Olympiad problems

2022 Caucasus Mathematical Olympiad, 4

Tags: geometry

Let $\omega$ is tangent to the sides of an acute angle with vertex $A$ at points $B$ and $C$. Let $D$ be an arbitrary point onn the major arc $BC$ of the circle $\omega$. Points $E$ and $F$ are chosen inside the angle $DAC$ so that quadrilaterals $ABDF$ and $ACED$ are inscribed and the points $A,E,F$ lie on the same straight line. Prove that lines $BE$ and $CF$ intersectat $\omega$.

2021/2022 Tournament of Towns, P2

Tags: board , chess rook , combinatorics

There was a rook at some square of a $10 \times 10{}$ chessboard. At each turn it moved to a square adjacent by side. It visited each square exactly once. Prove that for each main diagonal (the diagonal between the corners of the board) the following statement is true: in the rook’s path there were two consecutive steps at which the rook first stepped away from the diagonal and then returned back to the diagonal. [i]Alexandr Gribalko[/i]

2014 India IMO Training Camp, 3

Tags: function , number theory

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that $f^{k}((n,m))=(m,n)$,where $f^{k}(x)=f(f(f(...f(x))))$,$f$ being composed with itself $k$ times.

1975 All Soviet Union Mathematical Olympiad, 211

Tags: combinatorics , combinatorial geometry , points in space

Given a finite set of polygons in the plane. Every two of them have a common point. Prove that there exists a straight line, that crosses all the polygons.

2020-21 KVS IOQM India, 7

Tags:

$a,b,c$ are positive real numbers such that $a^2+b^2=c^2$ and $ab=c$. Determine the value of $\left\lvert{\frac{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}{c^2}}\right\rvert$

PEN L Problems, 3

Tags: induction , linear recurrence

The Fibonacci sequence $\{F_{n}\}$ is defined by \[F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.\] Show that $F_{mn-1}-F_{n-1}^{m}$ is divisible by $F_{n}^{2}$ for all $m \ge 1$ and $n>1$.

1970 AMC 12/AHSME, 31

Tags: probability

If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to $43$, what is the probability that this number is divisible by $11$? $\textbf{(A) }2/5\qquad\textbf{(B) }1/5\qquad\textbf{(C) }1/6\qquad\textbf{(D) }1/11\qquad \textbf{(E) }1/15$

1999 Austrian-Polish Competition, 2

Tags: inequalities

Find the best possible $k,k'$ such that \[k<\frac{v}{v+w}+\frac{w}{w+x}+\frac{x}{x+y}+\frac{y}{y+z}+\frac{z}{z+v}<k'\] for all positive reals $v,w,x,y,z$.

2012 Iran MO (2nd Round), 3

Tags: geometry , incenter , geometric transformation , reflection , circumcircle , angle bisector , congruent triangles

The incircle of triangle $ABC$, is tangent to sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. The reflection of $F$ with respect to $B$ and the reflection of $E$ with respect to $C$ are $T$ and $S$ respectively. Prove that the incenter of triangle $AST$ is inside or on the incircle of triangle $ABC$. [i]Proposed by Mehdi E'tesami Fard[/i]

2005 Slovenia National Olympiad, Problem 1

Tags: floor function , function

Evaluate the sum $\left\lfloor\log_21\right\rfloor+\left\lfloor\log_22\right\rfloor+\left\lfloor\log_23\right\rfloor+\ldots+\left\lfloor\log_2256\right\rfloor$.

2023 Indonesia TST, 3

Tags: number theory

Prove that $5^n-3^n$ is not divisible by $2^n+65$ for any positive integer $n$.

2020/2021 Tournament of Towns, P1

Tags: geometry

[list=a] [*]A convex pentagon is partitioned into three triangles by nonintersecting diagonals. Is it possible for centroids of these triangles to lie on a common straight line? [*]The same question for a non-convex pentagon. [/list] [i]Alexandr Gribalko[/i]

1979 IMO Longlists, 46

Tags: combinatorics , set systems , intersection

Let $K$ denote the set $\{a, b, c, d, e\}$. $F$ is a collection of $16$ different subsets of $K$, and it is known that any three members of $F$ have at least one element in common. Show that all $16$ members of $F$ have exactly one element in common.

2018 Romanian Master of Mathematics Shortlist, C4

Tags: combinatorics

Let $k$ and $s$ be positive integers such that $s<(2k + 1)^2$. Initially, one cell out of an $n \times n$ grid is coloured green. On each turn, we pick some green cell $c$ and colour green some $s$ out of the $(2k + 1)^2$ cells in the $(2k + 1) \times (2k + 1)$ square centred at $c$. No cell may be coloured green twice. We say that $s$ is $k-sparse$ if there exists some positive number $C$ such that, for every positive integer $n$, the total number of green cells after any number of turns is always going to be at most $Cn$. Find, in terms of $k$, the least $k$-sparse integer $s$. [I]Proposed by Nikolai Beluhov.[/i]

2006 APMO, 4

Tags: trigonometry , circumcircle , geometry

Let $A,B$ be two distinct points on a given circle $O$ and let $P$ be the midpoint of the line segment AB. Let $O_1$ be the circle tangent to the line $AB$ at $P$ and tangent to the circle $O$. Let $l$ be the tangent line, different from the line $AB$, to $O_1$ passing through $A$. Let $C$ be the intersection point, different from $A$, of $l$ and $O$. Let $Q$ be the midpoint of the line segment $BC$ and $O_2$ be the circle tangent to the line $BC$ at $Q$ and tangent to the line segment $AC$. Prove that the circle $O_2$ is tangent to the circle $O$.

2018 CMIMC Combinatorics, 2

Tags: combinatorics

Compute the number of ways to rearrange nine white cubes and eighteen black cubes into a $3\times 3\times 3$ cube such that each $1\times1\times3$ row or column contains exactly one white cube. Note that rotations are considered $\textit{distinct}$.

2006 AMC 12/AHSME, 2

Tags:

For real numbers $ x$ and $ y$, define $ x\spadesuit y \equal{} (x \plus{} y)(x \minus{} y)$. What is $ 3\spadesuit(4\spadesuit 5)$? $ \textbf{(A) } \minus{} 72 \qquad \textbf{(B) } \minus{} 27 \qquad \textbf{(C) } \minus{} 24 \qquad \textbf{(D) } 24 \qquad \textbf{(E) } 72$

2017 Mathematical Talent Reward Programme, MCQ: P 9

Tags: geometry , tangent , circles

From a point $P$ outside of a circle with centre $O$, tangent segments $PA$ and $PB$ are drawn. $\frac{1}{OA^2}+\frac{1}{PA^2}=\frac{1}{16}$ then $AB=$ [list=1] [*] 4 [*] 6 [*] 8 [*] 10 [/list]

2022 Chile National Olympiad, 5

Tags: combinatorics , combinatorial geometry

Is it possible to divide a polygon with $21$ sides into $2022$ triangles in such a way that among all the vertices there are not three collinear?

2020 Miklós Schweitzer, 2

Tags: analysis , function , real analysis

Prove that if $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous periodic function and $\alpha \in \mathbb{R}$ is irrational, then the sequence $\{n\alpha+f(n\alpha)\}_{n=1}^{\infty}$ modulo 1 is dense in $[0,1]$.

2013 IMAC Arhimede, 1

Tags: number theory , multiple , divisible

Show that in any set of three distinct integers there are two of them, say $a$ and $b$ such that the number $a^5b^3-a^3b^5$ is a multiple of $10$.

2019 AMC 10, 23

Tags: geometry

Points $A(6,13)$ and $B(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$? $\textbf{(A) }\frac{83\pi}{8}\qquad\textbf{(B) }\frac{21\pi}{2}\qquad\textbf{(C) } \frac{85\pi}{8}\qquad\textbf{(D) }\frac{43\pi}{4}\qquad\textbf{(E) }\frac{87\pi}{8}$

VMEO III 2006, 10.1

Tags: geometry , projection , ratio , angle

Given a triangle $ABC$ ($AB \ne AC$). Let $ P$ be a point in the plane containing triangle $ABC$ satisfying the following property: If the projections of $ P$ onto $AB$,$AC$ are $C_1$,$B_1$ respectively, then $\frac{PB}{PC}=\frac{PC_1}{PB_1}=\frac{AB}{AC}$ or $\frac{PB}{PC}=\frac{PB_1}{PC_1}=\frac{AB}{AC}$. Prove that $\angle PBC + \angle PCB = \angle BAC$.

2008 Mexico National Olympiad, 3

Tags: circumcircle , power of a point , radical axis , geometry

The internal angle bisectors of $A$, $B$, and $C$ in $\triangle ABC$ concur at $I$ and intersect the circumcircle of $\triangle ABC$ at $L$, $M$, and $N$, respectively. The circle with diameter $IL$ intersects $BC$ at $D$ and $E$; the circle with diameter $IM$ intersects $CA$ at $F$ and $G$; the circle with diameter $IN$ intersects $AB$ at $H$ and $J$. Show that $D$, $E$, $F$, $G$, $H$, and $J$ are concyclic.

2021 Indonesia TST, A

Tags: inequality , inequalities , algebra

A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying $$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$ then the following inequality holds: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$ (a) Prove that $M=20-\frac{1}{20}$ is not $strong$. (b) Prove that $M=20-\frac{1}{21}$ is $strong$.