Olympiad problems

1976 Miklós Schweitzer, 8

Prove that the set of all linearly combinations (with real coefficients) of the system of polynomials $ \{ x^n\plus{}x^{n^2} \}_{n\equal{}0}^{\infty}$ is dense in $ C[0,1]$. [i]J. Szabados[/i]

1999 Harvard-MIT Mathematics Tournament, 6

Tags: geometry

A sphere of radius $1$ is covered in ink and rolling around between concentric spheres of radii $3$ and $5$. If this process traces a region of area 1 on the larger sphere, what is the area of the region traced on the smaller sphere?

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

Tags:

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 $4ABC.$