Olympiad problems

2004 Austria Beginners' Competition, 3

Determine the value of the parameter $m$ such that the equation $(m-2)x^2+(m^2-4m+3)x-(6m^2-2)=0$ has real solutions, and the sum of the third powers of these solutions is equal to zero.

1992 Bundeswettbewerb Mathematik, 3

Tags: geometry , 3d geometry , sphere

Given is a triangle $ABC$ with side lengths $a, b,c$. Three spheres touch each other in pairs and also touch the plane of the triangle at points $A,B$ and $C$, respectively. Determine the radii of these spheres.

2025 AIME, 15

Tags: abstract algebra

Let $N$ denote the numbers of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \le 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.

1970 IMO Longlists, 44

Tags: inequalities

If $a, b, c$ are side lengths of a triangle, prove that \[(a + b)(b + c)(c + a) \geq 8(a + b - c)(b + c - a)(c + a - b).\]

2022 Auckland Mathematical Olympiad, 11

Tags: number theory , prime

For which $k$ the number $N = 101 ... 0101$ with $k$ ones is a prime?

2006 Baltic Way, 13

Tags: circumcircle , geometry

In a triangle $ABC$, points $D,E$ lie on sides $AB,AC$ respectively. The lines $BE$ and $CD$ intersect at $F$. Prove that if $\color{white}\ .\quad\ \color{black}\ \quad BC^2=BD\cdot BA+CE\cdot CA,$ then the points $A,D,F,E$ lie on a circle.

2019 Yasinsky Geometry Olympiad, p4

Tags: geometry , perimeter , equal angles , incenter

In the triangle $ABC$, the side $BC$ is equal to $a$. Point $F$ is midpoint of $AB$, $I$ is the point of intersection of the bisectors of triangle $ABC$. It turned out that $\angle AIF = \angle ACB$. Find the perimeter of the triangle $ABC$. (Grigory Filippovsky)

2018 Romania National Olympiad, 2

Tags: geometry , circumcircle , complex number

Let $ABC$ be a triangle, $O$ its circumcenter and $R=1$ its circumradius. Let $G_1,G_2,G_3$ be the centroids of the triangles $OBC, OAC$ and $OAB.$ Prove that the triangle $ABC$ is equilateral if and only if $$AG_1+BG_2+CG_3=4$$

2012 India IMO Training Camp, 3

Tags: algorithm , combinatorics

In a $2\times n$ array we have positive reals s.t. the sum of the numbers in each of the $n$ columns is $1$. Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most $\frac{n+1}4$.

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 $X$.