Olympiad problems

2016 Azerbaijan Balkan MO TST, 3

Tags: combinatorics

$k$ is a positive integer. $A$ company has a special method to sell clocks. Every customer can reason with two customers after he has bought a clock himself $;$ it's not allowed to reason with an agreed person. These new customers can reason with other two persons and it goes like this.. If both of the customers agreed by a person could play a role (it can be directly or not) in buying clocks by at least $k$ customers, this person gets a present. Prove that, if $n$ persons have bought clocks, then at most $\frac{n}{k+2}$ presents have been accepted.

2015 Purple Comet Problems, 8

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In the ﬁgure below $\angle$LAM = $\angle$LBM = $\angle$LCM = $\angle$LDM, and $\angle$AEB = $\angle$BFC = $\angle$CGD = 34 degrees. Given that $\angle$KLM = $\angle$KML, ﬁnd the degree measure of $\angle$AEF. This is #8 on the 2015 Purple comet High School. For diagram go to http://www.purplecomet.org/welcome/practice

2022 HMNT, 5

Tags: number theory

A triple of positive integers $(a, b, c)$ is [i]tasty [/i] if $lcm (a, b, c) | a + b + c - 1$ and $a < b < c$. Find the sum of $a + b + c$ across all tasty triples.

2003 Polish MO Finals, 3

Tags: polynomial , algebra

Find all polynomials $W$ with integer coefficients satisfying the following condition: For every natural number $n, 2^n - 1$ is divisible by $W(n).$

2016 NIMO Problems, 3

Tags: geometry , incenter

Right triangle $ABC$ has hypotenuse $AB = 26$, and the inscribed circle of $ABC$ has radius $5$. The largest possible value of $BC$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are both positive integers. Find $100m + n$. [i]Proposed by Jason Xia[/i]

1980 Bundeswettbewerb Mathematik, 2

Tags: number theory , linear algebra , combinatorics

Prove that from every set of $n+1$ natural numbers, whose prime factors are in a given set of $n$ prime numbers, one can select several distinct numbers whose product is a perfect square.

1971 IMO Shortlist, 3

Tags: polynomial , algebra , system of equations , vieta

Knowing that the system \[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\] has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.

1987 AMC 8, 18

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Half the people in a room left. One third of those remaining started to dance. There were then $12$ people who were not dancing. The original number of people in the room was $\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 42 \qquad \text{(E)}\ 72$

2004 Vietnam National Olympiad, 1

Tags: system of equations , algebra

Solve the system of equations $ \begin{cases} x^3 \plus{} x(y \minus{} z)^2 \equal{} 2\\ y^3 \plus{} y(z \minus{} x)^2 \equal{} 30\\ z^3 \plus{} z(x \minus{} y)^2 \equal{} 16\end{cases}$.

2008 Tournament Of Towns, 2

Tags: segment , ratio , combinatorial geometry , combinatorics

There are ten congruent segments on a plane. Each intersection point divides every segment passing through it in the ratio $3:4$. Find the maximum number of intersection points.

1983 IMO Shortlist, 17

Tags: geometry , geometric inequality , optimization , minimization , maximization , point set

Let $P_1, P_2, \dots , P_n$ be distinct points of the plane, $n \geq 2$. Prove that \[ \max_{1\leq i<j\leq n} P_iP_j > \frac{\sqrt 3}{2}(\sqrt n -1) \min_{1\leq i<j\leq n} P_iP_j \]

2001 Mongolian Mathematical Olympiad, Problem 5

Tags: geometry , hexagon , projective geometry

Let $A,B,C,D,E,F$ be the midpoints of consecutive sides of a hexagon with parallel opposite sides. Prove that the points $AB\cap DE$, $BC\cap EF$, $AC\cap DF$ lie on a line.

2023 Princeton University Math Competition, 5

Tags: number theory

5. Let $S$ denote the set of all positive integers whose prime factors are elements of $\{2,3,5,7,11\}$. (We include 1 in the set $S$.) If $$ \sum_{q \in S} \frac{\varphi(q)}{q^{2}} $$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, find $a+b$. (Here $\varphi$ denotes Euler's totient function.)

2015 Singapore MO Open, 1

Tags: geometry

In an acute-angled triangle $\triangle ABC$, D is the point on BC such that AD bisects ∠BAC, E and F are the feet of the perpendiculars from D onto AB and AC respectively. The segments BF and CE intersect at K. Prove that AK is perpendicular to BC.

2007 ITAMO, 1

Tags: inequalities , circumcircle , triangle inequality , geometry

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

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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: $\text{(A)} \ x^2+12x+100=0 ~~ \text{(B)} \ x^2+6x+100=0 ~~ \text{(C)} \ x^2-12x-10=0$ $\text{(D)} \ x^2-12x+100=0 \qquad \text{(E)} \ x^2-6x+100=0$