Olympiad problems

Fractal Edition 1, P1

Is the number $1234567890987654321$ prime?

There are four boats on one of the river banks; their names are Eight, Four, Two and One, because that is the number of hours it takes each of them to cross the river. One boat can be tied to another, but not more than one, and then the time it takes to cross is equal to that of the slower of the two boats. A single sailor must take all the boats to the other shore. What is the least amount of time you need to complete the move?

1999 Abels Math Contest (Norwegian MO), 1b

Tags: inequalities , algebra

If $a,b,c,d,e$ are real numbers, prove the inequality $a^2 +b^2 +c^2 +d^2+e^2 \ge a(b+c+d+e)$.

1995 IMO, 1

Tags: geometry , circumcircle , concurrency

Let $ A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $ AC$ and $ BD$ intersect at $ X$ and $ Y$. The line $ XY$ meets $ BC$ at $ Z$. Let $ P$ be a point on the line $ XY$ other than $ Z$. The line $ CP$ intersects the circle with diameter $ AC$ at $ C$ and $ M$, and the line $ BP$ intersects the circle with diameter $ BD$ at $ B$ and $ N$. Prove that the lines $ AM,DN,XY$ are concurrent.

2021 Turkey Junior National Olympiad, 3

Tags: inequalities , maximum value , find maximum

Let $x, y, z$ be real numbers such that $$x+y+z=2, \;\;\;\; xy+yz+zx=1$$ Find the maximum possible value of $x-y$.

2018 NZMOC Camp Selection Problems, 10

Tags: functional equation , functional , algebra

Find all functions $f : R \to R$ such that $$f(x)f(y) = f(xy + 1) + f(x - y) - 2$$ for all $x, y \in R$.

1998 IMO Shortlist, 5

Tags: geometry , reflection , homothety , complex bash

Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.

1984 National High School Mathematics League, 5

Tags: function

If $a>0,a\neq1$, $F(x)$ is an odd function. $G(x)=F(x)\cdot(\frac{1}{a^x-1}+\frac{1}{2})$, then $G(x)$ is $\text{(A)}$ odd function $\text{(B)}$ even function $\text{(C)}$ not odd or even function $\text{(D)}$ not sure

2023 Brazil EGMO Team Selection Test, 3

Tags: geometry , circumcircle , circumcenter , incenter , incircle , concurrency

Let $\Delta ABC$ be a triangle and $L$ be the foot of the bisector of $\angle A$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle ABL$ and $\triangle ACL$ respectively and let $B_1$ and $C_1$ be the projections of $C$ and $B$ through the bisectors of the angles $\angle B$ and $\angle C$ respectively. The incircle of $\Delta ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively and the bisectors of angles $\angle B$ and $\angle C$ meet the perpendicular bisector of $AL$ at points $Q$ and $P$ respectively. Prove that the five lines $PC_0, QB_0, O_1C_1, O_2B_1$ and $BC$ are all concurrent.

2013 Taiwan TST Round 1, 6

Tags: geometry , homothety , incenter

Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.

2024 Portugal MO, 2

Tags: geometry

Let $ABC$ be a triangle and $D,E$ and $F$ the midpoints of sides $BC, AC$ and $BC$. Medians $AD$ and $BE$ are perpendicular, $AD = 12$ and $BE = 9$. What is the value of $CF$?

1980 Austrian-Polish Competition, 9

Tags: tangent , geometry , equal segments , circles

Through the endpoints $A$ and $B$ of a diameter $AB$ of a given circle, the tangents $\ell$ and $m$ have been drawn. Let $C\ne A$ be a point on $\ell$ and let $q_1,q_2$ be two rays from $C$. Ray $q_i$ cuts the circle in $D_i$ and $E_i$ with $D_i$ between $C$ and $E_i, i = 1,2$. Rays $AD_1,AD_2,AE_1,AE_2$ meet $m$ in the respective points $M_1,M_2,N_1,N_2$. Prove that $M_1M_2 = N_1N_2$.

1987 China Team Selection Test, 1

Tags: combinatorics , number theoretic functions , set

a.) For all positive integer $k$ find the smallest positive integer $f(k)$ such that $5$ sets $s_1,s_2, \ldots , s_5$ exist satisfying: [b]i.[/b] each has $k$ elements; [b]ii.[/b] $s_i$ and $s_{i+1}$ are disjoint for $i=1,2,...,5$ ($s_6=s_1$) [b]iii.[/b] the union of the $5$ sets has exactly $f(k)$ elements. b.) Generalisation: Consider $n \geq 3$ sets instead of $5$.

2010 Laurențiu Panaitopol, Tulcea, 2

Tags: freshman s dream , frobenius , nilpotence

Let be two $ n\times n $ complex matrices $ A,B $ satisfying the equations $ (A+B)^2=A^2+B^2 $ and $ (A+B)^4=A^4+B^4. $ Show that $ (AB)^2=0. $

2008 Peru IMO TST, 6

Tags: number theory

We say that a positive integer is happy if can expressed in the form $ (a^{2}b)/(a \minus{} b)$ where $ a > b > 0$ are integers. We also say that a positive integer $ m$ is evil if it doesn't a happy integer $ n$ such that $ d(n) \equal{} m$. Prove that all integers happy and evil are a power of $ 4$.

2021 MOAA, 1

Tags: team

The value of \[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\] can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

2017 Regional Olympiad of Mexico West, 6

Tags: number theory , divide

A [i]change [/i] in a natural number $n$ consists of adding a pair of zeros between two digits or at the end of the decimal representation of $n$. A [i]countryman [/i] of $n$ is a number that can be obtained from one or more changes in $n$. For example. $40041$, $4410000$ and $4004001$ are all countrymen from $441$. Determine all the natural numbers $n$ for which there is a natural number m with the property that $n$ divides $m$ and all the countrymen of $m$.

2018 Dutch BxMO TST, 5

Tags: equation , algebra

Let $n$ be a positive integer. Determine all positive real numbers $x$ satisfying $nx^2 +\frac{2^2}{x + 1}+\frac{3^2}{x + 2}+...+\frac{(n + 1)^2}{x + n}= nx + \frac{n(n + 3)}{2}$

2013 Sharygin Geometry Olympiad, 6

Tags: geometry , orthocenter , circles

A line $\ell$ passes through the vertex $B$ of a regular triangle $ABC$. A circle $\omega_a$ centered at $I_a$ is tangent to $BC$ at point $A_1$, and is also tangent to the lines $\ell$ and $AC$. A circle $\omega_c$ centered at $I_c$ is tangent to $BA$ at point $C_1$, and is also tangent to the lines $\ell$ and $AC$. Prove that the orthocenter of triangle $A_1BC_1$ lies on the line $I_aI_c$.

2014 Korea National Olympiad, 1

Tags: geometry

There is a convex quadrilateral $ ABCD $ which satisfies $ \angle A=\angle D $. Let the midpoints of $ AB, AD, CD $ be $ L,M,N $. Let's say the intersection point of $ AC, BD $ be $ E $ . Let's say point $ F $ which lies on $ \overrightarrow{ME} $ satisfies $ \overline{ME}\times \overline{MF}=\overline{MA}^{2} $. Prove that $ \angle LFM=\angle MFN $. :)

2023 Costa Rica - Final Round, 3.3

Tags: geometry , polygon , concurrency

Let $ABCD \dots KLMN$ be a regular polygon with $14$ sides. Show that the diagonals $AE$, $BG$, and $CK$ are concurrent.

2008 Germany Team Selection Test, 1

Tags: inequalities , algebra

Determine $ Q \in \mathbb{R}$ which is so big that a sequence with non-negative reals elements $ a_1 ,a_2, \ldots$ which satisfies the following two conditions: [b](i)[/b] $ \forall m,n \geq 1$ we have $ a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right)$ [b](ii)[/b] $ \forall k \geq 0$ we have $ a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}}$ such that for each sequence element we have the inequality $ a_n \leq Q.$

2016 Harvard-MIT Mathematics Tournament, 8

Tags: hmmt

Let $P_1P_2 \ldots P_8$ be a convex octagon. An integer $i$ is chosen uniformly at random from $1$ to $7$, inclusive. For each vertex of the octagon, the line between that vertex and the vertex $i$ vertices to the right is painted red. What is the expected number times two red lines intersect at a point that is not one of the vertices, given that no three diagonals are concurrent?

2021 BMT, 8

Tags: expected value

Consider the randomly generated base 10 real number $r = 0.\overline{p_0p_1p_2\ldots}$, where each $p_i$ is a digit from $0$ to $9$, inclusive, generated as follows: $p_0$ is generated uniformly at random from $0$ to $9$, inclusive, and for all $i \geq 0$, $p_{i + 1}$ is generated uniformly at random from $p_i$ to $9$, inclusive. Compute the expected value of $r$.

1998 Belarus Team Selection Test, 2

Tags: algebra , set

Find all finite sets $M \subset R$ containing at least two elements such that $(2a/3 -b^2) \in M$ for any two different elements $a,b \in M$.