Found problems: 3632
2019 AMC 10, 14
The base-ten representation for $19!$ is $121,6T5,100,40M,832,H00$, where $T$, $M$, and $H$ denote digits that are not given. What is $T+M+H$?
$\textbf{(A) }3
\qquad\textbf{(B) }8
\qquad\textbf{(C) }12
\qquad\textbf{(D) }14
\qquad\textbf{(E) } 17 $
2000 AMC 8, 12
A block wall $100$ feet long and $7$ feet high will be constructed using blocks that are $1$ foot high and either $2$ feet long or $1$ foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall?
[asy]
draw((0,0)--(6,0)--(6,1)--(5,1)--(5,2)--(0,2)--cycle);
draw((0,1)--(5,1));
draw((1,1)--(1,2));
draw((3,1)--(3,2));
draw((2,0)--(2,1));
draw((4,0)--(4,1));
[/asy]
$\text{(A)}\ 344 \qquad \text{(B)}\ 347 \qquad \text{(C)}\ 350 \qquad \text{(D)}\ 353 \qquad \text{(E)}\ 356$
2006 AIME Problems, 8
There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color.
Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed?
1961 AMC 12/AHSME, 34
Let $S$ be the set of values assumed by the fraction \[\frac{2x+3}{x+2}\] when $x$ is any member of the interval $x \ge 0$. If there exists a number $M$ such that no number of the set $S$ is greater than $M$, then $M$ is an upper bound of $S$. If there exists a number $m$ such that such that no number of the set $S$ is less than $m$, then $m$ is a lower bound of $S$. We may then say:
$ \textbf{(A)}\ \text{m is in S, but M is not in S} $
$\textbf{(B)}\ \text{M is in S, but m is not in S}$
$\textbf{(C)}\ \text{Both m and M are in S} $
$\textbf{(D)}\ \text{Neither m nor M are in S}$
$\textbf{(E)}\ \text{M does not exist either in or outside S} $
2025 USAMO, 1
Let $k$ and $d$ be positive integers. Prove that there exists a positive integer $N$ such that for every odd integer $n>N$, the digits in the base-$2n$ representation of $n^k$ are all greater than $d$.
1963 AMC 12/AHSME, 11
The arithmetic mean of a set of $50$ numbers is $38$. If two numbers of the set, namely $45$ and $55$, are discarded, the arithmetic mean of the remaining set of numbers is:
$\textbf{(A)}\ 38.5 \qquad
\textbf{(B)}\ 37.5 \qquad
\textbf{(C)}\ 37 \qquad
\textbf{(D)}\ 36.5 \qquad
\textbf{(E)}\ 36$
2020 AIME Problems, 3
A positive integer $N$ has base-eleven representation $\underline{a}\,\underline{b}\,\underline{c}$ and base-eight representation $\underline{1}\,\underline{b}\,\underline{c}\,\underline{a}$, where $a$, $b$, and $c$ represent (not necessarily distinct) digits. Find the least such $N$ expressed in base ten.
2024 AMC 12/AHSME, 7
In the figure below $WXYZ$ is a rectangle with $WX=4$ and $WZ=8$. Point $M$ lies $\overline{XY}$, point $A$ lies on $\overline{YZ}$, and $\angle WMA$ is a right angle. The areas of $\triangle WXM$ and $\triangle WAZ$ are equal. What is the area of $\triangle WMA$?
[asy]
pair X = (0, 0);
pair W = (0, 4);
pair Y = (8, 0);
pair Z = (8, 4);
label("$X$", X, dir(180));
label("$W$", W, dir(180));
label("$Y$", Y, dir(0));
label("$Z$", Z, dir(0));
draw(W--X--Y--Z--cycle);
dot(X);
dot(Y);
dot(W);
dot(Z);
pair M = (2, 0);
pair A = (8, 3);
label("$A$", A, dir(0));
dot(M);
dot(A);
draw(W--M--A--cycle);
markscalefactor = 0.05;
draw(rightanglemark(W, M, A));
label("$M$", M, dir(-90));
[/asy]
$
\textbf{(A) }13 \qquad
\textbf{(B) }14 \qquad
\textbf{(C) }15 \qquad
\textbf{(D) }16 \qquad
\textbf{(E) }17 \qquad
$
2021 AIME Problems, 4
Find the number of ways $66$ identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.
2020 AMC 10, 25
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7$. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
$\textbf{(A) } \frac{7}{36} \qquad\textbf{(B) } \frac{5}{24} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{17}{72} \qquad\textbf{(E) } \frac{1}{4}$
2025 USAMO, 5
Determine, with proof, all positive integers $k$ such that $$\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k$$ is an integer for every positive integer $n.$
1976 AMC 12/AHSME, 18
[asy]
//size(100);//local
size(200);
real r1=2;
pair
O=(0,0),
D=(.5,.5*sqrt(3)),
C=(D.x+.5*3,D.y),
B,
B_prime=endpoint(arc(D, 3, 0,-2));
B=B_prime;
path
c1=circle(O, r1);
pair C=midpoint(D--B_prime);
path arc2=arc(B_prime, 6/2, 158.25,250);
draw(c1);
draw(O--D);
draw(D--C);
draw(C--B_prime);
pair A=beginpoint(arc2);
draw(B_prime--A);
//dot(O^^D^^C^^A);
//dot(B_prime);
label("\scriptsize{$O$}",O,.6dir(D--O));
label("\scriptsize{$C$}",C,.5dir(-55));
label("\scriptsize{$D$}", D,.2NW);
//label("\scriptsize{$B$}",B,S);
label("\scriptsize{$B$}", B_prime, .5*dir(D--B_prime));
label("\scriptsize{$A$}",A,.5dir(NE));
label("\tiny{2}", O--D, .45*LeftSide);
label("\tiny{3}", D--C, .45*LeftSide);
label("\tiny{6}", B_prime--A, .45*RightSide);
label("\tiny{3}", waypoint(C--B_prime,.1), .45*N);
//Credit to Klaus-Anton for the diagram[/asy]
In the adjoining figure, $AB$ is tangent at $A$ to the circle with center $O$; point $D$ is interior to the circle; and $DB$ intersects the circle at $C$. If $BC=DC=3$, $OD=2$, and $AB=6$, then the radius of the circle is
$\textbf{(A) }3+\sqrt{3}\qquad\textbf{(B) }15/\pi\qquad\textbf{(C) }9/2\qquad\textbf{(D) }2\sqrt{6}\qquad \textbf{(E) }\sqrt{22}$
2002 AIME Problems, 13
In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
2019 AIME Problems, 12
For $n \ge 1$ call a finite sequence $(a_1, a_2 \ldots a_n)$ of positive integers [i]progressive[/i] if $a_i < a_{i+1}$ and $a_i$ divides $a_{i+1}$ for all $1 \le i \le n-1$. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360$.
2007 AMC 10, 18
Consider the $ 12$-sided polygon $ ABCDEFGHIJKL$, as shown. Each of its sides has length $ 4$, and each two consecutive sides form a right angle. Suppose that $ \overline{AG}$ and $ \overline{CH}$ meet at $ M$. What is the area of quadrilateral $ ABCM$?
[asy]unitsize(13mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;
pair A=(1,3), B=(2,3), C=(2,2), D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1,1), J=(0,1), K=(0,2), L=(1,2);
pair M=intersectionpoints(A--G,H--C)[0];
draw(A--B--C--D--Ep--F--G--H--I--J--K--L--cycle);
draw(A--G);
draw(H--C);
dot(M);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,NE);
label("$D$",D,NE);
label("$E$",Ep,SE);
label("$F$",F,SE);
label("$G$",G,SE);
label("$H$",H,SW);
label("$I$",I,SW);
label("$J$",J,SW);
label("$K$",K,NW);
label("$L$",L,NW);
label("$M$",M,W);[/asy]$ \textbf{(A)}\ \frac {44}{3}\qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ \frac {88}{5}\qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ \frac {62}{3}$
1963 AMC 12/AHSME, 32
The dimensions of a rectangle $R$ are $a$ and $b$, $a < b$. It is required to obtain a rectangle with dimensions $x$ and $y$, $x < a$, $y < a$, so that its perimeter is one-third that of $R$, and its area is one-third that of $R$. The number of such (different) rectangles is:
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ \text{infinitely many}$
2010 AMC 10, 24
The number obtained from the last two nonzero digits of $ 90!$ is equal to $ n$. What is $ n$?
$ \textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 32 \qquad
\textbf{(C)}\ 48 \qquad
\textbf{(D)}\ 52 \qquad
\textbf{(E)}\ 68$
2010 AMC 10, 11
The length of the interval of solutions of the inequality $ a\le 2x\plus{}3\le b$ is $ 10$. What is $ b\minus{}a$?
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 10 \qquad
\textbf{(C)}\ 15 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 30$
2024 AMC 12/AHSME, 7
In $\Delta ABC$, $\angle ABC = 90^\circ$ and $BA = BC = \sqrt{2}$. Points $P_1, P_2, \dots, P_{2024}$ lie on hypotenuse $\overline{AC}$ so that $AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C$. What is the length of the vector sum
\[ \overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}? \]
$
\textbf{(A) }1011 \qquad
\textbf{(B) }1012 \qquad
\textbf{(C) }2023 \qquad
\textbf{(D) }2024 \qquad
\textbf{(E) }2025 \qquad
$
1975 AMC 12/AHSME, 29
What is the smallest integer larger than $(\sqrt{3}+\sqrt{2})^6$?
$ \textbf{(A)}\ 972 \qquad\textbf{(B)}\ 971 \qquad\textbf{(C)}\ 970 \qquad\textbf{(D)}\ 969 \qquad\textbf{(E)}\ 968 $
2020 AMC 10, 23
Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations:
[list=]
[*]$L,$ a rotation of $90^{\circ}$ counterclockwise around the origin;
[*]$R,$ a rotation of $90^{\circ}$ clockwise around the origin;
[*]$H,$ a reflection across the $x$-axis; and
[*]$V,$ a reflection across the $y$-axis.
[/list]
Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\{L, R, H, V\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.)
$\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\ 2^{38} \qquad\textbf{(D)}\ 3\cdot 2^{37} \qquad\textbf{(E)}\ 2^{39}$
2024 AMC 10, 7
What is the remainder when $7^{2024}+7^{2025}+7^{2026}$ is divided by $19$?
$
\textbf{(A) }0 \qquad
\textbf{(B) }1 \qquad
\textbf{(C) }7 \qquad
\textbf{(D) }11 \qquad
\textbf{(E) }18 \qquad
$
1988 USAMO, 1
By a [i]pure repeating decimal[/i] (in base $10$), we mean a decimal $0.\overline{a_1\cdots a_k}$ which repeats in blocks of $k$ digits beginning at the decimal point. An example is $.243243243\cdots = \tfrac{9}{37}$. By a [i]mixed repeating decimal[/i] we mean a decimal $0.b_1\cdots b_m\overline{a_1\cdots a_k}$ which eventually repeats, but which cannot be reduced to a pure repeating decimal. An example is $.011363636\cdots = \tfrac{1}{88}$.
Prove that if a mixed repeating decimal is written as a fraction $\tfrac pq$ in lowest terms, then the denominator $q$ is divisible by $2$ or $5$ or both.
2010 AMC 10, 15
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in the swamp, and they make the following statements:
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs?
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$
2016 AMC 10, 3
For every dollar Ben spent on bagels, David spent $25$ cents less. Ben paid $\$12.50$ more than David. How much did they spend in the bagel store together?
$\textbf{(A)}\ \$37.50 \qquad\textbf{(B)}\ \$50.00\qquad\textbf{(C)}\ \$87.50\qquad\textbf{(D)}\ \$90.00\qquad\textbf{(E)}\ \$92.50$