Olympiad problems

2012 Czech-Polish-Slovak Match, 3

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$. Let $I, J$ and $K$ be the incentres of the triangles $ABC, ACD$ and $ABD$ respectively. Let $E$ be the midpoint of the arc $DB$ of circle $\omega$ containing the point $A$. The line $EK$ intersects again the circle $\omega$ at point $F$ $(F \neq E)$. Prove that the points $C, F, I, J$ lie on a circle.

2014 Contests, 4

Tags: inequalities , calculus , three variable inequality

Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that: \[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]

2008 ISI B.Math Entrance Exam, 1

Tags: calculus , integration , function , calculus computations

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function . Suppose \[f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy\] $\forall x\in \mathbb{R}$ and all $t>0$ . Then show that there exists a constant $c$ such that $f(x)=cx\ \forall x$

1996 German National Olympiad, 2

Tags: inequalities , algebra

Let $a$ and $b$ be positive real numbers smaller than $1$. Prove that the following two statements are equivalent: (i) $a+b = 1$, (ii) Whenever $x,y$ are positive real numbers such that $x < 1, y < 1, ax+by < 1$, the following inequlity holds: $$\frac{1}{1-ax-by} \le \frac{a}{1-x} + \frac{b}{1-y}$$

1999 Mexico National Olympiad, 6

Tags: calculus , integration , combinatorics

A polygon has each side integral and each pair of adjacent sides perpendicular (it is not necessarily convex). Show that if it can be covered by non-overlapping $2 x 1$ dominos, then at least one of its sides has even length.

2012 Philippine MO, 3

Tags: trigonometry , inequalities

If $ab>0$ and $\displaystyle 0<x<\frac{\pi}{2}$, prove that \[ \left ( 1+\frac{a^2}{\sin x} \right ) \left ( 1+\frac{b^2}{\cos x} \right ) \geq \frac{(1+\sqrt{2}ab)^2 \sin 2x}{2}. \]

2014 Balkan MO Shortlist, G4

Tags: analytic geometry , geometry

Let $A_0B_0C_0$ be a triangle with area equal to $\sqrt 2$. We consider the excenters $A_1$,$B_1$ and $C_1$ then we consider the excenters ,say $A_2,B_2$ and $C_2$,of the triangle $A_1B_1C_1$. By continuing this procedure ,examine if it is possible to arrive to a triangle $A_nB_nC_n$ with all coordinates rational.

2018 BMT Spring, 2

Tags:

For how many values of $x$ does $20^x \cdot 18^x = 2018^x$?

Kyiv City MO Juniors 2003+ geometry, 2009.89.5

Tags: chord , geometry , orthocenter , circles

A chord $AB$ is drawn in the circle, on which the point $P$ is selected in such a way that $AP = 2PB$. The chord $DE$ is perpendicular to the chord $AB $ and passes through the point $P$. Prove that the midpoint of the segment $AP$ is the orthocener of the triangle $AED$.

2011 Turkey Junior National Olympiad, 1

Tags: algebra , inequalities

Show that \[1 \leq \frac{(x+y)(x^3+y^3)}{(x^2+y^2)^2} \leq \frac98\] holds for all positive real numbers $x,y$.

Kvant 2024, M2813

Tags: geometry

The quadrilateral $ABCD$ is described around a circle centered on $I$. Let the diagonals $AC$ and $BD$ intersect at point $E$. The perpendicular bisectors to the segments $AC$ and $BD$ intersect at the point $P$ lying inside the triangle $BEC$. The circumscribed circles of the triangles $APC$ and $BPD$ intersect at points $P$ and $Q$. Prove that $I$ lies on the line $PQ$. [i] Proposed by Tran Quang Hung [/i]

2014 Federal Competition For Advanced Students, P2, 1

Tags: number of divisors , divisor , number theory

For each positive natural number $n$ let $d (n)$ be the number of its divisors including $1$ and $n$. For which positive natural numbers $n$, for every divisor $t$ of $n$, that $d (t)$ is a divisor of $d (n)$?

2023 Yasinsky Geometry Olympiad, 3

Tags: geometry , incircle

Points $K$ and $N$ are the midpoints of sides $AC$ and $AB$ of triangle $ABC$. The inscribed circle $\omega$ of the triangle $AKN$ is tangent to $BC$. Find $BC$ if $AC + AB = n$. (Oleksii Karliuchenko)

2007 Singapore Junior Math Olympiad, 5

Tags: algebra , function

For any positive integer $n$, let $f(n)$ denote the $n$- th positive nonsquare integer, i.e., $f(1) = 2, f(2) = 3, f(3) = 5, f(4) = 6$, etc. Prove that $f(n)=n +\{\sqrt{n}\}$ where $\{x\}$ denotes the integer closest to $x$. (For example, $\{\sqrt{1}\} = 1, \{\sqrt{2}\} = 1, \{\sqrt{3}\} = 2, \{\sqrt{4}\} = 2$.)

2008 Postal Coaching, 1

Tags: incenter , circumcircle , right angle , arc midpoint , angle chasing , angle , geometry

In triangle $ABC,\angle B > \angle C, T$ is the midpoint of arc $BAC$ of the circumcicle of $ABC$, and $I$ is the incentre of $ABC$. Let $E$ be point such that $\angle AEI = 90^0$ and $AE$ is parallel to $BC$. If $TE$ intersects the circumcircle of $ABC$ at $P(\ne T)$ and $\angle B = \angle IPB$, determine $\angle A$.

2018 Costa Rica - Final Round, 4

Tags: functional , functional inequality , algebra

Determine if there exists a function f: $N^*\to N^*$ that satisfies that for all $n \in N^*$, $$10^{f (n)} <10n + 1 <10^{f (n) +1}.$$ Justify your answer. Note: $N^*$ denotes the set of positive integers.

1980 IMO Shortlist, 11

Tags: combinatorics , system of equations , counting

Ten gamblers started playing with the same amount of money. Each turn they cast (threw) five dice. At each stage the gambler who had thrown paid to each of his 9 opponents $\frac{1}{n}$ times the amount which that opponent owned at that moment. They threw and paid one after the other. At the 10th round (i.e. when each gambler has cast the five dice once), the dice showed a total of 12, and after payment it turned out that every player had exactly the same sum as he had at the beginning. Is it possible to determine the total shown by the dice at the nine former rounds ?

2015 Kazakhstan National Olympiad, 6

Tags: geometry

The quadrilateral $ABCD$ has an incircle of diameter $d$ which touches $BC$ at $K$ and touches $DA$ at $L$. Is it always true that the harmonic mean of $AB$ and $CD$ is equal to $KL$ if and only if the geometric mean of $AB$ and $CD$ is equal to $d$?

1965 All Russian Mathematical Olympiad, 058

Tags: geometry , incenter , circumcenter

A circle is circumscribed around the triangle $ABC$. Chords, from the midpoint of the arc $AC$ to the midpoints of the arcs $AB$ and $BC$, intersect sides $[AB]$ and $[BC]$ in the points $D$ and $E$. Prove that $(DE)$ is parallel to $(AC)$ and passes through the centre of the inscribed circle.

V Soros Olympiad 1998 - 99 (Russia), 10.2

Tags: algebra , trigonometry

Solve the equation $$ |\cos 3x - tgt| + |\cos 3x + tgt| = |tg^2t -3|.$$

2017 Switzerland - Final Round, 8

Tags: midpoint , isosceles , equal segments , geometry

Let $ABC$ be an isosceles triangle with vertex $A$ and $AB> BC$. Let $k$ be the circle with center $A$ passsing through $B$ and $C$. Let $H$ be the second intersection of $k$ with the altitude of the triangle $ABC$ through $B$. Further let $G$ be the second intersection of $k$ with the median through $B$ in triangle $ABC$. Let $X$ be the intersection of the lines $AC$ and $GH$. Show that $C$ is the midpoint of $AX$.

2022 Bulgarian Spring Math Competition, Problem 8.3

Tags: algebra , inequality , inequalities

Given the inequalities: $a)$ $\left(\frac{2a}{b+c}\right)^2+\left(\frac{2b}{a+c}\right)^2+\left(\frac{2c}{a+b}\right)^2\geq \frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ $b)$ $\left(\frac{a+b}{c}\right)^2+\left(\frac{b+c}{a}\right)^2+\left(\frac{c+a}{b}\right)^2\geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+9$ For each of them either prove that it holds for all positive real numbers $a$, $b$, $c$ or present a counterexample $(a,b,c)$ which doesn't satisfy the inequality.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.7

Tags: inequalities , algebra

Without using a calculator, prove that $$2^{1995} >5^{854},$$

2001 Singapore MO Open, 3

Tags: combinatorics , difference

Suppose that there are $2001$ golf balls which are numbered from $1$ to $2001$ respectively, and some of these golf balls are placed inside a box. It is known that the difference between the two numbers of any two golf balls inside the box is neither $5$ nor $8$. How many such golf balls the box can contain at most? Justify your answer.

1961 Czech and Slovak Olympiad III A, 1

Tags: sequence , consecutive , positive integer

Consider an infinite sequence $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, \ldots, \underbrace{n,\ldots,n}_{n\text{ times}},\ldots.$$ Find the 1000th term of the sequence.