Olympiad problems

2011 Brazil National Olympiad, 2

33 friends are collecting stickers for a 2011-sticker album. A distribution of stickers among the 33 friends is incomplete when there is a sticker that no friend has. Determine the least $m$ with the following property: every distribution of stickers among the 33 friends such that, for any two friends, there are at least $m$ stickers both don't have, is incomplete.

2008 Cuba MO, 7

Tags: algebra , inequalities

For non negative reals $a,b$ we know that $a^2+a+b^2\ge a^4+a^3+b^4$. Prove that $$\frac{1-a^4}{a^2}\ge \frac{b^2-1}{b}$$

2009 Peru IMO TST, 2

Tags: combinatorics

300 bureaucrats are split into three comissions of 100 people. Each two bureaucrats are either familiar to each other or non familiar to each other. Prove that there exists two bureaucrats from two distinct commissions such that the third commission contains either 17 bureaucrats familiar to both of them, or 17 bureaucrats familiar to none of them. _________________________________________ This problem is taken from Russian Olympiad 2007-2008 district round 9.8 $ Tipe$

2017 JBMO Shortlist, C3

Tags: coin , game strategy , combinatorics

We have two piles with $2000$ and $2017$ coins respectively. Ann and Bob take alternate turns making the following moves: The player whose turn is to move picks a pile with at least two coins, removes from that pile $t$ coins for some $2\le t \le 4$, and adds to the other pile $1$ coin. The players can choose a different $t$ at each turn, and the player who cannot make a move loses. If Ann plays first determine which player has a winning strategy.

1985 Vietnam Team Selection Test, 3

Tags: geometry , trigonometry

Does there exist a triangle $ ABC$ satisfying the following two conditions: (a) ${ \sin^2A + \sin^2B + \sin^2C = \cot A + \cot B + \cot C}$ (b) $ S\ge a^2 - (b - c)^2$ where $ S$ is the area of the triangle $ ABC$.

2001 Federal Math Competition of S&M, Problem 4

Tags: game , combinatorics

There are $n$ coins in the pile. Two players play a game by alternately performing a move. A move consists of taking $5,7$ or $11$ coins away from the pile. The player unable to perform a move loses the game. Which player - the one playing first or second - has the winning strategy if: (a) $n=2001$; (b) $n=5000$?

1994 Tournament Of Towns, (403)

Tags: digit , number theory

A schoolgirl forgot to write a multiplication sign between two $3$-digit numbers and wrote them as one number. This $6$-digit result proved to be $3$ times greater than the product (obtained by multiplication). Find these numbers. (A Kovaldzhi,

2012 AMC 10, 5

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Last year $100$ adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was $4$. What was the total number of cats and kittens received by the shelter last year? $ \textbf{(A)}\ 150 \qquad\textbf{(B)}\ 200 \qquad\textbf{(C)}\ 250 \qquad\textbf{(D)}\ 300 \qquad\textbf{(E)}\ 400 $

2020 JBMO Shortlist, 6

Tags: diophantine , shortlist , number theory

Are there any positive integers $m$ and $n$ satisfying the equation $m^3 = 9n^4 + 170n^2 + 289$ ?

1963 AMC 12/AHSME, 32

Tags: perimeter , geometry , rectangle

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: $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \text{infinitely many}$

2001 Tuymaada Olympiad, 1

Tags: inequalities , combinatorics

Ten volleyball teams played a tournament; every two teams met exactly once. The winner of the play gets 1 point, the loser gets 0 (there are no draws in volleyball). If the team that scored $n$-th has $x_{n}$ points ($n=1, \dots, 10$), prove that $x_{1}+2x_{2}+\dots+10x_{10}\geq 165$. [i]Proposed by D. Teryoshin[/i]

2013 IMAC Arhimede, 3

Tags: geometry , angle bisector , concyclic , incenter

Let $ABC$ be a triangle with $\angle ABC=120^o$ and triangle bisectors $(AA_1),(BB_1),(CC_1)$, respectively. $B_1F \perp A_1C_1$, where $F\in (A_1C_1)$. Let $R,I$ and $S$ be the centers of the circles which are inscribed in triangles $C_1B_1F,C_1B_1A_1, A_1B_1F$, and $B_1S\cap A_1C_1=\{Q\}$. Show that $R,I,S,Q$ are on the same circle.

2012 Iran Team Selection Test, 3

Tags: number theory

Find all integer numbers $x$ and $y$ such that: \[(y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y).\] [i]Proposed by Mahyar Sefidgaran[/i]

1976 USAMO, 2

Tags: derivative , function , graphing lines , calculus , slope , analytic geometry

If $ A$ and $ B$ are fixed points on a given circle and $ XY$ is a variable diameter of the same circle, determine the locus of the point of intersection of lines $ AX$ and $ BY$. You may assume that $ AB$ is not a diameter.

Russian TST 2018, P3

Tags: polynomial , algebra

Let $a < b$ be positive integers. Prove that there is a positive integer $n{}$ and a polynomial of the form \[\pm1\pm x\pm x^2\pm\cdots\pm x^n,\]divisible by the polynomial $1+x^a+x^b$.

2011 Dutch BxMO TST, 1

Tags: coloring , red , green , combinatorics

All positive integers are coloured either red or green, such that the following conditions are satisfied: - There are equally many red as green integers. - The sum of three (not necessarily distinct) red integers is red. - The sum of three (not necessarily distinct) green integers is green. Find all colourings that satisfy these conditions.

2012 China Western Mathematical Olympiad, 1

Tags: prime number , number theory , modular arithmetic

Find the smallest positive integer $m$ satisfying the following condition: for all prime numbers $p$ such that $p>3$,have $105|9^{ p^2}-29^p+m.$ (September 28, 2012, Hohhot)

LMT Team Rounds 2010-20, 2020.S23

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Let $\triangle ABC$ be a triangle such that $AB=AC=40$ and $BC=79.$ Let $X$ and $Y$ be the points on segments $AB$ and $AC$ such that $AX=5, AY=25.$ Given that $P$ is the intersection of lines $XY$ and $BC,$ compute $PX\cdot PY-PB\cdot PC.$

2014 Belarus Team Selection Test, 3

Tags: floor function , combinatorics

Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)

2015 ASDAN Math Tournament, 30

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Suppose that $10$ mathematics teachers gather at a circular table with $25$ seats to discuss the upcoming mathematics competition. Each teacher is assigned a unique integer ID number between $1$ and $10$, and the teachers arrange themselves in such a way that teachers with consecutive ID numbers are not separated by any other teacher (IDs $1$ and $10$ are considered consecutive). In addition, each pair of teachers is separated by at least one empty seat. Given that seating arrangements obtained by rotation are considered identical, how many ways are there for the teachers to sit at the table?

2009 China Western Mathematical Olympiad, 3

Tags: geometry , circumcircle

Let $H$ be the orthocenter of acute triangle $ABC$ and $D$ the midpoint of $BC$. A line through $H$ intersects $AB,AC$ at $F,E$ respectively, such that $AE=AF$. The ray $DH$ intersects the circumcircle of $\triangle ABC$ at $P$. Prove that $P,A,E,F$ are concyclic.

2007 Korea - Final Round, 2

Tags: combinatorics

Given a $ 4\times 4$ squares table. How many ways that we can fill the table with $ \{0,1\}$ such that two neighbor squares (have one common side) have product which is equal to $ 0$?

1955 Putnam, B4

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Do there exist $1,000,000$ consecutive integers each of which contains a repeated prime factor?

2021 Vietnam TST, 6

Tags: prime number , number theory

Let $n \geq 3$ be a positive integers and $p$ be a prime number such that $p > 6^{n-1} - 2^n + 1$. Let $S$ be the set of $n$ positive integers with different residues modulo $p$. Show that there exists a positive integer $c$ such that there are exactly two ordered triples $(x,y,z) \in S^3$ with distinct elements, such that $x-y+z-c$ is divisible by $p$.

2011 Today's Calculation Of Integral, 709

Tags: function , calculus computations , integration , calculus , trigonometry

Evaluate $ \int_0^1 \frac{x}{1\plus{}x}\sqrt{1\minus{}x^2}\ dx$.