Olympiad problems

2019 Macedonia Junior BMO TST, 5

Tags: JMMO , 2019 , Junior , Macedonia , number theory , prime numbers

Let $p_{1}$, $p_{2}$, ..., $p_{k}$ be different prime numbers. Determine the number of positive integers of the form $p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}...p_{k}^{\alpha_{k}}$, $\alpha_{i}$ $\in$ $\mathbb{N}$ for which $\alpha_{1} \alpha_{2}...\alpha_{k}=p_{1}p_{2}...p_{k}$.

2012 USA Team Selection Test, 2

Tags: trigonometry , geometry , reflection , cyclic quadrilateral , angle bisector , geometry solved , projective geometry

In cyclic quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $P$. Let $E$ and $F$ be the respective feet of the perpendiculars from $P$ to lines $AB$ and $CD$. Segments $BF$ and $CE$ meet at $Q$. Prove that lines $PQ$ and $EF$ are perpendicular to each other.

2020 AIME Problems, 14

Tags: algebra , 2020 AIME I , AIME , AMC , 2020 AMC , AIME I , polynomial

Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1$. Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b$. Find the sum of all possible values of $(a+b)^2$.

1989 Czech And Slovak Olympiad IIIA, 2

Tags: combinatorics , combinatorial geometry , graph theory

There are $mn$ line segments in a plane that connect $n$ given points. Prove that a sequence $V_0$, $V_1$, $...$, $V_m$ of different points can be selected from them such that $V_{i-1}$ and $V_i$ are connected by a line ($1 \le i \le m$).

2004 Iran MO (3rd Round), 30

Tags: algebra , polynomial , number theory proposed , number theory

Find all polynomials $ p\in\mathbb Z[x]$ such that $ (m,n)\equal{}1\Rightarrow (p(m),p(n))\equal{}1$

1981 Romania Team Selection Tests, 2.

Tags: geometry , 3D geometry , tetrahedron

Consider a tetrahedron $OABC$ with $ABC$ equilateral. Let $S$ be the area of the triangle of sides $OA$, $OB$ and $OC$. Show that $V\leqslant \dfrac12 RS$ where $R$ is the circumradius and $V$ is the volume of the tetrahedron. [i]Stere Ianuș[/i]

2011 South africa National Olympiad, 5

Tags: function , inequalities , number theory , relatively prime , number theory unsolved

Let $\mathbb{N}_0$ denote the set of all nonnegative integers. Determine all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ with the following two properties: [list] [*] $0\le f(x)\le x^2$ for all $x\in\mathbb{N}_0$ [*] $x-y$ divides $f(x)-f(y)$ for all $x,y\in\mathbb{N}_0$ with $x>y$[/list]

2012 Thailand Mathematical Olympiad, 10

Tags: algebra , number theory , irrational , inequalities

Let $x$ be an irrational number. Show that there are integers $m$ and $n$ such that $\frac{1}{2555}< mx + n <\frac{1}{2012}$

1947 Moscow Mathematical Olympiad, 135-

Tags: geometry , combinatorial geometry

Position the $4$ points on plane so that when measuring of all pairwise distances between them, it turned out only two different numbers. Find all such locations.

1997 Kurschak Competition, 3

Tags: combinatorics unsolved , combinatorics

Prove that the vertices of any planar graph can be colored with $3$ colors such that there is no monochromatic cycle.

2007 F = Ma, 25

Tags:

Find the period of small oscillations of a water pogo, which is a stick of mass m in the shape of a box (a rectangular parallelepiped.) The stick has a length $L$, a width $w$ and a height $h$ and is bobbing up and down in water of density $\rho$ . Assume that the water pogo is oriented such that the length $L$ and width $w$ are horizontal at all times. Hint: The buoyant force on an object is given by $F_{buoy} = \rho Vg$, where $V$ is the volume of the medium displaced by the object and $\rho$ is the density of the medium. Assume that at equilibrium, the pogo is floating. $ \textbf{(A)}\ 2\pi \sqrt{\frac{L}{g}} $ $ \textbf{(B)}\ \pi \sqrt{\frac{\rho w^2L^2 g}{mh^2}} $ $ \textbf{(C)}\ 2\pi \sqrt{\frac{mh^2}{\rho L^2w^2 g}} $ $\textbf{(D)}\ 2\pi \sqrt{\frac{m}{\rho wLg}}$ $\textbf{(E)}\ \pi \sqrt{\frac{m}{\rho wLg}}$

2011 Tournament of Towns, 4

Tags: Digits , number theory

Positive integers $a < b < c$ are such that $b + a$ is a multiple of $b - a$ and $c + b$ is a multiple of $c-b$. If $a$ is a $2011$-digit number and $b$ is a $2012$-digit number, exactly how many digits does $c$ have?

2018 IMO Shortlist, G3

Tags: IMO Shortlist , geometry

A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold: [list=1] [*] each triangle from $T$ is inscribed in $\omega$; [*] no two triangles from $T$ have a common interior point. [/list] Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.

2008 Brazil Team Selection Test, 2

Tags: combinatorics , modular arithmetic , counting , IMO Shortlist , triplets , Hi

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]

2012 Online Math Open Problems, 16

Tags: Online Math Open , geometry , geometric transformation , reflection , circumcircle , trigonometry , trig identities

Let $ABC$ be a triangle with $AB = 4024$, $AC = 4024$, and $BC=2012$. The reflection of line $AC$ over line $AB$ meets the circumcircle of $\triangle{ABC}$ at a point $D\ne A$. Find the length of segment $CD$. [i]Ray Li.[/i]

1995 Canada National Olympiad, 4

Tags: number theory , Diophantine equation , number theory unsolved

Let $n$ be a constant positive integer. Show that for only non-negative integers $k$, the Diophantine equation $\sum_{i=1 }^{n}{ x_i ^3}=y^{3k+2}$ has infinitely many solutions in the positive integers $x_i, y$.

2016 Romania National Olympiad, 1

Tags: number theory , Integer

Find all non-negative integers $n$ so that $\sqrt{n + 3}+ \sqrt{n +\sqrt{n + 3}} $ is an integer.

2024 Polish MO Finals, 1

Tags: geometry , geometry proposed , rectangle , angles

Let $X$ be an interior point of a rectangle $ABCD$. Let the bisectors of $\angle DAX$ and $\angle CBX$ intersect in $P$. A point $Q$ satisfies $\angle QAP=\angle QBP=90^\circ$. Show that $PX=QX$.

2021 Israel National Olympiad, P4

Tags: combinatorics , Digits

Danny likes seven-digit numbers with the following property: the 1's digit is divisible by the 10's digit, the 10's digit is divisible by the 100's digit, and so on. For example, Danny likes the number $1133366$ but doesn't like $9999993$. Is the amount of numbers Danny likes divisible by $7$?

2019 Online Math Open Problems, 9

Tags: Online Math Open

Convex equiangular hexagon $ABCDEF$ has $AB=CD=EF=1$ and $BC = DE = FA = 4$. Congruent and pairwise externally tangent circles $\gamma_1$, $\gamma_2$, and $\gamma_3$ are drawn such that $\gamma_1$ is tangent to side $\overline{AB}$ and side $\overline{BC}$, $\gamma_2$ is tangent to side $\overline{CD}$ and side $\overline{DE}$, and $\gamma_3$ is tangent to side $\overline{EF}$ and side $\overline{FA}$. Then the area of $\gamma_1$ can be expressed as $\frac{m\pi}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Sean Li[/i]

2000 Baltic Way, 20

Tags: integration , trigonometry , logarithms , algebra proposed , algebra

For every positive integer $n$, let \[x_n=\frac{(2n+1)(2n+3)\cdots (4n-1)(4n+1)}{(2n)(2n+2)\cdots (4n-2)(4n)}\] Prove that $\frac{1}{4n}<x_n-\sqrt{2}<\frac{2}{n}$.

2020 Durer Math Competition Finals, 6

Tags: number theory , divides

Positive integers $a, b$ and $c$ are all less than $2020$. We know that $a$ divides $b + c$, $b$ divides $a + c$ and $c$ divides $a + b$. How many such ordered triples $(a, b, c)$ are there? Note: In an ordered triple, the order of the numbers matters, so the ordered triple $(0, 1, 2)$ is not the same as the ordered triple $(2, 0, 1)$.

2013 Harvard-MIT Mathematics Tournament, 21

Tags: HMMT

Find the number of positive integers $j\leq 3^{2013}$ such that \[j=\sum_{k=0}^m\left((-1)^k\cdot 3^{a_k}\right)\] for some strictly increasing sequence of nonnegative integers $\{a_k\}$. For example, we may write $3=3^1$ and $55=3^0-3^3+3^4$, but $4$ cannot be written in this form.

2014 BAMO, 1

Tags: B8 , cube , geometry , 3D geometry , counting , combinatorics , Arrangements

The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.

2018 Cyprus IMO TST, 1

Tags: number theory , number base

Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square.