Olympiad problems

2013 IPhOO, 1

A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$. The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant velocity, this time $v_3$. The value of $\frac{v_1}{v_3}+\frac{v_3}{v_1}$ can be expressed as a positive number $\frac{m\sqrt{r}}{n}$, where $m$ and $n$ are relatively prime, and $r$ is not divisible by the square of any prime. Find $m+n+r$. If the number is rational, let $r=1$. [i](Ahaan Rungta, 2 points)[/i]

2011 Akdeniz University MO, 3

Tags: inequalities

Let $a,b,c$ positive reals such that $a+b+c=3$. Show that following expression's minimum value is $2$. $$\frac{\sqrt a +\sqrt b +\sqrt c}{ab+bc+ca} + \frac{1}{1+2\sqrt {ab}} + \frac {1}{1+ 2\sqrt {bc}} + \frac{1}{1+ 2\sqrt {ca}}$$

2013 Sharygin Geometry Olympiad, 6

Tags: geometry

Dear Mathlinkers, 1. A, B the end points of an arch circle 2. (O) a circle tangent to AB intersecting the arch in question 3. T the point of contact of (O) and AB 4. C, D the points of intersection of (O) with the arch in the order A, D, C, B 5. E, F the points of intersection of AC and DT, BD and CT. Prove : EF is parallel to AB. Sincerely Jean-Louis

2010 Contests, 4

Tags: inequalities

Let's consider the inequality $ a^3\plus{}b^3\plus{}c^3<k(a\plus{}b\plus{}c)(ab\plus{}bc\plus{}ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number. [b]a)[/b] Prove the inequality for $ k\equal{}1$. [b]b) [/b]Find the smallest value of $ k$ such that the inequality holds for all triangles.

1988 Tournament Of Towns, (196) 3

Tags: polyhedron , combinatorial geometry , combinatorics

Prove that for each vertex of a polyhedron it is possible to attach a natural number so that for each pair of vertices with a common edge, the attached numbers are not relatively prime (i.e. they have common divisors), and with each pair of vertices without a common edge the attached numbers are relatively prime. (Note: there are infinitely many prime numbers.)

1969 Czech and Slovak Olympiad III A, 5

Tags: geometry , conic , locus

Two perpendicular lines $p,q$ and a point $A\notin p\cup q$ are given in plane. Find locus of all points $X$ such that \[XA=\sqrt{|Xp|\cdot|Xq|\,},\] where $|Xp|$ denotes the distance of $X$ from $p.$

2011 Today's Calculation Of Integral, 712

Tags: trigonometry , logarithm , integration , calculus , calculus computations

Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \left\{\frac{1}{\tan x\ (\ln \sin x)}+\frac{\tan x}{\ln \cos x}\right\}\ dx.$

2019 IMEO, 2

Tags: graph theory , combinatorics

Consider some graph $G$ with $2019$ nodes. Let's define [i]inverting[/i] a vertex $v$ the following process: for every other vertex $u$, if there was an edge between $v$ and $u$, it is deleted, and if there wasn't, it is added. We want to minimize the number of edges in the graph by several [i]invertings[/i] (we are allowed to invert the same vertex several times). Find the smallest number $M$ such that we can always make the number of edges in the graph not larger than $M$, for any initial choice of $G$. [i]Proposed by Arsenii Nikolaev, Anton Trygub (Ukraine)[/i]

1992 APMO, 5

Tags: inequalities , algebra

Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.

2007 Sharygin Geometry Olympiad, 2

Tags: diagonal , isosceles , isosceles triangle , quadrilateral , geometry

Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?

2019 Peru IMO TST, 4

Tags: sequence , floor function , number theory

Let $k\geq 0$ an integer. The sequence $a_0,\ a_1,\ a_2, \ a_3, \ldots$ is defined as follows: [LIST] [*] $a_0=k$ [/*] [*] For $n\geq 1$, we have that $a_n$ is the smallest integer greater than $a_{n-1}$ so that $a_n+a_{n-1}$ is a perfect square. [/*] [/LIST] Prove that there are exactly $\left \lfloor{\sqrt{2k}} \right \rfloor$ positive integers that cannot be written as the difference of two elements of such a sequence. [i]Note.[/i] If $x$ is a real number, $\left \lfloor{x} \right \rfloor$ denotes the greatest integer smaller or equal than $x$.

2009 All-Russian Olympiad, 6

Tags: geometry , geometric transformation , combinatorics

There are $ k$ rooks on a $ 10 \times 10$ chessboard. We mark all the squares that at least one rook can capture (we consider the square where the rook stands as captured by the rook). What is the maximum value of $ k$ so that the following holds for some arrangement of $ k$ rooks: after removing any rook from the chessboard, there is at least one marked square not captured by any of the remaining rooks.

2020-2021 OMMC, 1

Tags: number theory

Find the remainder when $$20^{20}+21^{21}-21^{20}-20^{21}$$ is divided by $100$.

2010 Thailand Mathematical Olympiad, 6

Tags: prime , divide , number theory

Show that no triples of primes $p, q, r$ satisfy $p > r, q > r$, and $pq | r^p + r^q$

KoMaL A Problems 2017/2018, A. 711

Tags: combinatorial geometry , combinatorics

For which pairs $(m,n)$ does there exist an injective function $f:\mathbb{R}^2\to\mathbb{R}^2$ under which the image of every regular $m$-gon is a regular $n$-gon. (Note that $m,n\geq 3$, and that by a regular $N$-gon we mean the union of the boundary segments, not the closed polygonal region.) [i]Proposed by Sutanay Bhattacharya, Bishnupur, India[/i]

2019 Jozsef Wildt International Math Competition, W. 27

Tags: functional equation , function , integration

Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ such that$$f(-x)+\int \limits_0^xtf(x-t)dt=x,\ \forall\ x\in \mathbb{R}$$

2016 China Northern MO, 6

Tags: geometry

Four points $B,E,A,F$ lie on line $AB$ in order, four points $C,G,D,H$ lie on line $CD$ in order, satisfying: $$\frac{AE}{EB}=\frac{AF}{FB}=\frac{DG}{GC}=\frac{DH}{HC}=\frac{AD}{BC}.$$ Prove that $FH\perp EG$.

2023 239 Open Mathematical Olympiad, 8

Tags: derivative , inequalities , algebra

Let $r\geqslant 0$ be a real number and define $f(x)=1/(1+x^2)^r$. Prove that \[|f^{(k)}(x)|\leqslant\frac{2r\cdot(2r+1)\cdots(2r+k-1)}{(1+x^2)^{r+k/2}},\]for every natural number $k{}$. Here, $f^{(k)}(x)$ denotes the $k^{\text{th}}$ derivative of $f$.

1999 IMO Shortlist, 6

Tags: ramsey theory , combinatorics , function , linear algebra

Suppose that every integer has been given one of the colours red, blue, green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$. Show that there are two integers of the same colour whose difference has one of the following values: $x,y,x+y$ or $x-y$.

2024 Kosovo Team Selection Test, P1

Tags: prime , number theory

Find all prime numbers $p$ and $q$ such that $p^q + 5q - 2$ is also a prime number.

2007 Switzerland - Final Round, 2

Tags: divisible , divide , number theory

Let $a, b, c$ be three integers such that $a + b + c$ is divisible by $13$. Prove that $$a^{2007}+b^{2007}+c^{2007}+2 \cdot 2007abc$$ is divisible by $13$.

1964 Putnam, A2

Tags: system of equations , integration

Find all continuous positive functions $f(x)$, for $0\leq x \leq 1$, such that $$\int_{0}^{1} f(x)\; dx =1, $$ $$\int_{0}^{1} xf(x)\; dx =\alpha,$$ $$\int_{0}^{1} x^2 f(x)\; dx =\alpha^2, $$ where $\alpha$ is a given real number.

1999 Tournament Of Towns, 6

Tags: rectangle , geometry

Inside a rectangular piece of paper $n$ rectangular holes with sides parallel to the sides of the paper have been cut out. Into what minimal number of rectangular pieces (without holes) is it always possible to cut this piece of paper? (A Shapovalov)

2000 Moldova National Olympiad, Problem 8

Tags: geometric inequality , geometry

A circle with radius $r$ touches the sides $AB,BC,CD,DA$ of a convex quadrilateral $ABCD$ at $E,F,G,H$, respectively. The inradii of the triangles $EBF,FCG,GDH,HAE$ are equal to $r_1,r_2,r_3,r_4$. Prove that $$r_1+r_2+r_3+r_4\ge2\left(2-\sqrt2\right)r.$$

1998 Harvard-MIT Mathematics Tournament, 7

Tags: vieta , algebra , polynomial

Given that three roots of $f(x)=x^4+ax^2+bx+c$ are $2$, $-3$, and $5$, what is the value of $a+b+c$?