Olympiad problems

2019 SAFEST Olympiad, 5

Tags: combinatorics

There are $25$ IMO participants attending a party. Every two of them speak to each other in some language, and they use only one language even if they both know some other language as well. Among every three participants there is a person who uses the same language to speak to the other two (in that group of three). Prove that there is an IMO participant who speaks the same language to at least $10$ other participants

2020 IMO Shortlist, N1

Tags: number theory , sequence , divisibility

Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

2020 Memorial "Aleksandar Blazhevski-Cane", 1

Tags: geometry , concurrency

A convex quadrilateral $ABCD$ is given in which the bisectors of the interior angles $\angle ABC$ and $\angle ADC$ have a common point on the diagonal $AC$. Prove that the bisectors of the interior angles $\angle BAD$ and $\angle BCD$ have a common point on the diagonal $BD$.

2010 National Chemistry Olympiad, 8

Tags:

Which compound contains the highest percentage of nitrogen by mass? $ \textbf{(A)} \text{NH}_2\text{OH} (M=33.0) \qquad\textbf{(B)}\text{NH}_4\text{NO}_2 (M=64.1)\qquad$ $\textbf{(C)}\text{N}_2\text{O}_3 (M=76.0)\qquad\textbf{(D)}\text{NH}_4\text{NH}_2\text{CO}_2 (M=78.1)\qquad $

1974 Polish MO Finals, 1

Tags: geometry , 3d geometry , tetrahedron

In a tetrahedron $ABCD$ the edges $AB$ and $CD$ are perpendicular and $\angle ACB =\angle ADB$. Prove that the plane through $AB$ and the midpoint of the edge $CD$, is perpendicular to $CD$.

2023 Azerbaijan IZhO TST, 2

Tags: algebra , polynomial

P(x) is polynomial such that, polynomial P(P(x)) is strictly monotone in all real number line. Prove that polynomial P(x) is also strictly monotone in all real number line.

2011 Morocco National Olympiad, 4

Tags: modular arithmetic , number theory

Let $a, b, c, d, m, n$ be positive integers such that $a^{2}+b^{2}+c^{2}+d^{2}=1989$, $n^{2}=max\left \{ a,b,c,d \right \}$ and $a+b+c+d=m^{2}$. Find the values of $m$ and $n$.

2016 Peru IMO TST, 10

Tags: geometry

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

2006 Tournament of Towns, 4

Tags: number theory

Is there exist some positive integer $n$, such that the first decimal of $2^n$ (from left to the right) is $5$ while the first decimal of $5^n$ is $2$? [i](5 points)[/i]

2021 Thailand Mathematical Olympiad, 10

Tags: algebra , polynomial

Let $d\geq 13$ be an integer, and let $P(x) = a_dx^d + a_{d-1}x^{d-1} + \dots + a_1x+a_0$ be a polynomial of degree $d$ with complex coefficients such that $a_n = a_{d-n}$ for all $n\in\{0,1,\dots,d\}$. Prove that if $P$ has no double roots, then $P$ has two distinct roots $z_1$ and $z_2$ such that $|z_1-z_2|<1$.

2023 Brazil Cono Sur TST, 1

Tags:

A quadrilateral $ABCD$ is inscribed in a circle and the lenght of side $AD$ equals the sum of the lenghts of the sides $AB$ and $CD$. Prove that the angle bisectors of $\angle ABC$ and $\angle BCD$ meet on the side $AD$.

Russian TST 2020, P3

Tags: geometry

Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$. (Slovakia)

2003 All-Russian Olympiad Regional Round, 8.5

Tags: combinatorics , combinatorial geometry , absolute value

Numbers from$ 1$ to $8$ were written at the vertices of the cube, and on each edge the absolute value of the difference between the numbers at its ends.. What is the smallest number of different numbers that can be written on the edges?

2016-2017 SDML (Middle School), 12

Tags:

What is the area of the region enclosed by the graph of the equations $x^2 - 14x + 3y + 70 = 21 + 11y - y^2$ that lies below the line $y = x-3$? $\text{(A) }6\pi\qquad\text{(B) }7\pi\qquad\text{(C) }8\pi\qquad\text{(D) }9\pi\qquad\text{(E) }10\pi$

2004 Estonia National Olympiad, 4

Tags: inequalities , algebra

Let $a, b, c$ be positive real numbers such that $a^2 + b^2 + c^2 = 3$. Prove that $$\frac{1}{1+2ab}+\frac{1}{1+2bc}+\frac{1}{1+2ca}\ge 1$$

2015 China Northern MO, 6

Tags: combinatorics , combinatorial geometry , tiles , tiling

The figure obtained by removing one small unit square from the $2\times 2$ grid table is called an $L$ ''shape". .Put $k$ L-shapes in an $8\times 8$ grid table. Each $L$-shape can be rotated, but each $L$ shape is required to cover exactly three small unit squares in the grid table, and the common area covered by any two $L$ shapes is $0$, and except for these $k$ $L$ shapes, no other $L$ shapes can be placed. Find the minimum value of $k$.

Indonesia MO Shortlist - geometry, g5

Tags: midpoint , geometry , equal angles

Given a cyclic quadrilateral $ABCD$. Suppose $E, F, G, H$ are respectively the midpoint of the sides $AB, BC, CD, DA$. The line passing through $G$ and perpendicular on $AB$ intersects the line passing through $H$ and perpendicular on $BC$ at point $K$. Prove that $\angle EKF = \angle ABC$.

2015 Greece National Olympiad, 2

Tags: polynomial , algebra

Let $P(x)=ax^3+(b-a)x^2-(c+b)x+c$ and $Q(x)=x^4+(b-1)x^3+(a-b)x^2-(c+a)x+c$ be polynomials of $x$ with $a,b,c$ non-zero real numbers and $b>0$.If $P(x)$ has three distinct real roots $x_0,x_1,x_2$ which are also roots of $Q(x)$ then: A)Prove that $abc>28$, B)If $a,b,c$ are non-zero integers with $b>0$,find all their possible values.

2016 Estonia Team Selection Test, 6

Tags: arc , polynomial , coloring , combinatorics , algebra

A circle is divided into arcs of equal size by $n$ points ($n \ge 1$). For any positive integer $x$, let $P_n(x)$ denote the number of possibilities for colouring all those points, using colours from $x$ given colours, so that any rotation of the colouring by $ i \cdot \frac{360^o}{n}$ , where i is a positive integer less than $n$, gives a colouring that differs from the original in at least one point. Prove that the function $P_n(x)$ is a polynomial with respect to $x$.

2009 AMC 10, 21

Tags: modular arithmetic , geometric series

What is the remainder when $ 3^0\plus{}3^1\plus{}3^2\plus{}\ldots\plus{}3^{2009}$ is divided by $ 8$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 6$

2002 Junior Balkan Team Selection Tests - Moldova, 1

Tags: number theory , greatest common divisor

For any integer $n$ we define the numbers $a = n^5 + 6n^3 + 8n$ ¸ $b = n^4 + 4n^2 + 3$. Prove that the numbers $a$ and $b$ are relatively prime or have the greatest common factor of $3$.

2011 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: circumcircle , power of a point , radical axis , geometric transformation , angle bisector , geometry

Let $ABC$ an acute triangle and $H$ its orthocenter. Let $E$ and $F$ be the intersection of lines $BH$ and $CH$ with $AC$ and $AB$ respectively, and let $D$ be the intersection of lines $EF$ and $BC$. Let $\Gamma_1$ be the circumcircle of $AEF$, and $\Gamma_2$ the circumcircle of $BHC$. The line $AD$ intersects $\Gamma_1$ at point $I \neq A$. Let $J$ be the feet of the internal bisector of $\angle{BHC}$ and $M$ the midpoint of the arc $\stackrel{\frown}{BC}$ from $\Gamma_2$ that contains the point $H$. The line $MJ$ intersects $\Gamma_2$ at point $N \neq M$. Show that the triangles $EIF$ and $CNB$ are similar.

2020 Tuymaada Olympiad, 4

Tags: ratio , geometry , perimeter

Points $D$ and $E$ lie on the lines $BC$ and $AC$ respectively so that $B$ is between $C$ and $D$, $C$ is between $A$ and $E$, $BC = BD$ and $\angle BAD = \angle CDE$. It is known that the ratio of the perimeters of the triangles $ABC$ and $ADE$ is $2$. Find the ratio of the areas of these triangles.

2018 239 Open Mathematical Olympiad, 10-11.3

Tags: number theory

Given a prime number $p>5$. It is known that the length of the smallest period of the fraction $1/p$ is a multiple of three. This period (including possible leading zeros) was written on a strip of paper and cut into three equal-length parts $a$, $b$, $c$ (they may also have leading zeros). What could be the sum of the three periodic fractions: $0.(a)$, $0.(b)$, and $0.(c)$? [i]Proposed by A. Khrabrov[/i]

1989 China Team Selection Test, 3

Tags: combinatorics

$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.