Found problems: 634
1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 3
A bowling contest consists of several series. Mary got 185 points in her previous series and thereby increased her average score per series from 176 to 177 points. How many points would Mary need in her next series to increase her average to 178?
$ \text{(A)}\ 184 \qquad \text{(B)}\ 185 \qquad \text{(C)}\ 186 \qquad \text{(D)}\ 187 \qquad \text{(E)}\ 188$
1985 AIME Problems, 9
In a circle, parallel chords of lengths 2, 3, and 4 determine central angles of $\alpha$, $\beta$, and $\alpha + \beta$ radians, respectively, where $\alpha + \beta < \pi$. If $\cos \alpha$, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?
1988 AIME Problems, 9
Find the smallest positive integer whose cube ends in 888.
2023 AIME, 15
Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying [list]
[*] the real and imaginary part of $z$ are both integers;
[*] $|z|=\sqrt{p}$, and
[*] there exists a triangle whose three side lengths are $p$, the real part of $z^{3}$, and the imaginary part of $z^{3}$.
[/list]
1992 AIME Problems, 8
For any sequence of real numbers $A=(a_1,a_2,a_3,\ldots)$, define $\Delta A$ to be the sequence $(a_2-a_1,a_3-a_2,a_4-a_3,\ldots)$, whose $n^\text{th}$ term is $a_{n+1}-a_n$. Suppose that all of the terms of the sequence $\Delta(\Delta A)$ are $1$, and that $a_{19}=a_{92}=0$. Find $a_1$.
2021 AIME Problems, 12
Let $A_1A_2A_3...A_{12}$ be a dodecagon (12-gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2009 AIME Problems, 8
Let $ S \equal{} \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $ S$. Let $ N$ be the sum of all of these differences. Find the remainder when $ N$ is divided by $ 1000$.
2023 AIME, 7
Call a positive integer $n$ [i]extra-distinct[/i] if the remainders when $n$ is divided by $2, 3, 4, 5,$ and $6$ are distinct. Find the number of extra-distinct positive integers less than $1000$.
1991 AIME Problems, 15
For positive integer $n$, define $S_n$ to be the minimum value of the sum \[ \sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2}, \] where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is 17. There is a unique positive integer $n$ for which $S_n$ is also an integer. Find this $n$.
2011 AMC 10, 12
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width 6 meters, and it takes her 36 seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?
$ \textbf{(A)}\ \frac{\pi}{3} \qquad
\textbf{(B)}\ \frac{2\pi}{3} \qquad
\textbf{(C)}\ \pi \qquad
\textbf{(D)}\ \frac{4\pi}{3} \qquad
\textbf{(E)}\ \frac{5\pi}{3} $
2018 AIME Problems, 1
Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$.
1999 AIME Problems, 3
Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.
2023 AIME, 12
In $\triangle ABC$ with side lengths $AB=13$, $BC=14$, and $CA=15$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the point on the circumcircle of $\triangle ABC$ such that $M$ is on $\overline{AP}$. There exists a unique point $Q$ on segment $\overline{AM}$ such that $\angle PBQ = \angle PCQ$. Then $AQ$ can be written as $\frac{m}{\sqrt{n}}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2015 AIME Problems, 7
In the diagram below, $ABCD$ is a square. Point $E$ is the midpoint of $\overline{AD}$. Points $F$ and $G$ lie on $\overline{CE}$, and $H$ and $J$ lie on $\overline{AB}$ and $\overline{BC}$, respectively, so that $FGHJ$ is a square. Points $K$ and $L$ lie on $\overline{GH}$, and $M$ and $N$ lie on $\overline{AD}$ and $\overline{AB}$, respectively, so that $KLMN$ is a square. The area of $KLMN$ is 99. Find the area of $FGHJ$.
[asy]
pair A,B,C,D,E,F,G,H,J,K,L,M,N;
B=(0,0);
real m=7*sqrt(55)/5;
J=(m,0);
C=(7*m/2,0);
A=(0,7*m/2);
D=(7*m/2,7*m/2);
E=(A+D)/2;
H=(0,2m);
N=(0,2m+3*sqrt(55)/2);
G=foot(H,E,C);
F=foot(J,E,C);
draw(A--B--C--D--cycle);
draw(C--E);
draw(G--H--J--F);
pair X=foot(N,E,C);
M=extension(N,X,A,D);
K=foot(N,H,G);
L=foot(M,H,G);
draw(K--N--M--L);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,NE);
label("$E$",E,dir(90));
label("$F$",F,NE);
label("$G$",G,NE);
label("$H$",H,W);
label("$J$",J,S);
label("$K$",K,SE);
label("$L$",L,SE);
label("$M$",M,dir(90));
label("$N$",N,dir(180));
[/asy]
2014 JHMMC 7 Contest, 26
Alex is training to make $\text{MOP}$. Currently he will score a $0$ on $\text{the AMC,}\text{ the AIME,}\text{and the USAMO}$. He can expend $3$ units of effort to gain $6$ points on the $\text{AMC}$, $7$ units of effort to gain $10$ points on the $\text{AIME}$, and $10$ units of effort to gain $1$ point on the $\text{USAMO}$. He will need to get at least $200$ points on $\text{the AMC}$ and $\text{AIME}$ combined and get at least $21$ points on $\text{the USAMO}$ to make $\text{MOP}$. What is the minimum amount of effort he can expend to make $\text{MOP}$?
2017 AIME Problems, 2
Teams $T_1$, $T_2$, $T_3$, and $T_4$ are in the playoffs. In the semifinal matches, $T_1$ plays $T_4$ and $T_2$ plays $T_3$. The winners of those two matches will play each other in the final match to determine the champion. When $T_i$ plays $T_j$, the probability that $T_i$ wins is $\frac{i}{i+j}$, and the outcomes of all the matches are independent. The probability that $T_4$ will be the champion is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2008 AIME Problems, 10
Let $ ABCD$ be an isosceles trapezoid with $ \overline{AD}\parallel{}\overline{BC}$ whose angle at the longer base $ \overline{AD}$ is $ \dfrac{\pi}{3}$. The diagonals have length $ 10\sqrt {21}$, and point $ E$ is at distances $ 10\sqrt {7}$ and $ 30\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the foot of the altitude from $ C$ to $ \overline{AD}$. The distance $ EF$ can be expressed in the form $ 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$.
2010 Stanford Mathematics Tournament, 8
Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=3^k$ for $0\le k \le n$. Find $P(n+1)$
2011 India Regional Mathematical Olympiad, 5
Let $ABC$ be a triangle and let $BB_1,CC_1$ be respectively the bisectors of $\angle{B},\angle{C}$ with $B_1$ on $AC$ and $C_1$ on $AB$, Let $E,F$ be the feet of perpendiculars drawn from $A$ onto $BB_1,CC_1$ respectively. Suppose $D$ is the point at which the incircle of $ABC$ touches $AB$. Prove that $AD=EF$
2002 AIME Problems, 12
A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_{n}$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10}=.4$ and $a_{n}\le .4$ for all $n$ such that $1\le n \le 9$ is given to be $p^{a}q^{b}r/(s^{c}),$ where $p,$ $q,$ $r,$ and $s$ are primes, and $a,$ $b,$ and $c$ are positive integers. Find $(p+q+r+s)(a+b+c).$
2016 AIME Problems, 1
For $-1 < r < 1$, let $S(r)$ denote the sum of the geometric series \[12 + 12r + 12r^2 + 12r^3 + \ldots.\] Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2016$. Find $S(a) + S(-a)$.
2021 AIME Problems, 10
Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[img]https://imgur.com/1mfBNNL.png[/img]
2022 AIME Problems, 9
Ellina has twelve blocks, two each of red $\left({\bf R}\right),$ blue $\left({\bf B}\right),$ yellow $\left({\bf Y}\right),$ green $\left({\bf G}\right),$ orange $\left({\bf O}\right),$ and purple $\left({\bf P}\right).$ Call an arrangement of blocks [i]even[/i] if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement
$$ {\text {\bf R B B Y G G Y R O P P O}}$$is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2023 AIME, 9
Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are integers in $\{-20, -19,-18, \dots , 18, 19, 20\}$, such that there is a unique integer $m \neq 2$ with $p(m) = p(2)$.
2008 ITest, 90
For $a,b,c$ positive reals, let \[N=\dfrac{a^2+b^2}{c^2+ab}+\dfrac{b^2+c^2}{a^2+bc}+\dfrac{c^2+a^2}{b^2+ca}.\] Find the minimum value of $\lfloor 2008N\rfloor$.