Olympiad problems

2014 HMNT, 1

Tags: number theory

What is the smallest positive integer $n$ which cannot be written in any of the following forms? $\bullet$ $n = 1 + 2 +... + k$ for a positive integer $k$. $\bullet$ $n = p^k$ for a prime number $p$ and integer $k$. $\bullet$ $n = p + 1$ for a prime number $p$.

2016 India Regional Mathematical Olympiad, 2

Tags: counting , combinatorics

At an international event there are $100$ countries participating, each with its own flag. There are $10$ distinct flagpoles at the stadium, labelled 1,#2,...,#10 in a row. In how many ways can all the $100$ flags be hoisted on these $10$ flagpoles, such that for each $i$ from $1$ to $10$, the flagpole #i has at least $i$ flags? (Note that the vertical order of the flagpoles on each flag is important)

Indonesia MO Shortlist - geometry, g4

Tags: geometry , tangent circle

Given that two circles $\sigma_1$ and $\sigma_2$ internally tangent at $N$ so that $\sigma_2$ is inside $\sigma_1$. The points $Q$ and $R$ lies at $\sigma_1$ and $\sigma_2$, respectively, such that $N,R,Q$ are collinear. A line through $Q$ intersects $\sigma_2$ at $S$ and intersects $\sigma_1$ at $O$. The line through $N$ and $S$ intersects $\sigma_1$ at $P$. Prove that $$\frac{PQ^3}{PN^2} = \frac{PS \cdot RS}{NS}.$$

2008 ITest, 87

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Find the number of $12$-digit "words" that can be formed from the alphabet $\{0,1,2,3,4,5,6\}$ if neighboring digits must differ by exactly $2$.

2012 India IMO Training Camp, 2

Tags: number theory

Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.

2024 AIME, 14

Tags: aime 1 , 3b1b , puzzle

Let $ABCD$ be a tetrahedron such that $AB = CD = \sqrt{41}$, $AC = BD = \sqrt{80}$, and $BC = AD = \sqrt{89}$. There exists a point $I$ inside the tetrahedron such that the distances from $I$ to each of the faces of the tetrahedron are all equal. This distance can be written in the form $\frac{m \sqrt{n}}{p}$, when $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+p$.

2013 ELMO Shortlist, 1

Tags: function , algebra

Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) [i]Proposed by Evan Chen[/i]

2003 Bulgaria Team Selection Test, 2

Tags: functional equation , algebra

Find all $f:R-R$ such that $f(x^2+y+f(y))=2y+f(x)^2$

1992 Romania Team Selection Test, 8

Tags: combinatorial geometry , combinatorics

Let $m,n \ge 2$ be integers. The sides $A_{00}A_{0m}$ and $A_{nm}A_{n0}$ of a convex quadrilateral $A_{00}A_{0m}A_{nm}A_{n0}$ are divided into $m$ equal segments by points $A_{0j}$ and $A_{nj}$ respectively ($j = 1,...,m-1$). The other two sides are divided into $n$ equal segments by points $A_{i0}$ and $A_{im}$ ($i = 1,...,n -1$). Denote by $A_{ij}$ the intersection of lines $A_{0j}A{nj}$ and $A_{i0}A_{im}$, by $S_{ij}$ the area of quadrilateral $A_{ij}A_{i, j+1}A_{i+1, j+1}A_{i+1, j}$ and by $S$ the area of the big quadrilateral. Show that $S_{ij} +S_{n-1-i,m-1-j} = \frac{2S}{mn}$

2017 Polish Junior Math Olympiad Finals, 1.

Tags: number theory

Let $a$, $b$, and $c$ be positive integers for which the number \[\frac{a\sqrt2+b}{b\sqrt2+c}\] is rational. Show that the number $ab+bc+ca$ is divisible by $a+b+c$.

2022/2023 Tournament of Towns, P1

Tags: combinatorics

There are two letter sequences $A$ and $B$, both with length $100$ letters. In one move you can insert in any place of sequence ( possibly to start or to end) any number of same letters or remove any number of consecutive same letters. Prove that it is possible to make second sequence from first sequence using not more than $100$ moves.

2017 Taiwan TST Round 1, 3

Tags: functional equation , algebra

Find all injective functions $ f:\mathbb{N} \to \mathbb{N} $ such that $$ f^{f\left(a\right)}\left(b\right)f^{f\left(b\right)}\left(a\right)=\left(f\left(a+b\right)\right)^2 $$ holds for all $ a,b \in \mathbb{N} $. Note that $ f^{k}\left(n\right) $ means $ \underbrace{f(f(\ldots f}_{k}(n) \ldots )) $

1985 Iran MO (2nd round), 1

Tags: circumcircle , geometry , inequalities

Inscribe in the triangle $ABC$ a triangle with minimum perimeter.

2005 Purple Comet Problems, 18

Tags: trapezoid , number theory , ratio , geometry , relatively prime

The side lengths of a trapezoid are $\sqrt[4]{3}, \sqrt[4]{3}, \sqrt[4]{3}$, and $2 \cdot \sqrt[4]{3}$. Its area is the ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$.

2012 India IMO Training Camp, 1

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

The cirumcentre of the cyclic quadrilateral $ABCD$ is $O$. The second intersection point of the circles $ABO$ and $CDO$, other than $O$, is $P$, which lies in the interior of the triangle $DAO$. Choose a point $Q$ on the extension of $OP$ beyond $P$, and a point $R$ on the extension of $OP$ beyond $O$. Prove that $\angle QAP=\angle OBR$ if and only if $\angle PDQ=\angle RCO$.

2013 CIIM, Problem 1

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Given two natural numbers $m$ and $n$, denote by $\overline{m.n}$ the number obtained by writing $m$ followed by $n$ after the decimal dot. a) Prove that there are infinitely many natural numbers $k$ such that for any of them the equation $\overline{m.n} \times \overline{n.m} = k$ has no solution. b) Prove that there are infinitely many natural numbers $k$ such that for any of them the equation $\overline{m.n} \times \overline{n.m} = k$ has a solution.

2018 German National Olympiad, 3

Tags: coloring , integer , combinatorics

Given a positive integer $n$, Susann fills a square of $n \times n$ boxes. In each box she inscribes an integer, taking care that each row and each column contains distinct numbers. After this an imp appears and destroys some of the boxes. Show that Susann can choose some of the remaining boxes and colour them red, satisfying the following two conditions: 1) There are no two red boxes in the same column or in the same row. 2) For each box which is neither destroyed nor coloured, there is a red box with a larger number in the same row or a red box with a smaller number in the same column. [i]Proposed by Christian Reiher[/i]

2007 Harvard-MIT Mathematics Tournament, 3

Tags: ratio

Three real numbers $x$, $y$, and $z$ are such that $(x+4)/2=(y+9)/(z-3)=(x+5)/(z-5)$. Determine the value of $x/y$.

2024 China Team Selection Test, 10

Tags: polynomial , algebra

Let $M$ be a positive integer. $f(x):=x^3+ax^2+bx+c\in\mathbb Z[x]$ satisfy $|a|,|b|,|c|\le M.$ $x_1,x_2$ are different roots of $f.$ Prove that $$|x_1-x_2|>\frac 1{M^2+3M+1}.$$ [i]Created by Jingjun Han[/i]

1988 Tournament Of Towns, (184) 1

Tags: algebra

It is known that the proportion of people with fair hair among people with blue eyes is more than the proportion of people with fair hair among all people. Which is greater , the proportion of people with blue eyes among people with fair hair, or the proportion of people with blue eyes among all people? (Folklore)

LMT Team Rounds 2010-20, A7 B15

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Let $S$ denote the sum of all rational numbers of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive divisors of $1300$. If $S$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then find $m+n$. [i]Proposed by Ephram Chun[/i]

2024 CMIMC Algebra and Number Theory, 2

Tags: algebra

Suppose $P(x)=x^2+Ax+B$ for real $A$ and $B$. If the sum of the roots of $P(2x)$ is $\tfrac 12$ and the product of the roots of $P(3x)$ is $\tfrac 13$, find $A+B$. [i]Proposed by Connor Gordon[/i]

2005 China Girls Math Olympiad, 3

Tags: geometry , 3d geometry , tetrahedron

Determine if there exists a convex polyhedron such that (1) it has 12 edges, 6 faces and 8 vertices; (2) it has 4 faces with each pair of them sharing a common edge of the polyhedron.

2012 AMC 10, 7

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For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid $3$ acorns in each of the holes it dug. The squirrel hid $4$ acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed $4$ fewer holes. How many acorns did the chipmunk hide? ${{ \textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48}\qquad\textbf{(E)}\ 54} $

2024 Junior Balkan Team Selection Tests - Moldova, 6

Tags: geometry

In the isosceles triangle $ABC$, with $AB=BC$, points $X$ and $Y$ are the midpoints of the sides $AB$ and $AC$, respectively. Point $Z$ is the foot of the perpendicular from $B$ to $CX$. Prove that the circumcenter of the triangle $XYZ$ is of the line $AC$.