Olympiad problems

2022 Czech and Slovak Olympiad III A, 3

Given a scalene acute triangle $ABC$, let M be the midpoints of its side $BC$ and $N$ the midpoint of the arc $BAC$ of its circumcircle. Let $\omega$ be the circle with diameter $BC$ and $D$, $E$ its intersections with the bisector of angle $\angle BAC$. Points $D'$, $E'$ lie on $\omega$ such that $DED'E' $ is a rectangle. Prove that $D'$, $E'$, $M$, $N$ lie on a single circle. [i] (Patrik Bak)[/i]

2010 Germany Team Selection Test, 1

Tags: algebra , polynomial

Let $a \in \mathbb{R}.$ Show that for $n \geq 2$ every non-real root $z$ of polynomial $X^{n+1}-X^2+aX+1$ satisfies the condition $|z| > \frac{1}{\sqrt[n]{n}}.$

2018 India IMO Training Camp, 3

Tags: convex polygon , combinatorics

A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the [b]interior[/b] of the polygon, in such a way that all the resulting triangles have vertices of all three colours.

2009 Grand Duchy of Lithuania, 1

Tags: multiple , number theory , sum of digits

The natural number $N$ is a multiple of $2009$ and the sum of its (decimal) digits equals $2009$. (a) Find one such number. (b) Find the smallest such number.

1993 Iran MO (2nd round), 1

Tags: combinatorics

$G$ is a graph with $n$ vertices $A_1,A_2,\ldots,A_n,$ such that for each pair of non adjacent vertices $A_i$ and $A_j$ , there exist another vertex $A_k$ that is adjacent to both $A_i$ and $A_j .$ [b](a) [/b]Find the minimum number of edges in such a graph. [b](b) [/b]If $n = 6$ and $A_1,A_2,A_3,A_4,A_5,$ and $A_6$ form a cycle of length $6,$ find the number of edges that must be added to this cycle such that the above condition holds.

1998 Turkey Team Selection Test, 1

Tags: combinatorics

Suppose $n$ houses are to be assigned to $n$ people. Each person ranks the houses in the order of preference, with no ties. After the assignment is made, it is observed that every other assignment would assign to at least one person a less preferred house. Prove that there is at least one person who received the house he/she preferred most under this assignment.

2002 China Team Selection Test, 2

Tags: combinatorics

$ m$ and $ n$ are positive integers. In a $ 8 \times 8$ chessboard, $ (m,n)$ denotes the number of grids a Horse can jump in a chessboard ($ m$ horizontal $ n$ vertical or $ n$ horizontal $ m$ vertical ). If a $ (m,n) \textbf{Horse}$ starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a $ \textbf{H Route}$. For example, the $ (1,2) \textbf{Horse}$ has its $ \textbf{H Route}$. Find the smallest positive integer $ t$, such that from any grid of the chessboard, the $ (t,t\plus{}1) \textbf{Horse}$ does not has any $ \textbf{H Route}$.

2022 BMT, 1

Tags: easy , algebra

Define an operation $\Diamond$ as $ a \Diamond b = 12a - 10b.$ Compute the value of $((((20 \Diamond 22) \Diamond 22) \Diamond 22) \Diamond22).$

1993 National High School Mathematics League, 2

Tags: function

$f(x)=a\sin x+b\sqrt[3]{x}+4$. If $f(\lg\log_{3}10)=5$, then the value of $f(\lg\lg 3)$ is $\text{(A)}-5\qquad\text{(B)}-3\qquad\text{(C)}3\qquad\text{(D)}$ not sure

2010 Thailand Mathematical Olympiad, 1

Tags: combinatorics

Find the number of ways to distribute $11$ balls into $5$ boxes with different sizes, so that each box receives at least one ball, and the total number of balls in the largest and smallest boxes is more than the total number of balls in the remaining boxes.

2019 Turkey EGMO TST, 5

Tags: geometry

Let $D$ be the midpoint of $\overline{BC}$ in $\Delta ABC$. Let $P$ be any point on $\overline{AD}$. If the internal angle bisector of $\angle ABP$ and $\angle ACP$ intersect at $Q$. Prove that, if $BQ \perp QC$, then $Q$ lies on $AD$

MathLinks Contest 6th, 5.3

Tags: geometric inequality , geometry

Let $ABC$ be a triangle, and let $ABB_2A_3$, $BCC_3B_1$ and $CAA_1C_2$ be squares constructed outside the triangle. Denote with $S$ the area of the triangle $ABC$ and with s the area of the triangle formed by the intersection of the lines $A_1B_1$, $B_2C_2$ and $C_3A_3$. Prove that $s \le (4 - 2\sqrt3)S$.

2015 Junior Regional Olympiad - FBH, 1

Tags: counting years

Father is $42$ years old, and son has $14$ years. In how many years father will be twice as old as his son?

2004 National Olympiad First Round, 14

Tags:

What is $o-w$, if $gun^2 = wowgun$ where $g,n,o,u,w \in \{0,1,2,\dots, 9\}$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None of above} $

2017 Taiwan TST Round 2, 1

Tags: geometry

Given a circle and four points $B,C,X,Y$ on it. Assume $A$ is the midpoint of $BC$, and $Z$ is the midpoint of $XY$. Let $L_1,L_2$ be lines perpendicular to $BC$ and pass through $B,C$ respectively. Let the line pass through $X$ and perpendicular to $AX$ intersects $L_1,L_2$ at $X_1,X_2$ respectively. Similarly, let the line pass through $Y$ and perpendicular to $AY$ intersects $L_1,L_2$ at $Y_1,Y_2$ respectively. Assume $X_1Y_2$ intersects $X_2Y_1$ at $P$. Prove that $\angle AZP=90^o.$ [i]Proposed by William Chao[/i]

2007 Harvard-MIT Mathematics Tournament, 20

Tags: algebra , vieta , polynomial

For $a$ a positive real number, let $x_1$, $x_2$, $x_3$ be the roots of the equation $x^3-ax^2+ax-a=0$. Determine the smallest possible value of $x_1^3+x_2^3+x_3^3-3x_1x_2x_3$.

2011 JBMO Shortlist, 5

Tags: number theory , sum of digits

Find the least positive integer such that the sum of its digits is $2011$ and the product of its digits is a power of $6$.

2005 Georgia Team Selection Test, 5

Tags: geometry , trigonometry , circumcircle

Let $ ABCD$ be a convex quadrilateral. Points $ P,Q$ and $ R$ are the feets of the perpendiculars from point $ D$ to lines $ BC, CA$ and $ AB$, respectively. Prove that $ PQ\equal{}QR$ if and only if the bisectors of the angles $ ABC$ and $ ADC$ meet on segment $ AC$.

2013 Taiwan TST Round 1, 2

Tags: polyomino , combinatorics

A V-tromino is a diagram formed by three unit squares.(As attachment.) (a)Is it possible to cover a $3\times 2013$ table by $3\times 671$ V-trominoes? (b)Is it possible to cover a $5\times 2013$ table by $5\times 671$ V-trominoes?

2014 Kyiv Mathematical Festival, 1a

Tags:

a) 2 white and 2 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 4, to the other black cat is 8. The sum of distances from the black cats to one white cat is 3, to the other white cat is 9. What cats are sitting on the edges? b) 2 white and 3 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 11, to another black cat is 7 and to the third black cat is 9. The sum of distances from the black cats to one white cat is 12, to the other white cat is 15. What cats are sitting on the edges? [size=85](Kyiv mathematical festival 2014)[/size]

2020/2021 Tournament of Towns, P4

Tags: geometry

The sides of a triangle are divided by the angle bisectors into two segments each. Is it always possible to form two triangles from the obtained six segments? [i]Lev Emelyanov[/i]

1997 Slovenia National Olympiad, Problem 1

Tags: algebra

Prove that if real numbers $a,b,c,d$ satisfy $a^2+b^2+(a+b)^2=c^2+d^2+(c+d)^2$, then they also satisfy $a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4$.

2003 Romania National Olympiad, 2

Tags: pigeonhole principle

In a meeting there are 6 participants. It is known that among them there are seven pairs of friends and in any group of three persons there are at least two friends. Prove that: (a) there exists a person who has at least three friends; (b) there exists three persons who are friends with each other. [i]Valentin Vornicu[/i]

2022 Belarusian National Olympiad, 11.1

Tags: sequence , number theory

A sequence of positive integer numbers $a_1,a_2,\ldots$ for $i \geq 3$ satisfies $$a_{i+1}=a_i+gcd(a_{i-1},a_{i-2})$$ Prove that there exist two positive integer numbers $N, M$, such that $a_{n+1}-a_n=M$ for all $n \geq N$

2016 Hong Kong TST, 1

Tags: combinatorics

During a school year 44 competitions were held. Exactly 7 students won in each of the competition. For any two competitions, there exists exactly 1 student who won both competitions. Is it true that there exists a student who won all the competitions?