Found problems: 3632
1979 AMC 12/AHSME, 15
Two identical jars are filled with alcohol solutions, the ratio of the volume of alcohol to the volume of water being $p : 1$ in one jar and $q : 1$ in the other jar. If the entire contents of the two jars are mixed together, the ratio of the volume of alcohol to the volume of water in the mixture is
$\textbf{(A) }\frac{p+q}{2}\qquad\textbf{(B) }\frac{p^2+q^2}{p+q}\qquad\textbf{(C) }\frac{2pq}{p+q}\qquad\textbf{(D) }\frac{2(p^2+pq+q^2)}{3(p+q)}\qquad\textbf{(E) }\frac{p+q+2pq}{p+q+2}$
2024 AIME, 10
Let $\triangle ABC$ have side lengths $AB = 5, BC = 9,$ and $CA = 10.$ The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ intersect at point $D,$ and $\overline{AD}$ intersects the circumcircle at $P \ne A.$ The length of $\overline{AP}$ is equal to $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
1972 AMC 12/AHSME, 12
The number of cubic feet in the volume of a cube is the same as the number of square inches in its surface area. The length of the edge expressed as a number of feet is
$\textbf{(A) }6\qquad\textbf{(B) }864\qquad\textbf{(C) }1728\qquad\textbf{(D) }6\times 1728\qquad \textbf{(E) }2304$
1969 AMC 12/AHSME, 2
If an item is sold for $x$ dollars, there is a loss of $15\%$ based on the cost. If, however, the same item is sold for $y$ dollars, there is a profit of $15\%$ based on the cost. The ratio $y:x$ is:
$\textbf{(A) }23:17\qquad
\textbf{(B) }17y:23\qquad
\textbf{(C) }23x:17\qquad$
$\textbf{(D) }\text{dependent upon the cost}\qquad
\textbf{(E) }\text{none of these.}$
2011 AMC 10, 3
At a store, when a length is reported as $x$ inches that means the length is at least $x-0.5$ inches and at most $x+0.5$ inches. Suppose the dimensions of a rectangular tile are reported as $2$ inches by $3$ inches. In square inches, what is the minimum area for the rectangle?
$ \textbf{(A)}\ 3.75 \qquad
\textbf{(B)}\ 4.5 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 8.75 $
2018 AMC 10, 19
A number $m$ is randomly selected from the set $\{11,13,15,17,19\}$, and a number $n$ is randomly selected from $\{1999,2000,2001,\ldots,2018\}$. What is the probability that $m^n$ has a units digit of $1$?
$\textbf{(A) } \frac{1}{5} \qquad \textbf{(B) } \frac{1}{4} \qquad \textbf{(C) } \frac{3}{10} \qquad \textbf{(D) } \frac{7}{20} \qquad \textbf{(E) } \frac{2}{5} $
2022 AMC 10, 7
For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?
$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }14 \qquad \textbf{(E) }16$
1986 AMC 12/AHSME, 9
The product \[\left(1 - \frac{1}{2^{2}}\right)\left(1 - \frac{1}{3^{2}}\right)\ldots\left(1 - \frac{1}{9^{2}}\right)\left(1 - \frac{1}{10^{2}}\right)\] equals
$ \textbf{(A)}\ \frac{5}{12}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{11}{20}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{7}{10} $
1979 AMC 12/AHSME, 19
Find the sum of the squares of all real numbers satisfying the equation \[x^{256}-256^{32}=0.\]
$\textbf{(A) }8\qquad\textbf{(B) }128\qquad\textbf{(C) }512\qquad\textbf{(D) }65,536\qquad\textbf{(E) }2(256^{32})$
2008 AMC 10, 5
For real numbers $ a$ and $ b$, define $ a\$b\equal{}(a\minus{}b)^2$. What is $ (x\minus{}y)^2\$(y\minus{}x)^2$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ x^2\plus{}y^2 \qquad
\textbf{(C)}\ 2x^2 \qquad
\textbf{(D)}\ 2y^2 \qquad
\textbf{(E)}\ 4xy$
2007 AMC 12/AHSME, 17
Suppose that $ \sin a \plus{} \sin b \equal{} \sqrt {\frac {5}{3}}$ and $ \cos a \plus{} \cos b \equal{} 1.$ What is $ \cos(a \minus{} b)?$
$ \textbf{(A)}\ \sqrt {\frac {5}{3}} \minus{} 1 \qquad \textbf{(B)}\ \frac {1}{3}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {2}{3}\qquad \textbf{(E)}\ 1$
2021 AIME Problems, 5
For positive real numbers $s$, let $\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\tau(s)$ is nonempty, but all triangles in $\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$.
2023 AMC 12/AHSME, 19
Each of $2023$ balls is placed in on of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?
$\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}$
2003 USAMO, 6
At the vertices of a regular hexagon are written six nonnegative integers whose sum is $2003^{2003}$. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.
1988 AMC 12/AHSME, 18
At the end of a professional bowling tournament, the top 5 bowlers have a playoff. First #5 bowls #4. The loser receives 5th prize and the winner bowls #3 in another game. The loser of this game receives 4th prize and the winner bowls #2. The loser of this game receives 3rd prize and the winner bowls #1. The winner of this game gets 1st prize and the loser gets 2nd prize. In how many orders can bowlers #1 through #5 receive the prizes?
$ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ \text{none of these} $
2022 AMC 10, 24
Consider functions $f$ that satisfy $|f(x)-f(y)|\leq \frac{1}{2}|x-y|$ for all real numbers $x$ and $y$. Of all such functions that also satisfy the equation $f(300) = f(900)$, what is the greatest possible value of
$$f(f(800))-f(f(400))?$$
$ \textbf{(A)}\ 25 \qquad
\textbf{(B)}\ 50 \qquad
\textbf{(C)}\ 100 \qquad
\textbf{(D)}\ 150 \qquad
\textbf{(E)}\ 200$
1997 USAMO, 5
Prove that, for all positive real numbers $ a$, $ b$, $ c$, the inequality
\[ \frac {1}{a^3 \plus{} b^3 \plus{} abc} \plus{} \frac {1}{b^3 \plus{} c^3 \plus{} abc} \plus{} \frac {1}{c^3 \plus{} a^3 \plus{} abc} \leq \frac {1}{abc}
\]
holds.
1991 AMC 12/AHSME, 11
Jack and Jill run $10$ kilometers. They start at the same point, run $5$ kilometers up a hill, and return to the starting point by the same route. Jack has a $10$ minute head start and runs at the rate of $15$ km/hr uphill and $20$ km/hr downhill. Jill runs $16$ km/hr uphill and $22$ km/hr downhill. How far from the top of the hill are they when they pass going in opposite directions?
$ \textbf{(A)}\ \frac{5}{4}\ km\qquad\textbf{(B)}\ \frac{35}{27}\ km\qquad\textbf{(C)}\ \frac{27}{20}\ km\qquad\textbf{(D)}\ \frac{7}{3}\ km\qquad\textbf{(E)}\ \frac{28}{9}\ km $
2012 AIME Problems, 2
Two geometric sequences $ a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3\ldots $have the same common ratio, with $a_1=27$,$b_1=99$, and $a_{15}=b_{11}$. Find $a_9.$
2000 AMC 8, 6
Figure $ABCD$ is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded L-shaped region is
[asy]
pair A,B,C,D;
A = (5,5); B = (5,0); C = (0,0); D = (0,5);
fill((0,0)--(0,4)--(1,4)--(1,1)--(4,1)--(4,0)--cycle,gray);
draw(A--B--C--D--cycle);
draw((4,0)--(4,4)--(0,4));
draw((1,5)--(1,1)--(5,1));
label("$A$",A,NE);
label("$B$",B,SE);
label("$C$",C,SW);
label("$D$",D,NW);
label("$1$",(1,4.5),E);
label("$1$",(0.5,5),N);
label("$3$",(1,2.5),E);
label("$3$",(2.5,1),N);
label("$1$",(4,0.5),E);
label("$1$",(4.5,1),N);
[/asy]
$\text{(A)}\ 7 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12.5 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15$
1977 AMC 12/AHSME, 24
Find the sum \[\frac{1}{1(3)}+\frac{1}{3(5)}+\dots+\frac{1}{(2n-1)(2n+1)}+\dots+\frac{1}{255(257)}.\]
$\textbf{(A) }\frac{127}{255}\qquad\textbf{(B) }\frac{128}{255}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{128}{257}\qquad \textbf{(E) }\frac{129}{257}$
2021 AMC 10 Spring, 1
What is the value of $$(2^2-2) - (3^2-3) + (4^2-4)?$$
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 12$
1992 AMC 12/AHSME, 16
If $\frac{y}{x - z} = \frac{x + y}{z} = \frac{x}{y}$ for three positive numbers $x$, $y$ and $z$, all different, then $\frac{x}{y} =$
$ \textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{3}{5}\qquad\textbf{(C)}\ \frac{2}{3}\qquad\textbf{(D)}\ \frac{5}{3}\qquad\textbf{(E)}\ 2 $
2009 AMC 12/AHSME, 11
On Monday, Millie puts a quart of seeds, $ 25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $ 25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
$ \textbf{(A)}\ \text{Tuesday}\qquad \textbf{(B)}\ \text{Wednesday}\qquad \textbf{(C)}\ \text{Thursday} \qquad \textbf{(D)}\ \text{Friday}\qquad \textbf{(E)}\ \text{Saturday}$
2015 USAMO, 6
Consider $0<\lambda<1$, and let $A$ be a multiset of positive integers. Let $A_n=\{a\in A: a\leq n\}$. Assume that for every $n\in\mathbb{N}$, the set $A_n$ contains at most $n\lambda$ numbers. Show that there are infinitely many $n\in\mathbb{N}$ for which the sum of the elements in $A_n$ is at most $\frac{n(n+1)}{2}\lambda$. (A multiset is a set-like collection of elements in which order is ignored, but repetition of elements is allowed and multiplicity of elements is significant. For example, multisets $\{1, 2, 3\}$ and $\{2, 1, 3\}$ are equivalent, but $\{1, 1, 2, 3\}$ and $\{1, 2, 3\}$ differ.)