Olympiad problems

2015 Indonesia MO Shortlist, A5

Let $a,b,c$ be positive real numbers. Prove that $\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$

2023 Middle European Mathematical Olympiad, 1

Tags: algebra

(a) A function $f:\mathbb{Z} \rightarrow \mathbb{Z}$ is called $\mathbb{Z}$-good if $f(a^2+b)=f(b^2+a)$ for all $a, b \in \mathbb{Z}$. What is the largest possible number of distinct values that can occur among $f(1), \ldots, f(2023)$, where $f$ is a $\mathbb{Z}$-good function? (b) A function $f:\mathbb{N} \rightarrow \mathbb{N}$ is called $\mathbb{N}$-good if $f(a^2+b)=f(b^2+a)$ for all $a, b \in \mathbb{N}$. What is the largest possible number of distinct values that can occur among $f(1), \ldots, f(2023)$, where $f$ is a $\mathbb{N}$-good function?

2010 Iran MO (3rd Round), 5

Tags: inequalities

$x,y,z$ are positive real numbers such that $xy+yz+zx=1$. prove that: $3-\sqrt{3}+\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge(x+y+z)^2$ (20 points) the exam time was 6 hours.

2015 BMT Spring, 7

Tags: incircle , angle , geometry

In $ \vartriangle ABC$, $\angle B = 46^o$ and $\angle C = 48^o$ . A circle is inscribed in $ \vartriangle ABC$ and the points of tangency are connected to form $PQR$. What is the measure of the largest angle in $\vartriangle P QR$?

2025 Poland - First Round, 12

Tags: combinatorics

We will say that a subset $A$ of the set of non-negative integers is $cool$, if there exist an integer $k$, such that for every integer $n\geq k$ there exists exactly one pair of integers $a>b$ from $A$ such that $n=a+b$. Decide, if there exists a $cool$ set.

2021 Belarusian National Olympiad, 11.5

Tags: combinatorics

$n_1<n_2<\ldots<n_k$ are all positive integer numbers $n$, that have the following property: In a square $n \times n$ one can mark $50$ cells so that in any square $3 \times 3$ an odd number of cells are marked. Find $n_{k-2}$

2016 Sharygin Geometry Olympiad, 4

Tags: combinatorics , geometry , area , game , game strategy

The Devil and the Man play a game. Initially, the Man pays some cash $s$ to the Devil. Then he lists some $97$ triples $\{i,j,k\}$ consisting of positive integers not exceeding $100$. After that, the Devil draws some convex polygon $A_1A_2...A_{100}$ with area $100$ and pays to the Man, the sum of areas of all triangles $A_iA_jA_k$. Determine the maximal value of $s$ which guarantees that the Man receives at least as much cash as he paid. [i]Proposed by Nikolai Beluhov, Bulgaria[/i]

2021/2022 Tournament of Towns, P1

Tags: combinatorics , algebra

Alice wrote a sequence of $n > 2$ nonzero nonequal numbers such that each is greater than the previous one by the same amount. Bob wrote the inverses of those n numbers in some order. It so happened that each number in his row also is greater than the previous one by the same amount, possibly not the same as in Alice’s sequence. What are the possible values of $n{}$? [i]Alexey Zaslavsky[/i]

2003 China Team Selection Test, 2

Tags: ratio , inequalities , analytic geometry , linear algebra , matrix , geometry

In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.

2020 Harvard-MIT Mathematics Tournament, 8

Tags:

Let $ABC$ be a scalene triangle with angle bisectors $AD$, $BE$, and $CF$ so that $D$, $E$, and $F$ lie on segments $BC$, $CA$, and $AB$ respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$ respectively. Prove that line $AN$ and the line through $M$ parallel to $AD$ intersect on the circumcircle of $ABC$ if and only if $DE=DF$. [i]Proposed by Michael Ren.[/i]

2010 Indonesia TST, 1

Tags: geometry , trigonometry , algebra , polynomial , induction , function , ratio

Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number? [i]Nanang Susyanto, Jogjakarta[/i]

2021 Austrian MO Regional Competition, 4

Tags: number theory , divide , divisible

Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$. (Walther Janous)

2020 DMO Stage 1, 2.

Tags: combinatorics , dmo day 2 , p2

[b]Q[/b] On a $10 \times 10$ chess board whose colors of square are green and blue in an arbitrary way and we are simultaneously allowed to switch all the colors of all squares in any $(2 \times 2)$ and $(5\times 5)$ region. Can we transform any coloring of the board into one where all squares are blue ? Give a proper explanation of your answer. Note. that if a unit square is part of both the $2\times 2$ and $5\times 5$ region,then its color switched is twice(i.e switching is additive) [i]Proposed by Aritra12[/i]

2010 Romania Team Selection Test, 2

Tags: ceiling function , algebra , polynomial , number theory

(a) Given a positive integer $k$, prove that there do not exist two distinct integers in the open interval $(k^2, (k + 1)^2)$ whose product is a perfect square. (b) Given an integer $n > 2$, prove that there exist $n$ distinct integers in the open interval $(k^n, (k + 1)^n)$ whose product is the $n$-th power of an integer, for all but a finite number of positive integers $k$. [i]AMM Magazine[/i]

2003 Purple Comet Problems, 9

Tags: function

Let $f$ be a real-valued function of real and positive argument such that $f(x) + 3xf(\tfrac1x) = 2(x + 1)$ for all real numbers $x > 0$. Find $f(2003)$.

2017 India National Olympiad, 4

Tags: geometry

Let $ABCDE$ be a convex pentagon in which $\angle{A}=\angle{B}=\angle{C}=\angle{D}=120^{\circ}$ and the side lengths are five [i]consecutive integers[/i] in some order. Find all possible values of $AB+BC+CD$.

2004 Cuba MO, 5

Tags: geometry , cyclic quadrilateral

Consider a circle $K$ and an inscribed quadrilateral $ABCD$, such that the diagonal $BD$ is not the diameter of the circle. Prove that the intersection of the lines tangent to $K$ through the points $B$ and $D$ lies on the line $AC$ if and only if $AB \cdot CD = AD \cdot BC$.

1994 North Macedonia National Olympiad, 5

Tags: tiling , combinatorics , combinatorial geometry

A square with the dimension $ 1 \times1 $ has been removed from a square board $ 3 ^n \times 3 ^n $ ($ n \in \mathbb {N}, $ $ n> 1 $). a) Prove that any defective board with the dimension $ 3 ^ n \times3 ^ n $ can be covered with shaped figures of shape 1 (the 3 squares' one) and of shape 2 (the 5 squares' one). Figures covering the board must not overlap each other and must not cross the edge of the board. Also the squares removed from the board must not be covered. (b) How many small figures in shape 2 must be used to cover the board? [img]https://cdn.artofproblemsolving.com/attachments/4/7/e970fadd7acc7fd6f5897f1766a84787f37acc.png[/img]

2004 Vietnam National Olympiad, 3

Tags:

Let $A$ be the set of the $16$ first positive integers. Find the least positive integer $k$ satisfying the condition: In every $k$-subset of $A$, there exist two distinct $a, b \in A$ such that $a^2 + b^2$ is prime.

1970 All Soviet Union Mathematical Olympiad, 130

Tags: algebra , inequalities

The product of three positive numbers equals to one, their sum is strictly greater than the sum of the inverse numbers. Prove that one and only one of them is greater than one.

2012 Today's Calculation Of Integral, 830

Tags: calculus , integration , limit , function , calculus computations , logarithm

Find $\lim_{n\to\infty} \frac{1}{(\ln n)^2}\sum_{k=3}^n \frac{\ln k}{k}.$

1993 French Mathematical Olympiad, Problem 5

Tags: geometry , 3d geometry , tetrahedron , inequalities

(a) Let there be two given points $A,B$ in the plane. i. Find the triangles $MAB$ with the given area and the minimal perimeter. ii. Find the triangles $MAB$ with a given perimeter and the maximal area. (b) In a tetrahedron of volume $V$, let $a,b,c,d$ be the lengths of its four edges, no three of which are coplanar, and let $L=a+b+c+d$. Determine the maximum value of $\frac V{L^3}$.