Olympiad problems

1984 IMO Longlists, 15

Tags: number theory

Consider all the sums of the form \[\displaystyle\sum_{k=1}^{1985} e_kk^5=\pm 1^5\pm 2^5\pm\cdots\pm1985^5\] where $e_k=\pm 1$. What is the smallest nonnegative value attained by a sum of this type?

1998 Canada National Olympiad, 1

Tags: number theory

Determine the number of real solutions $a$ to the equation: \[ \left[\,\frac{1}{2}\;a\,\right]+\left[\,\frac{1}{3}\;a\,\right]+\left[\,\frac{1}{5}\;a\,\right] = a. \] Here, if $x$ is a real number, then $[\,x\,]$ denotes the greatest integer that is less than or equal to $x$.

2017 NMTC Junior, 1

Tags: number theory , algebra

(a) Find all prime numbers $p$ such that $4p^2+1$ and $6p^2+1$ are also primes. (b)Find real numbers $x,y,z,u$ such that \[xyz+xy+yz+zx+x+y+z=7\]\[yzu+yz+zu+uy+y+z+u=10\]\[zux+zu+ux+xz+z+u+x=10\]\[uxy+ux+xy+yu+u+x+y=10\]

2007 Princeton University Math Competition, 6

Tags: geometry , 3d geometry , probability , expected value

Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of sidelengths $9$ and $10$ or of sidelengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of sidelengths $1$ and $12$ or $9$ and $10$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?

2006 Denmark MO - Mohr Contest, 2

Tags: algebra , system of equations , system

Determine all sets of real numbers $(x,y,z)$ which fulfills $$\begin{cases} x + y =2 \\ xy -z^2= 1\end{cases}$$

2010 Tournament Of Towns, 2

Tags: parallelogram , geometry

In a quadrilateral $ABCD$ with an incircle, $AB = CD; BC < AD$ and $BC$ is parallel to $AD$. Prove that the bisector of $\angle C$ bisects the area of $ABCD$.

2011 Korea National Olympiad, 1

Tags: number theory

Find the number of positive integer $ n < 3^8 $ satisfying the following condition. "The number of positive integer $k (1 \leq k \leq \frac {n}{3})$ such that $ \frac{n!}{(n-3k)! \cdot k! \cdot 3^{k+1}} $ is not a integer" is $ 216 $.

2023 Sharygin Geometry Olympiad, 10.8

Tags: geometry

A triangle $ABC$ is given. Let $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ be circles centered at points $X$, $Y$, $Z$, $T$ respectively such that each of lines $BC$, $CA$, $AB$ cuts off on them four equal chords. Prove that the centroid of $ABC$ divides the segment joining $X$ and the radical center of $\omega_2$, $\omega_3$, $\omega_4$ in the ratio $2:1$ from $X$.

2017 Thailand Mathematical Olympiad, 1

Tags: algebra , irrational number , irrational

Let $p$ be a prime. Show that $\sqrt[3]{p} +\sqrt[3]{p^5} $ is irrational.

2012 Tournament of Towns, 6

Tags: geometry , combinatorial geometry , perpendicular , centroid , regular polygon

(a) A point $A$ is marked inside a circle. Two perpendicular lines drawn through $A$ intersect the circle at four points. Prove that the centre of mass of these four points does not depend on the choice of the lines. (b) A regular $2n$-gon ($n \ge 2$) with centre $A$ is drawn inside a circle (A does not necessarily coincide with the centre of the circle). The rays going from $A$ to the vertices of the $2n$-gon mark $2n$ points on the circle. Then the $2n$-gon is rotated about $A$. The rays going from $A$ to the new locations of vertices mark new $2n$ points on the circle. Let $O$ and $N$ be the centres of gravity of old and new points respectively. Prove that $O = N$.

2025 Poland - First Round, 10

Tags: geometry

An acute triangle $ABC$ is given, in which $AB<AC$. Let $\Omega$ be the circumcircle of $ABC$. Points $M$ and $N$ are the midpoints of the longer arc $BC$ and shorter arc $BC$ of $\Omega$ respectively. Points $X\ne M$ and $Y\ne N$ lie on the line $AM$ and satisfy $BX=BM=CM=CY$. Let $E$ be a point on $AC$ such that $BE$ and $AC$ are perpendicular. Prove that $\angle FNX=\angle YNE$.

2011 Purple Comet Problems, 18

Tags:

Find the positive integer $n$ so that $n^2$ is the perfect square closest to $8 + 16 + 24 + \cdots + 8040.$

2023 Taiwan TST Round 3, G

Tags: geometry

Let $ABC$ be a scalene triangle with circumcenter $O$ and orthocenter $H$. Let $AYZ$ be another triangle sharing the vertex $A$ such that its circumcenter is $H$ and its orthocenter is $O$. Show that if $Z$ is on $BC$, then $A,H,O,Y$ are concyclic. [i]Proposed by usjl[/i]

2020 JBMO TST of France, 1

Tags: geometry

Given are four distinct points $A, B, E, P$ so that $P$ is the middle of $AE$ and $B$ is on the segment $AP$. Let $k_1$ and $k_2$ be two circles passing through $A$ and $B$. Let $t_1$ and $t_2$ be the tangents of $k_1$ and $k_2$, respectively, to $A$.Let $C$ be the intersection point of $t_2$ and $k_1$ and $Q$ be the intersection point of $t_2$ and the circumscribed circle of the triangle $ECB$. Let $D$ be the intersection posit of $t_1$ and $k_2$ and $R$ be the intersection point of $t_1$ and the circumscribed circle of triangle $BDE$. Prove that $P, Q, R$ are collinear.

2006 All-Russian Olympiad, 2

Tags: number theory

The sum and the product of two purely periodic decimal fractions $a$ and $b$ are purely periodic decimal fractions of period length $T$. Show that the lengths of the periods of the fractions $a$ and $b$ are not greater than $T$. [i]Note.[/i] A [i]purely periodic decimal fraction[/i] is a periodic decimal fraction without a non-periodic starting part.

2000 Tournament Of Towns, 1

Tags: area , geometry , midpoint

The diagonals of a convex quadrilateral $ABCD$ meet at $P$. The sum of the areas of triangles $PAB$ and $PCD$ is equal to the sum of areas of triangles $PAD$ and $PCB$. Prove that $P$ is the midpoint of either $AC$ or $BD$. (Folklore)

2023 USA IMOTST, 3

Tags: function , algebra

Let $\mathbb{N}$ denote the set of positive integers. Fix a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and for any $m,n \in \mathbb{N}$ define $$\Delta(m,n)=\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(m)\ldots))-\underbrace{f(f(\ldots f}_{f(m)\text{ times}}(n)\ldots)).$$ Suppose $\Delta(m,n) \neq 0$ for any distinct $m,n \in \mathbb{N}$. Show that $\Delta$ is unbounded, meaning that for any constant $C$ there exists $m,n \in \mathbb{N}$ with $\left|\Delta(m,n)\right| > C$.

2011 China Team Selection Test, 1

Tags: circumcircle , geometric transformation , reflection , perpendicular bisector , geometry

Let $AA',BB',CC'$ be three diameters of the circumcircle of an acute triangle $ABC$. Let $P$ be an arbitrary point in the interior of $\triangle ABC$, and let $D,E,F$ be the orthogonal projection of $P$ on $BC,CA,AB$, respectively. Let $X$ be the point such that $D$ is the midpoint of $A'X$, let $Y$ be the point such that $E$ is the midpoint of $B'Y$, and similarly let $Z$ be the point such that $F$ is the midpoint of $C'Z$. Prove that triangle $XYZ$ is similar to triangle $ABC$.

Russian TST 2014, P3

Tags: geometry , angle

On the sides $AB{}$ and $AC{}$ of the acute-angled triangle $ABC{}$ the points $M{}$ and $N{}$ are chosen such that $MN$ passes through the circumcenter of $ABC.$ Let $P{}$ and $Q{}$ be the midpoints of the segments $CM{}$ and $BN{}.$ Prove that $\angle POQ=\angle BAC.$

2020 CCA Math Bonanza, T4

Tags:

Compute \[ \left(\frac{4-\log_{36} 4 - \log_6 {18}}{\log_4 3} \right) \cdot \left( \log_8 {27} + \log_2 9 \right). \] [i]2020 CCA Math Bonanza Team Round #4[/i]

2006 District Olympiad, 1

Tags: logarithm , inequalities

Let $ a,b,c\in (0,1)$ and $ x,y,z\in (0, \plus{} \infty)$ be six real numbers such that \[ a^x \equal{} bc , \quad b^y \equal{} ca , \quad c^z \equal{} ab .\] Prove that \[ \frac 1{2 \plus{} x} \plus{} \frac 1{2 \plus{} y} \plus{} \frac 1{2 \plus{} z} \leq \frac 34 .\] [i]Cezar Lupu[/i]

2010 Harvard-MIT Mathematics Tournament, 4

Tags: geometry

Let $ABCD$ be an isosceles trapezoid such that $AB=10$, $BC=15$, $CD=28$, and $DA=15$. There is a point $E$ such that $\triangle AED$ and $\triangle AEB$ have the same area and such that $EC$ is minimal. Find $EC$.

2013 Chile TST Ibero, 3

Tags: geometry

The incircle of triangle $\triangle ABC$ touches $AC$ and $BC$ at $E$ and $D$ respectively. The excircle corresponding to $A$ touches the extensions of $BC$ at $A_1$, $CA$ at $B_1$, and $AB$ at $C_1$. Let $DE \cap A_1B_1 = L$. Prove that $L$ belongs to the circumcircle of triangle $\triangle A_1B_1C_1$.

India EGMO 2023 TST, 4

Tags: algebra , function , identity , periodic function

Let $f, g$ be functions $\mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y$, $$f(g(x) + y) = g(x + y)$$ Prove that either $f$ is the identity function or $g$ is periodic. [i]Proposed by Pranjal Srivastava[/i]

2014 Middle European Mathematical Olympiad, 6

Tags: geometric transformation , reflection , homothety , perpendicular bisector , geometry

Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$. Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$. Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively. Prove that the points $B, C, N,$ and $L$ are concyclic.