Olympiad problems

2012 NZMOC Camp Selection Problems, 3

Tags: combinatorics

Two courier companies offer services in the country of Old Aland. For any two towns, at least one of the companies offers a direct link in both directions between them. Additionally, each company is willing to chain together deliveries (so if they offer a direct link from $A$ to $B$, and $B$ to $C$, and $C$ to $D$ for instance, they will deliver from $A$ to $D$.) Show that at least one of the two companies must be able to deliver packages between any two towns in Old Aland.

2017 Korea Winter Program Practice Test, 1

Tags: number theory

For every positive integers $n,m$, show that there exist two sets $A,B$ which satisfy the following. [list] [*]$A$ is a set of $n$ successive positive integers, and $B$ is a set of $m$ successive positive integers. [*]$A\cup B = \phi$ [*]For every $a\in A$ and $b\in B$, $a$ and $b$ are not relatively prime. [/list]

2018 European Mathematical Cup, 3

Tags: algebra

For which real numbers $k > 1$ does there exist a bounded set of positive real numbers $S$ with at least $3$ elements such that $$k(a - b)\in S$$ for all $a,b\in S $ with $a > b?$ Remark: A set of positive real numbers $S$ is bounded if there exists a positive real number $M$ such that $x < M$ for all $x \in S.$

2009 Romanian Masters In Mathematics, 3

Tags: circumcircle , 3d geometry , tetrahedron , prism , rhombus , geometry

Given four points $ A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that \[ A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3, \] denote by $ O_i$ the circumcenter of $ \triangle A_j A_k A_l$ with $ \{i,j,k,l\} \equal{} \{1,2,3,4\}.$ Assuming $ \forall i A_i \neq O_i ,$ prove that the four lines $ A_iO_i$ are concurrent or parallel. [i]Nikolai Ivanov Beluhov, Bulgaria[/i]

2021 Austrian MO Regional Competition, 1

Tags: inequalities , algebra

Let $a$ and $b$ be positive integers and $c$ be a positive real number satisfying $$\frac{a + 1}{b + c}=\frac{b}{a}.$$ Prove that $c \ge 1$ holds. (Karl Czakler)

2023 Kurschak Competition, 3

Tags: geometry , circumcircle , tangent circle

Given is a convex cyclic pentagon $ABCDE$ and a point $P$ inside it, such that $AB=AE=AP$ and $BC=CE$. The lines $AD$ and $BE$ intersect in $Q$. Points $R$ and $S$ are on segments $CP$ and $BP$ such that $DR=QR$ and $SR||BC$. Show that the circumcircles of $BEP$ and $PQS$ are tangent to each other.

2018 Ukraine Team Selection Test, 4

Tags: coloring , combinatorics

Let $n$ be an odd integer. Consider a square lattice of size $n \times n$, consisting of $n^2$ unit squares and $2n(n +1)$ edges. All edges are painted in red or blue so that the number of red edges does not exceed $n^2$. Prove that there is a cell that has at least three blue edges.

2020 Korea - Final Round, P2

Tags: tournament graphs , hamiltonian path , combinatorics

There are $2020$ groups, each of which consists of a boy and a girl, such that each student is contained in exactly one group. Suppose that the students shook hands so that the following conditions are satisfied: [list] [*] boys didn't shake hands with boys, and girls didn't shake hands with girls; [*] in each group, the boy and girl had shake hands exactly once; [*] any boy and girl, each in different groups, didn't shake hands more than once; [*] for every four students in two different groups, there are at least three handshakes. [/list] Prove that one can pick $4038$ students and arrange them at a circular table so that every two adjacent students had shake hands.

Ukrainian TYM Qualifying - geometry, 2010.16

Tags: geometry , 3d geometry , tetrahedron , sphere , regular

Points $A, B, C, D$ lie on the sphere of radius $1$. It is known that $AB\cdot AC\cdot AD\cdot BC\cdot BD\cdot CD=\frac{512}{27}$. Prove that $ABCD$ is a regular tetrahedron.

2004 Cuba MO, 1

Tags: algebra , system of equations , system

Determine all real solutions to the system of equations: $$x_1 + x_2 +...+ x_{2004 }= 2004$$ $$x^4_1+ x^4_2+ ... + x^4_{2004} = x^3_1+x^3_2+... + x^3_{2004}$$

2021 DIME, 1

Tags:

Find the remainder when the number of positive divisors of the value $$(3^{2020}+3^{2021})(3^{2021}+3^{2022})(3^{2022}+3^{2023})(3^{2023}+3^{2024})$$ is divided by $1000$. [i]Proposed by pog[/i]

2022 CHMMC Winter (2022-23), 1

Tags: combinatorics

Yor and Fiona are playing a match of tennis against each other. The first player to win $6$ games wins the match (while the other player loses the match). Yor has currently won $2$ games, while Fiona has currently won $0$ games. Each game is won by one of the two players: Yor has a probability of $\frac23$ to win each game, while Fiona has a probability of $\frac13$ to win each game. Then, $\frac{m}{n}$ is the probability Fiona wins the tennis match, for relatively prime integers $m,n$. Compute $m$.

2020 ISI Entrance Examination, 5

Tags: geometry

Prove that the largest pentagon (in terms of area) that can be inscribed in a circle of radius $1$ is regular (i.e., has equal sides).

2024 IFYM, Sozopol, 4

Tags: number theory

At the wedding of two Bulgarian nationals in mathematics, every guest who gave a positive integer $n$, not yet given by another guest, which divides $3^n-3$ but does not divide $2^n-2$, received a prize. If there were an infinite number of guests, would the newlyweds theoretically need an infinite number of gifts?

2023 Indonesia TST, C

Tags:

There are $2023$ distinct points on a plane, which are coloured in white or red, such that for each white point, there are exactly two red points whose distance is $2023$ to that white point. Find the minimum number of red points.

2009 Kyiv Mathematical Festival, 1

Tags: number theory , sum of divisors , last digit

Let $X$ be the sum of all divisors of the number $(3\cdot 2009)^{((2\cdot 2009)^{2009}-1)}$ . Find the last digit of $X$.

2017 China Northern MO, 5

Tags: geometry , spiral similarity , circumcircle

Triangle $ABC$ has $AB > AC$ and $\angle A = 60^\circ $. Let $M$ be the midpoint of $BC$, $N$ be the point on segment $AB$ such that $\angle BNM = 30^\circ$. Let $D,E$ be points on $AB, AC$ respectively. Let $F, G, H$ be the midpoints of $BE, CD, DE$ respectively. Let $O$ be the circumcenter of triangle $FGH$. Prove that $O$ lies on line $MN$.

2020 Chile National Olympiad, 3

Tags: ratio , geometry , equal segments , isosceles

Given the isosceles triangle $ABC$ with $| AB | = | AC | = 10$ and $| BC | = 15$. Let points $P$ in $BC$ and $Q$ in $AC$ chosen such that $| AQ | = | QP | = | P C |$. Calculate the ratio of areas of the triangles $(PQA): (ABC)$.

2021 AMC 12/AHSME Fall, 21

Tags:

For real numbers $x$, let \[P(x)=1+\cos (x)+i \sin (x)-\cos (2 x)-i \sin (2 x)+\cos (3 x)+i \sin (3 x)\] where $i=\sqrt{-1}$. For how many values of $x$ with $0 \leq x<2 \pi$ does $P(x)=0 ?$ $\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 4$

2020 USMCA, 10

Tags:

Let $ABCD$ be a unit square, and let $E$ be a point on segment $AC$ such that $AE = 1$. Let $DE$ meet $AB$ at $F$ and $BE$ meet $AD$ at $G$. Find the area of $CFG$.

1995 Israel Mathematical Olympiad, 8

Tags: functional equation , functional , function , algebra

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

2019 India Regional Mathematical Olympiad, 2

Tags: geometry , p2

Let $ABC$ be a triangle with circumcircle $\Omega$ and let $G$ be the centroid of triangle $ABC$. Extend $AG, BG$ and $CG$ to meet the circle $\Omega$ again in $A_1, B_1$ and $C_1$. Suppose $\angle BAC = \angle A_1 B_1 C_1, \angle ABC = \angle A_1 C_1 B_1$ and $ \angle ACB = B_1 A_1 C_1$. Prove that $ABC$ and $A_1 B_1 C_1$ are equilateral triangles.

2019 AIME Problems, 15

Tags: geometry , circumcircle , altitude , power of a point

In acute triangle $ABC$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\overline{AB}$ and from $B$ to $\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\triangle ABC$ in two distinct points, $X$ and $Y$. Suppose $XP=10$, $PQ=25$, and $QY=15$. The value of $AB\cdot AC$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

1969 Putnam, A5

Tags: function , differential equation

Let $u(t)$ be a continuous function in the system of differential equations $$\frac{dx}{dt} =-2y +u(t),\;\;\; \frac{dy}{dt}=-2x+ u(t).$$ Show that, regardless of the choice of $u(t)$, the solution of the system which satisfies $x=x_0 , y=y_0$ at $t=0$ will never pass through $(0, 0)$ unless $x_0 =y_0.$ When $x_0 =y_0 $, show that, for any positive value $t_0$ of $t$, it is possible to choose $u(t)$ so the solution is equal to $(0,0)$ when $t=t_0 .$

2013 Sharygin Geometry Olympiad, 7

Tags: 3d geometry , sphere , ratio , geometry

Given five fixed points in the space. It is known that these points are centers of five spheres, four of which are pairwise externally tangent, and all these point are internally tangent to the fifth one. It turns out that it is impossible to determine which of the marked points is the center of the largest sphere. Find the ratio of the greatest and the smallest radii of the spheres.