Olympiad problems

2004 Miklós Schweitzer, 1

The Lindelöf number $L(X)$ of a topological space $X$ is the least infinite cardinal $\lambda$ with the property that every open covering of $X$ has a subcovering of cardinality at most $\lambda$. Prove that if evert non-countably infinite subset of a first countable space $X$ has a point of condensation, then $L(X)=\sup L(A)$, where $A$ runs over the separable closed subspaces of $X$. (A point of condensation of a subset $H\subseteq X$ is a point $x\in X$ such that any neighbourhood of $x$ intersects $H$ in a non-countably infinite set.)

2005 Serbia Team Selection Test, 5

Tags: inequalities

Let $a,b,c$ be positive reals such that $abc=1$ .Prove the inequality $\frac{a}{a^2+2}+\frac{b}{b^2+2}+\frac{c}{c^2+2}\leq 1$

2007 Iran MO (3rd Round), 1

Tags: rotation , geometry , geometric transformation , reflection , rectangle , geometry proposed

Consider two polygons $ P$ and $ Q$. We want to cut $ P$ into some smaller polygons and put them together in such a way to obtain $ Q$. We can translate the pieces but we can not rotate them or reflect them. We call $ P,Q$ equivalent if and only if we can obtain $ Q$ from $ P$(which is obviously an equivalence relation). [img]http://i3.tinypic.com/4lrb43k.png[/img] a) Let $ P,Q$ be two rectangles with the same area(their sides are not necessarily parallel). Prove that $ P$ and $ Q$ are equivalent. b) Prove that if two triangles are not translation of each other, they are not equivalent. c) Find a necessary and sufficient condition for polygons $ P,Q$ to be equivalent.

2011 AMC 10, 21

Tags: probability , conditional probability , AMC

Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine? $ \textbf{(A)}\ \frac{7}{11}\qquad\textbf{(B)}\ \frac{9}{13}\qquad\textbf{(C)}\ \frac{11}{15}\qquad\textbf{(D)}\ \frac{15}{19}\qquad\textbf{(E)}\ \frac{15}{16} $

2016 239 Open Mathematical Olympiad, 4

Tags: number theory , Sequence

The sequences of natural numbers $p_n$ and $q_n$ are given such that $$p_1 = 1,\ q_1 = 1,\ p_{n + 1} = 2q_n^2-p_n^2,\ q_{n + 1} = 2q_n^2+p_n^2 $$ Prove that $p_n$ and $q_m$ are coprime for any m and n.

2024 Miklos Schweitzer, 9

Tags: number theory , finite fields

Let $q > 1$ be a power of $2$. Let $f: \mathbb{F}_{q^2} \to \mathbb{F}_{q^2}$ be an affine map over $\mathbb{F}_2$. Prove that the equation \[ f(x) = x^{q+1} \] has at most $2q - 1$ solutions.

1998 Tournament Of Towns, 2

Tags: number theory proposed , number theory

The units-digit of the square of an integer is 9 and the tens-digit of this square is 0. Prove that the hundreds-digit is even.

2022 Bangladesh Mathematical Olympiad, 4

Tags: combinatorics

Pratyya and Payel have a number each, $n$ and $m$ respectively, where $n>m.$ Everyday, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, and Payel multiplies his number by $2$ and then add $2$ to it. In other words, on the first day their numbers will be $(2n-2)$ and $(2m+2)$ respectively. Find minimum integer $x$ with proof such that if $n-m\geq x,$ then Pratyya's number will be larger than Payel's number everyday.

Ukrainian TYM Qualifying - geometry, III.13

Tags: geometry , tangent circles , regular polygon , Ukrainian TYM

Inside the regular $n$ -gon $M$ with side $a$ there are $n$ equal circles so that each touches two adjacent sides of the polygon $M$ and two other circles. Inside the formed "star", which is bounded by arcs, these $n$ equal circles are reconstructed so that each touches the two adjacent circles built in the previous step, and two more newly built circles. This process will take $k$ steps. Find the area $S_n (k)$ of the "star", which is formed in the center of the polygon $M$. Consider the spatial analogue of this problem.

2008 China Team Selection Test, 1

Tags: geometry , circumcircle , perpendicular bisector , power of a point , radical axis , geometry proposed

Let $ P$ be the the isogonal conjugate of $ Q$ with respect to triangle $ ABC$, and $ P,Q$ are in the interior of triangle $ ABC$. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ PBC,PCA,PAB$, $ O'_{1},O'_{2},O'_{3}$ the circumcenters of triangle $ QBC,QCA,QAB$, $ O$ the circumcenter of triangle $ O_{1}O_{2}O_{3}$, $ O'$ the circumcenter of triangle $ O'_{1}O'_{2}O'_{3}$. Prove that $ OO'$ is parallel to $ PQ$.

2012 China Team Selection Test, 2

Tags: geometry , geometry unsolved

Given a scalene triangle $ABC$. Its incircle touches $BC,AC,AB$ at $D,E,F$ respectvely. Let $L,M,N$ be the symmetric points of $D$ with $EF$,of $E$ with $FD$,of $F$ with $DE$,respectively. Line $AL$ intersects $BC$ at $P$,line $BM$ intersects $CA$ at $Q$,line $CN$ intersects $AB$ at $R$. Prove that $P,Q,R$ are collinear.

2017 Tournament Of Towns, 6

Tags: combinatorics , Russia

A grasshopper can jump along a checkered strip for $8, 9$ or $10$ cells in any direction. A natural number $n$ is called jumpable if the grasshopper can start from some cell of a strip of length $n$ and visit every cell exactly once. Find at least one non-jumpable number $n > 50$. [i](Egor Bakaev)[/i]

1956 Moscow Mathematical Olympiad, 327

Tags: combinatorics , maximum , table , Average

On an infinite sheet of graph paper a table is drawn so that in each square of the table stands a number equal to the arithmetic mean of the four adjacent numbers. Out of the table a piece is cut along the lines of the graph paper. Prove that the largest number on the piece always occurs at an edge, where $x = \frac14 (a + b + c + d)$.

2018 CMIMC Combinatorics, 10

Tags: 2018 , combinatorics

Call a set $S \subseteq \{0,1,\dots,14\}$ $\textit{sparse}$ if $x+1 \pmod{15}$ is not in $S$ whenever $x \in S$. Find the number of sparse sets $T$ such that the sum of the elements of $T$ is a multiple of 15.

2014 Iran Team Selection Test, 5

Tags: inequalities , inequalities proposed

if $x,y,z>0$ are postive real numbers such that $x^{2}+y^{2}+z^{2}=x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}$ prove that \[((x-y)(y-z)(z-x))^{2}\leq 2((x^{2}-y^{2})^{2}+(y^{2}-z^{2})^{2}+(z^{2}-x^{2})^{2})\]

JOM 2015 Shortlist, G4

Tags: inequalities

Let $ ABC $ be a triangle and let $ AD, BE, CF $ be cevians of the triangle which are concurrent at $ G $. Prove that if $ CF \cdot BE \ge AF \cdot EC + AE \cdot BF + BC \cdot FE $ then $ AG \le GD $.

2003 All-Russian Olympiad Regional Round, 11.5

Tags: algebra , trinomial

Square trinomials $P(x) = x^2 + ax + b$ and $Q(x) = x^2 + cx + d$ are such that the equation $P(Q(x)) = Q(P(x))$ has no real roots. Prove that $b \ne d$.

2023 CMIMC Geometry, 8

Tags: geometry

Let $\omega$ be a unit circle with center $O$ and diameter $AB$. A point $C$ is chosen on $\omega$. Let $M$, $N$ be the midpoints of arc $AC$, $BC$, respectively, and let $AN,BM$ intersect at $I$. Suppose that $AM,BC,OI$ concur at a point. Find the area of $\triangle ABC$. [i]Proposed by Kevin You[/i]

2018 India IMO Training Camp, 1

Tags: combinatorics , IMO Shortlist , Strings , sorting , distance , Extremal combinatorics , radius

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

1984 Bulgaria National Olympiad, Problem 2

Tags: geometry , trapezoid

The diagonals of a trapezoid $ABCD$ with bases $AB$ and $CD$ intersect in a point $O$, and $AB/CD=k>1$. The bisectors of the angles $AOB,BOC,COD,DOA$ intersect $AB,BC,CD,DA$ respectively at $K,L,M,N$. The lines $KL$ and $MN$ meet at $P$, and the lines $KN$ and $LM$ meet at $Q$. If the areas of $ABCD$ and $OPQ$ are equal, find the value of $k$.

2020 India National Olympiad, 2

Tags: polynomial , trigonometry , INMO 2020

Suppose $P(x)$ is a polynomial with real coefficients, satisfying the condition $P(\cos \theta+\sin \theta)=P(\cos \theta-\sin \theta)$, for every real $\theta$. Prove that $P(x)$ can be expressed in the form$$P(x)=a_0+a_1(1-x^2)^2+a_2(1-x^2)^4+\dots+a_n(1-x^2)^{2n}$$for some real numbers $a_0, a_1, \dots, a_n$ and non-negative integer $n$. [i]Proposed by C.R. Pranesacher[/i]

2005 Baltic Way, 18

Tags: modular arithmetic , number theory proposed , number theory

Let $x$ and $y$ be positive integers and assume that $z=\frac{4xy}{x+y}$ is an odd integer. Prove that at least one divisor of $z$ can be expressed in the form $4n-1$ where $n$ is a positive integer.

2010 Albania Team Selection Test, 1

Tags: ratio , geometry , circumcircle , parallelogram , incenter , rhombus , geometry unsolved

$ABC$ is an acute angle triangle such that $AB>AC$ and $\hat{BAC}=60^{\circ}$. Let's denote by $O$ the center of the circumscribed circle of the triangle and $H$ the intersection of altitudes of this triangle. Line $OH$ intersects $AB$ in point $P$ and $AC$ in point $Q$. Find the value of the ration $\frac{PO}{HQ}$.

2014 ELMO Shortlist, 13

Tags: geometry , circumcircle , ratio , geometric transformation , reflection , geometry proposed

Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$. [i]Proposed by David Stoner[/i]

2020 Regional Olympiad of Mexico West, 4

Tags: number theory , product of digits

Given a positive integer $ n $, we denote by $ P(n) $ the result of multiplying all the digits of $ n $. Find a number $ m $ with ten digits, none of them zero, with the following property: $$P\left(m+P(m)\right)= P (m)$$