Olympiad problems

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: $$P\left(m+P(m)\right)= P (m)$$

1997 Finnish National High School Mathematics Competition, 5

Tags: combinatorics unsolved , combinatorics

For an integer $n\geq 3$, place $n$ points on the plane in such a way that all the distances between the points are at most one and exactly $n$ of the pairs of points have the distance one.

2005 Vietnam National Olympiad, 1

Tags: inequalities proposed , inequalities , algebra , Vietnam

Let $x,y$ be real numbers satisfying the condition: \[x-3\sqrt {x+1}=3\sqrt{y+2} -y\] Find the greatest value and the smallest value of: \[P=x+y\]

2022 Bulgarian Spring Math Competition, Problem 11.4

Tags: Combinatorial Number Theory , Subsets , number theory , combinatorics , Probabilistic Method , Bulgaria

Let $n \geq 2$ be a positive integer. The set $M$ consists of $2n^2-3n+2$ positive rational numbers. Prove that there exists a subset $A$ of $M$ with $n$ elements with the following property: $\forall$ $2 \leq k \leq n$ the sum of any $k$ (not necessarily distinct) numbers from $A$ is not in $A$.

2024 IMC, 4

Tags: group theory , abstract algebra , Combinatorics of words , subgroup , Sequences

Let $g$ and $h$ be two distinct elements of a group $G$, and let $n$ be a positive integer. Consider a sequence $w=(w_1,w_2,\dots)$ which is not eventually periodic and where each $w_i$ is either $g$ or $h$. Denote by $H$ the subgroup of $G$ generated by all elements of the form $w_kw_{k+1}\dotsc w_{k+n-1}$ with $k \ge 1$. Prove that $H$ does not depend on the choice of the sequence $w$ (but may depend on $n$).

2011 JBMO Shortlist, 1

Tags: geometry , JBMO

Let $ABC$ be an isosceles triangle with $AB=AC$. On the extension of the side ${CA}$ we consider the point ${D}$ such that ${AD<AC}$. The perpendicular bisector of the segment ${BD}$ meets the internal and the external bisectors of the angle $\angle BAC$ at the points ${E}$and ${Z}$, respectively. Prove that the points ${A, E, D, Z}$ are concyclic.

JOM 2015 Shortlist, C2

Tags: combinatorics

Cauchy the magician has a new card trick. He takes a standard deck(which consists of 52 cards with 13 denominations in each 4 suits) and let Schwartz to shuffle randomly. Schwartz is told to take $ m $ cards not more than $ \frac{1}{3} $ form the top of the deck. Then, Cauchy take $ 18 $ cards one by one from the top of the remaining deck and show it to Schwartz with the second card is placed in front of the first card (from Schwartz view) and so on. He ask Schwartz to memorize the $ m-th $ card when showing the cards. Let it be $ C_1 $. After that, Cauchy places the $ 18 $ cards and the $ m $ cards on the bottom of the deck with the $ m $ cards are placed lower than the $ 18 $ cards. Now, Cauchy distributes and flip the cards on the table from the top of the deck while shouting the numbers $ 10 $ until $ 1 $ with the following operation: a) When a card flipped has the number which is same as the number shouted by Cauchy, stop the distribution and continue with another set.\\ b) When $ 10 $ cards are flipped and none of the cards flipped has the number which is same as the number shouted by Cauchy, take a card from the top of the deck and place it on top of the set with backside(the site which has no value) facing up. Then continue with another set.\\ Cauchy stops when 3 sets of cards are placed. Then, he adds up all the numbers on top of each sets of cards( backside is consider $ 0 $ ). Let $ k $ be the sum. He placed another $ k $ cards to the table from the top of the remaining deck. Finally, he shows the first card on top of the remaining deck to Schwartz. Let it be $ C_2 $. Show that $ C_1 = C_2 $.

2023 MOAA, 4

Tags: MOAA 2023

A number is called \textit{super odd} if it is an odd number divisible by the square of an odd prime. For example, $2023$ is a \textit{super odd} number because it is odd and divisible by $17^2$. Find the sum of all \textit{super odd} numbers from $1$ to $100$ inclusive. [i]Proposed by Andy Xu[/i]