Olympiad problems

2004 239 Open Mathematical Olympiad, 4

Tags: inequalities

Let the sum of positive reals $a,b,c$ be equal to 1. Prove an inequality \[ \sqrt{{ab}\over {c+ab}}+\sqrt{{bc}\over {a+bc}}+\sqrt{{ac}\over {b+ac}}\le 3/2 \]. [b]proposed by Fedor Petrov[/b]

2019 Saudi Arabia JBMO TST, 2

Tags: tiling , rectangle , tiles , combinatorics

We call a tiling of an $m\times$ n rectangle with arabos (see figure below) [i]regular[/i] if there is no sub-rectangle which is tiled with arabos. Prove that if for some $m$ and $n$ there exists a [i]regular[/i] tiling of the $m\times n$ rectangle then there exists a [i]regular[/i] tiling also for the $2m \times 2n$ rectangle. [img]https://cdn.artofproblemsolving.com/attachments/1/1/2ab41cc5107a21760392253ed52d9e4ecb22d1.png[/img]

2013 Balkan MO Shortlist, N5

Tags: number theory , diophantine equation , diophantine , prime

Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$

2014 India IMO Training Camp, 2

Tags: inequalities

Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.

2010 Hanoi Open Mathematics Competitions, 1

Tags: compare , algebra

Compare the numbers: $P = 888...888 \times 333 .. 333$ ($2010$ digits of $8$ and $2010$ digits of $3$) and $Q = 444...444\times 666...6667$ ($2010$ digits of $4$ and $2009$ digits of $6$) (A): $P = Q$, (B): $P > Q$, (C): $P < Q$.

2019 Regional Olympiad of Mexico West, 2

Tags: parallel , square , geometry

Given a square $ABCD$, points $E$ and $F$ are taken inside the segments $BC$ and $CD$ so that $\angle EAF = 45^o$. The lines $AE$ and $AF$ intersect the circle circumscribed to the square at points $G$ and $H$ respectively. Prove that lines $EF$ and $GH$ are parallel.

2018 ASDAN Math Tournament, 7

Tags:

Nathan starts with the number $0$, and randomly adds either $1$ or $2$ with equal probability until his number reaches or exceeds $2018$. What is the probability his number ends up being exactly $2018$?

2011 Kyrgyzstan National Olympiad, 7

Tags: induction , function , continued fraction , algebra

Given that $g(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1}}}}}}}}$ and $k(n) = \frac{1}{{2 + \frac{1}{{3 + \frac{1}{{... + \frac{1}{{n - 1 + \frac{1}{n}}}}}}}}}$, for natural $n$. Prove that $\left| {g(n) - k(n)} \right| \le \frac{1}{{(n - 1)!n!}}$.

2024 All-Russian Olympiad Regional Round, 11.6

Tags: weight , algebra , easy

Teacher has 100 weights with masses $1$ g, $2$ g, $\dots$, $100$ g. He wants to give 30 weights to Petya and 30 weights to Vasya so that no 11 Petya's weights have the same total mass as some 12 Vasya's weights, and no 11 Vasya's weights have the same total mass as some 12 Petya's weights. Can the teacher do that?

2003 Tournament Of Towns, 2

Tags: combinatorics

Smallville is populated by unmarried men and women, some of them are acquainted. Two city’s matchmakers are aware of all acquaintances. Once, one of matchmakers claimed: “I could arrange that every brunette man would marry a woman he was acquainted with”. The other matchmaker claimed “I could arrange that every blonde woman would marry a man she was acquainted with”. An amateur mathematician overheard their conversation and said “Then both arrangements could be done at the same time! ” Is he right?

2024 ELMO Shortlist, N2

Tags: number theory

Call a positive integer [i]emphatic[/i] if it can be written in the form $a^2+b!$, where $a$ and $b$ are positive integers. Prove that there are infinitely many positive integers $n$ such that $n$, $n+1$, and $n+2$ are all [i]emphatic[/i]. [i]Allen Wang[/i]

2021-IMOC, G6

Tags: geometry , circumcircle , concurrency

Let $\Omega$ be the circumcircle of triangle $ABC$. Suppose that $X$ is a point on the segment $AB$ with $XB=XC$, and the angle bisector of $\angle BAC$ intersects $BC$ and $\Omega$ at $D$, $M$, respectively. If $P$ is a point on $BC$ such that $AP$ is tangent to $\Omega$ and $Q$ is a point on $DX$ such that $CQ$ is tangent to $\Omega$, show that $AB$, $CM$, $PQ$ are concurrent.

2014 Contests, 2

Tags: probability , ratio , algebra , linear equation

An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$.

2006 Sharygin Geometry Olympiad, 10

Tags: diagonal , regular polygon , cut , isosceles , combinatorial geometry , geometry

At what $n$ can a regular $n$-gon be cut by disjoint diagonals into $n- 2$ isosceles (including equilateral) triangles?

2017 AIME Problems, 5

Tags:

A rational number written in base eight is $\underline{a} \underline{b} . \underline{c} \underline{d}$, where all digits are nonzero. The same number in base twelve is $\underline{b} \underline{b} . \underline{b} \underline{a}$. Find the base-ten number $\underline{a} \underline{b} \underline{c}$.

2024 IFYM, Sozopol, 7

Tags: number theory

A set $ S $ of two or more positive integers is called [i]almost closed under addition[/i] if the sum of any two distinct elements of $ S $ also belongs to $ S $. Let $ P(x) $ be a polynomial with integer coefficients for which there exists an almost closed under addition set $ S $, such that for any two distinct $ a $ and $ b $ from $ S $, the numbers $ P(a) $ and $ P(b) $ are coprime. Prove that $ P $ is a constant.

2016 USA TSTST, 5

Tags: combinatorics

In the coordinate plane are finitely many [i]walls[/i]; which are disjoint line segments, none of which are parallel to either axis. A bulldozer starts at an arbitrary point and moves in the $+x$ direction. Every time it hits a wall, it turns at a right angle to its path, away from the wall, and continues moving. (Thus the bulldozer always moves parallel to the axes.) Prove that it is impossible for the bulldozer to hit both sides of every wall. [i]Proposed by Linus Hamilton and David Stoner[/i]

Ukrainian TYM Qualifying - geometry, 2020.12

Tags: square , geometry , concyclic , equal circles

On the side $CD$ of the square $ABCD$, the point $F$ is chosen and the equal squares $DGFE$ and $AKEH$ are constructed ($E$ and $H$ lie inside the square). Let $M$ be the midpoint of $DF$, $J$ is the incenter of the triangle $CFH$. Prove that: a) the points $D, K, H, J, F$ lie on the same circle; b) the circles inscribed in triangles $CFH$ and $GMF$ have the same radii.

2013 Princeton University Math Competition, 4

Tags: factorial

Find the sum of all positive integers $m$ such that $2^m$ can be expressed as a sum of four factorials (of positive integers). Note: The factorials do not have to be distinct. For example, $2^4=16$ counts, because it equals $3!+3!+2!+2!$.

2019 Korea Winter Program Practice Test, 1

Tags: algebra , functional equation , function , geometry , incenter

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that if $a,b,c$ are the length sides of a triangle, and $r$ is the radius of its incircle, then $f(a),f(b),f(c)$ also form a triangle where its radius of the incircle is $f(r)$.

2022-IMOC, G5

Tags: geometry , midpoint , concyclic

$P$ is a point inside $ABC$. $BP$, $CP$ intersect $AC, AB$ at $E, F$, respectively. $AP$ intersect $\odot (ABC)$ again at X. $\odot (ABC)$ and $\odot (AEF)$ intersect again at $S$. $T$ is a point on $BC$ such that $P T \parallel EF$. Prove that $\odot (ST X)$ passes through the midpoint of $BC$. [i]proposed by chengbilly[/i]

2019 Oral Moscow Geometry Olympiad, 2

Tags: congruent , quadrilateral , congruence , diagonal , geometry

The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily similar?

2014 Math Prize For Girls Problems, 14

Tags: geometry , perimeter , inequalities , triangle inequality

A triangle has area 114 and sides of integer length. What is the perimeter of the triangle?

2015 British Mathematical Olympiad Round 1, 6

Tags: number theory

A positive integer is called [i]charming[/i] if it is equal to $2$ or is of the form $3^{i}5^{j}$ where $i$ and $j$ are non-negative integers. Prove that every positive integer can be written as a sum of different charming numbers.

2019 Korea National Olympiad, 2

Tags: geometry , circumcircle

Triangle $ABC$ is an scalene triangle. Let $I$ the incenter, $\Omega$ the circumcircle, $E$ the $A$-excenter of triangle $ABC$. Let $\Gamma$ the circle centered at $E$ and passes $A$. $\Gamma$ and $\Omega$ intersect at point $D(\neq A)$, and the perpendicular line of $BC$ which passes $A$ meets $\Gamma$ at point $K(\neq A)$. $L$ is the perpendicular foot from $I$ to $AC$. Now if $AE$ and $DK$ intersects at $F$, prove that $BE\cdot CI=2\cdot CF\cdot CL$.