Olympiad problems

2023 Malaysia IMONST 2, 5

Find the smallest positive $m$ such that if $a,b,c$ are three side lengths of a triangle with $a^2 +b^2 > mc^2$, then $c$ must be the length of shortest side.

2017 ASDAN Math Tournament, 3

Tags: geometry test

Line segment $AB$ has length $10$. A circle centered at $A$ has radius $5$, and a circle centered at $B$ has radius $5\sqrt{3}$. What is the area of the intersection of the two circles?

2021 Albanians Cup in Mathematics, 3

Tags: number theory , sequence

Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.

2006 Petru Moroșan-Trident, 3

Tags: indefinite integral , calculus , primitive

Determine the primitives of: [b]1)[/b] $ (0,\pi /2)\ni x\mapsto\frac{x^2}{-x+\tan x} $ [b]2)[/b] $ 1<x\mapsto \frac{-1+\ln x}{x^2-\ln^2 x} $ [i]Ion Nedelcu[/i]

1995 APMO, 4

Tags: rectangle , circumcircle , parallelogram , cyclic quadrilateral , geometry

Let $C$ be a circle with radius $R$ and centre $O$, and $S$ a fixed point in the interior of $C$. Let $AA'$ and $BB'$ be perpendicular chords through $S$. Consider the rectangles $SAMB$, $SBN'A'$, $SA'M'B'$, and $SB'NA$. Find the set of all points $M$, $N'$, $M'$, and $N$ when $A$ moves around the whole circle.

2010 NZMOC Camp Selection Problems, 1

Tags: recurrence relation , sequence , algebra

For any two positive real numbers $x_0 > 0$, $x_1 > 0$, a sequence of real numbers is defined recursively by $$x_{n+1} =\frac{4 \max\{x_n, 4\}}{x_{n-1}}$$ for $n \ge 1$. Find $x_{2010}$.

2010 All-Russian Olympiad, 1

Tags: algebra , induction , topology , geometry , 3d geometry , sphere , quadratic

Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.

2009 Belarus Team Selection Test, 2

Tags: geometry , circumcircle , reflection

In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$. [i]Proposed by Davood Vakili, Iran[/i]

2012 NZMOC Camp Selection Problems, 2

Tags: geometry , similar triangles , trapezoid

Let $ABCD$ be a trapezoid, with $AB \parallel CD$ (the vertices are listed in cyclic order). The diagonals of this trapezoid are perpendicular to one another and intersect at $O$. The base angles $\angle DAB$ and $\angle CBA$ are both acute. A point $M$ on the line sgement $OA$ is such that $\angle BMD = 90^o$, and a point $N$ on the line segment $OB$ is such that $\angle ANC = 90^o$. Prove that triangles $OMN$ and $OBA$ are similar.

OIFMAT III 2013, 2

Tags: combinatorial geometry , geometry

We will say that a set $ A $ of points is [i]disastrous [/i] if it meets the following conditions: $\bullet$ There are no $ 3 $ collinear points $\bullet$ There is not a trio of mutually equal distances between points. If $ P $ and $ Q $ are points in $ A $, then there are $ M $, $ N $, $ R $ and $ T $ in $ A $ such that: $$ d (P, Q) = \frac {d (M, N) + d (R, T)} {2} $$ Show that all disastrous sets are infinite. [hide=original wording of second condition]No existe ni un trío de distancias entre puntos mutuamente iguales. [/hide]

2021 Novosibirsk Oral Olympiad in Geometry, 4

Tags: geometry , angle bisector , angle

Angle bisectors $AD$ and $BE$ are drawn in triangle $ABC$. It turned out that $DE$ is the bisector of triangle $ADC$. Find the angle $BAC$.

2019 Germany Team Selection Test, 2

Tags: algebra

Does there exist a subset $M$ of positive integers such that for all positive rational numbers $r<1$ there exists exactly one finite subset of $M$ like $S$ such that sum of reciprocals of elements in $S$ equals $r$.

2017 Poland - Second Round, 6

Tags: prime number , number theory

A prime number $p > 2$ and $x,y \in \left\{ 1,2,\ldots, \frac{p-1}{2} \right\}$ are given. Prove that if $x\left( p-x\right)y\left( p-y\right)$ is a perfect square, then $x = y$.

2008 Junior Balkan Team Selection Tests - Romania, 1

Tags: projective geometry , geometry

Let $ ABCD$ be a convex quadrilateral with opposite side not parallel. The line through $ A$ parallel to $ BD$ intersect line $ CD$ in $ F$, but parallel through $ D$ to $ AC$ intersect line $ AB$ at $ E$. Denote by $ M,N,P,Q$ midpoints of the segments $ AC,BD,AF,DE$. Prove that lines $ MN,PQ$ and $ AD$ are concurrent.

2002 All-Russian Olympiad Regional Round, 11.6

Tags: vector , combinatorics , combinatorial geometry

There are $n > 1$ points on the plane. Two take turns connecting more an unconnected pair of points by a vector of one of two possible directions. If after the next move of a player the sum of all drawn vectors is zero, then the second one wins; if it's another move is impossible, and there was no zero sum, then the first one wins. Who wins when played correctly?

2006 Stanford Mathematics Tournament, 9

Tags: geometry , pythagorean theorem

$\triangle ABC$ has $AB=AC$. Points $M$ and $N$ are midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. The medians $\overline{MC}$ and $\overline{NB}$ intersect at a right angle. Find $(\tfrac{AB}{BC})^2$.

1939 Eotvos Mathematical Competition, 1

Tags: algebra , inequalities

Let $a_1$, $a_2$, $b_1$, $b_2$, $c_1$ and $c_2$ be real numbers for which $a_1a_2 > 0$, $a_1c_1 \ge b^2_1$ and $a_2c_2 > b^2_2$. Prove that $$(a_1 + a_2)(c_1 + c_2) \ge (b_1 + b_2)^2$$

2019 Irish Math Olympiad, 5

Tags: geometry , reflection , concyclic , circumcenter

Let $M$ be a point on the side $BC$ of triangle $ABC$ and let $P$ and $Q$ denote the circumcentres of triangles $ABM$ and $ACM$ respectively. Let $L$ denote the point of intersection of the extended lines $BP$ and $CQ$ and let $K$ denote the reflection of $L$ through the line $PQ$. Prove that $M, P, Q$ and $K$ all lie on the same circle.

2002 All-Russian Olympiad Regional Round, 11.2

Tags: geometry , 3d geometry , pyramid

The altitude of a quadrangular pyramid $SABCD$ passes through the intersection point of the diagonals of its base $ABCD$. From the tops of the base perpendiculars $AA_1$, $BB_1$, $CC_1$, $DD_1$ are dropped onto lines $SC$, $SD,$ $SA$ and $SB$ respectively. It turned out that the points $S$, $A_1$, $B_1$, $C_1$, $D_1$ are different and lie on the same sphere. Prove that lines $AA_1$, $ BB_1$, $CC_1$, $DD_1$ pass through one point.

2008 Baltic Way, 3

Tags: trigonometry , algebra , arithmetic sequence , relatively prime

Does there exist an angle $ \alpha\in(0,\pi/2)$ such that $ \sin\alpha$, $ \cos\alpha$, $ \tan\alpha$ and $ \cot\alpha$, taken in some order, are consecutive terms of an arithmetic progression?

1999 Baltic Way, 18

Tags: modular arithmetic , number theory

Let $m$ be a positive integer such that $m=2\pmod{4}$. Show that there exists at most one factorization $m=ab$ where $a$ and $b$ are positive integers satisfying \[0<a-b<\sqrt{5+4\sqrt{4m+1}}\]

1967 Polish MO Finals, 4

Tags: algebra , polynomial

Prove that the polynomial $ x^3 + x + 1 $ is a factor of the polynomial $ P_n(x) = x^{n + 2} + (x+1)^{2n+1} $ for every integer $ n \geq 0 $.

2006 Purple Comet Problems, 3

Tags:

Find the sum of all the prime numbers less than $100$ which are one more than a multiple of six.

2017 Math Prize for Girls Problems, 9

Tags:

Say that a positive integer $n$ is [i]smooth[/i] if $\frac{1}{n}$ has a terminating decimal expansion. (Note that 1 is smooth.) Compute the value of the infinite series \[ \sum_n \frac{1}{n^3} \, , \] where $n$ ranges over all smooth positive integers.

2010 Stanford Mathematics Tournament, 10

Tags: algebra , polynomial , vieta

Find the sum of all solutions of the equation $\frac{1}{x^2-1}+\frac{2}{x^2-2}+\frac{3}{x^2-3}+\frac{4}{x^2-4}=2010x-4$