Olympiad problems

2018 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , algebra

If $a, b, c$ are positive real numbers, prove that $$\frac{a}{\sqrt{(a + 2b)^3}}+\frac{b}{\sqrt{(b + 2c)^3}} +\frac{c} {\sqrt{(c + 2a)^3}} \ge \frac{1}{\sqrt{a + b + c}}$$ Alexandru Mihalcu

Putnam 1939, B3

Tags: Putnam

Given $a_n = (n^2 + 1) 3^n,$ find a recurrence relation $a_n + p a_{n+1} + q a_{n+2} + r a_{n+3} = 0.$ Hence evaluate $\sum_{n\geq0} a_n x^n.$

1991 Cono Sur Olympiad, 3

Tags: algebra unsolved , algebra

It is known that the number of real solutions of the following system if finite. Prove that this system has an even number of solutions: $(y^2+6)(x-1)=y(x^2+1)$ $(x^2+6)(y-1)=x(y^2+1)$

1968 Swedish Mathematical Competition, 3

Tags: geometry , Geometric Inequalities

Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When does equality hold?

2020 Brazil Team Selection Test, 6

Tags: geometry , cyberspace

Find all integers $n\geq 3$ for which the following statement is true: If $\mathcal{P}$ is a convex $n$-gon such that $n-1$ of its sides have equal length and $n-1$ of its angles have equal measure, then $\mathcal{P}$ is a regular polygon. (A [i]regular [/i]polygon is a polygon with all sides of equal length, and all angles of equal measure.) [i]Proposed by Ivan Borsenco and Zuming Feng[/i]

1984 AMC 12/AHSME, 18

Tags: analytic geometry , geometry , inradius , incenter

A point $(x,y)$ is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line $x+y = 2$. Then $x$ is A. $\sqrt{2} - 1$ B. $\frac{1}{2}$ C. $2 - \sqrt{2}$ D. 1 E. Not uniquely determined

2009 JBMO Shortlist, 4

Tags: JBMO , combinatorics

Determine all pairs of $(m, n)$ such that is possible to tile the table $ m \times n$ with figure ”corner” as in figure with condition that in that tilling does not exist rectangle (except $m \times n$) regularly covered with figures.

2007 IberoAmerican Olympiad For University Students, 7

Tags: number theory proposed , number theory

The [i]height[/i] of a positive integer is defined as being the fraction $\frac{s(a)}{a}$, where $s(a)$ is the sum of all the positive divisors of $a$. Show that for every pair of positive integers $N,k$ there is a positive integer $b$ such that the [i]height[/i] of each of $b,b+1,\cdots,b+k$ is greater than $N$.

2013 Romania Team Selection Test, 1

Tags: inequalities , trigonometry , inequalities unsolved

Let $n$ be a positive integer and let $x_1$, $\ldots$, $x_n$ be positive real numbers. Show that: \[ \min\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right )\leq 2\cos \frac{\pi}{n+2} \leq\max\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right ). \]

2019 Tuymaada Olympiad, 3

Tags: combinatorics , minimum value , minimum , Chessboard

The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to adjacent rooms. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess queen (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 2$)?

2021 Estonia Team Selection Test, 3

Tags: IMO Shortlist , number theory

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

2016 Saudi Arabia IMO TST, 2

Tags: combinatorics

Given a set of $2^{2016}$ cards with the numbers $1,2, ..., 2^{2016}$ written on them. We divide the set of cards into pairs arbitrarily, from each pair, we keep the card with larger number and discard the other. We now again divide the $2^{2015}$ remaining cards into pairs arbitrarily, from each pair, we keep the card with smaller number and discard the other. We now have $2^{2014}$ cards, and again divide these cards into pairs and keep the larger one in each pair. We keep doing this way, alternating between keeping the larger number and keeping the smaller number in each pair, until we have just one card left. Find all possible values of this final card.

2017 BMT Spring, 11

Tags: combinatorics

Naomi has a class of $100$ students who will compete with each other in five teams. Once the teams are made, each student will shake hands with every other student, except the students in his or her own team. Naomi chooses to partition the students into teams so as to maximize the number of handshakes. How many handshakes will there be?

2019 Korea USCM, 7

Tags: analysis , series , college contests

For a real number $a$ and an integer $n(\geq 2)$, define $$S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}}$$ Find every value of $a$ s.t. sequence $\{S_n(a)\}_{n\geq 2}$ converges to a positive real.

1995 Grosman Memorial Mathematical Olympiad, 3

Tags: combinatorics

Two thieves stole an open chain with $2k$ white beads and $2m$ black beads. They want to share the loot equally, by cutting the chain to pieces in such a way that each one gets $k$ white beads and $m$ black beads. What is the minimal number of cuts that is always sufficient?

2013 Korea National Olympiad, 1

Tags: geometry , circumcircle , geometry proposed

Let $P$ be a point on segment $BC$. $Q, R$ are points on $AC, AB$ such that $PQ \parallel AB $ and $ PR \parallel AC$. $O, O_{1}, O_{2} $ are the circumcenters of triangle $ABC, BPR, PCQ$. The circumcircles of $BPR, PCQ $ meet at point $K (\ne P)$. Prove that $OO_{1} = KO_{2} $.

2014 JBMO Shortlist, 4

Tags: number theory , prime numbers , AZE JBMO TST

Prove that there are not intgers $a$ and $b$ with conditions, i) $16a-9b$ is a prime number. ii) $ab$ is a perfect square. iii) $a+b$ is also perfect square.

2005 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: floor function , number theory unsolved , number theory

Find all numbers $n$ that can be expressed in the form $n=k+2\lfloor\sqrt{k}\rfloor+2$ for some nonnegative integer $k$.

2024 Tuymaada Olympiad, 8

Tags: inequalities , graph theory , graph

A graph $G$ has $n$ vertices ($n>1$). For each edge $e$ let $c(e)$ be the number of vertices of the largest complete subgraph containing $e$. Prove that the inequality (the summation is over all edges of $G$): \[\sum_{e} \frac{c(e)}{c(e)-1}\le \frac{n^2}{2}.\]

2024 Middle European Mathematical Olympiad, 8

Tags: number theory , number theory proposed , Sequence , Divisibility , constant

Let $k$ be a positive integer and $a_1,a_2,\dots$ be an infinite sequence of positive integers such that \[a_ia_{i+1} \mid k-a_i^2\] for all integers $i \ge 1$. Prove that there exists a positive integer $M$ such that $a_n=a_{n+1}$ for all integers $n \ge M$.

2017 Kyiv Mathematical Festival, 5

Tags: Kyiv mathematical festival , geometry , number theory , analytic geometry

A triangle $ABC$ is given on the plane, such that all its vertices have integer coordinates. Does there necessarily exist a straight line which intersects the straight lines $AB,$ $BC,$ and $AC$ at three distinct points with integer coordinates?

Denmark (Mohr) - geometry, 2011.2

Tags: geometry , Regular , octagon , distance

In the octagon below all sides have the length $1$ and all angles are equal. Determine the distance between the corners $A$ and $B$. [img]https://1.bp.blogspot.com/-i6TAFDvcQ8w/XzXCRhnV_kI/AAAAAAAAMVw/rKrQMfPYYJIaCwl8hhdVHdqO4fIn8O7cwCLcBGAsYHQ/s0/2011%2BMogh%2Bp2.png[/img]

2024 Saint Petersburg Mathematical Olympiad, 4

Tags: algebra , number theory , polynomial

Let's consider all possible quadratic trinomials of the form $x^2 + ax + b$, where $a$ and $b$ are positive integers not exceeding some positive integer $N$. Prove that the number of pairs of such trinomials having a common root does not exceed $N^2$.

2000 Saint Petersburg Mathematical Olympiad, 10.1

Tags: algebra , sequances

Sequences $x_1,x_2,\dots,$ and $y_1,y_2,\dots,$ are defined with $x_1=\dfrac{1}{8}$, $y_1=\dfrac{1}{10}$ and $x_{n+1}=x_n+x_n^2$, $y_{n+1}=y_n+y_n^2$. Prove that $x_m\neq y_n$ for all $m,n\in\mathbb{Z}^{+}$. [I]Proposed by A. Golovanov[/i]

1998 Bulgaria National Olympiad, 3

Tags: combinatorics proposed , combinatorics

The sides and diagonals of a regular $n$-gon $R$ are colored in $k$ colors so that: (i) For each color $a$ and any two vertices $A$,$B$ of $R$ , the segment $AB$ is of color $a$ or there is a vertex $C$ such that $AC$ and $BC$ are of color $a$. (ii) The sides of any triangle with vertices at vertices of $R$ are colored in at most two colors. Prove that $k\leq 2$.