Olympiad problems

2021 Brazil Team Selection Test, 3

Let $ABC$ be an acute triangle with $AC>CB$ and let $M$ be the midpoint of side $AB$. Denote by $Q$ the midpoint of the big arc $AB$ which cointais $C$ and by $B_1$ the point inside $AC$ such that $BC=CB_1$. $B_1Q$ touches $BC$ in $E$ and $K$ is the intersection of $(BB_1M)$ and $(ABC)$. Prove that $KC$ bissects $B_1E$.

1990 IMO Longlists, 60

Tags: geometry , 3d geometry , analytic geometry , euler , combinatorics

Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let $ A$ be the beginning vertex and $ B$ be the end vertex. Let there be $ p \times q \times r$ cubes on the string $ (p, q, r \geq 1).$ [i](a)[/i] Determine for which values of $ p, q,$ and $ r$ it is possible to build a block with dimensions $ p, q,$ and $ r.$ Give reasons for your answers. [i](b)[/i] The same question as (a) with the extra condition that $ A \equal{} B.$

2024 District Olympiad, P4

Tags: geometry , vector

Let $H{}$ be the orthocenter of the triangle $ABC{}$ and $X{}$ be the midpoint of the side $BC.$ The perpendicular at $H{}$ to $HX{}$ intersects the sides $(AB)$ and $(AC)$ at $Y{}$ and $Z{}$ respectively. Let $O{}$ be the circumcenter of $ABC{}$ and $O'$ be the circumcenter of $BHC.$ [list=a] [*]Prove that $HY=HZ.$ [*]Prove that $\overrightarrow{AY}+\overrightarrow{AZ}=2\overrightarrow{OO'}.$ [/list]

2018 Federal Competition For Advanced Students, P2, 6

Tags: decimal representation , number theory , digit

Determine all digits $z$ such that for each integer $k \ge 1$ there exists an integer $n\ge 1$ with the property that the decimal representation of $n^9$ ends with at least $k$ digits $z$. [i](Proposed by Walther Janous)[/i]

2017 Saudi Arabia BMO TST, 4

Tags: fibonacci number , number theory

Fibonacci sequences is defined as $f_1=1$,$f_2=2$, $f_{n+1}=f_{n}+f_{n-1}$ for $n \ge 2$. a) Prove that every positive integer can be represented as sum of several distinct Fibonacci number. b) A positive integer is called [i]Fib-unique[/i] if the way to represent it as sum of several distinct Fibonacci number is unique. Example: $13$ is not Fib-unique because $13 = 13 = 8 + 5 = 8 + 3 + 2$. Find all Fib-unique.

1988 IMO Longlists, 55

Tags: inequalities , trigonometry , induction , rearrangement inequality , algebra

Suppose $\alpha_i > 0, \beta_i > 0$ for $1 \leq i \leq n, n > 1$ and that \[ \sum^n_{i=1} \alpha_i = \sum^n_{i=1} \beta_i = \pi. \] Prove that \[ \sum^n_{i=1} \frac{\cos(\beta_i)}{\sin(\alpha_i)} \leq \sum^n_{i=1} \cot(\alpha_i). \]

OIFMAT II 2012, 3

Tags: equilateral , geometry , angle

In the interior of an equilateral triangle $ ABC $ a point $ P $ is chosen such that $ PA ^2 = PB ^2 + PC ^2 $. Find the measure of $ \angle BPC $.

1968 IMO, 2

Tags: number theory , decimal representation , digit , equation

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

2017 MIG, 6

Tags:

Thomas is to read $225$ pages of a book over the summer. He decides to read one page the first day, three pages the second day, five pages the third day, and so on, each day reading two more pages than the previous. How many days will it take Thomas to finish reading the book? $\textbf{(A) } 11\qquad\textbf{(B) } 12\qquad\textbf{(C) } 13\qquad\textbf{(D) } 14\qquad\textbf{(E) } 15$

2013 Romanian Master of Mathematics, 6

Tags: geometry , geometric transformation , reflection , combinatorics

A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.

2023 CMIMC Geometry, 6

Tags: geometry

Let $ABCD$ be a regular tetrahedron. Suppose points $X$, $Y$, and $Z$ lie on rays $AB$, $AC$, and $AD$ respectively such that $XY=YZ=7$ and $XZ=5$. Moreover, the lengths $AX$, $AY$, and $AZ$ are all distinct. Find the volume of tetrahedron $AXYZ$. [i]Proposed by Connor Gordon[/i]

2021 Miklós Schweitzer, 7

Tags: complex analysis , binary

If the binary representations of the positive integers $k$ and $n$ are $k = \sum_{i=0}^{\infty} k_i 2^i$ and $n = \sum_{i=0}^{\infty} n_i 2^i$, then the logical sum of these numbers is \[ k \oplus n =\sum_{i=0}^{\infty} |k_i-n_i|2^i. \] Let $N$ be an arbitrary positive integer and $(c_k)_{k \in \mathbb{N}}$ be a sequence of complex numbers such that for all $k \in \mathbb{N}$, $ |c_k| \le 1$. Prove that there exist positive constants $C$ and $\delta$ such that \[ \int_{[-\pi,\pi] \times [-\pi, \pi]} \sup_{n<N, n \in \mathbb{N}} \frac{1}{N} \Big| \sum_{k=1}^{n} c_k e^{i(kx+(k \oplus n) y)} \Big| \mathrm d(x,y) \le C \cdot N^{-\delta} \] holds.

2015 Baltic Way, 16

Tags: polynomial , algebra , number theory

Denote by $P(n)$ the greatest prime divisor of $n$. Find all integers $n\geq 2$ for which \[P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor\]

2019 District Olympiad, 3

Tags: group theory , abstract algebra , superior algebra

Let $G$ be a finite group and let $x_1,…,x_n$ be an enumeration of its elements. We consider the matrix $(a_{ij})_{1 \le i,j \le n},$ where $a_{ij}=0$ if $x_ix_j^{-1}=x_jx_i^{-1},$ and $a_{ij}=1$ otherwise. Find the parity of the integer $\det(a_{ij}).$

2022 Sharygin Geometry Olympiad, 9.6

Tags: circumcircle , geometry

Lateral sidelines $AB$ and $CD$ of a trapezoid $ABCD$ ($AD >BC$) meet at point $P$. Let $Q$ be a point of segment $AD$ such that $BQ = CQ$. Prove that the line passing through the circumcenters of triangles $AQC$ and $BQD$ is perpendicular to $PQ$.

2019 PUMaC Geometry B, 8

Tags: geometry

Let $ABCD$ be a trapezoid such that $AB||CD$ and let $P=AC\cap BD,AB=21,CD=7,AD=13,[ABCD]=168.$ Let the line parallel to $AB$ through $P$ intersect the circumcircle of $BCP$ in $X.$ Circumcircles of $BCP$ and $APD$ intersect at $P,Y.$ Let $XY\cap BC=Z.$ If $\angle ADC$ is obtuse, then $BZ=\frac{a}{b},$ where $a,b$ are coprime positive integers. Compute $a+b.$

2010 Morocco TST, 4

Tags: trigonometry , algebra , triangle , triangle inequality

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

1989 IMO Longlists, 7

Tags: floor function , prime number , algebra , number theory

For every sequence $ (x_1, x_2, \ldots, x_n)$ of non-zero natural prime numbers, $ \{1, 2, \ldots, n\}$ arranged in any order, denote by $ f(s)$ the sum of absolute values of the differences between two consecutive members of $ s.$ Find the maximum value of $ f(s)$ where $ s$ runs through the set of all such sequences, i.e. for all sequences $ s$ with the given properties.

2007 Sharygin Geometry Olympiad, 5

Tags: geometry , centroid , acute , median

Medians $AA'$ and $BB'$ of triangle $ABC$ meet at point $M$, and $\angle AMB = 120^o$. Prove that angles $AB'M$ and $BA'M$ are neither both acute nor both obtuse.

2016 Federal Competition For Advanced Students, P2, 4

Tags: algebra , polynomial , inequalities

Let $a,b,c\ge-1$ be real numbers with $a^3+b^3+c^3=1$. Prove that $a+b+c+a^2+b^2+c^2\le4$, and determine the cases of equality. (Proposed by Karl Czakler)

1991 IMO Shortlist, 11

Tags: binomial coefficient , combinatorics , series summation , summation , combinatorial identity

Prove that $ \sum_{k \equal{} 0}^{995} \frac {( \minus{} 1)^k}{1991 \minus{} k} {1991 \minus{} k \choose k} \equal{} \frac {1}{1991}$

2021 Princeton University Math Competition, 11

Tags: geometry

$ABC$ is a triangle where $AB = 10$, $BC = 14$, and $AC = 16$. Let $DEF$ be the triangle with smallest area so that $DE$ is parallel to $AB$, $EF$ is parallel to $BC$, $DF$ is parallel to $AC$, and the circumcircle of $ABC$ is $DEF$’s inscribed circle. Line $DA$ meets the circumcircle of $ABC$ again at a point $X$. Find $AX^2$ .

2025 Romania National Olympiad, 1

Tags: combinatorics , minimum value , coloring

Let $N \geq 1$ be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer $k$ (not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are $x$ and $y$ respectively, we replace them with $x+k+1$ and $y+k^2+2$, which we color blue and red (in this order). We keep doing this procedure until the blue number is at least $N$. Determine the minimum value of the red number at the end of this procedure.

2018 Saudi Arabia IMO TST, 2

Tags: number theory

A non-empty subset of $\{1,2, ..., n\}$ is called [i]arabic [/i] if arithmetic mean of its elements is an integer. Show that the number of arabic subsets of $\{1,2, ..., n\}$ has the same parity as $n$.

2009 Italy TST, 1

Tags: vector , linear algebra , matrix , combinatorics

Let $n,k$ be positive integers such that $n\ge k$. $n$ lamps are placed on a circle, which are all off. In any step we can change the state of $k$ consecutive lamps. In the following three cases, how many states of lamps are there in all $2^n$ possible states that can be obtained from the initial state by a certain series of operations? i)$k$ is a prime number greater than $2$; ii) $k$ is odd; iii) $k$ is even.