Olympiad problems

2019 MMATHS, 1

Tags: number theory

$S$ is a set of positive integers with the following properties: (a) There are exactly $3$ positive integers missing from $S$. (b) If $a$ and $b$ are elements of $S$, then $a + b$ is an element of $S$. (We allow a and b to be the same.) Find all possibilities for the set $S$ (with proof).

2017 AIME Problems, 10

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Let $z_1 = 18 + 83i$, $z_2 = 18 + 39i, $ and $z_3 = 78 + 99i,$ where $i = \sqrt{-1}$. Let $z$ be the unique complex number with the properties that $\frac{z_3 - z_1}{z_2 - z_1} \cdot \frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.

1990 Tournament Of Towns, (259) 3

Tags: combinatorics , combinatorial geometry

A cake is prepared for a dinner party to which only $p$ or $q$ persons will come ($p$ and $q$ are given co-prime integers). Find the minimum number of pieces (not necessarily equal) into which the cake must be cut in advance so that the cake may be equally shared between the persons in either case. (D. Fomin, Leningrad)

2007 Estonia Math Open Junior Contests, 2

Tags: midpoint , equal area , geometry , parallelogram

The sides $AB, BC, CD$ and $DA$ of the convex quadrilateral $ABCD$ have midpoints $E, F, G$ and $H$. Prove that the triangles $EFB, FGC, GHD$ and $HEA$ can be put together into a parallelogram equal to $EFGH$.

2000 Croatia National Olympiad, Problem 4

Tags: number theory , floor function

Let $S$ be the set of all squarefree numbers and $n$ be a natural number. Prove that $$\sum_{k\in S}\left\lfloor\sqrt{\frac nk}\right\rfloor=n.$$

1994 IMO Shortlist, 3

Tags: modular arithmetic , prime number , combinatorial number theory , combinatorics , number theory

Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.

2013 Moldova Team Selection Test, 1

Tags: inequalities

For any positive real numbers $x,y,z$, prove that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x} \geq \frac{z(x+y)}{y(y+z)} + \frac{x(z+y)}{z(x+z)} + \frac{y(x+z)}{x(x+y)}$

2017 Puerto Rico Team Selection Test, 4

Tags: combinatorics , game , strategy , winning strategy , domino

Alberto and Bianca play a game on a square board. Alberto begins. On their turn, players place a $1 \times 2$ or $2 \times 1$ domino on two empty squares on the board. The player who cannot put a domino loses. Determine who has a winning strategy (and prove it) if the board is: i) $3 \times 3$ ii) $3 \times 4$

2017 ASDAN Math Tournament, 6

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You roll three six-sided dice. If the three dice and indistinguishable, how many combinations of numbers can result?

1990 Greece National Olympiad, 4

Tags: function , algebra , functional

Find all functions $f: \mathbb{R}^+\to\mathbb{R}$ such that $f(x+y)=f(x^2)+f(y^2)$ for any $x,y \in\mathbb{R}^+$

Estonia Open Junior - geometry, 2018.1.5

Tags: geometry , cyclic quadrilateral

Let $M$ be the intersection of the diagonals of a cyclic quadrilateral $ABCD$. Find the length of $AD$, if it is known that $AB=2$ mm , $BC = 5$ mm, $AM = 4$ mm, and $\frac{CD}{CM}= 0.6$.

2022 Austrian MO National Competition, 6

Tags: combinatorics , tiles , tiling

(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$. (b) Show that a corresponding decomposition into $30$ squares is also possible. [i](Walther Janous)[/i]

1991 ITAMO, 1

Tags: geometry , angle bisector , circumcircle

For every triangle $ABC$ inscribed in a circle $\Gamma$ , let $A',B',C'$ be the intersections of the bisectors of the angles at $A,B,C$ with $\Gamma$ . Consider the triangle $A'B'C'$ . (a) Do triangles $A'B'C'$ go over all possible triangles inscribed in $\Gamma$ as $\vartriangle ABC$ varies? If not, what are the constraints? (b) Prove that the angle bisectors of $\vartriangle ABC$ are the altitudes of $\vartriangle A',B',C'$ .

2015 Puerto Rico Team Selection Test, 2

Tags: geometry , angle , equal segments

In the triangle $ABC$, let $P$, $Q$, and $R$ lie on the sides $BC$, $AC$, and $AB$ respectively, such that $AQ = AR$, $BP = BR$ and $CP = CQ$. Let $\angle PQR=75^o$ and $\angle PRQ=35^o$. Calculate the measures of the angles of the triangle $ABC$.

Oliforum Contest IV 2013, 6

Tags: 3d geometry , sphere , geometry

Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.

LMT Speed Rounds, 2016.22

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Albert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous number he rolled, he gains a point, where his first roll does not gain him a point. Find the expected number of points that Albert receives. [i]Proposed by Nathan Ramesh

2025 Serbia Team Selection Test for the IMO 2025, 2

Tags: geometry

Let $ABC$ be an acute triangle. Let $A'$ be the reflection of point $A$ over the line $BC$. Let $O$ and $H$ be the circumcenter and the orthocenter of triangle $ABC$, respectively, and let $E$ be the midpoint of segment $OH$. Let $D$ and $L$ be the points where the reflection of line $AA'$ with respect to line $OA'$ intersects the circumcircle of triangle $ABC$, where point $D$ lies on the arc $BC$ not containing $A$. If $ M $ is a point on the line $ BC $ such that $ OM \perp AD $, prove that $ \angle MAD = \angle EAL $. [i]Proposed by Strahinja Gvozdić[/i]

2005 IberoAmerican, 3

Tags: modular arithmetic , algebra , number theory , divisibility

Let $p > 3$ be a prime. Prove that if \[ \sum_{i=1 }^{p-1}{1\over i^p} = {n\over m}, \] with $\gdc(n,m) = 1$, then $p^3$ divides $n$.

2025 CMIMC Algebra/NT, 8

Tags: algebra , number theory

Let $P(x)=x^4+20x^3+29x^2-666x+2025.$ It is known that $P(x)>0$ for every real $x.$ There is a root $r$ for $P$ in the first quadrant of the complex plane that can be expressed as $r=\frac{1}{2}(a+bi+\sqrt{c+di}),$ where $a,b,c,d$ are integers. Find $a+b+c+d.$

2016 Postal Coaching, 5

Tags: geometry , circumcenter , incenter

Let $I$ and $O$ be respectively the incentre and circumcentre of a triangle $ABC$. If $AB = 2$, $AC = 3$ and $\angle AIO = 90^{\circ}$, find the area of $\triangle ABC$.

2008 Junior Balkan Team Selection Tests - Moldova, 12

Tags: combinatorics

Natural nonzero numder, which consists of $ m$ digits, is called hiperprime, if its any segment, which consists $ 1,2,...,m$ digits is prime (for example $ 53$ is hiperprime, because numbers $ 53,3,5$ are prime). Find all hiperprime numbers.

2006 Germany Team Selection Test, 3

Tags: modular arithmetic , number theory

Is the following statement true? For each positive integer $n$, we can find eight nonnegative integers $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$ such that $n=\frac{2^a-2^b}{2^c-2^d}\cdot\frac{2^e-2^f}{2^g-2^h}$.

1997 National High School Mathematics League, 1

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Squence $(x_n)$ satisfies that $x_{n+1}=x_n-x_{n-1}(n\geq2)$. If $x_1=a,x_2=b$, $S_n=x_1+x_2+\cdots+x_n$. Wich one is correct? $\text{(A)}x_{100}=-a,S_{100}=2b-a$ $\text{(B)}x_{100}=-b,S_{100}=2b-a$ $\text{(C)}x_{100}=-a,S_{100}=b-a$ $\text{(D)}x_{100}=-b,S_{100}=b-a$

1997 Korea - Final Round, 2

Tags: trigonometry , perimeter , inequalities , geometry

The incircle of a triangle $ A_1A_2A_3$ is centered at $ O$ and meets the segment $ OA_j$ at $ B_j$ , $ j \equal{} 1, 2, 3$. A circle with center $ B_j$ is tangent to the two sides of the triangle having $ A_j$ as an endpoint and intersects the segment $ OB_j$ at $ C_j$. Prove that \[ \frac{OC_1\plus{}OC_2\plus{}OC_3}{A_1A_2\plus{}A_2A_3\plus{}A_3A_1} \leq \frac{1}{4\sqrt{3}}\] and find the conditions for equality.

2020 LMT Spring, 25

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Let $\triangle ABC$ be a triangle such that $AB=5,AC=8,$ and $\angle BAC=60^{\circ}$. Let $\Gamma$ denote the circumcircle of $ABC$, and let $I$ and $O$ denote the incenter and circumcenter of $\triangle ABC$, respectively. Let $P$ be the intersection of ray $IO$ with $\Gamma$, and let $X$ be the intersection of ray $BI$ with $\Gamma$. If the area of quadrilateral $XICP$ can be expressed as $\frac{a\sqrt{b}+c\sqrt{d}}{e}$, where $a$ and $d$ are squarefree positive integers and $\gcd(a,c,e)=1$, compute $a+b+c+d+e$.