Olympiad problems

2003 CentroAmerican, 4

$S_1$ and $S_2$ are two circles that intersect at two different points $P$ and $Q$. Let $\ell_1$ and $\ell_2$ be two parallel lines such that $\ell_1$ passes through the point $P$ and intersects $S_1,S_2$ at $A_1,A_2$ respectively (both distinct from $P$), and $\ell_2$ passes through the point $Q$ and intersects $S_1,S_2$ at $B_1,B_2$ respectively (both distinct from $Q$). Show that the triangles $A_1QA_2$ and $B_1PB_2$ have the same perimeter.

2018 South East Mathematical Olympiad, 7

Tags: number theory

For positive integers $m,n,$ define $f(m,n)$ as the number of ordered triples $(x,y,z)$ of integers such that $$ \begin{cases} xyz=x+y+z+m, \\ \max\{|x|,|y|,|z|\} \leq n \end{cases} $$ Does there exist positive integers $m,n,$ such that $f(m,n)=2018?$ Please prove your conclusion.

2004 Bosnia and Herzegovina Team Selection Test, 4

Tags: game , combinatorics

On competition which has $16$ teams, it is played $55$ games. Prove that among them exists $3$ teams such that they have not played any matches between themselves.

2018 Thailand Mathematical Olympiad, 9

Tags: geometry , circumcircle , incenter , equal segments

In $\vartriangle ABC$ the incircle is tangent to $AB$ at $D$. Let $P$ be a point on $BC$ different from $B$ and $C$, and let $K$ and $L$ be incenters of $\vartriangle ABP$ and $\vartriangle ACP$ respectively. Suppose that the circumcircle of $\vartriangle KP L$ cuts $AP$ again at $Q$. Prove that $AD = AQ$.

2022 Purple Comet Problems, 21

Tags: combinatorics

Find the number of sequences of 10 letters where all the letters are either $A$ or $B$, the first letter is $A$, the last letter is $B$, and the sequence contains no three consecutive letters reading $ABA$. For example, count $AAABBABBAB$ and $ABBBBBBBAB$ but not $AABBAABABB$ or $AAAABBBBBA$.

1991 Baltic Way, 18

Tags: geometry , 3d geometry , sphere

Is it possible to place two non-intersecting tetrahedra of volume $\frac{1}{2}$ into a sphere with radius $1$?

2024 MMATHS, 2

Tags:

Grant has a box with $6$ red balls, $5$ blue balls, $4$ green balls, $3$ yellow balls, $2$ orange balls, and $1$ purple ball. Grant selects $6$ balls at random, without replacement. Let $P$ be the probability that Grant selects six balls of different colors, and let $Q$ be the probability that Grant selects six balls of the same color. What is $\tfrac{P}{Q}$?

2014 NIMO Problems, 4

Tags: modular arithmetic , euler , floor function , logarithm

Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer. Compute the remainder when $3^n$ is divided by $1000$.

2003 Romania National Olympiad, 3

Tags: limit , function , continuity , real analysis

Let be two functions $ f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ having that properties that $ f $ is continuous, $ g $ is nondecreasing and unbounded, and for any sequence of rational numbers $ \left( x_n \right)_{n\ge 1} $ that diverges to $ \infty , $ we have $$ 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . $$ Prove that $1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) . $ [i]Radu Gologan[/i]

2015 Postal Coaching, Problem 6

Tags: number theory , function

Let $k \in \mathbb{N}$, let $x_k$ denote the nearest integer to $\sqrt k$. Show that for each $m \in \mathbb {N}$, $$\sum_{k=1}^{m} \frac{1}{x_k} = f(m)+ \frac{m}{f(m)+1}$$, where $f(m)$ is the integer part of $\frac{\sqrt{4m-3}-1}{2}$

1969 Czech and Slovak Olympiad III A, 1

Tags: rational , algebra , equation , quadratic polynomial

Find all rational numbers $x,y$ such that \[\left(x+y\sqrt5\right)^2=7+3\sqrt5.\]

2022 German National Olympiad, 5

Tags: tangent line , tangent , touching circles , geometry

Let $ABC$ be an equilateral triangle with circumcircle $k$. A circle $q$ touches $k$ from outside in a point $D$, where the point $D$ on $k$ is chosen so that $D$ and $C$ lie on different sides of the line $AB$. We now draw tangent lines from $A,B$ and $C$ to the circle $q$ and denote the lengths of the respective tangent line segments by $a,b$ and $c$. Prove that $a+b=c$.

1991 Swedish Mathematical Competition, 1

Tags: diophantine equation , diophantine , number theory

Find all positive integers $m, n$ such that $\frac{1}{m} + \frac{1}{n} - \frac{1}{mn} =\frac{2}{5}$.

2006 Estonia Math Open Junior Contests, 4

Tags: relatively prime , number theory

Does there exist a natural number with the sum of digits of its $ kth$ power being equal to $ k$, if a) $ k \equal{} 2004$; b) $ k \equal{} 2006?$

2014 South East Mathematical Olympiad, 8

Tags: geometry , rectangle , combinatorics

Define a figure which is constructed by unit squares "cross star" if it satisfies the following conditions: $(1)$Square bar $AB$ is bisected by square bar $CD$ $(2)$At least one square of $AB$ lay on both sides of $CD$ $(3)$At least one square of $CD$ lay on both sides of $AB$ There is a rectangular grid sheet composed of $38\times 53=2014$ squares,find the number of such cross star in this rectangle sheet

1972 Poland - Second Round, 2

Tags: geometry , rectangle , combinatorics , combinatorial geometry

In a rectangle with sides of length 20 and 25 there are 120 squares of side length 1. Prove that there is a circle with a diameter of 1 contained in this rectangle and having no points in common with any of these squares.

2014 Paenza, 1

Tags: limit , induction , logarithm

Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation: \[a_{n+1}=10^n a_n^2\] (a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$. (b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.

2002 South africa National Olympiad, 6

Tags: number theory

Find all rational numbers $a$, $b$, $c$ and $d$ such that \[ 8a^2 - 3b^2 + 5c^2 + 16d^2 - 10ab + 42cd + 18a + 22b - 2c - 54d = 42, \] \[ 15a^2 - 3b^2 + 21c^2 - 5d^2 + 4ab +32cd - 28a + 14b - 54c - 52d = -22. \]

1997 Singapore MO Open, 2

Tags: binomial , divide , divisible , number theory

Observe that the number $4$ is such that $4 \choose k$ $= \frac{4!}{k!(4-k)!}$ divisible by $k + 1$ for $k = 0,1,2,3$. Find all the natural numbers $n$ between $50$ and $90$ such that $n \choose k$ is divisible by $k + 1$ for $k = 0,1,2,..., n - 1$. Justify your answers.

1980 IMO, 10

Tags: function , algebra

The function f is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1)=2$ and $f(xy)=f(x)f(y)-f(x+y)+1$ for all $x,y \in \mathbb{Q}$. Determine f (with proof)

2020 Harvard-MIT Mathematics Tournament, 6

Tags:

A polynomial $P(x)$ is a \emph{base-$n$ polynomial} if it is of the form $a_dx^d+a_{d-1}x^{d-1}+\cdots + a_1x+a_0$, where each $a_i$ is an integer between $0$ and $n-1$ inclusive and $a_d>0$. Find the largest positive integer $n$ such that for any real number $c$, there exists at most one base-$n$ polynomial $P(x)$ for which $P(\sqrt 2+\sqrt 3)=c$. [i]Proposed by James Lin.[/i]

2021 Sharygin Geometry Olympiad, 10-11.8

Tags: locus , arc , geometry

On the attraction "Merry parking", the auto has only two position* of a steering wheel: "right", and "strongly right". So the auto can move along an arc with radius $r_1$ or $r_2$. The auto started from a point $A$ to the Nord, it covered the distance $\ell$ and rotated to the angle $a < 2\pi$. Find the locus of its possible endpoints.

2016 HMNT, 7

Tags: geometry

Let ABC be a triangle with $AB = 13, BC = 14, CA = 15$. The altitude from $A$ intersects $BC$ at $D$. Let $\omega_1$ and $\omega_2$ be the incircles of $ABD$ and $ACD$, and let the common external tangent of $\omega_1$ and $\omega_2$ (other than $BC$) intersect $AD$ at $E$. Compute the length of $AE$.

PEN H Problems, 6

Tags: diophantine equation , number theory

Show that there are infinitely many pairs $(x, y)$ of rational numbers such that $x^3 +y^3 =9$.

2007 Sharygin Geometry Olympiad, 5

Tags: combinatorial geometry , convex , polyhedron , 3-dimensional geometry , geometry

Each edge of a convex polyhedron is shifted such that the obtained edges form the frame of another convex polyhedron. Are these two polyhedra necessarily congruent?