Found problems: 85335
2017 Thailand TSTST, 1
In $\vartriangle ABC, D, E, F$ are the midpoints of $AB, BC, CA$ respectively. Denote by $O_A, O_B, O_C$ the incenters of $\vartriangle ADF, \vartriangle BED, \vartriangle CFE$ respectively. Prove that $O_AE, O_BF, O_CD$ are concurrent.
2011 IMO, 3
Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies
\[f(x + y) \leq yf(x) + f(f(x))\]
for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x \leq 0$.
[i]Proposed by Igor Voronovich, Belarus[/i]
2014 May Olympiad, 3
There are nine boxes. In the first there is $1$ stone, in the second there are $2$ stones, in the third there are $3$ stones, and thus continuing, in the eighth there are $8$ stones and in the ninth there are $9$ stones. The allowed operation is to remove the same number of stones from two different boxes and place them in a third box. The goal is that all stones are in a single box. Describe how to do it with the minimum number of operations allowed.
Explain why it is impossible to achieve it with fewer operations.
2024 India National Olympiad, 2
All the squares of a $2024 \times 2024$ board are coloured white. In one move, Mohit can select one row or column whose every square is white, choose exactly $1000$ squares in that row or column, and colour all of them red. Find maximum number of squares Mohit can colour in a finite number of moves.
$\quad$
[i]Proposed[/i] by Pranjal Srivastava
2004 Estonia Team Selection Test, 5
Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.
2000 AMC 10, 21
If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) [b]must[/b] be true?
I. All alligators are creepy crawlers.
II. Some ferocious creatures are creepy crawlers.
III. Some alligators are not creepy crawlers.
$\text{(A)}\ \text{I only}\qquad\text{(B)}\ \text{II only}\qquad\text{(C)}\ \text{III only}\qquad\text{(D)}\ \text{II and III only}\qquad\text{(E)}\ \text{None must be true}$
1994 AMC 8, 17
Pauline Bunyan can shovel snow at the rate of $20$ cubic yards for the first hour, $19$ cubic yards for the second, $18$ for the third, etc., always shoveling one cubic yard less per hour than the previous hour. If her driveway is $4$ yards wide, $10$ yards long, and covered with snow $3$ yards deep, then the number of hours it will take her to shovel it clean is closest to
$\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 12$
2004 Romania Team Selection Test, 13
Let $m\geq 2$ be an integer. A positive integer $n$ has the property that for any positive integer $a$ coprime with $n$, we have $a^m - 1\equiv 0 \pmod n$.
Prove that $n \leq 4m(2^m-1)$.
Created by Harazi, modified by Marian Andronache.
2022 Baltic Way, 15
Let $\Omega$ be a circle, and $B, C$ are two fixed points on $\Omega$. Given a third point $A$ on $\Omega$, let $X$ and $Y$ denote the feet of the altitudes from $B$ and $C$, respectively, in the triangle $ABC$. Prove that there exists a fixed circle $\Gamma$ such that $XY$ is tangent to $\Gamma$ regardless of the choice of the point $A$.
2013 Harvard-MIT Mathematics Tournament, 5
Rahul has ten cards face-down, which consist of five distinct pairs of matching cards. During each move of his game, Rahul chooses one card to turn face-up, looks at it, and then chooses another to turn face-up and looks at it. If the two face-up cards match, the game ends. If not, Rahul flips both cards face-down and keeps repeating this process. Initially, Rahul doesn't know which cards are which. Assuming that he has perfect memory, find the smallest number of moves after which he can guarantee that the game has ended.
MOAA Gunga Bowls, 2021.21
King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on $x+y = 4$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Andrew Wen[/i]
2022 Tuymaada Olympiad, 3
Bisectors of a right triangle $\triangle ABC$ with right angle $B$ meet at point $I.$ The perpendicular to $IC$ drawn from $B$ meets the line $IA$ at $D;$ the perpendicular to $IA$ drawn from $B$ meets the line $IC$ at $E.$ Prove that the circumcenter of the triangle $\triangle IDE$ lies on the line $AC.$
[i](A. Kuznetsov )[/i]
2023 AMC 10, 14
A number is chosen at random from among the first $100$ positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by $11$?
$\textbf{(A)}~\frac{4}{100}\qquad\textbf{(B)}~\frac{9}{200} \qquad \textbf{(C)}~\frac{1}{20} \qquad\textbf{(D)}~\frac{11}{200}\qquad\textbf{(E)}~\frac{3}{50}$
2015 AMC 8, 25
One-inch squares are cut from the corners of this 5 inch square. What is the area in square inches of the largest square that can be fitted into the remaining space?
$ \textbf{(A) } 9\qquad \textbf{(B) } 12\frac{1}{2}\qquad \textbf{(C) } 15\qquad \textbf{(D) } 15\frac{1}{2}\qquad \textbf{(E) } 17$
[asy]
draw((0,0)--(0,5)--(5,5)--(5,0)--cycle);
filldraw((0,4)--(1,4)--(1,5)--(0,5)--cycle, gray);
filldraw((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray);
filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, gray);
filldraw((4,4)--(4,5)--(5,5)--(5,4)--cycle, gray);
[/asy]
2021 AMC 10 Fall, 10
A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5, $ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$?
$\textbf{(A)}\ {-}18.5 \qquad\textbf{(B)}\ {-}13.5 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\
13.5 \qquad\textbf{(E)}\ 18.5$
2019 Thailand TSTST, 3
Find all function $f:\mathbb{Z}\to\mathbb{Z}$ satisfying
$\text{(i)}$ $f(f(m)+n)+2m=f(n)+f(3m)$ for every $m,n\in\mathbb{Z}$,
$\text{(ii)}$ there exists a $d\in\mathbb{Z}$ such that $f(d)-f(0)=2$, and
$\text{(iii)}$ $f(1)-f(0)$ is even.
1980 Yugoslav Team Selection Test, Problem 3
A sequence $(x_n)$ satisfies $x_{n+1}=\frac{x_n^2+a}{x_{n-1}}$ for all $n\in\mathbb N$. Prove that if $x_0,x_1$, and $\frac{x_0^2+x_1^2+a}{x_0x_1}$ are integers, then all the terms of sequence $(x_n)$ are integers.
2024 China Team Selection Test, 24
Let $N=10^{2024}$. $S$ is a square in the Cartesian plane with side length $N$ and the sides parallel to the coordinate axes. Inside there are $N$ points $P_1$, $P_2$, $\dots$, $P_N$ all of which have different $x$ coordinates, and the absolute value of the slope of any connected line between these points is at most $1$. Prove that there exists a line $l$ such that at least $2024$ of these points is at most distance $1$ away from $l$.
2017 NIMO Problems, 4
For how many positive integers $100 < n \le 10000$ does $\lfloor \sqrt{n-100} \rfloor$ divide $n$?
[i]Proposed by Michael Tang[/i]
1984 IMO, 1
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.
2016 Brazil National Olympiad, 6
Lei it \(ABCD\) be a non-cyclical, convex quadrilateral, with no parallel sides.
The lines \(AB\) and \(CD\) meet in \(E\).
Let it \(M \not= E\) be the intersection of circumcircles of \(ADE\) and \(BCE\).
The internal angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(I\).
The external angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(J\).
Show that \(I,J,M\) are colinear.
2021/2022 Tournament of Towns, P3
The hypotenuse of a right triangle has length 1. Consider the line passing through the points of tangency of the incircle with the legs of the triangle. The circumcircle of the triangle cuts out a segment of this line. What is the possible length of this segment?
[i]Maxim Volchkevich[/i]
1990 National High School Mathematics League, 7
If $n\in\mathbb{Z_+}$, positive real numbers $a+b=2$, then the minumum value of $\frac{1}{1+a^n}+\frac{1}{1+b^n}$ is________.
2006 AMC 12/AHSME, 8
The lines $ x \equal{} \frac 14y \plus{} a$ and $ y \equal{} \frac 14x \plus{} b$ intersect at the point $ (1,2)$. What is $ a \plus{} b$?
$ \textbf{(A) } 0 \qquad \textbf{(B) } \frac 34 \qquad \textbf{(C) } 1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } \frac 94$
1982 IMO Longlists, 11
A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers $a$ and $b$. A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between $a$ and $b$ be for this to happen?