Olympiad problems

2015 Romania Team Selection Tests, 2

Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.

2011 Balkan MO Shortlist, G3

Tags: circumcircle , ratio , geometry

Given a triangle $ABC$, let $D$ be the midpoint of the side $AC$ and let $M$ be the point that divides the segment $BD$ in the ratio $1/2$; that is, $MB/MD=1/2$. The rays $AM$ and $CM$ meet the sides $BC$ and $AB$ at points $E$ and $F$, respectively. Assume the two rays perpendicular: $AM\perp CM$. Show that the quadrangle $AFED$ is cyclic if and only if the median from $A$ in triangle $ABC$ meets the line $EF$ at a point situated on the circle $ABC$.

2008 Korea - Final Round, 3

Tags: function , algebra

Determine all functions $f : \mathbb{R}^+\rightarrow\mathbb{R}$ that satisfy the following $f(1)=2008$, $|{f(x)}| \le x^2+1004^2$, $f\left (x+y+\frac{1}{x}+\frac{1}{y}\right )=f\left (x+\frac{1}{y}\right )+f\left (y+\frac{1}{x}\right ).$

2024 IFYM, Sozopol, 5

Tags: combinatorics

An infinite grid with two rows is divided into unit squares. One of the cells in the second row is colored red and all other cells in the grid are white. Initially, we are in the red cell. In one move, we can move from one cell to an adjacent cell (sharing a side). Find the number of sequences of $ n $ moves such that no cell is visited more than once. (In particular, it is not allowed to return to the red cell after several moves.)

2012 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory , diophantine , diophantine equation

Let $m$ and $n$ be two positive integers, $m, n \ge 2$. Solve in the set of the positive integers the equation $x^n + y^n = 3^m$.

2018 Belarus Team Selection Test, 1.4

Tags: inequalities , geometry

Let $A_1H_1,A_2H_2,A_3H_3$ be altitudes and $A_1L_1,A_2L_2,A_3L_3$ be bisectors of acute-angles triangle $A_1A_2A_3$. Prove the inequality $S(L_1L_2L_3)\ge S(H_1H_2H_3)$ where $S$ stands for the area of a triangle. [i](B. Bazylev)[/i]

2020-21 KVS IOQM India, 19

Tags: geometry , semicircle , paper folding , folding

A semicircular paper is folded along a chord such that the folded circular arc is tangent to the diameter of the semicircle. The radius of the semicircle is $4$ units and the point of tangency divides the diameter in the ratio $7 :1$. If the length of the crease (the dotted line segment in the figure) is $\ell$ then determine $ \ell^2$. [img]https://cdn.artofproblemsolving.com/attachments/5/6/63fed83742c8baa92d9e63962a77a57d43556f.png[/img]

2019 Turkey Junior National Olympiad, 3

Tags: incenter , circumcircle , geometry

In $ABC$ triangle $I$ is incenter and incircle of $ABC$ tangents to $BC,AC,AB$ at $D,E,F$, respectively. If $AI$ intersects $DE$ and $DF$ at $P$ and $Q$, prove that the circumcenter of $DPQ$ triangle is the midpoint of $BC$.

2013 Argentina National Olympiad Level 2, 6

Tags: geometry , rectangle , square , cover

Is there a square with side lenght $\ell < 1$ that can completely cover any rectangle of diagonal $1$?

2023 Belarus Team Selection Test, 1.1

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.

2006 BAMO, 2

Tags: number theory , consecutive , sum , algebra

Since $24 = 3+5+7+9$, the number $24$ can be written as the sum of at least two consecutive odd positive integers. (a) Can $2005$ be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not. (b) Can $2006$ be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not.

2025 Harvard-MIT Mathematics Tournament, 2

Tags: algebra , number theory

Mark writes the expression $\sqrt{\underline{abcd}}$ on the board, where $\underline{abcd}$ is a four-digit number and $a \neq 0.$ Derek, a toddler, decides to move the $a,$ changing Mark's expression to $a\sqrt{\underline{bcd}}.$ Surprisingly, these two expressions are equal. Compute the only possible four-digit number $\underline{abcd}.$

2004 Postal Coaching, 11

Tags: geometry , geometric transformation , rotation , homothety , rectangle , search , circumcircle

Three circles touch each other externally and all these cirlces also touch a fixed straight line. Let $A,B,C$ be the mutual points of contact of these circles. If $\omega$ denotes the Brocard angle of the triangle $ABC$, prove that $\cot{\omega}$ = 2.

2005 Croatia National Olympiad, 3

Tags: pigeonhole principle , induction , number theory

Show that there is a unique positive integer which consists of the digits $2$ and $5$, having $2005$ digits and divisible by $2^{2005}$.

2015 Bulgaria National Olympiad, 6

Tags: combinatorics , minimum

In a mathematical olympiad students received marks for any of the four areas: algebra, geometry, number theory and combinatorics. Any two of the students have distinct marks for all four areas. A group of students is called [i]nice [/i] if all students in the group can be ordered in increasing order simultaneously of at least two of the four areas. Find the least positive integer N, such that among any N students there exist a [i]nice [/i] group of ten students.

2009 Sharygin Geometry Olympiad, 5

Tags: geometry , angle , projection

Given triangle $ABC$. Point $M$ is the projection of vertex $B$ to bisector of angle $C$. $K$ is the touching point of the incircle with side $BC$. Find angle $\angle MKB$ if $\angle BAC = \alpha$ (V.Protasov)

2010 Grand Duchy of Lithuania, 5

Tags: number theory , divide

Find positive integers n that satisfy the following two conditions: (a) the quotient obtained when $n$ is divided by $9$ is a positive three digit number, that has equal digits. (b) the quotient obtained when $n + 36$ is divided by $4$ is a four digit number, the digits beeing $2, 0, 0, 9$ in some order.

2013 Baltic Way, 2

Tags: algebra , polynomial

Let $k$ and $n$ be positive integers and let $x_1, x_2, \cdots, x_k, y_1, y_2, \cdots, y_n$ be distinct integers. A polynomial $P$ with integer coefficients satisfies \[P(x_1)=P(x_2)= \cdots = P(x_k)=54\] \[P(y_1)=P(y_2)= \cdots = P(y_n)=2013.\] Determine the maximal value of $kn$.

2022 Oral Moscow Geometry Olympiad, 3

Tags: equal segments , geometry

In quadrilateral $ABCD$, sides $AB$ and $CD$ are equal (but not parallel), points $M$ and $N$ are the midpoints of $AD$ and $BC$. The perpendicular bisector of $MN$ intersects sides $AB$ and $CD$ at points $P$ and $Q$, respectively. Prove that $AP = CQ$. (M. Kungozhin)

2012 District Olympiad, 3

Tags: linear algebra , matrix

Let be a natural number $ n, $ and two matrices $ A,B\in\mathcal{M}_n\left(\mathbb{C}\right) $ with the property that $$ AB^2=A-B. $$ [b]a)[/b] Show that the matrix $ I_n+B $ is inversable. [b]b)[/b] Show that $ AB=BA. $

1993 Tournament Of Towns, (362) 1

Tags: number theory , algebra

One of two wizards, named Steven, was told the sum of three positive integers and the other, named Peter, their product. “If I knew”, said Steven, “that your number is greater than mine, I could find the integers”. “But my number is less than yours,” replied Peter, “and the integers are $X$, $Y$ and $Z$”. Find these numbers. (L Borisov)