Olympiad problems

1999 Polish MO Finals, 1

Tags: algebra

For which $n$ do the equations have a solution in integers: \begin{eqnarray*}x_1 ^2 + x_2 ^2 + 50 &=& 16x_1 + 12x_2 \\ x_2 ^2 + x_3 ^2 + 50 &=& 16x_2 + 12x_3 \\ \cdots \quad \cdots \quad \cdots & \cdots & \cdots \quad \cdots \\ x_{n-1} ^2 + x_n ^2 + 50 &=& 16x_{n-1} + 12x_n \\ x_n ^2 + x_1 ^2 + 50 &=& 16x_n + 12x_1 \end{eqnarray*}

2001 Stanford Mathematics Tournament, 6

Tags: college

Find the least $n$ such that any subset of ${1,2,\dots,100}$ with $n$ elements has 2 elements with a diﬀerence of 9.

2016 Moldova Team Selection Test, 8

Tags: combinatorics

Let us have $n$ ( $n>3$) balls with different rays. On each ball it is written an integer number. Determine the greatest natural number $d$ such that for any numbers written on the balls, we can always find at least 4 different ways to choose some balls with the sum of the numbers written on them divisible by $d$.

2001 National Olympiad First Round, 34

Tags: function

Let $f$ be a real-valued function defined over ordered pairs of integers such that \[f(x+3m-2n, y-4m+5n) = f(x,y)\] for every integers $x,y,m,n$. At most how many elements does the range set of $f$ have? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 49 \qquad\textbf{(E)}\ \text{Infinitely many} $

1969 IMO Shortlist, 59

Tags: function , algebra , continuous function

$(SWE 2)$ For each $\lambda (0 < \lambda < 1$ and $\lambda = \frac{1}{n}$ for all $n = 1, 2, 3, \cdots)$, construct a continuous function $f$ such that there do not exist $x, y$ with $0 < \lambda < y = x + \lambda \le 1$ for which $f(x) = f(y).$

1982 Canada National Olympiad, 3

Tags: trigonometry , combinatorics

Let $\mathbb{R}^n$ be the $n$-dimensional Euclidean space. Determine the smallest number $g(n)$ of a points of a set in $\mathbb{R}^n$ such that every point in $\mathbb{R}^n$ is an irrational distance from at least one point in that set.

2001 Czech And Slovak Olympiad IIIA, 6

Tags: functional equation , function , algebra

Let be given natural numbers $a_1,a_2,...,a_n$ and a function $f : Z \to R$ such that $f(x) = 1$ for all integers $x < 0$ and $f(x) = 1- f(x-a_1)f(x-a_2)... f(x-a_n)$ for all integers $x \ge 0$. Prove that there exist natural numbers $s$ and $t$ such that for all integers $x > s$ it holds that $f(x+t) = f(x)$.

2021 USAMTS Problems, 4

Tags:

Let $ABC$ be a scalene triangle, and let $X, Y , Z$ be points on sides $BC, CA, AB,$ respectively. Let $I$ and $O$ denote the incenter and circumcenter, respectively, of triangle $ABC.$ Suppose that\[ \frac{BX-CX}{BA-CA}=\frac{CY-AY}{CB-AB} = \frac{AZ-BZ}{AC-BC}.\] Prove that there exists a point $P$ on line $IO$ such that $PX \perp BC$, $PY \perp CA$, and $PZ \perp AB.$

2003 Kurschak Competition, 2

Tags: combinatorics

Prove that if a graph $\mathcal{G}$ on $n\ge 3$ vertices has a unique $3$-coloring, then $\mathcal{G}$ has at least $2n-3$ edges. (A graph is $3$-colorable when there exists a coloring of its vertices with $3$ colors such that no two vertices of the same color are connected by an edge. The graph can be $3$-colored uniquely if there do not exist vertices $u$ and $v$ of the graph that are painted different colors in one $3$-coloring, yet are colored the same in another.)

1971 IMO Shortlist, 7

Tags: geometry , 3d geometry , tetrahedron , trigonometry , distance

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$. [b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length; [b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.

1981 Bundeswettbewerb Mathematik, 2

Tags: geometry , classic , centroid , parallel

Prove that if the sides $a, b, c$ of a non-equilateral triangle satisfy $a + b = 2c$, then the line passing through the incenter and centroid is parallel to one of the sides of the triangle.

2016 ISI Entrance Examination, 1

Tags: combinatorics

In a sports tournament of $n$ players, each pair of players plays against each other exactly one match and there are no draws.Show that the players can be arranged in an order $P_1,P_2, .... , P_n$ such that $P_i$ defeats $P_{i+1}$ for all $1 \le i \le n-1$.

2017 Taiwan TST Round 2, 1

Tags: algebra , functional equation

Determine all surjective functions $ f: \mathbb{Z} \to \mathbb{Z} $ such that $$ f\left(xyz+xf\left(y\right)+yf\left(z\right)+zf\left(x\right)\right)=f\left(x\right)f\left(y\right)f\left(z\right) $$ for all $ x,y,z $ in $ \mathbb{Z} $

2009 IberoAmerican, 5

Tags: search , induction , algorithm , continued fraction , number theory , euclidean algorithm , algebra

Consider the sequence $ \{a_n\}_{n\geq1}$ defined as follows: $ a_1 \equal{} 1$, $ a_{2k} \equal{} 1 \plus{} a_k$ and $ a_{2k \plus{} 1} \equal{} \frac {1}{a_{2k}}$ for every $ k\geq 1$. Prove that every positive rational number appears on the sequence $ \{a_n\}$ exactly once.

2012 NZMOC Camp Selection Problems, 6

Tags: diophantine equation , diophantine , number theory

Let $a, b$ and $c$ be positive integers such that $a^{b+c} = b^{c} c$. Prove that b is a divisor of $c$, and that $c$ is of the form $d^b$ for some positive integer $d$.

2024 Thailand Mathematical Olympiad, 5

Tags: geometry

Let $ABC$ be a scalene triangle. Let $H$ be its orthocenter and $D$ is a foot of altitude from $A$ to $BC$. Also, let $S$ and $T$ be points on the circumcircle of triangle $ABC$ such that $\angle BSH=\angle CTH=90^{\circ}$. Given that $AH=2HD$, prove that $D,S,T$ are collinear.

CNCM Online Round 2, 6

Tags:

Let $S$ be the set of all ordered pairs $(x,y)$ of integer solutions to the equation $$6x^2+y^2+6x=3xy+6y+x^2y.$$ $S$ contains a unique ordered pair $(a,b)$ with a maximal value of $b$. Compute $a+b$. Proposed by Kenan Hasanaliyev (claserken)

2021 Science ON grade VII, 3

Tags: algebra , inequality , symmetric sums , newton sums

Are there any real numbers $a,b,c$ such that $a+b+c=6$, $ab+bc+ca=9$ and $a^4+b^4+c^4=260$? What about if we let $a^4+b^4+c^4=210$? [i] (Andrei Bâra)[/i]

2005 AMC 10, 4

Tags:

For real numbers $ a$ and $ b$, define $ a \diamond b \equal{} \sqrt{a^2 \plus{} b^2}$. What is the value of \[(5\diamond 12)\diamond ((\minus{}12) \diamond (\minus{}5))?\] $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac{17}{2}\qquad \textbf{(C)}\ 13\qquad \textbf{(D)}\ 13\sqrt{2}\qquad \textbf{(E)}\ 26$

2018 BMT Spring, 9

Tags:

Suppose $$\frac{1}{3}\frac{(x+1)(x-3)}{(x+2)(x-4)} + \frac{1}{4}\frac{(x+3)(x-5)}{(x+4)(x-6)} - \frac{2}{11}\frac{(x+5)(x-7)}{(x+6)(x-8)} = \frac{53}{132}.$$ Also, suppose $x > 0$. Then $x$ can be written as $a + \sqrt{b}$ where $a$ and $b$ are integers. Find $a + b$.

2022 Serbia JBMO TST, 4

Tags: combinatorics

Initially in every cell of a $5\times 5$ board is the number $0$. In one move you may take any cell of this board and add $1$ to it and all of its adjacent cells (two cells are adjacent if they share an edge). After a finite number of moves, number $n$ is written in all cells. Find all possible values of $n$.

2008 Thailand Mathematical Olympiad, 1

Tags: tangent , rhombus , geometry

Let $P$ be a point outside a circle $\omega$. The tangents from $P$ to $\omega$ are drawn touching $\omega$ at points $A$ and $B$. Let $M$ and $N$ be the midpoints of $AP$ and $AB$, respectively. Line $MN$ is extended to cut $\omega$ at $C$ so that $N$ lies between $M$ and $C$. Line $PC$ intersects $\omega$ again at $D$, and lines $ND$ and $PB$ intersect at $O$. Prove that $MNOP$ is a rhombus.

2019 Stanford Mathematics Tournament, 5

Tags: geometry , 3d geometry , prism

The bases of a right hexagonal prism are regular hexagons of side length $s > 0$, and the prism has height $h$. The prism contains some water, and when it is placed on a flat surface with a hexagonal face on the bottom, the water has depth $\frac{s\sqrt3}{4}$. The water depth doesn’t change when the prism is turned so that a rectangular face is on the bottom. Compute $\frac{h}{s}$.

2017 Flanders Math Olympiad, 2

Tags: geometry , circumcircle , angle chasing , angle

In triangle $\vartriangle ABC$, $\angle A = 50^o, \angle B = 60^o$ and $\angle C = 70^o$. The point $P$ is on the side $[AB]$ (with $P \ne A$ and $P \ne B$). The inscribed circle of $\vartriangle ABC$ intersects the inscribed circle of $\vartriangle ACP$ at points $U$ and $V$ and intersects the inscribed circle of $\vartriangle BCP$ at points $X$ and $Y$. The rights $UV$ and $XY$ intersect in $K$. Calculate the $\angle UKX$.

2015 Thailand TSTST, 1

Tags: inequalities

Let $x, y, z$ be positive real numbers satisfying $x + y + z =\frac{2}{x+y}+\frac{2}{y+z}+\frac{2}{z+x}$. Prove that $$\frac{3}{2}\leq\frac{3}{\sqrt[3]{xyz}(\sqrt[3]{xyz}+1)}\leq\frac{1}{x(y+1)}+\frac{1}{y(z+1)}+\frac{1}{z(x+1)}.$$