Olympiad problems

1999 Moldova Team Selection Test, 3

Tags: function

The fuction $f(0,\infty)\rightarrow\mathbb{R}$ verifies $f(x)+f(y)=2f(\sqrt{xy}), \forall x,y>0$. Show that for every positive integer $n>2$ the following relation takes place $$f(x_1)+f(x_2)+\ldots+f(x_n)=nf(\sqrt[n]{x_1x_2\ldots x_n}),$$ for every positive integers $x_1,x_2,\ldots,x_n$.

2007 Germany Team Selection Test, 2

Tags: geometry , trapezoid , circumcircle , ratio , IMO Shortlist , homothety , Hi

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

2007 Romania Team Selection Test, 1

Tags: geometry , incenter , induction , cyclic quadrilateral , geometry proposed

In a circle with center $O$ is inscribed a polygon, which is triangulated. Show that the sum of the squares of the distances from $O$ to the incenters of the formed triangles is independent of the triangulation.

1999 Tournament Of Towns, 1

Tags: algebra , geometry , speed

A father and his son are skating around a circular skating rink. From time to time, the father overtakes the son. After the son starts skating in the opposite direction, they begin to meet five times more often. What is the ratio of the skating speeds of the father and the son? (Tairova)

2010 Baltic Way, 8

Tags: combinatorics proposed , combinatorics

In a club with $30$ members, every member initially had a hat. One day each member sent his hat to a different member (a member could have received more than one hat). Prove that there exists a group of $10$ members such that no one in the group has received a hat from another one in the group.

2017 Online Math Open Problems, 23

Tags: Online Math Open

Determine the number of ordered quintuples $(a,b,c,d,e)$ of integers with $0\leq a<$ $b<$ $c<$ $d<$ $e\leq 30$ for which there exist polynomials $Q(x)$ and $R(x)$ with integer coefficients such that \[x^a+x^b+x^c+x^d+x^e=Q(x)(x^5+x^4+x^2+x+1)+2R(x).\] [i]Proposed by Michael Ren[/i]

2000 Croatia National Olympiad, Problem 1

Tags: geometry

Let $B$ and $C$ be fixed points, and let $A$ be a variable point such that $\angle BAC$ is fixed. The midpoints of $AB$ and $AC$ are $D$ and $E$ respectively, and $F,G$ are points such that $DF\perp AB$, $EG\perp AC$ and $BF$ and $CG$ are perpendicular to $BC$. Prove that $BF\cdot CG$ remains constant as $A$ varies.

1964 Putnam, A1

Tags: inequalities , combinatorics , geometry

Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.

1987 Bulgaria National Olympiad, Problem 4

Tags: Sequences , algebra , recurrence relation

The sequence $(x_n)_{n\in\mathbb N}$ is defined by $x_1=x_2=1$, $x_{n+2}=14x_{n+1}-x_n-4$ for each $n\in\mathbb N$. Prove that all terms of this sequence are perfect squares.

Russian TST 2015, P2

Tags: geometry , circles

In the isosceles triangle $ABC$ where $AB = AC$, the point $I{}$ is the center of the inscribed circle. Through the point $A{}$ all the rays lying inside the angle $BAC$ are drawn. For each such ray, we denote by $X{}$ and $Y{}$ the points of intersection with the arc $BIC$ and the straight line $BC$ respectively. The circle $\gamma$ passing through $X{}$ and $Y{}$, which touches the arc $BIC$ at the point $X{}$ is considered. Prove that all the circles $\gamma$ pass through a fixed point.

2013 BMT Spring, P2

Tags: Polynomials , algebra , inequalities

If $f(x)=x^n-7x^{n-1}+17x^{n-2}+a_{n-3}x^{n-3}+\ldots+a_0$ is a real-valued function of degree $n>2$ with all real roots, prove that no root has value greater than $4$ and at least one root has value less than $0$ or greater than $2$.

2017 Mid-Michigan MO, 10-12

Tags: Mid-Michigan MO , algebra , geometry , combinatorics , number theory

[b]p1.[/b] In the group of five people any subgroup of three persons contains at least two friends. Is it possible to divide these five people into two subgroups such that all members of any subgroup are friends? [b]p2.[/b] Coefficients $a,b,c$ in expression $ax^2+bx+c$ are such that $b-c>a$ and $a \ne 0$. Is it true that equation $ax^2+bx+c=0$ always has two distinct real roots? [b]p3.[/b] Point $D$ is a midpoint of the median $AF$ of triangle $ABC$. Line $CD$ intersects $AB$ at point $E$. Distances $|BD|=|BF|$. Show that $|AE|=|DE|$. [b]p4.[/b] Real numbers $a,b$ satisfy inequality $a+b^5>ab^5+1$. Show that $a+b^7>ba^7+1$. [b]p5.[/b] A positive number was rounded up to the integer and got the number that is bigger than the original one by $28\%$. Find the original number (find all solutions). [b]p6.[/b] Divide a $5\times 5$ square along the sides of the cells into $8$ parts in such a way that all parts are different. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

1983 All Soviet Union Mathematical Olympiad, 366

Tags: vector , geometry , areas

Given a point $O$ inside triangle $ABC$ . Prove that $$S_A * \overrightarrow{OA} + S_B * \overrightarrow{OB} + S_C * \overrightarrow{OC} = \overrightarrow{0}$$ where $S_A, S_B, S_C$ denote areas of triangles $BOC, COA, AOB$ respectively.

2022 JBMO Shortlist, G4

Tags: geometry , Junior , Balkan , shortlist , concurrent

Given is an equilateral triangle $ABC$ and an arbitrary point, denoted by $E$, on the line segment $BC$. Let $l$ be the line through $A$ parallel to $BC$ and let $K$ be the point on $l$ such that $KE$ is perpendicular to $BC$. The circle with centre $K$ and radius $KE$ intersects the sides $AB$ and $AC$ at $M$ and $N$, respectively. The line perpendicular to $AB$ at $M$ intersects $l$ at $D$, and the line perpendicular to $AC$ at $N$ intersects $l$ at $F$. Show that the point of intersection of the angle bisectors of angles $MDA$ and $NFA$ belongs to the line $KE$.

2024 Silk Road, 4

Tags: algebra

Let $a_1, a_2, \ldots$ be a strictly increasing sequence of positive integers, such that for any positive integer $n$, $a_n$ is not representable in the for $\sum_{i=1}^{n-1}c_ia_i$ for $c_i \in \{0, 1\}$. For every positive integer $m$, let $f(m)$ denote the number of $a_i$ that are at most $m$. Show that for any positive integers $m, k$, we have that $$f(m) \leq a_k+\frac{m} {k+1}.$$

2024 JHMT HS, 4

Tags: algebra , 2024

Let $N_3$ be the answer to problem 3. Compute the sum of all real solutions $x$ to the equation \[ 50^x+72^x+(N_3)^x=800^x. \]

2012 All-Russian Olympiad, 1

Tags: combinatorics proposed , combinatorics

$101$ wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself. The rest do not change. Prove that at one point they will all stop changing opinions.

MathLinks Contest 3rd, 3

Tags: number theory , 3rd edition

An integer $z$ is said to be a [i]friendly [/i] integer if $|z|$ is not the square of an integer. Determine all integers $n$ such that there exists an infinite number of triplets of distinct friendly integers $(a, b, c)$ such that $n = a+b+c$ and $abc$ is the square of an odd integer.

2023 Yasinsky Geometry Olympiad, 2

Tags: geometry , equal segments

Let $BC$ and $BD$ be the tangent lines to the circle with diameter $AC$. Let $E$ be the second point of intersection of line $CD$ and the circumscribed circle of triangle $ABC$. Prove that $CD= 2DE$. (Matthew Kurskyi)

2015 Peru IMO TST, 6

Tags: floor function , IMO Shortlist , number theory , Sequence , Parity , IMO Shortlist 2014 , Hi

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

2006 Bulgaria Team Selection Test, 1

Tags: linear algebra , matrix , induction , combinatorics unsolved , combinatorics

[b]Problem 1. [/b]In the cells of square table are written the numbers $1$, $0$ or $-1$ so that in every line there is exactly one $1$, amd exactly one $-1$. Each turn we change the places of two columns or two rows. Is it possible, from any such table, after finite number of turns to obtain its opposite table (two tables are opposite if the sum of the numbers written in any two corresponding squares is zero)? [i] Emil Kolev[/i]

PEN K Problems, 5

Tags: function , arithmetic sequence , Functional Equations , pen

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]

2024 Ecuador NMO (OMEC), 4

Tags: algebra

Danielle writes a sign '+' or '-' in each of the next $64$ spaces: $$\_\_1 \_\_2 \_\_3 \_\_4 \text{ }.... \text{ }\_\_63 \_\_64=2024$$ such that the equality holds. Find the largest number of negative signs Danielle can use.

2000 Brazil Team Selection Test, Problem 3

Tags: game , combinatorics , combinatorial geometry

Consider an equilateral triangle with every side divided by $n$ points into $n+1$ equal parts. We put a marker on every of the $3n$ division points. We draw lines parallel to the sides of the triangle through the division points, and this way divide the triangle into $(n+1)^2$ smaller ones. Consider the following game: if there is a small triangle with exactly one vertex unoccupied, we put a marker on it and simultaneously take markers from the two its occupied vertices. We repeat this operation as long as it is possible. (a) If $n\equiv1\pmod3$, show that we cannot manage that only one marker remains. (b) If $n\equiv0$ or $n\equiv2\pmod3$, prove that we can finish the game leaving exactly one marker on the triangle.

1999 Belarusian National Olympiad, 8

Tags: combinatorics , invariant , algorithm , IMO Shortlist

Let $n$ be an integer greater than 2. A positive integer is said to be [i]attainable [/i]if it is 1 or can be obtained from 1 by a sequence of operations with the following properties: 1.) The first operation is either addition or multiplication. 2.) Thereafter, additions and multiplications are used alternately. 3.) In each addition, one can choose independently whether to add 2 or $n$ 4.) In each multiplication, one can choose independently whether to multiply by 2 or by $n$. A positive integer which cannot be so obtained is said to be [i]unattainable[/i]. [b]a.)[/b] Prove that if $n\geq 9$, there are infinitely many unattainable positive integers. [b]b.)[/b] Prove that if $n=3$, all positive integers except 7 are attainable.