Olympiad problems

2021 Ukraine National Mathematical Olympiad, 2

Denote by $P^{(n)}$ the set of all polynomials of degree $n$ the coefficients of which is a permutation of the set of numbers $\{2^0, 2^1,..., 2^n\}$. Find all pairs of natural numbers $(k,d)$ for which there exists a $n$ such that for any polynomial $p \in P^{(n)}$, number $P(k)$ is divisible by the number $d$. (Oleksii Masalitin)

2020 LMT Fall, B2

Tags: geometry

The area of a square is $144$. An equilateral triangle has the same perimeter as the square. The area of a regular hexagon is $6$ times the area of the equilateral triangle. What is the perimeter of the hexagon?

2019 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , circumcircle , orthocenter , isosceles

Let $d$ be the tangent at $B$ to the circumcircle of the acute scalene triangle $ABC$. Let $K$ be the orthogonal projection of the orthocenter, $H$, of triangle $ABC$ to the line $d$ and $L$ the midpoint of the side $AC$. Prove that the triangle $BKL$ is isosceles.

1965 Putnam, A5

Tags:

In how many ways can the integers from $1$ to $n$ be ordered subject to the condition that, except for the first integer on the left, every integer differs by $1$ from some integer to the left of it?

2019 Moldova Team Selection Test, 2

Tags: trigonometry , easy , geometry

Prove that $E_n=\frac{\arccos {\frac{n-1}{n}} } {\text{arccot} {\sqrt{2n-1} }}$ is a natural number for any natural number $n$. (A natural number is a positive integer)

2024 All-Russian Olympiad Regional Round, 9.4

Tags: combinatorics

The positive integers $1, 2, \ldots, 1000$ are written in some order on one line. Show that we can find a block of consecutive numbers, whose sum is in the interval $(100000; 100500]$.

2022 Girls in Math at Yale, 2

Tags: college

How many ways are there to fill in a $2\times 2$ square grid with the numbers $1,2,3,$ and $4$ such that the numbers in any two grid squares that share an edge have an absolute difference of at most $2$? [i]Proposed by Andrew Wu[/i]

2017 Balkan MO Shortlist, C1

Tags: combinatorics , game strategy , combinatorial game

A grasshopper is sitting at an integer point in the Euclidean plane. Each second it jumps to another integer point in such a way that the jump vector is constant. A hunter that knows neither the starting point of the grasshopper nor the jump vector (but knows that the jump vector for each second is constant) wants to catch the grasshopper. Each second the hunter can choose one integer point in the plane and, if the grasshopper is there, he catches it. Can the hunter always catch the grasshopper in a finite amount of time?

2011 VJIMC, Problem 3

Tags: summation

Prove that $$\sum_{k=0}^\infty x^k\frac{1+x^{2k+2}}{(1-x^{2k+2})^2}=\sum_{k=0}^\infty(-1)^k\frac{x^k}{(1-x^{k+1})^2}$$for all $x\in(-1,1)$.

2003 Denmark MO - Mohr Contest, 3

Tags: number theory , prime

Determine the integers $n$ where $$|2n^2+9n+4|$$ is a prime number.

2023 ISI Entrance UGB, 6

Tags: inequalities , recurrence relation

Let $\{u_n\}_{n \ge 1}$ be a sequence of real numbers defined as $u_1 = 1$ and \[ u_{n+1} = u_n + \frac{1}{u_n} \text{ for all $n \ge 1$.}\] Prove that $u_n \le \frac{3\sqrt{n}}{2}$ for all $n$.

2018 District Olympiad, 4

Tags: complex number

Let $n\ge 2$ be a natural number. Find all complex numbers $z$ which simultaneously satisfy the relations $\text{a)}\ z^n + z^{n - 1} + \ldots + z^2 + |z| = n;$ $\text{b)}\ |z|^{n- 1} + |z|^{n - 2} + \ldots + |z|^2 + z = n z^n.$

2006 Victor Vâlcovici, 1

Tags: inequalities

Let be two nonnegative real numbers $ a,b, $ not both $ 0, $ satisfying $ \frac{1}{2a+b} +\frac{1}{a+2b} =1. $ Prove the following inequalities and explain the equality cases for [b]a),b).[/b] [b]a)[/b] $ 4/3\le a+b\le 3/2 $ [b]b)[/b] $ 8/9\le a^2+b^2\le 9/4 $ [b]c)[/b] $ ab<1/2 $ [i]Laurențiu Panaitopol[/i]

Ukrainian TYM Qualifying - geometry, II.16

Tags: symmetry , symmetric , geometry

Inside the circle are given three points that do not belong to one line. In one step it is allowed to replace one of the points with a symmetric one wrt the line containing the other two points. Is it always possible for a finite number of these steps to ensure that all three points are outside the circle?

2003 Tournament Of Towns, 7

Tags: combinatorics

A square is triangulated in such way that no three vertices are collinear. For every vertex (including vertices of the square) the number of sides issuing from it is counted. Can it happen that all these numbers are even?

2011 Math Prize For Girls Problems, 17

Tags: algebra , polynomial

There is a polynomial $P$ such that for every real number $x$, \[ x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x). \] When $P$ is written in standard polynomial form, how many of its coefficients are nonzero?

2015 HMNT, 10

Tags:

Let $N$ be the number of functions $f$ from $\{1, 2, \dots, 101 \} \rightarrow \{1, 2, \dots, 101 \}$ such that $f^{101}(1) = 2.$ Find the remainder when $N$ is divided by $103.$

2024 Ukraine National Mathematical Olympiad, Problem 1

Tags: greatest common divisor , least common multiple , number theory

Find all pairs $a, b$ of positive integers, for which $$(a, b) + 3[a, b] = a^3 - b^3$$ Here $(a, b)$ denotes the greatest common divisor of $a, b$, and $[a, b]$ denotes the least common multiple of $a, b$. [i]Proposed by Oleksiy Masalitin[/i]

1999 Singapore Team Selection Test, 3

Tags: divide , number theory , prime number

Let $f(x) = x^{1998} - x^{199}+x^{19}+ 1$. Prove that there is an infinite set of prime numbers, each dividing at least one of the integers $f(1), f(2), f(3), f(4), ...$

2008 May Olympiad, 5

Tags: combinatorics

On a $16 x 16$ board, $25$ coins are placed, as in the figure. It is allowed to select $8$ rows and $8$ columns and remove from the board all the coins that are in those $16$ lines. Determine if it is possible to remove all coins from the board. [img]https://cdn.artofproblemsolving.com/attachments/1/5/e2c7379a6f47e2e8b8c9b989b85b96454a38e1.gif[/img] If the answer is yes, indicate the $8$ rows and $8$ columns selected, and if no, explain why.

ICMC 6, 3

Tags: probability , combinatorics , random process

The numbers $1, 2, \dots , n$ are written on a blackboard and then erased via the following process:[list] [*] Before any numbers are erased, a pair of numbers is chosen uniformly at random and circled. [*] Each minute for the next $n -1$ minutes, a pair of numbers still on the blackboard is chosen uniformly at random and the smaller one is erased. [*] In minute $n$, the last number is erased. [/list] What is the probability that the smaller circled number is erased before the larger? [i]Proposed by Ethan Tan[/i]

2012 Princeton University Math Competition, B1

Tags: combinatorics

Your friend sitting to your left (or right?) is unable to solve any of the eight problems on his or her Combinatorics $B$ test, and decides to guess random answers to each of them. To your astonishment, your friend manages to get two of the answers correct. Assuming your friend has equal probability of guessing each of the questions correctly, what is the average possible value of your friend’s score? Recall that each question is worth the point value shown at the beginning of each question.

2008 China Team Selection Test, 2

Tags: polynomial , modular arithmetic , absolute value , algebra

Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that (1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$; (2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees; (3) for any integers $ x, |f(x)|$ isn't prime numbers.

2013 Regional Competition For Advanced Students, 3

Tags: inequalities , arithmetic mean-geometric mean , sequence

For non-negative real numbers $a,$ $b$ let $A(a, b)$ be their arithmetic mean and $G(a, b)$ their geometric mean. We consider the sequence $\langle a_n \rangle$ with $a_0 = 0,$ $a_1 = 1$ and $a_{n+1} = A(A(a_{n-1}, a_n), G(a_{n-1}, a_n))$ for $n > 0.$ (a) Show that each $a_n = b^2_n$ is the square of a rational number (with $b_n \geq 0$). (b) Show that the inequality $\left|b_n - \frac{2}{3}\right| < \frac{1}{2^n}$ holds for all $n > 0.$

1990 IMO Longlists, 31

Tags: symmetry , combinatorics

Let $S = \{1, 2, \ldots, 1990\}$. A $31$-element subset of $S$ is called "good" if the sum of its elements is divisible by $5$. Find the number of good subsets of $S.$