Olympiad problems

2007 ITAMO, 1

It is given a regular hexagon in the plane. Let P be a point of the plane. Define s(P) as the sum of the distances from P to each side of the hexagon, and v(P) as the sum of the distances from P to each vertex. a) Find the locus of points P that minimize s(P) b) Find the locus of points P that minimize v(P)

2019 Dutch IMO TST, 3

Tags: maximum , gcd , greatest common divisor , number theory

Let $n$ be a positive integer. Determine the maximum value of $gcd(a, b) + gcd(b, c) + gcd(c, a)$ for positive integers $a, b, c$ such that $a + b + c = 5n$.

2019 SIMO, Q2

Tags: combinatorics

Fix a convex $n > 3$ gon $A_{1}A_{2}...A_{n} $ and connect every two points with a road. Call this $n$-gon [i]crossy[/i] if no three roads intersect at a point inside the polygon. This $n$-gon is partitioned into a set $S$ of disjoint polygons formed by the roads. Label every intersection with an integer such that $A_{1}$ is non-zero. Call the labelling [i]basic[/i] if for every polygon in $S$, the sum of the labels of its vertices is $0$. $(a)$ Prove that there is a [i]basic[/i] labelling of a crossy $n$-gon when $n$ is even. $(b)$ Prove that there is no [i]basic[/i] labelling of a crossy $n$-gon when $n$ is odd.

2003 Polish MO Finals, 1

Tags: geometry , circumcircle , geometric transformation , rotation , homothety , trapezoid , symmetry

In an acute-angled triangle $ABC, CD$ is the altitude. A line through the midpoint $M$ of side $AB$ meets the rays $CA$ and $CB$ at $K$ and $L$ respectively such that $CK = CL.$ Point $S$ is the circumcenter of the triangle $CKL.$ Prove that $SD = SM.$

2014 Argentina National Olympiad Level 2, 6

Tags: greatest common divisor , number theory

Let $a, b, c$ be distinct positive integers with sum $547$ and let $d$ be the greatest common divisor of the three numbers $ab+1, bc+1, ca+1$. Find the maximal possible value of $d$.

2016 Germany Team Selection Test, 1

Tags: floor function , number theory , sequence

Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

2017 India IMO Training Camp, 1

Tags: combinatorics

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

1986 Vietnam National Olympiad, 3

Tags: combinatorics

A sequence of positive integers is constructed as follows: the first term is $ 1$, the following two terms are $ 2$, $ 4$, the following three terms are $ 5$, $ 7$, $ 9$, the following four terms are $ 10$, $ 12$, $ 14$, $ 16$, etc. Find the $ n$-th term of the sequence.

2013 Romania National Olympiad, 3

Tags: inradius , circumcircle , analytic geometry , perimeter , geometry

Given $P$ a point m inside a triangle acute-angled $ABC$ and $DEF$ intersections of lines with that $AP$, $BP$, $CP$ with$\left[ BC \right],\left[ CA \right],$respective $\left[ AB \right]$ a) Show that the area of the triangle $DEF$ is at most a quarter of the area of the triangle $ABC$ b) Show that the radius of the circle inscribed in the triangle $DEF$ is at most a quarter of the radius of the circle circumscribed on triangle $4ABC.$

2014 Chile National Olympiad, 3

Tags: point , combinatorics , combinatorial geometry

In the plane there are $2014$ plotted points, such that no $3$ are collinear. For each pair of plotted points, draw the line that passes through them. prove that for every three of marked points there are always two that are separated by an amount odd number of lines.

1966 AMC 12/AHSME, 3

Tags:

If the arithmetic mean of two numbers is $6$ and thier geometric mean is $10$, then an equation with the given two numbers as roots is: $\text{(A)} \ x^2+12x+100=0 ~~ \text{(B)} \ x^2+6x+100=0 ~~ \text{(C)} \ x^2-12x-10=0$ $\text{(D)} \ x^2-12x+100=0 \qquad \text{(E)} \ x^2-6x+100=0$

2015 Iran Team Selection Test, 5

Tags: number theory , combinatorics

We call a permutation $(a_1, a_2,\cdots , a_n)$ of the set $\{ 1,2,\cdots, n\}$ "good" if for any three natural numbers $i <j <k$, $n\nmid a_i+a_k-2a_j$ find all natural numbers $n\ge 3$ such that there exist a "good" permutation of a set $\{1,2,\cdots, n\}$.

2017 Switzerland - Final Round, 1

Tags: geometry , angle bisector , circles , equal segments

Let $A$ and $B$ be points on the circle $k$ with center $O$, so that $AB> AO$. Let $C$ be the intersection of the bisectors of $\angle OAB$ and $k$, different from $A$. Let $D$ be the intersection of the straight line $AB$ with the circumcircle of the triangle $OBC$, different from $B$. Show that $AD = AO$ .

2017 Math Prize for Girls Problems, 19

Tags:

Up to similarity, there is a unique nondegenerate convex equilateral 13-gon whose internal angles have measures that are multiples of 20 degrees. Find it. Give your answer by listing the degree measures of its 13 [i]external[/i] angles in clockwise or counterclockwise order. Start your list with the biggest external angle. You don't need to write the degree symbol $^\circ$.

2023 Iran MO (3rd Round), 2

Tags: combinatorics

Find the number of permutations of $\{1,2,...,n\}$ like $\{a_1,...,a_n\}$ st for each $1 \leq i \leq n$: $$a_i | 2i$$

2025 Bangladesh Mathematical Olympiad, P9

Tags: combinatorics , construction , game

Suppose there are several juice boxes, one of which is poisoned. You have $n$ guinea pigs to test the boxes. The testing happens in the following way: [list] [*] At each round, you can have the guinea pigs taste any number of juice boxes. [*] Conversely, a juice box can be tasted by any number of guinea pigs. [*] After the round ends, any guinea pigs who tasted the poisoned juice die. [/list] Suppose you have to find the poisoned juice box in at most $k$ rounds. What is the maximum number of juice boxes such that it is possible?

2012 NIMO Problems, 8

Tags: ratio , asymptote , geometric series

Points $A$, $B$, and $O$ lie in the plane such that $\measuredangle AOB = 120^\circ$. Circle $\omega_0$ with radius $6$ is constructed tangent to both $\overrightarrow{OA}$ and $\overrightarrow{OB}$. For all $i \ge 1$, circle $\omega_i$ with radius $r_i$ is constructed such that $r_i < r_{i - 1}$ and $\omega_i$ is tangent to $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\omega_{i - 1}$. If \[ S = \sum_{i = 1}^\infty r_i, \] then $S$ can be expressed as $a\sqrt{b} + c$, where $a, b, c$ are integers and $b$ is not divisible by the square of any prime. Compute $100a + 10b + c$. [i]Proposed by Aaron Lin[/i]

2012 Math Prize For Girls Problems, 5

Tags: similar triangles , geometry

The figure below shows a semicircle inscribed in a right triangle. [asy] draw((0, 0) -- (15, 0) -- (0, 8) -- cycle); real r = 120 / 23; real theta = -aTan(8/15); draw(arc((r, r), r, theta + 180, theta + 360)); [/asy] The triangle has legs of length 8 and 15. The semicircle is tangent to the two legs, and its diameter is on the hypotenuse. What is the radius of the semicircle?

2021 AMC 10 Fall, 19

Tags: function

Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. What is$$f(2) + f(3) + f(4) + f(5)+ f(6)?$$ $(\textbf{A})\: 8\qquad(\textbf{B}) \: 9\qquad(\textbf{C}) \: 11\qquad(\textbf{D}) \: 22\qquad(\textbf{E}) \: 29$

1987 All Soviet Union Mathematical Olympiad, 444

Tags: square table , game , game strategy , combinatorics

The "Sea battle" game. a) You are trying to find the $4$-field ship -- a rectangle $1x4$, situated on the $7x7$ playing board. You are allowed to ask a question, whether it occupies the particular field or not. How many questions is it necessary to ask to find that ship surely? b) The same question, but the ship is a connected (i.e. its fields have common sides) set of $4$ fields.

2021 LMT Fall, 3

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Two circles with radius $2$, $\omega_1$ and $\omega_2$, are centered at $O_1$ and $O_2$ respectively. The circles $\omega_1$ and $\omega_2$ are externally tangent to each other and internally tangent to a larger circle $\omega$ centered at $O$ at points $A$ and $B$, respectively. Let $M$ be the midpoint of minor arc $AB$. Let $P$ be the intersection of $\omega_1$ and $O_1M$, and let $Q$ be the intersection of $\omega_2$ and $O_2M$. Given that there is a point $R$ on $\omega$ such that $\triangle PQR$ is equilateral, the radius of $\omega$ can be written as $\frac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers and $a$ and $c$ are relatively prime. Find $a+b+c$.

1999 IMC, 1

Tags: superior algebra

Let $R$ be a ring where $\forall a\in R: a^2=0$. Prove that $abc+abc=0$ for all $a,b,c\in R$.

2021 Israel TST, 2

Tags: combinatorial geometry , combinatorics , distance

Let $n>1$ be an integer. Hippo chooses a list of $n$ points in the plane $P_1, \dots, P_n$; some of these points may coincide, but not all of them can be identical. After this, Wombat picks a point from the list $X$ and measures the distances from it to the other $n-1$ points in the list. The average of the resulting $n-1$ numbers will be denoted $m(X)$. Find all values of $n$ for which Hippo can prepare the list in such a way, that for any point $X$ Wombat may pick, he can point to a point $Y$ from the list such that $XY=m(X)$.

2014 Swedish Mathematical Competition, 3

Tags: algebra , functional equation , functional

Determine all functions $f: \mathbb R \to \mathbb R$, such that $$ f (f (x + y) - f (x - y)) = xy$$ for all real $x$ and $y$.

2014 ITAMO, 3

Tags: greatest common divisor , modular arithmetic , number theory

For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers. (a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$. (b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.