693 Answers]<br /> | n/a<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts'''<br /> | BOGTRO<br /> | 2014<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&amp;t=587950&amp;hilit=Mock+Test+BOGTRO Initial Discussion]<br /> | [https://www.dropbox.com/s/objrd6j21bcej2t/MockMATHCOUNTS.pdf Problems]<br /> | n/a<br /> | n/a<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts National'''<br /> | joey8189681<br /> | 2014<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&amp;t=590225&amp;hilit=Mock+Mathcounts Initial Discussion]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3504732#p3504732 Problems]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3519020#p3519020 Answers]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3519032#p3519032 Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts C'''<br /> | efang<br /> | 2014<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&amp;t=580260 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3494201#p3494201 Problems]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3494201#p3494201 Answers]<br /> | [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3448027#p3448027 Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts'''<br /> | mudkipswims42<br /> | 2015<br /> | [http://www.artofproblemsolving.com/community/c3h1117951p5126600 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c3h1117951p5126600 Problems]<br /> | [http://www.artofproblemsolving.com/community/c3h1117951p5364623 Answers]<br /> | [http://www.artofproblemsolving.com/community/c3h1117951p5363824 Results] / [http://artofproblemsolving.com/community/c139972 Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Crowd Sourced Mock Mathcounts National'''<br /> | DeathLlama9<br /> | 2015<br /> | [http://www.artofproblemsolving.com/community/c3h1072164_lets_crowdsource_a_mathcounts_nationals_test_discussion Initial Discussion]<br /> | [http://artofproblemsolving.com/community/c3h1081992_crowdsourced_mock_mathcounts Problems]<br /> | [http://artofproblemsolving.com/community/c3h1081992p4925914 Answers]<br /> | [http://artofproblemsolving.com/community/c3h1081992p4925914 Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts State B'''<br /> | Th3Numb3rThr33<br /> | 2015<br /> | [http://www.artofproblemsolving.com/community/c3h1059651p4584203 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c3h1059651p4584203 Problems]<br /> | [http://www.artofproblemsolving.com/community/c3h1059651p4584203 Answers]<br /> | [http://www.artofproblemsolving.com/community/c3h1059651p4584203 Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts National'''<br /> | Th3Numb3rThr33<br /> | 2015<br /> | [http://www.artofproblemsolving.com/community/c3h1076701p4708562 Initial Discussion]<br /> | [http://artofproblemsolving.com/community/u159196h1076701p4911404 Problems]<br /> | [http://artofproblemsolving.com/community/u159196h1076701p4911404 Answers]<br /> | [http://artofproblemsolving.com/community/u159196h1076701p4796869 Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts National Target'''<br /> | nosaj<br /> | 2015<br /> | [http://www.artofproblemsolving.com/community/c3h1072142p4666782 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c3h1072142p4666782 Problems]<br /> | [http://www.artofproblemsolving.com/community/c3h1072142p4709422 Answers]<br /> | [http://www.artofproblemsolving.com/community/c3h1072142p4709422 Results/Discussion]<br /> |-<br /> ! scope=&quot;row&quot; | '''Mock Mathcounts Nationals'''<br /> | Mudkipswims42<br /> | 2016<br /> | [http://www.artofproblemsolving.com/community/c5h1226862p6173481 Initial Discussion]<br /> | [http://www.artofproblemsolving.com/community/c5h1226862p6173481 Problems]<br /> | [http://www.artofproblemsolving.com/community/c5h1226862p6421011 Answers]<br /> | [http://www.artofproblemsolving.com/community/c275950_pwg_mock_mathcounts_nats_solutionsdiscussion Discussion]<br /> |}<br /> <br /> More Links:<br /> [http://agmath.com/57427/index.html Link 1]<br /> [http://agmath.com/169763.html Link 2]<br /> <br /> == See Also ==<br /> <br /> * [[Mock AIME]]<br /> * [[Mock AMC]]<br /> * [[Mock USAMO]]<br /> * [[AoPS Past Contests]]</div> Makorn https://artofproblemsolving.com/wiki/index.php?title=2017_AIME_I_Problems/Problem_3&diff=84478 2017 AIME I Problems/Problem 3 2017-03-08T21:37:37Z <p>Makorn: Created page with &quot;We see that &lt;math&gt;d(n)&lt;/math&gt; appears in cycles of &lt;math&gt;20&lt;/math&gt;, adding a total of &lt;math&gt;70&lt;/math&gt; each cycle. Since &lt;math&gt;\lfloor\frac{2017}{20}\rfloor=100&lt;/math&gt;, we know...&quot;</p> <hr /> <div>We see that &lt;math&gt;d(n)&lt;/math&gt; appears in cycles of &lt;math&gt;20&lt;/math&gt;, adding a total of &lt;math&gt;70&lt;/math&gt; each cycle.<br /> Since &lt;math&gt;\lfloor\frac{2017}{20}\rfloor=100&lt;/math&gt;, we know that by &lt;math&gt;2017&lt;/math&gt;, there have been &lt;math&gt;100&lt;/math&gt; cycles, or &lt;math&gt;7000&lt;/math&gt; has been added. This can be discarded, as we're just looking for the last three digits.<br /> Adding up the first &lt;math&gt;17&lt;/math&gt; of the cycle of &lt;math&gt;20&lt;/math&gt;, we get that the answer is &lt;math&gt;\boxed{069}&lt;/math&gt;.</div> Makorn https://artofproblemsolving.com/wiki/index.php?title=2014_USAJMO_Problems/Problem_2&diff=81998 2014 USAJMO Problems/Problem 2 2016-12-24T02:03:23Z <p>Makorn: /* Solution */</p> <hr /> <div>==Problem==<br /> Let &lt;math&gt;\triangle{ABC}&lt;/math&gt; be a non-equilateral, acute triangle with &lt;math&gt;\angle A=60^{\circ}&lt;/math&gt;, and let &lt;math&gt;O&lt;/math&gt; and &lt;math&gt;H&lt;/math&gt; denote the circumcenter and orthocenter of &lt;math&gt;\triangle{ABC}&lt;/math&gt;, respectively.<br /> <br /> (a) Prove that line &lt;math&gt;OH&lt;/math&gt; intersects both segments &lt;math&gt;AB&lt;/math&gt; and &lt;math&gt;AC&lt;/math&gt;.<br /> <br /> (b) Line &lt;math&gt;OH&lt;/math&gt; intersects segments &lt;math&gt;AB&lt;/math&gt; and &lt;math&gt;AC&lt;/math&gt; at &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;Q&lt;/math&gt;, respectively. Denote by &lt;math&gt;s&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt; the respective areas of triangle &lt;math&gt;APQ&lt;/math&gt; and quadrilateral &lt;math&gt;BPQC&lt;/math&gt;. Determine the range of possible values for &lt;math&gt;s/t&lt;/math&gt;.<br /> <br /> ==Solution==<br /> &lt;asy&gt;<br /> import olympiad;<br /> unitsize(1inch);<br /> pair A,B,C,O,H,P,Q,i1,i2,i3,i4;<br /> <br /> //define dots<br /> A=3*dir(50);<br /> B=(0,0);<br /> C=right*2.76481496;<br /> <br /> O=circumcenter(A,B,C);<br /> H=orthocenter(A,B,C);<br /> <br /> i1=2*O-H;<br /> i2=2*i1-O;<br /> i3=2*H-O;<br /> i4=2*i3-H;<br /> //These points are for extending PQ. DO NOT DELETE!<br /> <br /> P=intersectionpoint(i2--i4,A--B);<br /> Q=intersectionpoint(i2--i4,A--C);<br /> <br /> //draw<br /> dot(P);<br /> dot(Q);<br /> draw(P--Q);<br /> dot(A);<br /> dot(B);<br /> dot(C);<br /> dot(O);<br /> dot(H);<br /> draw(A--B--C--cycle);<br /> <br /> //label<br /> label(&quot;$A$&quot;,A,N);<br /> label(&quot;$B$&quot;,B,SW);<br /> label(&quot;$C$&quot;,C,SE);<br /> label(&quot;$P$&quot;,P,NW);<br /> label(&quot;$Q$&quot;,Q,NE);<br /> label(&quot;$O$&quot;,O,N);<br /> label(&quot;$H$&quot;,H,N);<br /> //change O and H label positions if interfering with other lines to be added<br /> <br /> //further editing: ABCPQOH are the dots to be further used. i1,i2,i3,i4 are for drawing assistance and are not to be used<br /> &lt;/asy&gt;<br /> <br /> Lemma: &lt;math&gt;H&lt;/math&gt; is the reflection of &lt;math&gt;O&lt;/math&gt; over the angle bisector of &lt;math&gt;\angle BAC&lt;/math&gt; (henceforth 'the' reflection)<br /> <br /> Proof: Let &lt;math&gt;H'&lt;/math&gt; be the reflection of &lt;math&gt;O&lt;/math&gt;, and let &lt;math&gt;B'&lt;/math&gt; be the reflection of &lt;math&gt;B&lt;/math&gt;. <br /> <br /> Then reflection takes &lt;math&gt;\angle ABH'&lt;/math&gt; to &lt;math&gt;\angle AB'O&lt;/math&gt;.<br /> <br /> &lt;math&gt;\Delta ABB'&lt;/math&gt; is equilateral, and &lt;math&gt;O&lt;/math&gt; lies on the perpendicular bisector of &lt;math&gt;\overline{AB}&lt;/math&gt; <br /> <br /> It's well known that &lt;math&gt;O&lt;/math&gt; lies strictly inside &lt;math&gt;\Delta ABC&lt;/math&gt; (since it's acute), meaning that &lt;math&gt;\angle ABH' = \angle AB'O = 30^{\circ},&lt;/math&gt; from which it follows that &lt;math&gt;\overline{BH'} \perp \overline{AC}&lt;/math&gt; . Similarly, &lt;math&gt;\overline{CH'} \perp \overline{AB}&lt;/math&gt;. Since &lt;math&gt;H'&lt;/math&gt; lies on two altitudes, &lt;math&gt;H&lt;/math&gt; is the orthocenter, as desired.<br /> <br /> So &lt;math&gt;\overline{OH}&lt;/math&gt; is perpendicular to the angle bisector of &lt;math&gt;\angle OAH&lt;/math&gt;, which is the same line as the angle bisector of &lt;math&gt;\angle BAC&lt;/math&gt;, meaning that &lt;math&gt;\Delta APQ&lt;/math&gt; is equilateral. <br /> <br /> Let its side length be &lt;math&gt;s&lt;/math&gt;, and let &lt;math&gt;PH=t&lt;/math&gt;, where &lt;math&gt;0 &lt; t &lt; s, t \neq s/2&lt;/math&gt; because &lt;math&gt;O&lt;/math&gt; lies strictly within &lt;math&gt;\angle BAC&lt;/math&gt;, as must &lt;math&gt;H&lt;/math&gt;, the reflection of &lt;math&gt;O&lt;/math&gt;. Also, it's easy to show that if &lt;math&gt;O=H&lt;/math&gt; in a general triangle, it's equilateral, and we know &lt;math&gt;\Delta ABC&lt;/math&gt; is not equilateral. Hence H is not on the bisector of &lt;math&gt;\angle BAC \implies t \neq s/2&lt;/math&gt;. Let &lt;math&gt;\overrightarrow{BH}&lt;/math&gt; intersect &lt;math&gt;\overline{AC}&lt;/math&gt; at &lt;math&gt;P_B&lt;/math&gt;. <br /> <br /> Since &lt;math&gt;\Delta HP_BQ&lt;/math&gt; and &lt;math&gt;BP_BA&lt;/math&gt; are 30-60-90 triangles, &lt;math&gt;AB=2AP_B=2(s-QP_B)=2(s-HQ/2)=2s-HQ=2s-(s-t)=s+t&lt;/math&gt;<br /> <br /> Similarly, &lt;math&gt;AC=2s-t&lt;/math&gt;<br /> <br /> The ratio &lt;math&gt;\frac{[APQ]}{[ABC]-[APQ]}&lt;/math&gt; is &lt;math&gt;\frac{AP \cdot AQ}{AB \cdot AC - AP \cdot AQ} = \frac{s^2}{(s+t)(2s-t)-s^2}&lt;/math&gt;<br /> The denominator equals &lt;math&gt;(1.5s)^2-(.5s-t)^2-s^2&lt;/math&gt; where &lt;math&gt;.5s-t&lt;/math&gt; can equal any value in &lt;math&gt;(-.5s, .5s)&lt;/math&gt; except &lt;math&gt;0&lt;/math&gt;. Therefore, the denominator can equal any value in &lt;math&gt;(s^2, 5s^2/4)&lt;/math&gt;, and the ratio is any value in &lt;math&gt;\boxed{\left(\frac{4}{5},1\right)}.&lt;/math&gt;<br /> <br /> Note: It's easy to show that for any point &lt;math&gt;H&lt;/math&gt; on &lt;math&gt;\overline{PQ}&lt;/math&gt; except the midpoint, Points B and C can be validly defined to make an acute, non-equilateral triangle.<br /> <br /> ==Solution 2==<br /> Let &lt;math&gt;J&lt;/math&gt; be the farthest point on the circumcircle of &lt;math&gt;\triangle{ABC}&lt;/math&gt; from line &lt;math&gt;BC&lt;/math&gt;.<br /> Lemma: Line &lt;math&gt;AJ&lt;/math&gt;||Line &lt;math&gt;OH&lt;/math&gt;<br /> Proof: Set &lt;math&gt;b=-\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}&lt;/math&gt; and &lt;math&gt;c=-\dfrac{1}{2}-i\dfrac{\sqrt{3}}{2}&lt;/math&gt;, and &lt;math&gt;a&lt;/math&gt; on the unit circle. It is well known that &lt;math&gt;o=0&lt;/math&gt; and &lt;math&gt;h=a+b+c&lt;/math&gt;, so we have &lt;math&gt;h-o=a+b+c=a-1=a-j&lt;/math&gt;, so &lt;math&gt;\dfrac{a-j}{h-o}&lt;/math&gt; is real and thus the 2 lines are parallel.<br /> <br /> WLOG let &lt;math&gt;A&lt;/math&gt; be in the first quadrant. Clearly by the above lemma &lt;math&gt;OH&lt;/math&gt; must intersect line &lt;math&gt;BC&lt;/math&gt; closer to &lt;math&gt;B&lt;/math&gt; than to &lt;math&gt;C&lt;/math&gt;. Intersect &lt;math&gt;AJ&lt;/math&gt; and &lt;math&gt;BC&lt;/math&gt; at &lt;math&gt;D&lt;/math&gt; and &lt;math&gt;OH&lt;/math&gt; and &lt;math&gt;BC&lt;/math&gt; at &lt;math&gt;E&lt;/math&gt;. We clearly have &lt;math&gt;0 &lt; \dfrac{\overarc{JC}-\overarc{AB}}{2} = \dfrac{120-\overarc{AC}}{2} = \angle{ADB} = \angle{OEC} &lt; 30 = \angle{OBC}&lt;/math&gt;, &lt;math&gt;OH&lt;/math&gt; must intersect &lt;math&gt;AB&lt;/math&gt;. We also have, letting the intersection of line &lt;math&gt;OH&lt;/math&gt; and line &lt;math&gt;AC&lt;/math&gt; be &lt;math&gt;Q&lt;/math&gt;, and letting intersection of &lt;math&gt;OH&lt;/math&gt; and &lt;math&gt;AB&lt;/math&gt; be &lt;math&gt;P&lt;/math&gt;, &lt;math&gt;\angle{OQA} = \angle{JAB} = \dfrac{\overarc{JB}}{2} = 60&lt;/math&gt;. Since &lt;math&gt;\angle{OCA}=\angle{BCA}-\angle{BCO} &lt; 90-30 =\angle{OQA}&lt;/math&gt;, and &lt;math&gt;\angle{OQC} = 120&gt;\angle{OAC}&lt;/math&gt;, &lt;math&gt;OH&lt;/math&gt; also intersects &lt;math&gt;AC&lt;/math&gt;. We have &lt;math&gt;\angle{OPA}=\angle{PAJ}=60&lt;/math&gt;, so &lt;math&gt;\triangle{APQ}&lt;/math&gt; is equilateral. Letting &lt;math&gt;AJ=2x&lt;/math&gt;, and letting the foot of the perpendicular from &lt;math&gt;O&lt;/math&gt; to &lt;math&gt;AJ&lt;/math&gt; be &lt;math&gt;L&lt;/math&gt;, we have &lt;math&gt;OL=\sqrt{1-x^{2}}&lt;/math&gt;, and since &lt;math&gt;OL&lt;/math&gt; is an altitude of &lt;math&gt;\triangle{APQ}&lt;/math&gt;, we have &lt;math&gt;[APQ]=\dfrac{OL^{2}\sqrt{3}}{3} = \dfrac{\sqrt{3}(1-x^{2})}{3}&lt;/math&gt;. Letting the foot of the perpendicular from &lt;math&gt;A&lt;/math&gt; to &lt;math&gt;OJ&lt;/math&gt; be &lt;math&gt;K&lt;/math&gt;, we have &lt;math&gt;\triangle{JKA}\sim \triangle{JLO}&lt;/math&gt; by AA with ratio &lt;math&gt;\dfrac{JA}{JO} = 2x&lt;/math&gt;. Therefore, &lt;math&gt;JK = 2x(JL) = 2x^{2}&lt;/math&gt;. Letting &lt;math&gt;D&lt;/math&gt; be the foot of the altitude from &lt;math&gt;J&lt;/math&gt; to &lt;math&gt;BC&lt;/math&gt;, we have &lt;math&gt;KD=JD-JK=\dfrac{3}{2}-2x^{2}&lt;/math&gt;, since &lt;math&gt;re(B)=re(C) \implies JD=re(j)-(-\dfrac{1}{2})=\dfrac{3}{2}&lt;/math&gt;. Thus, since &lt;math&gt;BC=\sqrt{3}&lt;/math&gt; we have &lt;math&gt;[ABC]=\dfrac{(\dfrac{3}{2}-2x^{2})(\sqrt{3})}{2} = \dfrac{(3-4x^{2})(\sqrt{3})}{2}&lt;/math&gt;, so &lt;math&gt;[PQCB]=[ABC]-[APQ]=\dfrac{(5-8x^{2})(\sqrt{3})}{12}&lt;/math&gt;, so &lt;math&gt;\dfrac{[APQ]}{[PQCB]} = \dfrac{4-4x^{2}}{5-8x^{2}} = \dfrac{1}{2}+\dfrac{3}{10-16x^{2}}&lt;/math&gt;. We have &lt;math&gt;x=\sin(\dfrac{JOA}{2})&lt;/math&gt;, with &lt;math&gt;0&lt;120-2\angle{ACB}=\angle{JOA}&lt;60&lt;/math&gt;, so &lt;math&gt;x&lt;/math&gt; can be anything in the interval &lt;math&gt;(0, \dfrac{1}{2})&lt;/math&gt;. Therefore, the desired range is &lt;math&gt;(\dfrac{4}{5}, 1)&lt;/math&gt;.<br /> <br /> Solution by Shaddoll<br /> <br /> ==See Also==<br /> <br /> {{USAJMO box|year=2014|num-b=1|num-a=3}}</div> Makorn https://artofproblemsolving.com/wiki/index.php?title=AMC_8&diff=81946 AMC 8 2016-12-20T00:30:48Z <p>Makorn: /* Schedule */</p> <hr /> <div>The '''AMC 8''' is an exam for students in grades 8 and below, administered annually by the [[American Mathematics Competitions]] (AMC) to students all over the United States.<br /> <br /> Usually, high scoring students will be given a chance by their school to take the more challenging [[AMC 10]] exam. However, there is no requirement for the AMC 10 besides the fact that you have to be 10&lt;math&gt;^\text{th}&lt;/math&gt; grade or below. <br /> <br /> The AMC 8 is administered by the [[American Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC. The test is intended to foster interest in mathematics and also to help middle school students learn [[mathematical problem solving]].<br /> <br /> <br /> == Format ==<br /> <br /> The AMC 8 is a 25 question, 40 minute multiple choice test. Problems generally increase in difficulty as the exam progresses. Through 2007, calculators were permitted; though now, they are not. A correct answer scores 1 point, but unlike the AMC 10 and 12, no points are given for blank answers.<br /> <br /> ==Schedule==<br /> <br /> The AMC 8 is usually administered on the third Tuesday of November. Some schools may take it as late as the fourth Tuesday.<br /> <br /> == Curriculum ==<br /> The AMC 8 tests [[mathematical problem solving]] with [[algebra]], [[arithmetic]], [[counting]], [[geometry]], [[logic]], [[number theory]], and [[probability]].<br /> <br /> == Resources ==<br /> === Links ===<br /> * [http://www.unl.edu/amc/ AMC homepage] and their [http://www.unl.edu/amc/e-exams/e4-amc08/amc8.shtml AMC 8 page]<br /> * [[AMC 8 Problems and Solutions]]<br /> Use the link above to access many years' worth of competition questions.<br /> <br /> === Recommended reading ===<br /> * Introduction to Counting &amp; Probability by Dr. [[David Patrick]]. [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=3 Information]<br /> * Introduction to Geometry by [[Richard Rusczyk]]. [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=9 Information]<br /> * The Art of Problem Solving Volume I by [[Sandor Lehoczky]] and [[Richard Rusczyk]]. [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=1 Information].<br /> <br /> <br /> === Preparation Classes ===<br /> * [[Art of Problem Solving]] offers many [http://www.artofproblemsolving.com/Classes/AoPS_C_About.php helpful online classes] on topics covered by the AMC 8.<br /> * [[AoPS]] holds many free [[Math Jams]], some of which include problems and concepts at the level tested by the AMC 8. [http://www.artofproblemsolving.com/Community/AoPS_Y_Math_Jams.php Math Jam Schedule]<br /> * [[EPGY]] offers some contest preparation classes.<br /> <br /> <br /> == See also ==<br /> * [[Mathematics competitions]]<br /> * [[MathCounts]]<br /> * [[MOEMS]]<br /> [[Category:Mathematics competitions]]</div> Makorn