Olympiad problems

2018 Kyiv Mathematical Festival, 2

Let $M$ be the intersection point of the medians $AD$ and $BE$ of a right triangle $ABC$ ($\angle C=90^\circ$), $\omega_1$ and $\omega_2$ be the circumcircles of triangles $AEM$ and $CDM.$ It is known that the circles $\omega_1$ and $\omega_2$ are tangent. Find the ratio in which the circle $\omega_2$ divides $AC.$

2016 Iran MO (3rd Round), 1

Tags: geometry

Let $ABC$ be an arbitrary triangle,$P$ is the intersection point of the altitude from $C$ and the tangent line from $A$ to the circumcircle. The bisector of angle $A$ intersects $BC$ at $D$ . $PD$ intersects $AB$ at $K$, if $H$ is the orthocenter then prove : $HK\perp AD$

2009 Today's Calculation Of Integral, 439

Tags: calculus , analytic geometry , integration , logarithm , inequalities , calculus computations

Find the volume of the solid defined by the inequality $ x^2 \plus{} y^2 \plus{} \ln (1 \plus{} z^2)\leq \ln 2$. Note that you may not directively use double integral here for Japanese high school students who don't study it.

2017-IMOC, N5

Tags: functional equation , fe , number theory

Find all functions $f:\mathbb N\to\mathbb N$ such that $$f(x)+f(y)\mid x^2-y^2$$holds for all $x,y\in\mathbb N$.

2017 CCA Math Bonanza, T3

Tags:

The operation $*$ is defined by $a*b=a+b+ab$, where $a$ and $b$ are real numbers. Find the value of \[\frac{1}{2}*\bigg(\frac{1}{3}*\Big(\cdots*\big(\frac{1}{9}*(\frac{1}{10}*\frac{1}{11})\big)\Big)\bigg).\] [i]2017 CCA Math Bonanza Team Round #3[/i]

1998 Singapore Team Selection Test, 3

Tags: integer polynomial , algebra , polynomial

Suppose $f(x)$ is a polynomial with integer coefficients satisfying the condition $0 \le f(c) \le 1997$ for each $c \in \{0, 1, ..., 1998\}$. Is is true that $f(0) = f(1) = ... = f(1998)$? (variation of [url=https://artofproblemsolving.com/community/c6h49788p315649]1997 IMO Shortlist p12[/url])

1990 National High School Mathematics League, 6

Tags: conic , ellipse

An ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ passes point $(2,1)$, then all points $(x,y)$ on the ellipse that $|y|>1$ are (shown as shadow) [img]https://graph.baidu.com/resource/122481219e60931bb707101582696834.jpg[/img]

2012 Dutch BxMO/EGMO TST, 3

Tags: number theory , diophantine equation

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

1961 AMC 12/AHSME, 36

Tags: pythagorean theorem , geometry

In triangle $ABC$ the median from $A$ is given perpendicular to the median from $B$. If $BC=7$ and $AC=6$, find the length of $AB$. ${{ \textbf{(A)}\ 4\qquad\textbf{(B)}\ \sqrt{17} \qquad\textbf{(C)}\ 4.25\qquad\textbf{(D)}\ 2\sqrt{5} }\qquad\textbf{(E)}\ 4.5} $

2019 Abels Math Contest (Norwegian MO) Final, 3b

Tags: functional equation , functional , function , algebra

Find all real functions $f$ defined on the real numbers except zero, satisfying $f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$

2018 Dutch IMO TST, 2

Tags: divisor , number theory

Find all positive integers $n$, for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.

PEN G Problems, 4

Tags: irrational number , absolute value

Let $a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that \[\left\vert a+b\sqrt{2}+c\sqrt{3}\right\vert > \frac{1}{10^{21}}.\]

MIPT student olimpiad spring 2024, 3

Tags: function , calculus

Is it true that if a function $f: R \to R$ is continuous and takes rational values at rational points, then at least at one point it is differentiable?

2020 Balkan MO Shortlist, C1

Tags: combinatorics

Let $s \geq 2$ and $n \geq k \geq 2$ be integes, and let $A$ be a subset of $\{1, 2, . . . , n\}^k$ of size at least $2sk^2n^{k-2}$ such that any two members of $A$ share some entry. Prove that there are an integer $p \leq k$ and $s+2$ members $A_1, A_2, . . . , A_{s+2}$ of $A$ such that $A_i$ and $A_j$ share the $p$-th entry alone, whenever $i$ and $j$ are distinct. [i]Miroslav Marinov, Bulgaria[/i]

1991 National High School Mathematics League, 1

Tags: arithmetic sequence

Set $S=\{1,2,\cdots,n\}$. $A$ is an increasing arithmetic sequence (at least two numbers), and all numbers are in $S$. Also, we can't add any number in $S$ to $A$ without changing its tolerance. Find the number of such sequence $A$.

2012 Flanders Math Olympiad, 2

Tags: perfect square , number theory

Let $n$ be a natural number. Call $a$ the smallest natural number you need to subtract from $n$ to get a perfect square. Call $b$ the smallest natural number that you must add to $n$ to get a perfect square. Prove that $n - ab$ is a perfect square.

2011 Macedonia National Olympiad, 1

Tags: inequalities

Let $~$ $ a,\,b,\,c,\,d\, >\, 0$ $~$ and $~$ $a+b+c+d\, =\, 1\, .$ $~$ Prove the inequality \[ \frac{1}{4a+3b+c}+\frac{1}{3a+b+4d}+\frac{1}{a+4c+3d}+\frac{1}{4b+3c+d}\; \ge\; 2\, . \]

2015 Turkey Team Selection Test, 3

Tags: algebra , set partition , combinatorics

Let $m, n$ be positive integers. Let $S(n,m)$ be the number of sequences of length $n$ and consisting of $0$ and $1$ in which there exists a $0$ in any consecutive $m$ digits. Prove that \[S(2015n,n).S(2015m,m)\ge S(2015n,m).S(2015m,n)\]

2011 NIMO Summer Contest, 4

Tags: inequalities

Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality \[ 1 < a < b+2 < 10. \] [i]Proposed by Lewis Chen [/i]

2014 Abels Math Contest (Norwegian MO) Final, 1b

Tags: functional , functional equation , function , algebra

Find all functions $f : R-\{0\} \to R$ which satisfy $(1 + y)f(x) - (1 + x)f(y) = yf(x/y) - xf(y/x)$ for all real $x, y \ne 0$, and which take the values $f(1) = 32$ and $f(-1) = -4$.

1999 Greece National Olympiad, 3

Tags: geometry , circumcircle

In an acute-angled triangle $ABC$, $AD,BE$ and $CF$ are the altitudes and $H$ the orthocentre. Lines $EF$ and $BC$ meet at $N$. The line passing through $D$ and parallel to $FE$ meets lines $AB$ and $AC$ at $K$ and $L$, respectively. Prove that the circumcircle of the triangle $NKL$ bisects the side $BC$.

2014 Portugal MO, 3

Tags: combinatorics

Amélia and Beatriz play battleship on a $2n\times2n$ board, using very peculiar rules. Amélia begins by choosing $n$ lines and $n$ columns of the board, placing her $n^2$ submarines on the cells that lie on their intersections. Next, Beatriz chooses a set of cells that will explode. Which is the least number of cells that Beatriz has to choose in order to assure that at least a submarine will explode?

Kvant 2021, M2663

Tags: inequality , algebra

For every positive integer $m$ prove the inquality $|\{\sqrt{m}\} - \frac{1}{2}| \geq \frac{1}{8(\sqrt m+1)} $ (The integer part $[x]$ of the number $x$ is the largest integer not exceeding $x$. The fractional part of the number $x$ is a number $\{x\}$ such that $[x]+\{x\}=x$.) A. Golovanov

2015 Balkan MO Shortlist, A6

Tags: polynomial , algebra

For a polynomials $ P\in \mathbb{R}[x]$, denote $f(P)=n$ if $n$ is the smallest positive integer for which is valid $$(\forall x\in \mathbb{R})(\underbrace{P(P(\ldots P}_{n}(x))\ldots )>0),$$ and $f(P)=0$ if such n doeas not exist. Exists polyomial $P\in \mathbb{R}[x]$ of degree $2014^{2015}$ such that $f(P)=2015$? (Serbia)

2013 Princeton University Math Competition, 8

Tags:

Let $k$ be a positive integer with the following property: For every subset $A$ of $\{1,2,\ldots, 25\}$ with $|A|=k$, we can find distinct elements $x$ and $y$ of $A$ such that $\tfrac23\leq\tfrac xy\leq\tfrac 32$. Find the smallest possible value of $k$.