Found problems: 138
2021 AIME Problems, 8
An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, the ant does not return to its previous vertex, but chooses to move to one of the other two adjacent vertices. All choices are selected at random so that each of the possible moves is equally likely. The probability that after exactly 8 moves that ant is at a vertex of the top face on the cube is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
2019 AIME Problems, 9
Call a positive integer $n$ $k$[i]-pretty[/i] if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$[i]-pretty[/i]. Let $S$ be the sum of positive integers less than $2019$ that are $20$[i]-pretty[/i]. Find $\tfrac{S}{20}$.
2024 AIME, 13
Let $\omega \ne 1$ be a $13$th root of unity. Find the remainder when \[ \prod_{k=0}^{12} \left(2 - 2\omega^k + \omega^{2k} \right) \] is divided by $1000$.
2024 AIME, 15
Find the number of rectangles that can be formed inside a fixed regular dodecagon ($12$-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles.
[asy]
unitsize(40);
real r = pi/6;
pair A1 = (cos(r),sin(r));
pair A2 = (cos(2r),sin(2r));
pair A3 = (cos(3r),sin(3r));
pair A4 = (cos(4r),sin(4r));
pair A5 = (cos(5r),sin(5r));
pair A6 = (cos(6r),sin(6r));
pair A7 = (cos(7r),sin(7r));
pair A8 = (cos(8r),sin(8r));
pair A9 = (cos(9r),sin(9r));
pair A10 = (cos(10r),sin(10r));
pair A11 = (cos(11r),sin(11r));
pair A12 = (cos(12r),sin(12r));
draw(A1--A2--A3--A4--A5--A6--A7--A8--A9--A10--A11--A12--cycle);
filldraw(A2--A1--A8--A7--cycle, mediumgray, linewidth(1.2));
draw(A4--A11);
draw(0.365*A3--0.365*A12, linewidth(1.2));
dot(A1);
dot(A2);
dot(A3);
dot(A4);
dot(A5);
dot(A6);
dot(A7);
dot(A8);
dot(A9);
dot(A10);
dot(A11);
dot(A12);
[/asy]
CIME II 2018, 11
Let $\mathcal{P}$ be a set of monic polynomials with integer coefficients of the least degree, with root $k \cdot \cos\left(\frac{4\pi}{7}\right)$, as $k$ spans over the positive integers. Let $P(x) \in \mathcal{P}$ be the polynomial so that $|P(1)|$ is minimized. Find the remainder when $P(2017)$ is divided by $1000$.
[i]Proposed by [b] eisirrational [/b][/i]
CIME II 2018, 6
Define $f(x)=-\frac{2x}{4x+3}$ and $g(x)=\frac{x+2}{2x+1}$. Moreover, let $h^{n+1} (x)=g(f(h^n(x)))$, where $h^1(x)=g(f(x))$. If the value of $\sum_{k=1}^{100} (-1)^k\cdot h^{100}(k)$ can be written in the form $ab^c$, for some integers $a,b,c$ where $c$ is as maximal as possible and $b\ne 1$, find $a+b+c$.
[i]Proposed by [b]AOPS12142015[/b][/i]
2021 AIME Problems, 14
Let $\triangle ABC$ be an acute triangle with circumcenter $O$ and centroid $G$. Let $X$ be the intersection of the line tangent to the circumcircle of $\triangle ABC$ at $A$ and the line perpendicular to $GO$ at $G$. Let $Y$ be the intersection of lines $XG$ and $BC$. Given that the measures of $\angle ABC, \angle BCA, $ and $\angle XOY$ are in the ratio $13 : 2 : 17, $ the degree measure of $\angle BAC$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
unitsize(5mm);
pair A,B,C,X,G,O,Y;
A = (2,8);
B = (0,0);
C = (15,0);
dot(A,5+black); dot(B,5+black); dot(C,5+black);
draw(A--B--C--A,linewidth(1.3));
draw(circumcircle(A,B,C));
O = circumcenter(A,B,C);
G = (A+B+C)/3;
dot(O,5+black); dot(G,5+black);
pair D = bisectorpoint(O,2*A-O);
pair E = bisectorpoint(O,2*G-O);
draw(A+(A-D)*6--intersectionpoint(G--G+(E-G)*15,A+(A-D)--A+(D-A)*10));
draw(intersectionpoint(G--G+(G-E)*10,B--C)--intersectionpoint(G--G+(E-G)*15,A+(A-D)--A+(D-A)*10));
X = intersectionpoint(G--G+(E-G)*15,A+(A-D)--A+(D-A)*10);
Y = intersectionpoint(G--G+(G-E)*10,B--C);
dot(Y,5+black);
dot(X,5+black);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$O$",O,ESE);
label("$G$",G,W);
label("$X$",X,dir(0));
label("$Y$",Y,NW);
draw(O--G--O--X--O--Y);
markscalefactor = 0.07;
draw(rightanglemark(X,G,O));
[/asy]
2017 AIME Problems, 7
Find the number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\log(kx)=2\log(x+2)$ has exactly one real solution.
2022 AIME Problems, 9
Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$. Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i=1,2,3,\ldots, m$ and $j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$. The figure shows that there are 8 regions when $m=3$ and $n=2$.
[asy]
import geometry;
size(10cm);
draw((-2,0)--(13,0));
draw((0,4)--(10,4));
label("$\ell_A$",(-2,0),W);
label("$\ell_B$",(0,4),W);
point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2);
draw(B1--A1--B2);
draw(B1--A2--B2);
draw(B1--A3--B2);
label("$A_1$",A1,S);
label("$A_2$",A2,S);
label("$A_3$",A3,S);
label("$B_1$",B1,N);
label("$B_2$",B2,N);
label("1",centroid(A1,B1,I1));
label("2",centroid(B1,I1,I3));
label("3",centroid(B1,B2,I3));
label("4",centroid(A1,A2,I1));
label("5",(A2+I1+I2+I3)/4);
label("6",centroid(B2,I2,I3));
label("7",centroid(A2,A3,I2));
label("8",centroid(A3,B2,I2));
dot(A1);
dot(A2);
dot(A3);
dot(B1);
dot(B2);
[/asy]
2015 AIME Problems, 12
There are $2^{10}=1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical.
2025 AIME, 12
Let $A_1A_2\dots A_{11}$ be a non-convex $11$-gon such that
- The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$,
- $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$,
- The perimeter of $A_1A_2\dots A_{11}$ is $20$.
If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive integers $m,n,p,q$ with $n$ squarefree and $\gcd(m,p,q)=1$, find $m+n+p+q$.
2019 AIME Problems, 10
There is a unique angle $\theta$ between $0^{\circ}$ and $90^{\circ}$ such that for nonnegative integers $n$, the value of $\tan{\left(2^{n}\theta\right)}$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\theta$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.
2023 AIME, 11
Find the number of collections of $16$ distinct subsets of $\{1, 2, 3, 4, 5\}$ with the property that for any two subsets $X$ and $Y$ in the collection, $X\cap Y \neq \emptyset$.
CIME II 2018, 14
Positive rational numbers $x<y<z$ sum to $1$ and satisfy the equation
$$(x^2+y^2+z^2-1)^3+8xyz=0.$$
Given that $\sqrt{z}$ is also rational, it can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. If $m+n < 1000$, find the maximum value of $m+n$.
[I]Proposed by [b] Th3Numb3rThr33 [/b][/I]
2017 AIME Problems, 6
Find the sum of all positive integers $n$ such that $\sqrt{n^2+85n+2017}$ is an integer.
2017 AIME Problems, 5
A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189$, $320$, $287$, $234$, $x$, and $y$. Find the greatest possible value of $x+y$.
2023 AIME, 13
Let $A$ be an acute angle such that $\tan A = 2\cos A$. Find the number of positive integers $n$ less than or equal to $1000$ such that $\sec^n A + \tan^n A$ is a positive integer whose units digit is $9$.
2025 AIME, 8
From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $N$ cents, where $N$ is a positive integer. He uses the so-called greedy algorithm, successively choosing the coin of greatest value that does not cause the value of his collection to exceed $N.$ For example, to get 42 cents, Silas will choose a 25-cent coin, then a 10-cent coin, then 7 1-cent coins. However, this collection of 9 coins uses more coins than necessary to get a total of 42 cents; indeed, choosing 4 10-cent coins and 2 1-cent coins achieves the same total value with only 6 coins. In general, the greedy algorithm succeeds for a given $N$ if no other collection of 1-cent, 10-cent, and 25-cent coins gives a total value of $N$ cents using strictly fewer coins than the collection given by the greedy algorithm. Find the number of values of $N$ between $1$ and $1000$ inclusive for which the greedy algorithm succeeds.
2019 AIME Problems, 5
Four ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order from $1$ to $12$. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the $8$ people to be seated at the table under these conditions. Find the remainder when $N$ is divided by $1000$.
CIME II 2018, 1
Find the largest positive integer value of $n<1000$ such that $\phi(36n)=\phi(25n)+\phi(16n)$, where $\phi(n)$ denotes the number of positive integers less than $n$ that are relatively prime to $n$.
[i]Proposed by [b]AOPS12142015[/b][/i]
2019 AIME Problems, 4
A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2021 AIME Problems, 7
Let $a, b, c,$ and $d$ be real numbers that satisfy the system of equations
\begin{align*} a+b&=-3\\ ab+bc+ca&= -4\\ abc+bcd+cda+dab&=14\\ abcd&=30. \end{align*}
There exist relatively prime positive integers $m$ and $n$ such that
$$a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.$$
Find $m + n$.
CIME II 2018, 4
Three fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.
[i]Proposed by [b]AOPS12142015[/b][/i]
2007 National Olympiad First Round, 19
If $x_1=5, x_2=401$, and
\[
x_n=x_{n-2}-\frac 1{x_{n-1}}
\]
for every $3\leq n \leq m$, what is the largest value of $m$?
$
\textbf{(A)}\ 406
\qquad\textbf{(B)}\ 2005
\qquad\textbf{(C)}\ 2006
\qquad\textbf{(D)}\ 2007
\qquad\textbf{(E)}\ \text{None of the above}
$
2022 AIME Problems, 13
There is a polynomial $P(x)$ with integer coefficients such that $$P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}$$ holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$