Olympiad problems

2001 Austria Beginners' Competition, 3

Tags: algebra , inequalities

Find all real numbers $x$ such that $(x-1)^2(x-4)^2<(x-2)^2$.

2007 Junior Macedonian Mathematical Olympiad, 2

Tags: Junior , Macedonia , JMMO , 2007 , geometry , parallelogram

Let $ABCD$ be a parallelogram and let $E$ be a point on the side $AD$, such that $\frac{AE}{ED} = m$. Let $F$ be a point on $CE$, such that $BF \perp CE$, and the point $G$ is symmetrical to $F$ with respect to $AB$. If point $A$ is the circumcenter of triangle $BFG$, find the value of $m$.

1972 Bundeswettbewerb Mathematik, 3

Tags: number theory proposed , number theory

The arithmetic mean of two different positive integers $x,y$ is a two digit integer. If one interchanges the digits, the geometric mean of these numbers is archieved. a) Find $x,y$. b) Show that a)'s solution is unique up to permutation if we work in base $g=10$, but that there is no solution in base $g=12$. c) Give more numbers $g$ such that a) can be solved; give more of them such that a) can't be solved, too.

2012 Tournament of Towns, 3

Tags: trigonometry , Sum , coordinates , algebra

Consider the points of intersection of the graphs $y = \cos x$ and $x = 100 \cos (100y)$ for which both coordinates are positive. Let $a$ be the sum of their $x$-coordinates and $b$ be the sum of their $y$-coordinates. Determine the value of $\frac{a}{b}$.

2005 Federal Math Competition of S&M, Problem 4

Tags: game , combinatorics

On each cell of a $2005\times2005$ chessboard, there is a marker. In each move, we are allowed to remove a marker that is a neighbor to an even number of markers (but at least one). Two markers are considered neighboring if their cells share a vertex. (a) Find the least number $n$ of markers that we can end up with on the chessboard. (b) If we end up with this minimum number $n$ of markers, prove that no two of them will be neighboring.

2009 Today's Calculation Of Integral, 475

Tags: calculus , integration , calculus computations

For a positive constant number $ t$, let denote $ D$ the region surrounded by the curve $ y \equal{} e^{x}$, the line $ x \equal{} t$, the $ x$ axis and the $ y$ axis. Let $ V_x,\ V_y$ be the volumes of the solid obtained by rotating $ D$ about the $ x$ axis and the $ y$ axis respectively. Compare the size of $ V_x,\ V_y.$

2017 QEDMO 15th, 1

Tags: diophantine , Diophantine equation , number theory

Find all integers $x, y, z$ satisfy the $x^4-10y^4 + 3z^6 = 21$.

2018 India PRMO, 26

Tags: combinatorics , Chessboard

What is the number of ways in which one can choose $60$ unit squares from a $11 \times 11$ chessboard such that no two chosen squares have a side in common?

2010 China Team Selection Test, 2

Tags: algebra , polynomial , inequalities , floor function , triangle inequality , inequalities unsolved

Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$, and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.

2015 Princeton University Math Competition, B2

Tags:

Jonathan has a magical coin machine which takes coins in amounts of $7, 8$, and $9$. If he puts in $7$ coins, he gets $3$ coins back; if he puts in $8$, he gets $11$ back; and if he puts in $9$, he gets $4$ back. The coin machine does not allow two entries of the same amount to happen consecutively. Starting with $15$ coins, what is the minimum number of entries he can make to end up with $4$ coins?

1996 IMO Shortlist, 7

Tags: inequalities , geometry , circumcircle , trigonometry , trig identities , IMO Shortlist , Triangle

Let $ABC$ be an acute triangle with circumcenter $O$ and circumradius $R$. $AO$ meets the circumcircle of $BOC$ at $A'$, $BO$ meets the circumcircle of $COA$ at $B'$ and $CO$ meets the circumcircle of $AOB$ at $C'$. Prove that \[OA'\cdot OB'\cdot OC'\geq 8R^{3}.\] Sorry if this has been posted before since this is a very classical problem, but I failed to find it with the search-function.

Mathematical Minds 2024, P2

Tags: square , geometry , Olympiad Geometry

Let $ABCD$ be a square and $E$ a point on side $CD$ such that $\angle DAE = 30^{\circ}$. The bisector of angle $\angle AEC$ intersects line $BD$ at point $F$. Lines $FC$ and $AE$ intersect at $S$. Find $\angle SDC$. [i]Proposed by Ana Boiangiu[/i]

2001 Kazakhstan National Olympiad, 3

Tags: algebra , inequalities , n-variable inequality

For positive numbers $ x_1, x_2, \ldots, x_n $ $ (n \geq 1) $ the following equality holds $$ \frac {1} {{1 + x_1}} + \frac {1} {{1 + x_2}} + \ldots + \frac {1} {{1 + x_n}} = 1. $$ Prove that $ x_1 \cdot x_2 \cdot \ldots \cdot x_n \geq (n-1) ^ n. $

2024 South Africa National Olympiad, 6

Tags: number theory , functional equation in N

Let $f:\mathbb{N}\to\mathbb{N}_0$ be a function that satisfies \[ f(mn) = mf(n) + nf(m)\] for all positive integers $m,n$ and $f(2024)=10120$. Prove that there are two integers $m,n$ with $m\ne n$ such that $f(m)=f(n)$.

Gheorghe Țițeica 2024, P4

Tags: Olympiad Number Theory

A positive integer is called [i]joli[/i] if it can be written as the arithmetic mean of two or more (not necessarily distinct) powers of two, and [i]superjoli[/i] if it can be written as the arithmetic mean of two or more distinct powers of two. For instance $7$ and $92$ are superjoli because $7=\frac{2^4+2^2+1}{3}$ and $92=\frac{2^8+2^4+2^2}{3}$. a) Prove that every positive integer is joli. b) Prove that no power of two is superjoli. c) Find the smallest positive integer different from a power of two that is not superjoli. [i]France Olympiad[/i]

1999 Belarusian National Olympiad, 6

Tags: number theory , Diophantine equation

Solve in integer:$x^6+x^2y=y^3+2y^2$

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 $a + b + c = 5n$.

2005 District Olympiad, 3

Tags: geometry , 3D geometry , tetrahedron , circumcircle , geometry proposed

Let $O$ be a point equally distanced from the vertices of the tetrahedron $ABCD$. If the distances from $O$ to the planes $(BCD)$, $(ACD)$, $(ABD)$ and $(ABC)$ are equal, prove that the sum of the distances from a point $M \in \textrm{int}[ABCD]$, to the four planes, is constant.

2016 BMT Spring, 2

Tags: geometry

Cyclic quadrilateral $ABCD$ has side lengths $AB = 6$, $BC = 7$, $CD = 7$, $DA = 6$. What is the area of $ABCD$?

2017 Math Prize for Girls Problems, 4

Tags: Math Prize for Girls

If $\mathrm{MATH} + \mathrm{WITH} = \mathrm{GIRLS}$, compute the smallest possible value of $\mathrm{GIRLS}$. Here $\mathrm{MATH}$ and $\mathrm{WITH}$ are 4-digit numbers and $\mathrm{GIRLS}$ is a 5-digit number (all with nonzero leading digits). Different letters represent different digits.

2022 Balkan MO Shortlist, C4

Tags: combinatorics , Balkan Mathematics Olympiad

Consider an $n \times n$ grid consisting of $n^2$ until cells, where $n \geq 3$ is a given odd positive integer. First, Dionysus colours each cell either red or blue. It is known that a frog can hop from one cell to another if and only if these cells have the same colour and share at least one vertex. Then, Xanthias views the colouring and next places $k$ frogs on the cells so that each of the $n^2$ cells can be reached by a frog in a finite number (possible zero) of hops. Find the least value of $k$ for which this is always possible regardless of the colouring chosen by Dionysus. [i]Proposed by Tommy Walker Mackay, United Kingdom[/i]

2011 Putnam, A3

Tags: Putnam , limit , integration , trigonometry , calculus , ratio , inequalities

Find a real number $c$ and a positive number $L$ for which \[\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.\]

2011 Purple Comet Problems, 9

Tags: algebra , polynomial , quadratics , quadratic formula , sum of roots , product of roots

There are integers $m$ and $n$ so that $9 +\sqrt{11}$ is a root of the polynomial $x^2 + mx + n.$ Find $m + n.$

2016 Hanoi Open Mathematics Competitions, 11

Tags: geometry , circumcircle , incenter

Let $I$ be the incenter of triangle $ABC$ and $\omega$ be its circumcircle. Let the line $AI$ intersect $\omega$ at point $D \ne A$. Let $F$ and $E$ be points on side $BC$ and arc $BDC$ respectively such that $\angle BAF = \angle CAE < \frac12 \angle BAC$ . Let $X$ be the second point of intersection of line $EI$ with $\omega$ and $T$ be the point of intersection of segment $DX$ with line $AF$ . Prove that $TF \cdot AD = ID \cdot AT$ .

2010 LMT, 32

Tags:

Compute the infinite sum $\frac{1^3}{2^1}+\frac{2^3}{2^2}+\frac{3^3}{2^3}+\dots+\frac{n^3}{2^n}+\dots.$