Olympiad problems

2018 Dutch IMO TST, 4

Tags: circumcircle , tangent , geometry , incenter , arc midpoint

In a non-isosceles triangle $ABC$ the centre of the incircle is denoted by $I$. The other intersection point of the angle bisector of $\angle BAC$ and the circumcircle of $\vartriangle ABC$ is $D$. The line through $I$ perpendicular to $AD$ intersects $BC$ in $F$. The midpoint of the circle arc $BC$ on which $A$ lies, is denoted by $M$. The other intersection point of the line $MI$ and the circle through $B, I$ and $C$, is denoted by $N$. Prove that $FN$ is tangent to the circle through $B, I$ and $C$.

2005 Taiwan TST Round 1, 3

Tags: linear algebra , matrix , quadratics , combinatorics proposed , combinatorics

$n$ teams take part in a tournament, in which every two teams compete exactly once, and that no draws are possible. It is known that for any two teams, there exists another team which defeated both of the two teams. Find all $n$ for which this is possible.

2002 Austrian-Polish Competition, 9

Tags: combinatorics , graph theory

A set $P$ of $2002$ persons is given. The family of subsets of $P$ containing exactly $1001$ persons has the property that the number of acquaintance pairs in each such subset is the same. (It is assumed that the acquaintance relation is symmetric). Find the best lower estimation of the acquaintance pairs in the set $P$.

2017 ASDAN Math Tournament, 15

Tags: 2017 , Guts Round

Each face of a regular tetrahedron can be colored one of red, purple, blue, or orange. How many distinct ways can we color the faces of the tetrahedron? Colorings are considered distinct if they cannot reach one another by rotation.

2011 Bulgaria National Olympiad, 2

Tags: algebra , polynomial , induction , algebra proposed

Let $f_1(x)$ be a polynomial of degree $2$ with the leading coefficient positive and $f_{n+1}(x) =f_1(f_n(x))$ for $n\ge 1.$ Prove that if the equation $f_2(x)=0$ has four different non-positive real roots, then for arbitrary $n$ then $f_n(x)$ has $2^n$ different real roots.

PEN P Problems, 2

Tags: geometry , 3D geometry , modular arithmetic , Additive Number Theory

Show that each integer $n$ can be written as the sum of five perfect cubes (not necessarily positive).

1992 IMO Longlists, 45

Tags: function , number theory , Additive Number Theory , Additive combinatorics , counting , IMO Shortlist , IMO Longlist

Let $n$ be a positive integer. Prove that the number of ways to express $n$ as a sum of distinct positive integers (up to order) and the number of ways to express $n$ as a sum of odd positive integers (up to order) are the same.

2002 Denmark MO - Mohr Contest, 4

Tags: geometry , isosceles , right triangle

In triangle $ABC$ we have $\angle C = 90^o$ and $AC = BC$. Furthermore $M$ is an interior pont in the triangle so that $MC = 1 , MA = 2$ and $MB =\sqrt2$. Determine $AB$

1978 Austrian-Polish Competition, 5

Tags: combinatorics , combinatorics unsolved

We are given $1978$ sets of size $40$ each. The size of the intersection of any two sets is exactly $1$. Prove that all the sets have a common element.

2014 Contests, 3

Tags: geometry , convex , pentagon , diagonal

Is there a convex pentagon in which each diagonal is equal to a side?

1981 National High School Mathematics League, 11

Tags:

A billiards table is in the figure of regular hexagon $ABCDEF$. $P$ is the midpoint of $AB$. We shut the ball at $P$, then it touches $Q$ on side $BC$, then it touches side $CD,DE,EF,FA$. Finally, the ball touches side $AB$ again. Let $\theta=\angle BPQ$, find the value range of $\theta$.

PEN O Problems, 40

Tags: floor function , induction , logarithms

Let $X$ be a non-empty set of positive integers which satisfies the following: [list] [*] if $x \in X$, then $4x \in X$, [*] if $x \in X$, then $\lfloor \sqrt{x}\rfloor \in X$. [/list] Prove that $X=\mathbb{N}$.

2013 Korea National Olympiad, 3

Tags: algebra , polynomial , quadratics , induction , calculus , integration , modular arithmetic

Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that (1) For all integer $m$, $f(m) \ne 0$. (2) For all positive odd integer $n$, there exist positive integer $k$ such that $f(k)$ is divided by $n$.

2020 HK IMO Preliminary Selection Contest, 7

Tags: algebra

Solve the equation $\sqrt{7-x}=7-x^2$, where $x>0$.

2019 Jozsef Wildt International Math Competition, W. 69

Tags: geometry , Triangle , angle bisector , geometric inequality , inequalities , circumradius , inradius

Denote $\overline{w_a}, \overline{w_b}, \overline{w_c}$ the external angle-bisectors in triangle $ABC$, prove that $$\sum \limits_{cyc} \frac{1}{w_a}\leq \sqrt{\frac{(s^2 - r^2 - 4Rr)(8R^2 - s^2 - r^2 - 2Rr)}{8s^2R^2r}}$$

2021 BMT, T2

Tags: geometry , 3D geometry , geometric inequality , cube

Compute the radius of the largest circle that fits entirely within a unit cube.

2001 May Olympiad, 5

Tags: game , combinatorics , game strategy

In an $8$-square board -like the one in the figure- there is initially one checker in each square. $ \begin{tabular}{ | l | c | c |c | c| c | c | c | r| } \hline & & & & & & & \\ \hline \end{tabular} $ A move consists of choosing two tokens and moving one of them one square to the right and the other one one square to the left. If after $4$ moves the $8$ checkers are distributed in only $2$ boxes, determine what those boxes can be and how many checkers are in each one.

2008 Junior Balkan Team Selection Tests - Romania, 4

Tags: floor function , irrational number , number theory , number theory proposed

Let $ a,b$ be real nonzero numbers, such that number $ \lfloor an \plus{} b \rfloor$ is an even integer for every $ n \in \mathbb{N}$. Prove that $ a$ is an even integer.

2016 Singapore Junior Math Olympiad, 1

Tags: Perfect Square , number theory

Find all integers$ n$ such that $n^2 + 24n + 35$ is a square.

2023 ELMO Shortlist, G2

Tags: Elmo , geometry

Let $ABC$ be an acute scalene triangle with orthocenter $H$. Line $BH$ intersects $\overline{AC}$ at $E$ and line $CH$ intersects $\overline{AB}$ at $F$. Let $X$ be the foot of the perpendicular from $H$ to the line through $A$ parallel to $\overline{EF}$. Point $B_1$ lies on line $XF$ such that $\overline{BB_1}$ is parallel to $\overline{AC}$, and point $C_1$ lies on line $XE$ such that $\overline{CC_1}$ is parallel to $\overline{AB}$. Prove that points $B$, $C$, $B_1$, $C_1$ are concyclic. [i]Proposed by Luke Robitaille[/i]

2023 ITAMO, 1

Tags: number theory

Let $a, b$ be positive integers such that $54^a=a^b$. Prove that $a$ is a power of $54$.

2006 AMC 10, 13

Tags: probability , AMC

A player pays $ \$ 5$ to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the number on the die is even, the die is rolled again. In this case the player wins if the second number matches the first and loses otherwise. How much should the player win if the game is fair? (In a fair game the probability of winning times the amount won is what the player should pay.) $ \textbf{(A) } \$ 12 \qquad \textbf{(B) } \$ 30 \qquad \textbf{(C) } \$ 50\qquad \textbf{(D) } \$ 60 \qquad \textbf{(E) } \$ 100$