Olympiad problems

2010 Kosovo National Mathematical Olympiad, 2

Tags: algebra

The set $S\subseteq \mathbb{R}$ is given with the properties: $(a) \mathbb{Z}\subset S$, $(b) (\sqrt 2 +\sqrt 3)\in S$, $(c)$ If $x,y\in S$ then $x+y\in S$, and $(d)$ If $x,y\in S$ then $x\cdot y\in S$. Prove that $(\sqrt 2+\sqrt 3)^{-1}\in S$.

IV Soros Olympiad 1997 - 98 (Russia), 9.6

Tags: geometry , combinatorial geometry , combinatorics

Cut an acute triangle, one of whose sides is equal to the altitude drawn, by two straight cuts, into four parts, from which you can fold a square.

2014 Harvard-MIT Mathematics Tournament, 7

Tags: geometry , analytic geometry , circumcircle , perpendicular bisector

Triangle $ABC$ has sides $AB = 14$, $BC = 13$, and $CA = 15$. It is inscribed in circle $\Gamma$, which has center $O$. Let $M$ be the midpoint of $AB$, let $B'$ be the point on $\Gamma$ diametrically opposite $B$, and let $X$ be the intersection of $AO$ and $MB'$. Find the length of $AX$.

1996 Irish Math Olympiad, 1

Tags: number theory

The Fibonacci sequence is defined by $ F_0\equal{}0, F_1\equal{}1$ and $ F_{n\plus{}2}\equal{}F_n\plus{}F_{n\plus{}1}$ for $ n \ge 0$. Prove that: $ (a)$ The statement $ "F_{n\plus{}k}\minus{}F_n$ is divisible by $ 10$ for all $ n \in \mathbb{N}"$ is true if $ k\equal{}60$ but false for any positive integer $ k<60$. $ (b)$ The statement $ "F_{n\plus{}t}\minus{}F_n$ is divisible by $ 100$ for all $ n \in \mathbb{N}"$ is true if $ t\equal{}300$ but false for any positive integer $ t<300$.

2001 Slovenia National Olympiad, Problem 2

Tags: quadratic

Find all rational numbers $r$ such that the equation $rx^2 + (r + 1)x + r = 1$ has integer solutions.

2023 Bulgarian Autumn Math Competition, 10.3

Tags: number theory

Find all positive integers $k$, so that there exists a polynomial $f(x)$ with rational coefficients, such that for all sufficiently large $n$, $$f(n)=\text{lcm}(n+1, n+2, \ldots, n+k).$$

2004 JBMO Shortlist, 5

Tags: circumcircle , symmetry , angle bisector , perpendicular bisector , geometry

Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.

2003 France Team Selection Test, 2

Tags: combinatorics

$10$ cities are connected by one-way air routes in a way so that each city can be reached from any other by several connected flights. Let $n$ be the smallest number of flights needed for a tourist to visit every city and return to the starting city. Clearly $n$ depends on the flight schedule. Find the largest $n$ and the corresponding flight schedule.

2012 India IMO Training Camp, 2

Tags: greatest common divisor , number theory

Show that there exist infinitely many pairs $(a, b)$ of positive integers with the property that $a+b$ divides $ab+1$, $a-b$ divides $ab-1$, $b>1$ and $a>b\sqrt{3}-1$

2023 SG Originals, Q4

Tags: graph theory , combinatorics

On a connected graph $G$, one may perform the following operations: [list] [*]choose a vertice $v$, and add a vertice $v'$ such that $v'$ is connected to $v$ and all of its neighbours [*] choose a vertice $v$ with odd degree and delete it [/list] Show that for any connected graph $G$, we may perform a finite number of operations such that the resulting graph is a clique. Proposed by [i]idonthaveanaopsaccount[/i]

2000 Romania Team Selection Test, 3

Tags: geometry , 3d geometry , sphere , function , calculus , topology , analytic geometry

Let $S$ be the set of interior points of a sphere and $C$ be the set of interior points of a circle. Find, with proof, whether there exists a function $f:S\rightarrow C$ such that $d(A,B)\le d(f(A),f(B))$ for any two points $A,B\in S$ where $d(X,Y)$ denotes the distance between the points $X$ and $Y$. [i]Marius Cavachi[/i]

2013 Saudi Arabia BMO TST, 7

Tags: excircle , tangent , geometry

The excircle $\omega_B$ of triangle $ABC$ opposite $B$ touches side $AC$, rays $BA$ and $BC$ at $B_1, C_1$ and $A_1$, respectively. Point $D$ lies on major arc $A_1C_1$ of $\omega_B$. Rays $DA_1$ and $C_1B_1$ meet at $E$. Lines $AB_1$ and $BE$ meet at $F$. Prove that line $FD$ is tangent to $\omega_B$ (at $D$).

1954 Putnam, A2

Tags: square , distance

Consider any five points in the interior of square $S$ of side length $1$. Prove that at least one of the distances between these points is less than $\sqrt{2} \slash 2.$ Can this constant be replaced by a smaller number?

2022-2023 OMMC FINAL ROUND, 4

Tags: geometry

In $\triangle ABC$ points $D$, $E$ lie on segment $BC$ where $BD = DE = EC.$ Points $X$, $Y$ lie on the perpendicular bisectors of $AD$, $AE$ respectively. If $XE$, $YD$ are tangent to the circumcircles of $\triangle AEC$, $\triangle ADB$ respectively, prove $X,A,Y$ are collinear.

2016 JBMO Shortlist, 7

Tags: geometry

Let ${AB}$ be a chord of a circle ${(c)}$ centered at ${O}$, and let ${K}$ be a point on the segment ${AB}$ such that ${AK<BK}$. Two circles through ${K}$, internally tangent to ${(c)}$ at ${A}$ and ${B}$, respectively, meet again at ${L}$. Let ${P}$ be one of the points of intersection of the line ${KL}$ and the circle ${(c)}$, and let the lines ${AB}$ and ${LO}$ meet at ${M}$. Prove that the line ${MP}$ is tangent to the circle ${(c)}$. Theoklitos Paragyiou (Cyprus)

1978 IMO Longlists, 18

Tags: inequalities , analytic geometry , number theory

Given a natural number $n$, prove that the number $M(n)$ of points with integer coordinates inside the circle $(O(0, 0),\sqrt{n})$ satisfies \[\pi n - 5\sqrt{n} + 1<M(n) < \pi n+ 4\sqrt{n} + 1\]

2001 Austrian-Polish Competition, 6

Tags: floor function , sequence , integer , algebra , number theory

Let $k$ be a fixed positive integer. Consider the sequence definited by \[a_{0}=1 \;\; , a_{n+1}=a_{n}+\left\lfloor\root k \of{a_{n}}\right\rfloor \;\; , n=0,1,\cdots\] where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For each $k$ find the set $A_{k}$ containing all integer values of the sequence $(\sqrt[k]{a_{n}})_{n\geq 0}$.

2012 Chile National Olympiad, 2

Tags: number theory , sum , inequalities

Let $a_1,a_2,...,a_n$ be all positive integers with $2012$ digits or less, none of which is a $9$. Prove that $$ \frac{1}{a_1}+\frac{1}{a_2}+ ... +\frac{1}{a_{n}}\le 80.$$

VI Soros Olympiad 1999 - 2000 (Russia), 10.7

Tags: geometry , bisects segment

Let a line, perpendicular to side $AD$ of parallelogram $ABCD$ passing through point $B$, intersect line $CD$ at point $M$, and a line, passing through point $B$ and perpendicular to side $CD$, intersect line $AD$ at point $N$. Prove that the line passing through point $B$ perpendicular to the diagonal $AC$, passes through the midpoint of the segment $MN$.