Olympiad problems

1992 Polish MO Finals, 1

Segments $AC$ and $BD$ meet at $P$, and $|PA| = |PD|$, $|PB| = |PC|$. $O$ is the circumcenter of the triangle $PAB$. Show that $OP$ and $CD$ are perpendicular.

1998 Swedish Mathematical Competition, 6

Tags: inequalities , algebra

Show that for some $c > 0$, we have $\left|\sqrt[3]{2} - \frac{m}{n}\right | > \frac{c}{n^3}$ for all integers $m, n$ with $n \ge 1$.

2022 Cyprus JBMO TST, 3

Tags: algebra , inequalities

If $x,y$ are real numbers with $x+y\geqslant 0$, determine the minimum value of the expression \[K=x^5+y^5-x^4y-xy^4+x^2+4x+7\] For which values of $x,y$ does $K$ take its minimum value?

2023 SG Originals, Q4

Tags: algebra , functional equation

Find all functions $f: \mathbb{Z} \to \mathbb{Z}$, such that $$f(x+y)((f(x) - f(y))^2+f(xy))=f(x^3)+f(y^3)$$ for all integers $x, y$.

1989 IMO Longlists, 61

Tags: inequalities , calculus , derivative , function , inequalities unsolved

Prove for $ 0 < k \leq 1$ and $ a_i \in \mathbb{R}^\plus{},$ $ i \equal{} 1,2 \ldots, n$ the following inequality holds: \[ \left( \frac{a_1}{a_2 \plus{} \ldots \plus{} a_n} \right)^k \plus{} \ldots \plus{} \left( \frac{a_n}{a_1 \plus{} \ldots \plus{} a_{n\minus{}1}} \right)^k \geq \frac{n}{(n\minus{}1)^k}.\]

2013 Germany Team Selection Test, 3

Tags: geometry , circumcircle , geometric transformation , reflection

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. Prove that there exists a point $J$ such that for any point $X$ inside $ABC$ if $AX,BX,CX$ intersect $\omega$ in $A_1,B_1,C_1$ and $A_2,B_2,C_2$ be reflections of $A_1,B_1,C_1$ in midpoints of $BC,AC,AB$ respectively then $A_2,B_2,C_2,J$ lie on a circle.

2018 Sharygin Geometry Olympiad, 4

Tags: geometry , circumcircle

Let $BC$ be a fixed chord of a circle $\omega$. Let $A$ be a variable point on the major arc $BC$ of $\omega$. Let $H$ be the orthocenter of $ABC$. The points $D, E$ lie on $AB, AC$ such that $H$ is the midpoint of $DE$. $O_A$ is the circumcenter of $ADE$. Prove that as $A$ varies, $O_A$ lies on a fixed circle.

2018 AMC 10, 25

Tags: AMC , AMC 12 , AMC 10 , 2018 AMC , 2018 AMC 10B , 2018 AMC 12B

Let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$. How many real numbers $x$ satisfy the equation $x^2 + 10{,}000\lfloor x \rfloor = 10{,}000x$? $\textbf{(A) } 197 \qquad \textbf{(B) } 198 \qquad \textbf{(C) } 199 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 201$

2010 Turkey Team Selection Test, 3

Tags: combinatorics proposed , combinatorics

A teacher wants to divide the $2010$ questions she asked in the exams during the school year into three folders of $670$ questions and give each folder to a student who solved all $670$ questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations in which there are at most two students who did not solve any given question.

2022 Korea National Olympiad, 7

Tags: inequalities , algebra , Sequence

Suppose that the sequence $\{a_n\}$ of positive reals satisfies the following conditions: [list] [*]$a_i \leq a_j$ for every positive integers $i <j$. [*]For any positive integer $k \geq 3$, the following inequality holds: $$(a_1+a_2)(a_2+a_3)\cdots(a_{k-1}+a_k)(a_k+a_1)\leq (2^k+2022)a_1a_2\cdots a_k$$ [/list] Prove that $\{a_n\}$ is constant.

1967 IMO Shortlist, 4

Tags: number theory , approximation , irrational number

Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.

2021 Philippine MO, 1

Tags: geometry , PMO

In convex quadrilateral $ABCD$, $\angle CAB = \angle BCD$. $P$ lies on line $BC$ such that $AP = PC$, $Q$ lies on line $AP$ such that $AC$ and $DQ$ are parallel, $R$ is the point of intersection of lines $AB$ and $CD$, and $S$ is the point of intersection of lines $AC$ and $QR$. Line $AD$ meets the circumcircle of $AQS$ again at $T$. Prove that $AB$ and $QT$ are parallel.

2015 India IMO Training Camp, 3

Tags: graph theory , combinatorics

Let $G$ be a simple graph on the infinite vertex set $V=\{v_1, v_2, v_3,\ldots\}$. Suppose every subgraph of $G$ on a finite vertex subset is $10$-colorable, Prove that $G$ itself is $10$-colorable.

2022 Middle European Mathematical Olympiad, 4

Tags: number theory

Initially, two distinct positive integers $a$ and $b$ are written on a blackboard. At each step, Andrea picks two distinct numbers $x$ and $y$ on the blackboard and writes the number $gcd(x, y) + lcm(x, y)$ on the blackboard as well. Let $n$ be a positive integer. Prove that, regardless of the values of $a$ and $b$, Andrea can perform a finite number of steps such that a multiple of $n$ appears on the blackboard.

Russian TST 2018, P2

Tags: algebra , inequalities , IMO Shortlist , Hi

An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have $$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$ Find the largest constant $K = K(n)$ such that $$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$ holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.

2020 CMIMC Geometry, 10

Tags: geometry , 2020

Four copies of an acute scalene triangle $\mathcal T$, one of whose sides has length $3$, are joined to form a tetrahedron with volume $4$ and surface area $24$. Compute the largest possible value for the circumradius of $\mathcal T$.

2022 AMC 12/AHSME, 17

Tags: AMC , AMC 12 , 2022 AMC 12B , 2022 AMC

How many $4 \times 4$ arrays whose entries are $0$s and $1$s are there such that the row sums (the sum of the entries in each row) are $1,2,3,$ and $4,$ in some order, and the column sums (the sum of the entries in each column) are also $1,2,3,$ and $4,$ in some order? For example, the array $\begin{bmatrix} 1 & 1 & 1 & 0\\ 0 & 1 & 1 & 0\\ 1 & 1 & 1 & 1\\ 0 & 1 & 0 & 0 \end{bmatrix}$ satisfies the condition. $\textbf{(A)}144~\textbf{(B)}240~\textbf{(C)}336~\textbf{(D)}576~\textbf{(E)}624$

2022 Durer Math Competition Finals, 4

Tags: number theory , Perfect Squares , Divisors

Show that the divisors of a number $n \ge 2$ can only be divided into two groups in which the product of the numbers is the same if the product of the divisors of $n$ is a square number.

2007 Stanford Mathematics Tournament, 8

Tags: LaTeX

If $r+s+t=3$, $r^2+s^2+t^2=1$, and $r^3+s^3+t^3=3$, compute $rst$.

2009 Iran MO (2nd Round), 3

Tags: topology , geometry , circumcircle , calculus , integration , invariant , combinatorics proposed

$11$ people are sitting around a circle table, orderly (means that the distance between two adjacent persons is equal to others) and $11$ cards with numbers $1$ to $11$ are given to them. Some may have no card and some may have more than $1$ card. In each round, one [and only one] can give one of his cards with number $ i $ to his adjacent person if after and before the round, the locations of the cards with numbers $ i-1,i,i+1 $ don’t make an acute-angled triangle. (Card with number $0$ means the card with number $11$ and card with number $12$ means the card with number $1$!) Suppose that the cards are given to the persons regularly clockwise. (Mean that the number of the cards in the clockwise direction is increasing.) Prove that the cards can’t be gathered at one person.

1992 IMO Longlists, 77

Tags: combinatorics , Ramsey Theory , Additive Number Theory , Additive combinatorics , IMO Shortlist , Extremal combinatorics , IMO Longlist

Show that if $994$ integers are chosen from $1, 2,\cdots , 1992$ and one of the chosen integers is less than $64$, then there exist two among the chosen integers such that one of them is a factor of the other.

2017 BMT Spring, 6

Tags: number theory

For how many numbers $n$ does $2017$ divided by $n$ have a remainder of either $1$ or $2$?

Russian TST 2016, P3

Tags: geometry , Inequality

Prove that for any points $A,B,C,D$ in the plane, the following inequality holds \[\frac{AB}{DA+DB}+\frac{BC}{DB+DC}\geqslant\frac{AC}{DA+DC}.\]

2001 ITAMO, 2

Tags: combinatorics proposed , combinatorics

In a basketball tournament every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$, and there are no draws. A single team wins the tournament with $26$ points and exactly two teams share the last position with $20$ points. How many teams participated in the tournament?

2005 Estonia National Olympiad, 3

Tags: algebra , Percentage

Rein solved a test on mathematics that consisted of questions on algebra, geometry and logic. After checking the results, it occurred that Rein had answered correctly $50\%$ of questions on algebra, $70\%$ of questions on geometry and $80\%$ of questions on logic. Thereby, Rein had answered correctly altogether $62\%$ of questions on algebra and logic, and altogether $74\%$ of questions on geometry and logic. What was the percentage of correctly answered questions throughout all the test by Rein?