Olympiad problems

2017 Mediterranean Mathematics Olympiad, Problem 1

Tags: geometry

Let $ABC$ be an equilateral triangle, and let $P$ be some point in its circumcircle. Determine all positive integers $n$, for which the value of the sum $S_n (P) = |PA|^n + |PB|^n + |PC|^n$ is independent of the choice of point $P$.

2023 Macedonian Balkan MO TST, Problem 4

Tags: algebra , functional equation

Let $f$ be a non-zero function from the set of positive integers to the set of non-negative integers such that for all positive integers $a$ and $b$ we have $$2f(ab)=(b+1)f(a)+(a+1)f(b).$$ Prove that for every prime number $p$ there exists a prime $q$ and positive integers $x_{1}$, ..., $x_{n}$ and $m \geq 0$ so that $$\frac{f(q^{p})}{f(q)} = (px_{1}+1) \cdot ... \cdot (px_{n}+1) \cdot p^{m},$$ where the integers $px_{1}+1$,..., $px_{n}+1$ are all prime. [i]Authored by Nikola Velov[/i]

2015 Sharygin Geometry Olympiad, 8

Tags: geometry , rectangle

Does there exist a rectangle which can be divided into a regular hexagon with sidelength $1$ and several congruent right-angled triangles with legs $1$ and $\sqrt{3}$?

2008 China Team Selection Test, 2

Tags: combinatorics

Prove that for arbitary integer $ n > 16$, there exists the set $ S$ that contains $ n$ positive integers and has the following property:if the subset $ A$ of $ S$ satisfies for arbitary $ a,a'\in A, a\neq a', a \plus{} a'\notin S$ holds, then $ |A|\leq4\sqrt n.$

2017 ELMO Shortlist, 1

Tags: algebra

Let $0<k<\frac{1}{2}$ be a real number and let $a_0, b_0$ be arbitrary real numbers in $(0,1)$. The sequences $(a_n)_{n\ge 0}$ and $(b_n)_{n\ge 0}$ are then defined recursively by $$a_{n+1} = \dfrac{a_n+1}{2} \text{ and } b_{n+1} = b_n^k$$ for $n\ge 0$. Prove that $a_n<b_n$ for all sufficiently large $n$. [i]Proposed by Michael Ma

2018 CCA Math Bonanza, L5.2

Tags:

Two circles of equal radii are drawn to intersect at $X$ and $Y$. Suppose that the two circles bisect each other's areas. If the measure of minor arc $\widehat{XY}$ is $\theta$ degrees, estimate $\left\lfloor1000\theta\right\rfloor$. An estimate of $E$ earns $2e^{-\frac{\left|A-E\right|}{50000}}$ points, where $A$ is the actual answer. [i]2018 CCA Math Bonanza Lightning Round #5.2[/i]

2020 MBMT, 9

Tags:

Consider a regular pentagon $ABCDE$, and let the intersection of diagonals $\overline{CA}$ and $\overline{EB}$ be $F$. Find $\angle AFB$. [i]Proposed by Justin Chen[/i]

2011 Czech and Slovak Olympiad III A, 1

Tags: geometry

In a certain triangle $ABC$, there are points $K$ and $M$ on sides $AB$ and $AC$, respectively, such that if $L$ is the intersection of $MB$ and $KC$, then both $AKLM$ and $KBCM$ are cyclic quadrilaterals with the same size circumcircles. Find the measures of the interior angles of triangle $ABC$.

2019 Durer Math Competition Finals, 15

Tags: diophantine , diophantine equation , number theory , sum

The positive integer $m$ and non-negative integers $x_0, x_1,..., x_{1001}$ satisfy the following equation: $$m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.$$ How many possibilities are there for the value of $m$?

2006 Thailand Mathematical Olympiad, 6

Tags: algebra , inequalities

Let $a, b, c$ be positive reals. Show that $$1 +\frac{3}{ab + bc + ca}\ge \frac{6}{a + b + c}$$

2008 Swedish Mathematical Competition, 1

Tags: convex , geometry , rhombus , inscribed

A rhombus is inscribed in a convex quadrilateral. The sides of the rhombus are parallel with the diagonals of the quadrilateral, which have the lengths $d_1$ and $d_2$. Calculate the length of side of the rhombus , expressed in terms of $d_1$ and $d_2$.

2019 AMC 10, 3

Tags: percent

In a high school with $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument? $\textbf{(A) } 66 \qquad\textbf{(B) } 154 \qquad\textbf{(C) } 186 \qquad\textbf{(D) } 220 \qquad\textbf{(E) } 266$

1999 USAMTS Problems, 2

Tags:

The Fibonacci numbers are defined by $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n>2$. It is well-known that the sum of any $10$ consecutive Fibonacci numbers is divisible by $11$. Determine the smallest integer $N$ so that the sum of any $N$ consecutive Fibonacci numbers is divisible by $12$.

Ukraine Correspondence MO - geometry, 2007.11

Tags: ratio , circumcircle , geometry

Denote by $B_1$ and $C_1$, the midpoints of the sides $AB$ and $AC$ of the triangle $ABC$. Let the circles circumscribed around the triangles $ABC_1$ and $AB_1C$ intersect at points $A$ and $P$, and let the line $AP$ intersect the circle circumscribed around the triangle $ABC$ at points $A$ and $Q$. Find the ratio $\frac{AQ}{AP}$.

2016 Brazil Team Selection Test, 2

Tags: functional equation

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

2024 Korea National Olympiad, 1

Tags: geometry

Let there be a circle with center $O$, and three distinct points $A, B, X$ on the circle, where $A, B, O$ are not collinear. Let $\Omega$ be the circumcircle of triangle $ABO$. Segments $AX, BX$ intersect $\Omega$ at points $C(\neq A), D(\neq B)$, respectively. Prove that $O$ is the orthocenter of triangle $CXD$.

1983 Miklós Schweitzer, 10

Tags: algebra , function , domain , advanced fields , geometry

Let $ R$ be a bounded domain of area $ t$ in the plane, and let $ C$ be its center of gravity. Denoting by $ T_{AB}$ the circle drawn with the diameter $ AB$, let $ K$ be a circle that contains each of the circles $ T_{AB} \;(A,B \in R)$. Is it true in general that $ K$ contains the circle of area $ 2t$ centered at $ C$? [i]J. Szucs[/i]

2020 Romanian Master of Mathematics Shortlist, C1

Tags: combinatorics , grid

Bethan is playing a game on an $n\times n$ grid consisting of $n^2$ cells. A move consists of placing a counter in an unoccupied cell $C$ where the $2n-2$ other cells in the same row or column as $C$ contain an even number of counters. After making $M$ moves Bethan realises she cannot make any more moves. Determine the minimum value of $M$. [i]United Kingdom, Sam Bealing[/i]

2009 QEDMO 6th, 11

Tags: incircle , geometry , concurrency

The inscribed circle of a triangle $ABC$ has the center $O$ and touches the triangle sides $BC, CA$ and $AB$ at points $X, Y$ and $Z$, respectively. The parallels to the straight lines $ZX, XY$ and $YZ$ the straight lines $BC, CA$ and $AB$ (in this order!) intersect through the point $O$. Points $K, L$ and $M$. Then the parallels to the straight lines $CA, AB$ and $BC$ intersect through the points $K, L$ and $M$ in one point.

2022 Thailand Mathematical Olympiad, 5

Tags: function , functional equation , geometry , analytic geometry

Determine all functions $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ that satisfies the equation $$f\left(\frac{x+y+z}{3},\frac{a+b+c}{3}\right)=f(x,a)f(y,b)f(z,c)$$ for any real numbers $x,y,z,a,b,c$ such that $az+bx+cy\neq ay+bz+cx$.

2023 South East Mathematical Olympiad, 1

Tags: algebra , sequence

The positive sequence $\{a_n\}$ satisfies:$a_1=1$ and $$a_n=2+\sqrt{a_{n-1}}-2 \sqrt{1+\sqrt{a_{n-1}}}(n\geq 2)$$ Let $S_n=\sum\limits_{k=1}^{n}{2^ka_k}$. Find the value of $S_{2023}$.

1969 Canada National Olympiad, 8

Tags: function , induction , strong induction , algebra

Let $f$ be a function with the following properties: 1) $f(n)$ is defined for every positive integer $n$; 2) $f(n)$ is an integer; 3) $f(2)=2$; 4) $f(mn)=f(m)f(n)$ for all $m$ and $n$; 5) $f(m)>f(n)$ whenever $m>n$. Prove that $f(n)=n$.

KoMaL A Problems 2022/2023, A. 831

Tags: geometry

In triangle $ABC$ let $F$ denote the midpoint of side $BC$. Let the circle passing through point $A$ and tangent to side $BC$ at point $F$ intersect sides $AB$ and $AC$ at points $M$ and $N$, respectively. Let the line segments $CM$ and $BN$ intersect in point $X$. Let $P$ be the second point of intersection of the circumcircles of triangles $BMX$ and $CNX$. Prove that points $A, F$ and $P$ are collinear. Proposed by Imolay András, Budapest

2010 Romania Team Selection Test, 3

Tags: rectangle , geometry

Two rectangles of unit area overlap to form a convex octagon. Show that the area of the octagon is at least $\dfrac {1} {2}$. [i]Kvant Magazine [/i]

2007 Princeton University Math Competition, 2

Tags:

Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits.