Olympiad problems

2021 Bulgaria National Olympiad, 2

Tags: geometry

A point $T$ is given on the altitude through point $C$ in the acute triangle $ABC$ with circumcenter $O$, such that $\measuredangle TBA=\measuredangle ACB$. If the line $CO$ intersects side $AB$ at point $K$, prove that the perpendicular bisector of $AB$, the altitude through $A$ and the segment $KT$ are concurrent.

2020 LIMIT Category 1, 13

Tags: geometry , limit

On the side $AC$ of an acute triangle $\triangle ABC$, a point $D$ is taken such that $2AD=CD=2, BD\perp AC$. A circle of radius $2$ passes through $A$ and $D$ and is tangent to the circumcircle of $\triangle BDC$. Find $[\text{Area}(\triangle ABC)]$ where $[.]$ is the greatest integer function.

2015 BMT Spring, 6

Tags: geometry

Let $C$ be the sphere $x^2 + y^2 + (z -1)^2 = 1$. Point $P$ on $C$ is $(0, 0, 2)$. Let $Q = (14, 5, 0)$. If $PQ$ intersects $C$ again at $Q'$, then find the length $PQ'$ .

1983 Polish MO Finals, 3

Tags: chessboard , infinite chessboard , combinatorics

Consider the following one-player game on an infinite chessboard. If two horizontally or vertically adjacent squares are occupied by a pawn each, and a square on the same line that is adjacent to one of them is empty, then it is allowed to remove the two pawns and place a pawn on the third (empty) square. Prove that if in the initial position all the pawns were forming a rectangle with the number of squares divisible by $3$, then it is not possible to end the game with only one pawn left on the board.

2009 Saint Petersburg Mathematical Olympiad, 7

Tags: algebra

$f(x)=x^2+x$ $b_1,...,b_{10000}>0$ and $|b_{n+1}-f(b_n)|\leq \frac{1}{1000}$ for $n=1,...,9999$ Prove, that there is such $a_1>0$ that $a_{n+1}=f(a_n);n=1,...,9999$ and $|a_n-b_n|<\frac{1}{100}$

2003 Baltic Way, 9

Tags: induction , combinatorics

It is known that $n$ is a positive integer and $n \le 144$. Ten questions of the type “Is $n$ smaller than $a$?” are allowed. Answers are given with a delay: for $i = 1, \ldots , 9$, the $i$-th question is answered only after the $(i + 1)$-th question is asked. The answer to the tenth question is given immediately. Find a strategy for identifying $n$.

1991 India National Olympiad, 6

Tags: induction , algebra

(i) Determine the set of all positive integers $n$ for which $3^{n+1}$ divides $2^{3^n} + 1$; (ii) Prove that $3^{n+2}$ does not divide $2^{3^n} + 1$ for any positive integer $n$.

2023 AIME, 9

Tags: algebra , polynomial

Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are integers in $\{-20, -19,-18, \dots , 18, 19, 20\}$, such that there is a unique integer $m \neq 2$ with $p(m) = p(2)$.

1969 Canada National Olympiad, 7

Tags: quadratic , modular arithmetic , algebra , function , domain , polynomial , abstract algebra

Show that there are no integers $a,b,c$ for which $a^2+b^2-8c=6$.

2017 Harvard-MIT Mathematics Tournament, 20

Tags: number theory

For positive integers $a$ and $N$, let $r(a, N) \in \{0, 1, \dots, N - 1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \le 1000000$ for which \[r(n, 1000) > r(n, 1001).\]

2004 Greece National Olympiad, 3

Tags: symmetry , geometry

Consider a circle $K(O,r)$ and a point $A$ outside $K.$ A line $\epsilon$ different from $AO$ cuts $K$ at $B$ and $C,$ where $B$ lies between $A$ and $C.$ Now the symmetric line of $\epsilon$ with respect to axis of symmetry the line $AO$ cuts $K$ at $E$ and $D,$ where $E$ lies between $A$ and $D.$ Show that the diagonals of the quadrilateral $BCDE$ intersect in a fixed point.

2023 CCA Math Bonanza, L2.3

Tags:

A frog starts at origin (0,0). At each minute it picks a random integer $x$, turns $x$ degrees counterclockwise, and jumps exactly 1 unit. After 2 minutes what is the probability that the frog is exactly one unit from the origin? [i]Lightning 2.3[/i]

2014 Stars Of Mathematics, 4

Tags: combinatorics

At a point on the real line sits a greyhound. On one of the sides a hare runs, away from the hound. The only thing known is that the (maximal) speed of the hare is strictly less than the (maximal) speed of the greyhound (but not their precise ratio). Does the greyhound have a strategy for catching the hare in a finite amount of time? ([i]Dan Schwarz[/i])

2003 Cono Sur Olympiad, 3

Tags: geometry

Let $ABC$ be an acute triangle such that $\angle{B}=60$. The circle with diameter $AC$ intersects the internal angle bisectors of $A$ and $C$ at the points $M$ and $N$, respectively $(M\neq{A},$ $N\neq{C})$. The internal bisector of $\angle{B}$ intersects $MN$ and $AC$ at the points $R$ and $S$, respectively. Prove that $BR\leq{RS}$.

2004 Tournament Of Towns, 4

Tags: geometry , locus , circumcenter , circumcircle

A circle with the center $I$ is entirely inside of a circle with center $O$. Consider all possible chords $AB$ of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle $AIB$.

2009 China Northern MO, 7

Tags: number theory , algebra , floor function

Let $\lfloor m \rfloor$ be the largest integer smaller than $m$ . Assume $x,y \in \mathbb{R+}$ , For all positive integer $n$ , $\lfloor x \lfloor ny \rfloor \rfloor =n-1$ . Prove : $xy=1$ , $y$ is an irrational number larger than $ 1 $ .

1994 All-Russian Olympiad Regional Round, 10.1

Tags: algebra , combinatorics

We have seven equal pails with water, filled to one half, one third, one quarter, one fifth, one eighth, one ninth, and one tenth, respectively. We are allowed to pour water from one pail into another until the first pail empties or the second one fills to the brim. Can we obtain a pail that is filled to (a) one twelfth, (b) one sixth after several such steps?

2020 Canadian Mathematical Olympiad Qualification, 1

Tags: floor function , radical , algebra

Show that for all integers $a \ge 1$,$ \lfloor \sqrt{a}+\sqrt{a+1}+\sqrt{a+2}\rfloor = \lfloor \sqrt{9a+8}\rfloor$

2018 Oral Moscow Geometry Olympiad, 6

Tags: geometry , equilateral triangle , equilateral , combinatorial geometry

Cut each of the equilateral triangles with sides $2$ and $3$ into three parts and construct an equilateral triangle from all received parts.

2022 Oral Moscow Geometry Olympiad, 5

Tags: tangent , geometry , tangent circle

Circle $\omega$ is tangent to the interior of the circle $\Omega$ at the point C. Chord $AB$ of circle $\Omega$ is tangent to $\omega$. Chords $CF$ and $BG$ of circle $\Omega$ intersect at point $E$ lying on $\omega$. Prove that the circumcircle of triangle $CGE$ is tangent to straight line $AF$. (I. Kukharchuk)

2010 Dutch IMO TST, 1

Tags: minimum , algebra , sequence

Consider sequences $a_1, a_2, a_3,...$ of positive integers. Determine the smallest possible value of $a_{2010}$ if (i) $a_n < a_{n+1}$ for all $n\ge 1$, (ii) $a_i + a_l > a_j + a_k$ for all quadruples $ (i, j, k, l)$ which satisfy $1 \le i < j \le k < l$.

2005 China Western Mathematical Olympiad, 8

Tags: ramsey theory , combinatorics , graph theory

For $n$ people, if it is known that (a) there exist two people knowing each other among any three people, and (b) there exist two people not knowing each other among any four people. Find the maximum of $n$. Here, we assume that if $A$ knows $B$, then $B$ knows $A$.

2020 BMT Fall, 4

Tags: ratio , algebra

Let $\varphi$ be the positive solution to the equation $$x^2=x+1.$$ For $n\ge 0$, let $a_n$ be the unique integer such that $\varphi^n-a_n\varphi$ is also an integer. Compute $$\sum_{n=0}^{10}a_n.$$

2006 Vietnam Team Selection Test, 1

Tags: inequalities , function , convex-concave inequalities , three variable inequality

Prove that for all real numbers $x,y,z \in [1,2]$ the following inequality always holds: \[ (x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\geq 6(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}). \] When does the equality occur?

2019-2020 Fall SDPC, 4

Tags: geometry

Let $\triangle{ABC}$ be an acute, scalene triangle with orthocenter $H$, and let $AH$ meet the circumcircle of $\triangle{ABC}$ at a point $D \neq A$. Points $E$ and $F$ are chosen on $AC$ and $AB$ such that $DE \perp AC$ and $DF \perp AB$. Show that $BE$, $CF$, and the line through $H$ parallel to $EF$ concur.