Olympiad problems

2001 Czech-Polish-Slovak Match, 3

Let $ n$ and $ k$ be positive integers such that $ \frac{1}{2} n < k \leq \frac{2}{3} n.$ Find the least number $ m$ for which it is possible to place $ m$ pawns on $ m$ squares of an $ n \times n$ chessboard so that no column or row contains a block of $ k$ adjacent unoccupied squares.

2013 Dutch IMO TST, 1

Tags: factorial , number theory

Show that $\sum_{n=0}^{2013}\frac{4026!}{(n!(2013-n)!)^2}$ is a perfect square.

LMT Team Rounds 2021+, 4

Tags: geometry

Segment $AB$ of length $13$ is the diameter of a semicircle. Points $C$ and $D$ are located on the semicircle but not on segment $AB$. Segments $AC$ and $BD$ both have length $5$. Given that the length of $CD$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a +b$.

2012 India Regional Mathematical Olympiad, 2

Tags: number theory , divisibility , divide , divisor

Prove that for all positive integers $n$, $169$ divides $21n^2 + 89n + 44$ if $13$ divides $n^2 + 3n + 51$.

2023 MOAA, 5

Tags:

Andy creates a 3 sided dice with a side labeled $7$, a side labeled $17$, and a side labeled $27$. He then asks Anthony to roll the dice $3$ times. The probability that the product of Anthony's rolls is greater than $2023$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andy Xu[/i]

2014 Junior Balkan Team Selection Tests - Romania, 2

Tags: diophantine , diophantine equation , number theory

Solve, in the positive integers, the equation $5^m + n^2 = 3^p$ .

2023 VN Math Olympiad For High School Students, Problem 6

Tags: algebra , polynomial , number theory , prime number

Prove that these polynomials are irreducible in $\mathbb{Q}[x]:$ a) $\frac{{{x^p}}}{{p!}} + \frac{{{x^{p - 1}}}}{{(p - 1)!}} + ... + \frac{{{x^2}}}{2} + x + 1,$ with $p$ is a prime number. b) $x^{2^n}+1,$ with $n$ is a positive integer.

2011 China Second Round Olympiad, 8

Tags: number theory

Given that $a_{n}= \binom{200}{n} \cdot 6^{\frac{200-n}{3}} \cdot (\dfrac{1}{\sqrt{2}})^n$ ($ 1 \leq n \leq 95$). How many integers are there in the sequence $\{a_n\}$?

2014 VJIMC, Problem 2

Tags: group theory , abstract algebra

Let $p$ be a prime number and let $A$ be a subgroup of the multiplicative group $\mathbb F^*_p$ of the finite field $\mathbb F_p$ with $p$ elements. Prove that if the order of $A$ is a multiple of $6$, then there exist $x,y,z\in A$ satisfying $x+y=z$.

MBMT Team Rounds, 2020.27

Tags:

The perfect square game is played as follows: player 1 says a positive integer, then player 2 says a strictly smaller positive integer, and so on. The game ends when someone says 1; that player wins if and only if the sum of all numbers said is a perfect square. What is the sum of all $n$ such that, if player 1 starts by saying $n$, player 1 has a winning strategy? A winning strategy for player 1 is a rule player 1 can follow to win, regardless of what player 2 does. If player 1 wins, player 2 must lose, and vice versa. Both players play optimally. [i]Proposed by Jacob Stavrianos[/i]

2010 Princeton University Math Competition, 6

Tags: probability , expected value

A regular pentagon is drawn in the plane, along with all its diagonals. All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded. An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of the region it's in, with an equal probability of choosing each edge, and crosses that edge into another region. If the ant enters an unbounded region, it explodes. After first leaving the central region of the pentagon, let $x$ be the expected number of times the ant re-enters the central region before it explodes. Find the closest integer to $100x$.

1996 Romania National Olympiad, 3

Tags: rectangle , trigonometry , geometry

Let $AB CD$ be a rectangle with $AB=1$. If $m ( \angle BDC) = 82^o30'$, compute the length of$ BD$ and the cosine of $82^o30'$.

2004 Paraguay Mathematical Olympiad, 4

Tags: geometry , square , trisection

In a square $ABCD$, $E$ is the midpoint of $BC$ and $F$ is the midpoint of $CD$. Prove that $AF$ and $AE$ divide the diagonal $BD$ in three equal segments.

2000 Saint Petersburg Mathematical Olympiad, 10.5

Tags: grid , combinatorics , coloring , table , geometry , rectangle , invariant

Cells of a $2000\times2000$ board are colored according to the following rules: 1)At any moment a cell can be colored, if none of its neighbors are colored 2)At any moment a $1\times2$ rectangle can be colored, if exactly two of its neighbors are colored. 3)At any moment a $2\times2$ squared can be colored, if 8 of its neighbors are colored (Two cells are considered to be neighboring, if they share a common side). Can the entire $2000\times2000$ board be colored? [I]Proposed by K. Kohas[/i]

VMEO II 2005, 12

Tags: algebra , inequalities

a) Find all real numbers $k$ such that there exists a positive constant $c_k$ satisfying $$(x^2 + 1)(y^2 + 1)(z^2 + 1) \ge c_k(x + y + z)^k$$ for all positive real numbers. b) Given the numbers $k$ found, determine the largest number $c_k$.

2001 Nordic, 4

Tags: geometry , hexagon , concurrency , area

Let ${ABCDEF}$ be a convex hexagon, in which each of the diagonals ${AD, BE}$ , and ${CF}$ divides the hexagon into two quadrilaterals of equal area. Show that ${AD, BE}$ , and ${CF}$ are concurrent.

2006 Romania National Olympiad, 3

Tags: superior algebra , group theory , abstract algebra

Let $\displaystyle G$ be a finite group of $\displaystyle n$ elements $\displaystyle ( n \geq 2 )$ and $\displaystyle p$ be the smallest prime factor of $\displaystyle n$. If $\displaystyle G$ has only a subgroup $\displaystyle H$ with $\displaystyle p$ elements, then prove that $\displaystyle H$ is in the center of $\displaystyle G$. [i]Note.[/i] The center of $\displaystyle G$ is the set $\displaystyle Z(G) = \left\{ a \in G \left| ax=xa, \, \forall x \in G \right. \right\}$.

2007 Moldova National Olympiad, 12.3

Tags: inequalities , function , algebra , logarithm

For $a,b \in [1;\infty)$ show that \[ab\leq e^{a-1}+b\ln b\]

1936 Moscow Mathematical Olympiad, 031

Tags: combinatorial geometry , combinatorics , geometry , 3d geometry

Given three planes and a ball in space. In space, find the number of different ways of placing another ball so that it would be tangent the three given planes and the given ball. It is assumed that the balls can only touch externally.

2024 Malaysia IMONST 2, 3

Tags: number theory

Janson wants to find a sequence of positive integers $a_{1}, a_{2}, . . . , a_{2024}$ such that each term is at least $10$, and $a_{i}$ has exactly $a_{i+1}$ divisors for all $1 \leq i \leq 2023$. Can you help him find one such sequence, or is this task impossible?

2023 Sharygin Geometry Olympiad, 14

Tags: geometry , polygon , winding number

Suppose that a closed oriented polygonal line $\mathcal{L}$ in the plane does not pass through a point $O$, and is symmetric with respect to $O$. Prove that the winding number of $\mathcal{L}$ around $O$ is odd. The winding number of $\mathcal{L}$ around $O$ is defined to be the following sum of the oriented angles divided by $2\pi$: $$\deg_O\mathcal{L} := \dfrac{\angle A_1OA_2+\angle A_2OA_3+\dots+\angle A_{n-1}OA_n+\angle A_nOA_1}{2\pi}.$$

2005 AMC 12/AHSME, 7

Tags: geometry , rhombus

What is the area enclosed by the graph of $ |3x| \plus{} |4y| \equal{} 12$? $ \textbf{(A)}\ 6\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 24\qquad \textbf{(E)}\ 25$

2012 Bogdan Stan, 2

Tags: real analyssi , sequence

Let be a bounded sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ satisfying the recurrence: $$ x_{n+3} =\sqrt[3]{3x_n-2} . $$ Prove that $ \left( x_n \right)_{n\ge 1} $ is convergent. [i]Cristinel Mortici[/i]

2015 Thailand TSTST, 2

Tags: number theory , diophantine equation

Find all integer solutions to the equation $y^2=2x^4+17$.

2018 Belarusian National Olympiad, 9.3

Tags: geometry , perpendicularity

The bisector of angle $CAB$ of triangle $ABC$ intersects the side $CB$ at $L$. The point $D$ is the foot of the perpendicular from $C$ to $AL$ and the point $E$ is the foot of perpendicular from $L$ to $AB$. The lines $CB$ and $DE$ meet at $F$. Prove that $AF$ is an altitude of triangle $ABC$.