Olympiad problems

1979 IMO, 3

Tags: combinatorics , recurrence relation , Linear Recurrences , counting , IMO , IMO 1979

Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]

2024 Assara - South Russian Girl's MO, 2

Tags: geometry

Prove that in any described $8$-gon there is a side that does not exceed the diameter of the inscribed circle in length. [i]P.A.Kozhevnikov[/i]

2017 Flanders Math Olympiad, 4

Tags: number theory , algebra

For every natural number $n$ we define the derived number $n'$ as follows: $\bullet$ $0' = 1' = 0$ $\bullet$ if $n$ is prime, then $n' = 1$ $\bullet$ if $n = a \cdot b$, then $n' = a' b + a b'$ . For example: $15' = 3' 5 + 3 5' = 1\cdot 5 + 3\cdot 1 = 8$. Determine all natural numbers $n$ for which $n = n'$.

2017 Kosovo National Mathematical Olympiad, 1

Tags: number theory

The sequence ${a_{n}}$ $n\in \mathbb{N}$ is given in a recursive way with $a_{1}=1$, $a_{n}=\prod_{i=1}^{n-1} a_{i}+1$, for all $n\geq 2$. Determine the least number $M$, such that $\sum_{n=1}^{m} \frac{1}{a_{n}} <M$ for all $m\in \mathbb{N}$

2007 Poland - Second Round, 2

Tags: geometry , cyclic quadrilateral , power of a point , radical axis , geometry unsolved

We are given a cyclic quadrilateral $ABCD \quad AB\not=CD$. Quadrilaterals $AKDL$ and $CMBN$ are rhombuses with equal sides. Prove, that $KLMN$ is cyclic

2023 Junior Balkan Mathematical Olympiad, 1

Tags: number theory , JBMO , JBMO Shortlist

Find all pairs $(a,b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of $5$. [i]Nikola Velov, North Macedonia[/i]

2002 District Olympiad, 2

Tags: algebra , system of equations

Solve in $ \mathbb{C}^3 $ the following chain of equalities: $$ x(x-y)(x-z)=y(y-x)(y-z)=z(z-x)(z-y)=3. $$

2006 AIME Problems, 6

Tags: geometry , trigonometry , AMC , AIME , USA(J)MO , USAMO , similar triangles

Square $ABCD$ has sides of length 1. Points $E$ and $F$ are on $\overline{BC}$ and $\overline{CD}$, respectively, so that $\triangle AEF$ is equilateral. A square with vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\overline{AE}$. The length of a side of this smaller square is $\displaystyle \frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b+c$.

2001 AMC 10, 6

Tags: algebra

Let $ P(n)$ and $ S(n)$ denote the product and the sum, respectively, of the digits of the integer $ n$. For example, $ P(23) \equal{} 6$ and $ S(23) \equal{} 5$. Suppose $ N$ is a two-digit number such that $ N \equal{} P(N) \plus{} S(N)$. What is the units digit of $ N$? $ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

2023 Korea Junior Math Olympiad, 2

Tags: geometry

Quadrilateral $ABCD (\overline{AD} < \overline{BC})$ is inscribed in a circle, and $H(\neq A, B)$ is a point on segment $AB.$ The circumcircle of triangle $BCH$ meets $BD$ at $E(\neq B)$ and line $HE$ meets $AD$ at $F$. The circle passes through $C$ and tangent to line $BD$ at $E$ meets $EF$ at $G(\neq E).$ Prove that $\angle DFG = \angle FCG.$

2020 BMT Fall, 9

Tags: Bmt , algebra , Diophantine equation

There is a unique triple $(a,b,c)$ of two-digit positive integers $a,\,b,$ and $c$ that satisfy the equation $$a^3+3b^3+9c^3=9abc+1.$$ Compute $a+b+c$.

2010 Indonesia TST, 4

Tags: number theory

Let $n$ be a positive integer with $n = p^{2010}q^{2010}$ for two odd primes $p$ and $q$. Show that there exist exactly $\sqrt[2010]{n}$ positive integers $x \le n$ such that $p^{2010}|x^p - 1$ and $q^{2010}|x^q - 1$.

2012 AMC 12/AHSME, 3

Tags: AMC

A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$? $\textbf{(A)}\ 120\qquad\textbf{(B)}\ 160\qquad\textbf{(C)}\ 200\qquad\textbf{(D)}\ 240\qquad\textbf{(E)}\ 280$

2008 Greece JBMO TST, 4

Tags: number theory , Integers , Diophantine equation , Product , Sum

Product of two integers is $1$ less than three times of their sum. Find those integers.

2023 Abelkonkurransen Finale, 4a

Tags: inequalities , geometry

Assuming $a,b,c$ are the side-lengths of a triangle, show that \begin{align*} \frac{a^2+b^2-c^2}{ab} + \frac{b^2+c^2-a^2}{bc} + \frac{c^2+a^2-b^2}{ca} > 2. \end{align*} Also show that the inequality does not necessarily hold if you replace $2$ (on the right-hand side) by a bigger by a bigger number.

2022 Iran-Taiwan Friendly Math Competition, 6

Tags: function , number theory , algebra , Iran , Taiwan

Find all completely multipiclative functions $f:\mathbb{Z}\rightarrow \mathbb{Z}_{\geqslant 0}$ such that for any $a,b\in \mathbb{Z}$ and $b\neq 0$, there exist integers $q,r$ such that $$a=bq+r$$ and $$f(r)<f(b)$$ Proposed by Navid Safaei

1999 Slovenia National Olympiad, Problem 3

Tags: geometry , incenter

The incircle of a right triangle $ABC$ touches the hypotenuse $AB$ at a point $D$. Show that the area of $\triangle ABC$ equals $AD\cdot DB$.

2024 MMATHS, 12

Tags: Yale , MMATHS

$S_1,S_2,\ldots,S_n$ are subsets of $\{1,2,\ldots,10000\}$ which satisfy that, whenever $|S_i| > |S_j|$, the sum of all elements in $S_i$ is less than the sum of all elements in $S_j$. Let $m$ be the maximum number of distinct values among $|S_1|,\ldots,|S_n|$. Find $\left\lfloor\frac{m}{100}\right\rfloor$.

2023 Thailand TSTST, 6

Tags: geometry

Let $ABC$ be an acute triangle with orthocenter $H$. Let $D$ and $E$ be feet of the altitudes from $B$ and $C$ respectively. Let $M$ be the midpoint of segment $AH$ and $F$ be the intersection point of $AH$ and $DE$. Furthermore, let $P$ and $Q$ be the points inside triangle $ADE$ so that $P$ is an intersection of $CM$ and the circumcircle of $DFH$, and $Q$ is an intersection of $BM$ and the circumcircle of $EFH$. Prove that the intersection of lines $DQ$ and $EP$ lies on segment $AH$.

2014 Portugal MO, 4

Tags: number theory unsolved , number theory

Determine all natural numbers $x$, $y$ and $z$, such that $x\leq y\leq z$ and \[\left(1+\frac1x\right)\left(1+\frac1y\right)\left(1+\frac1z\right) = 3\text{.}\]

1963 AMC 12/AHSME, 36

Tags: LaTeX , AMC

A person starting with $64$ cents and making $6$ bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is: $\textbf{(A)}\ \text{a loss of } 27 \qquad \textbf{(B)}\ \text{a gain of }27 \qquad \textbf{(C)}\ \text{a loss of }37 \qquad$ $ \textbf{(D)}\ \text{neither a gain nor a loss} \qquad \textbf{(E)}\ \text{a gain or a loss depending upon the order in which the wins and losses occur}$ Note: Due to the lack of $\LaTeX$ packages, the numbers in the answer choices are in cents ¢

2010 Abels Math Contest (Norwegian MO) Final, 1a

Tags: geometry , angles , right angle , equal segments

The point $P$ lies on the edge $AB$ of a quadrilateral $ABCD$. The angles $BAD, ABC$ and $CPD$ are right, and $AB = BC + AD$. Show that $BC = BP$ or $AD = BP$.

2015 Germany Team Selection Test, 3

Tags: IMO Shortlist , combinatorics , algebra , prime numbers

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

2020 Thailand Mathematical Olympiad, 5

Tags: combinatorics

You have an $n\times n$ grid and want to remove all edges of the grid by the sequence of the following moves. In each move, you can select a cell and remove exactly three edges surrounding that cell; in particular, that cell must have at least three remaining edges for the operation to be valid. For which positive integers $n$ is this possible?

2018 CCA Math Bonanza, TB1

Tags:

What is the maximum number of diagonals of a regular $12$-gon which can be selected such that no two of the chosen diagonals are perpendicular? Note: sides are not diagonals and diagonals which intersect outside the $12$-gon at right angles are still considered perpendicular. [i]2018 CCA Math Bonanza Tiebreaker Round #1[/i]