Olympiad problems

2003 Finnish National High School Mathematics Competition, 5

Players Aino and Eino take turns choosing numbers from the set $\{0,..., n\}$ with $n\in \Bbb{N}$ being fixed in advance. The game ends when the numbers picked by one of the players include an arithmetic progression of length $4.$ The one who obtains the progression wins. Prove that for some $n,$ the starter of the game wins. Find the smallest such $n.$

2024 Canadian Mathematical Olympiad Qualification, 1

Tags: algebra , functional equation

Find all functions $f : R \to R$ that satisfy the functional equation $$f(x + f(xy)) = f(x)(1 + y).$$

2013 Middle European Mathematical Olympiad, 3

Tags: perpendicular bisector , angle chasing , geometry

Let $ABC$ be an isosceles triangle with $AC=BC$. Let $N$ be a point inside the triangle such that $2 \angle ANB = 180 ^\circ + \angle ACB $. Let $ D $ be the intersection of the line $BN$ and the line parallel to $AN$ that passes through $C$. Let $P$ be the intersection of the angle bisectors of the angles $CAN$ and $ABN$. Show that the lines $DP$ and $AN$ are perpendicular.

2016 ISI Entrance Examination, 2

Tags: algebra , polynomial

Consider the polynomial $ax^3+bx^2+cx+d$ where $a,b,c,d$ are integers such that $ad$ is odd and $bc$ is even.Prove that not all of its roots are rational..

2019 Bulgaria National Olympiad, 3

Tags: integer sequence , inequalities

Find all real numbers $a,$ which satisfy the following condition: For every sequence $a_1,a_2,a_3,\ldots$ of pairwise different positive integers, for which the inequality $a_n\leq an$ holds for every positive integer $n,$ there exist infinitely many numbers in the sequence with sum of their digits in base $4038,$ which is not divisible by $2019.$

2006 Bulgaria Team Selection Test, 3

Tags: function , inequalities , floor function , ceiling function , combinatorics

[b]Problem 3.[/b] Let $n\geq 3$ is given natural number, and $M$ is the set of the first $n$ primes. For any nonempty subset $X$ of $M$ with $P(X)$ denote the product of its elements. Let $N$ be a set of the kind $\ds\frac{P(A)}{P(B)}$, $A\subset M, B\subset M, A\cap B=\emptyset$ such that the product of any 7 elements of $N$ is integer. What is the maximal number of elements of $N$? [i]Alexandar Ivanov[/i]

1962 Miklós Schweitzer, 10

Tags: geometry , probability , probability and stats

From a given triangle of unit area, we choose two points independetly with uniform distribution. The straight line connecting these points divides the triangle. with probability one, into a triangle and a quadrilateral. Calculate the expected values of the areas of these two regions. [A. Renyi]

2018 Tuymaada Olympiad, 7

Tags: combinatorics

A school has three senior classes of $M$ students each. Every student knows at least $\frac{3}{4}M$ people in each of the other two classes. Prove that the school can send $M$ non-intersecting teams to the olympiad so that each team consists of $3$ students from different classes who know each other. [i]Proposed by C. Magyar, R. Martin[/i]

2006 ITAMO, 1

Tags: combinatorics

Two people play the following game: there are $40$ cards numbered from $1$ to $10$ with $4$ different signs. At the beginning they are given $20$ cards each. Each turn one player either puts a card on the table or removes some cards from the table, whose sum is $15$. At the end of the game, one player has a $5$ and a $3$ in his hand, on the table there's a $9$, the other player has a card in his hand. What is it's value?

2015 Middle European Mathematical Olympiad, 8

Tags: divisor , c.r.t , number theory

Let $n\ge 2$ be an integer. Determine the number of positive integers $m$ such that $m\le n$ and $m^2+1$ is divisible by $n$.

2010 Iran Team Selection Test, 9

Tags: induction , inequalities , concavity

Sequence of real numbers $a_0,a_1,\dots,a_{1389}$ are called concave if for each $0<i<1389$, $a_i\geq\frac{a_{i-1}+a_{i+1}}2$. Find the largest $c$ such that for every concave sequence of non-negative real numbers: \[\sum_{i=0}^{1389}ia_i^2\geq c\sum_{i=0}^{1389}a_i^2\]

2019 Brazil Undergrad MO, Problem 5

Tags: combinatorics , counting , limit , number theory

Let $M, k>0$ integers. Let $X(M,k)$ the (infinite) set of all integers that can be factored as ${p_1}^{e_1} \cdot {p_2}^{e_2} \cdot \ldots \cdot {p_r}^{e_r}$ where each $p_i$ is not smaller than $M$ and also each $e_i$ is not smaller than $k$. Let $Z(M,k,n)$ the number of elements of $X(M,k)$ not bigger than $n$. Show that there are positive reals $c(M,k)$ and $\beta(M,k)$ such that $$\lim_{n \rightarrow \infty}{\frac{Z(M,k,n)}{n^{\beta(M,k)}}} = c(M,k)$$ and find $\beta(M,k)$

2022 ELMO Revenge, 1

Tags: algebra

In terms of $p$ and $k$, compute the number of solutions in positive integers to the equation $ab+bc+ca=p^{2k}$ satisfying $a\leq b\leq c$ where $p$ is a fixed prime and $k$ is a fixed positive integer. [i]Proposed by Alexander Wang[/i]

2023 Kazakhstan National Olympiad, 4

Tags: inequalities , algebra

Given $x,y>0$ such that $x^2y^2+2x^3y=1$. Find the minimum value of sum $x+y$

2021-2022 OMMC, 13

Tags:

$ABCD$ is a rhombus where $\angle BAD = 60^\circ$. Point $E$ lies on minor arc $\widehat{AD}$ of the circumcircle of $ABD$, and $F$ is the intersection of $AC$ and the circle circumcircle of $EDC$. If $AF = 4$ and the circumcircle of $EDC$ has radius $14$, find the squared area of $ABCD$. [i]Proposed by Vivian Loh [/i]

2011 Spain Mathematical Olympiad, 2

Tags: function , calculus , derivative , algebra , inequalities

Let $a$, $b$, $c$ be positive real numbers. Prove that \[ \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge\frac52\] and determine when equality holds.

1981 Vietnam National Olympiad, 3

Tags: ratio , geometry

A plane $\rho$ and two points $M, N$ outside it are given. Determine the point $A$ on $\rho$ for which $\frac{AM}{AN}$ is minimal.

2008 Sharygin Geometry Olympiad, 9

Tags: geometry , geometric transformation , reflection , circumcircle , trigonometry , projective geometry , cyclic quadrilateral

(A.Zaslavsky, 9--10) The reflections of diagonal $ BD$ of a quadrilateral $ ABCD$ in the bisectors of angles $ B$ and $ D$ pass through the midpoint of diagonal $ AC$. Prove that the reflections of diagonal $ AC$ in the bisectors of angles $ A$ and $ C$ pass through the midpoint of diagonal $ BD$ (There was an error in published condition of this problem).

2021 Spain Mathematical Olympiad, 3

Tags: combinatorics , parity

We have $2021$ colors and $2021$ chips of each color. We place the $2021^2$ chips in a row. We say that a chip $F$ is [i]bad[/i] if there is an odd number of chips that have a different color to $F$ both to the left and to the right of $F$. (a) Determine the minimum possible number of bad chips. (b) If we impose the additional condition that each chip must have at least one adjacent chip of the same color, determine the minimum possible number of bad chips.

2001 Tournament Of Towns, 4

Tags: combinatorics

On an east-west shipping lane are ten ships sailing individually. The first five from the west are sailing eastwards while the other five ships are sailing westwards. They sail at the same constant speed at all times. Whenever two ships meet, each turns around and sails in the opposite direction. When all ships have returned to port, how many meetings of two ships have taken place?

2020 Princeton University Math Competition, 9

Tags: combinatorics

Consider a regular $2020$-gon circumscribed into a circle of radius $ 1$. Given three vertices of this polygon such that they form an isosceles triangle, let $X$ be the expected area of the isosceles triangle they create. $X$ can be written as $\frac{1}{m \tan((2\pi)/n)}$ where $m$ and $n$ are integers. Compute $m + n$.

2019 EGMO, 3

Tags: trigonometry , tangent , geometry

Let $ABC$ be a triangle such that $\angle CAB > \angle ABC$, and let $I$ be its incentre. Let $D$ be the point on segment $BC$ such that $\angle CAD = \angle ABC$. Let $\omega$ be the circle tangent to $AC$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the circumcircle of $ABC$. Prove that the angle bisectors of $\angle DAB$ and $\angle CXB$ intersect at a point on line $BC$.

1999 National Olympiad First Round, 4

Tags: inequalities , trigonometry

If inequality $ \frac {\sin ^{3} x}{\cos x} \plus{} \frac {\cos ^{3} x}{\sin x} \ge k$ is hold for every $ x\in \left(0,\frac {\pi }{2} \right)$, what is the largest possible value of $ k$? $\textbf{(A)}\ \frac {1}{2} \qquad\textbf{(B)}\ \frac {3}{4} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \frac {3}{2} \qquad\textbf{(E)}\ \text{None}$

1990 Flanders Math Olympiad, 2

Tags: number theory

Let $a$ and $b$ be two primes having at least two digits, such that $a > b$. Show that \[240|\left(a^4-b^4\right)\] and show that 240 is the greatest positive integer having this property.

1992 National High School Mathematics League, 3

Tags: geometry

In coordinate system, there are six points $P_i(x_i,y_i)(i=1,2,\cdots,6)$, satisfying: (1) $x_i,y_i\in\{-2,-1,0,1,2\}$. (2) For any three points, they are not collinear. Prove that there exists a triangle $\triangle P_iP_jP_k(1\leq i<j<k\leq6)$, its area is not larger than $2$.