Olympiad problems

1958 February Putnam, B5

Tags: combinatorics

$S$ is an infinite set of points in the plane. The distance between any two points of $S$ is integral. Prove that $S$ is a subset of a straight line.

1996 VJIMC, Problem 1

Tags: parabola , geometry , conic

Is it possible to cover the plane with the interiors of a finite number of parabolas?

Problems 15, 16, and 17 all refer to the following: In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles. 16. Estimate the year in which the population of Nisos will be approximately 6,000. $ \text{(A)}\ 2050\qquad\text{(B)}\ 2075\qquad\text{(C)}\ 2100\qquad\text{(D)}\ 2125\qquad\text{(E)}\ 2150 $

2018 Romania Team Selection Tests, 1

Tags: cyclic quadrilateral , incenter , geometry

Let $ABCD$ be a cyclic quadrilateral and let its diagonals $AC$ and $BD$ cross at $X$. Let $I$ be the incenter of $XBC$, and let $J$ be the center of the circle tangent to the side $BC$ and the extensions of sides $AB$ and $DC$ beyond $B$ and $C$. Prove that the line $IJ$ bisects the arc $BC$ of circle $ABCD$, not containing the vertices $A$ and $D$ of the quadrilateral.

1979 Spain Mathematical Olympiad, 3

Tags: binomial coefficient , binomial , algebra

Prove the equality $${n \choose 0}^2+ {n \choose 1}^2+ {n \choose 2}^2+...+{n \choose n}^2={2n \choose n}$$

2001 Czech-Polish-Slovak Match, 3

Tags: combinatorics , extremal combinatorics , graph theory , chessboard

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