Olympiad problems

2017 Yasinsky Geometry Olympiad, 5

$ABCD$ is a rectangle. The segment $MA$ is perpendicular to plane $ABC$ . $MB= 15$ , $MC=24$ , $MD=20$. Find the length of $MA$ .

1996 Irish Math Olympiad, 2

Tags: number theory

Let $ S(n)$ denote the sum of the digits of a natural number $ n$ (in base $ 10$). Prove that for every $ n$, $ S(2n) \le 2S(n) \le 10S(2n)$. Prove also that there is a positive integer $ n$ with $ S(n)\equal{}1996S(3n)$.

2001 National High School Mathematics League, 12

Tags:

Three main diagonals of a hegular hexagon divide the hegular hexagon into six regular triangles. Note them $A,B,C,D,E,F$. In one part, grow one kind of plant. Also, adjacent parts must be grown with different plants. If we are given four kinds of plants, then the number of wanys to grow plants is________.

2010 Morocco TST, 3

Tags: number theory , prime number

Any rational number admits a non-decimal representation unlimited decimal expansion. This development has the particularity of being periodic. Examples: $\frac{1}{7} = 0.142857142857…$ has a period $6$ while $\frac{1}{11}=0.0909090909 …$ $2$ periodic. What are the reciprocals of the prime integers with a period less than or equal to five?

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 $MN$.

1989 Greece National Olympiad, 3

Tags: limit , real analysis , sequence

Find the limit of the sequence $x_n$ defined by recurrence relation $$x_{n+2}=\frac{1}{12}x_{n+1}+\frac{1}{2}x_{n}+1$$ where $n=0,1,2,...$ for any initial values $x_2,x_1$.

2020/2021 Tournament of Towns, P4

Tags: geometry

[list=a] [*]Is it possible to split a square into 4 isosceles triangles such that no two are congruent? [*]Is it possible to split an equilateral triangle into 4 isosceles triangles such that no two are congruent? [/list] [i]Vladimir Rastorguev[/i]