Found problems: 85335
2014 India IMO Training Camp, 3
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that
\[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \]
Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.
2020 Thailand TSTST, 6
Prove that the unit square can be tiled with rectangles (not necessarily of the same size) similar to a rectangle of size $1\times(3+\sqrt[3]{3})$.
2021 Brazil National Olympiad, 5
Let $ABC$ be an acute-angled triangle. Let $A_1$ be the midpoint of the arc $BC$ which contain the point $A$. Let $A_2$ and $A_3$ be the foot(s) of the perpendicular(s) of the point $A_1$ to the lines $AB$ and $AC$, respectively. Define $B_2,B_3,C_2,C_3$ analogously.
a) Prove that the line $A_2A_3$ cuts $BC$ in the midpoint.
b) Prove that the lines $A_2A_3,B_2B_3$ and $C_2C_3$ are concurrents.
2023 Princeton University Math Competition, A3 / B5
Call an arrangement of n not necessarily distinct nonnegative integers in a circle [i]wholesome[/i] when, for any subset of the integers such that no pair of them is adjacent in the circle, their average is an integer. Over all wholesome arrangements of $n$ integers where at least two of them are distinct, let $M(n)$ denote the smallest possible value for the maximum of the integers in the arrangement. What is the largest integer $n < 2023$ such that $M(n+1)$ is strictly greater than $M(n)$?
2009 F = Ma, 21
What is the value of the gravitational potential energy of the two star system?
(A) $-\frac{GM^2}{d}$
(B) $\frac{3GM^2}{d}$
(C) $-\frac{GM^2}{d^2}$
(D) $-\frac{3GM^2}{d}$
(E) $-\frac{3GM^2}{d^2}$
BIMO 2021, 3
Let $ABC$ be an actue triangle with $AB<AC$. Let $\Gamma$ be its circumcircle, $I$ its incenter and $P$ is a point on $\Gamma$ such that $\angle API=90^{\circ}$. Let $Q$ be a point on $\Gamma$ such that $$QB\cdot\tan \angle B=QC\cdot \tan \angle C$$ Consider a point $R$ such that $PR$ is tangent to $\Gamma$ and $BR=CR$. Prove that the points $A, Q, R$ are colinear.
1987 Tournament Of Towns, (155) 6
There are $2000$ apples , contained in several baskets. One can remove baskets and /or remove apples from the baskets. Prove that it is possible to then have an equal number of apples in each of the remaining baskets, with the total number of apples being not less than $100$ .
(A. Razborov)
1997 Abels Math Contest (Norwegian MO), 4
Let $p(x)$ be a polynomial with integer coefficients. Suppose that there exist different integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$. Show that the equation $f(x) = x$ has at most one integer solution.
2019 Mexico National Olympiad, 6
Let $ABC$ be a triangle such that $\angle BAC = 45^{\circ}$. Let $H,O$ be the orthocenter and circumcenter of $ABC$, respectively. Let $\omega$ be the circumcircle of $ABC$ and $P$ the point on $\omega$ such that the circumcircle of $PBH$ is tangent to $BC$. Let $X$ and $Y$ be the circumcenters of $PHB$ and $PHC$ respectively. Let $O_1,O_2$ be the circumcenters of $PXO$ and $PYO$ respectively. Prove that $O_1$ and $O_2$ lie on $AB$ and $AC$, respectively.
2007 Croatia Team Selection Test, 1
Find integral solutions to the equation \[(m^{2}-n^{2})^{2}=16n+1.\]
2013 USAJMO, 3
In triangle $ABC$, points $P$, $Q$, $R$ lie on sides $BC$, $CA$, $AB$ respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\omega_A$, $\omega_B$, $\omega_C$ again at $X$, $Y$, $Z$, respectively, prove that $YX/XZ=BP/PC$.
PEN Q Problems, 10
Suppose that the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are distinct. Show that \[(x-a_{1})(x-a_{2}) \cdots (x-a_{n})-1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.
2018 PUMaC Live Round, 8.1
Let $a$, $b$, and $c$ be such that the coefficient of the $x^ay^bz^c$ term in the expansion of $(x+2y+3z)^{100}$ is maximal (no other term has a strictly larger coefficient). Find the sum of all possible values of $1,000,000a+1,000b+c$.
2020 LMT Spring, 14
Let $\triangle ABC$ be a triangle such that $AB=40$ and $AC=30.$ Points $X$ and $Y$ are on the segment $AB$ and $BC,$ respectively such that $AX:BX=3:2$ and $BY:CY=1:4.$ Given that $XY=12,$ the area of $\triangle ABC$ can be written as $a\sqrt{b}$ where $a$ and $b$ are positive integers and $b$ is squarefree. Compute $a+b.$
1990 Austrian-Polish Competition, 5
Let $n>1$ be an integer and let $f_1$, $f_2$, ..., $f_{n!}$ be the $n!$ permutations of $1$, $2$, ..., $n$. (Each $f_i$ is a bijective function from $\{1,2,...,n\}$ to itself.) For each permutation $f_i$, let us define $S(f_i)=\sum^n_{k=1} |f_i(k)-k|$. Find $\frac{1}{n!} \sum^{n!}_{i=1} S(f_i)$.
2013-2014 SDML (High School), 15
Right triangle $ABC$ has its right angle at $A$. A semicircle with center $O$ is inscribed inside triangle $ABC$ with the diameter along $AB$. Let $D$ be the point where the semicircle is tangent to $BC$. If $AD=4$ and $CO=5$, find $\cos\angle{ABC}$.
[asy]
import olympiad;
pair A, B, C, D, O;
A = (1,0);
B = origin;
C = (1,1);
O = incenter(C, B, (1,-1));
draw(A--B--C--cycle);
dot(O);
draw(arc(O, 0.41421356237,0,180));
D = O+0.41421356237*dir(135);
label("$A$",A,SE);
label("$B$",B,SW);
label("$C$",C,NE);
label("$D$",D,NW);
label("$O$",O,S);
[/asy]
$\text{(A) }\frac{\sqrt{5}}{4}\qquad\text{(B) }\frac{3}{5}\qquad\text{(C) }\frac{12}{25}\qquad\text{(D) }\frac{4}{5}\qquad\text{(E) }\frac{2\sqrt{5}}{5}$
2015 JHMT, 7
Triangle $ABC$ is isoceles with $AB = AC$. Point $D$ lies on $AB$ such that the inradius of $ADC$ and the inradius of $BDC$ both equal $\frac{3-\sqrt3}{2}$ . The inradius of $ABC$ equals $1$. What is the length of $BD$?
2011 AMC 12/AHSME, 1
A cell phone plan costs $\$20$ dollars each month, plus $5$ cents per text message sent, plus $10$ cents for each minute used over $30$ hours. In January Michelle sent $100$ text messages and talked for $30.5$ hours. How much did she have to pay?
$ \textbf{(A)}\ \$ 24.00 \qquad
\textbf{(B)}\ \$ 24.50\qquad
\textbf{(C)}\ \$ 25.50\qquad
\textbf{(D)}\ \$ 28.00\qquad
\textbf{(E)}\ \$ 30.00$
2022 Grosman Mathematical Olympiad, P7
Let $k\leq n$ be two positive integers. $n$ points are marked on a line. It is known that for each marked point, the number of marked points at a distance $\leq 1$ from it (including the point itself) is divisible by $k$.
Show that $k$ divides $n$ (without remainder).
2014 Canada National Olympiad, 1
Let $a_1,a_2,\dots,a_n$ be positive real numbers whose product is $1$. Show that the sum \[\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}\] is greater than or equal to $\frac{2^n-1}{2^n}$.
2023 HMNT, 5
A complex quartic polynomial $Q$ is [i]quirky [/i] if it has four distinct roots, one of which is the sum of the other three. There are four complex values of $k$ for which the polynomial $Q(x) = x^4-kx^3-x^2-x-45$ is quirky. Compute the product of these four values of $k$.
2024 pOMA, 6
Given a positive integer $n\ge 3$, Arándano and Banana play a game. Initially, numbers $1,2,3,\dots,n$ are written on the blackboard. Alternatingly and starting with Arándano, the players erase numbers from the board one at a time, until exactly three numbers remain on the board. Banana wins the game if the last three numbers on the board are the sides of a nondegenerate triangle, and Arándano wins otherwise.
Determine, in terms of $n$, who has a winning strategy.
2016 Sharygin Geometry Olympiad, 2
A circumcircle of triangle $ABC$ meets the sides $AD$ and $CD$ of a parallelogram $ABCD$ at points $K$ and $L$ respectively. Let $M$ be the midpoint of arc $KL$ not containing $B$. Prove that $DM \perp AC$.
by E.Bakaev
2021 AMC 10 Fall, 19
A disk of radius $1$ rolls all the way around the inside of a square of side length $s>4$ and sweeps out a region of area $A$. A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$. The value of $s$ can be written as $a+\frac{b\pi}{c}$, where $a,b$, and $c$ are positive integers and $b$ and $c$ are relatively prime. What is $a+b+c$?
$\textbf{(A)} ~10\qquad\textbf{(B)} ~11\qquad\textbf{(C)} ~12\qquad\textbf{(D)} ~13\qquad\textbf{(E)} ~14$
2017 Iran MO (2nd Round), 4
Let $x,y$ be two positive real numbers such that $x^4-y^4=x-y$. Prove that
$$\frac{x-y}{x^6-y^6}\leq \frac{4}{3}(x+y).$$