Olympiad problems

2017 Turkey EGMO TST, 6

Tags: Turkey , EGMO , TST , number theory , contest problem , order of an element

Find all pairs of prime numbers $(p,q)$, such that $\frac{(2p^2-1)^q+1}{p+q}$ and $\frac{(2q^2-1)^p+1}{p+q}$ are both integers.

2017 Turkey EGMO TST, 2

Tags: Turkey , EGMO , TST , combinatorics , contest problem

At the beginning there are $2017$ marbles in each of $1000$ boxes. On each move Aybike chooses a box, grabs some of the marbles from that box and delivers them one for each to the boxes she wishes. At least how many moves does Aybike have to make to have different number of marbles in each box?

2019 Romanian Master of Mathematics Shortlist, original P5

Tags: combinatorics , polyhedron , edge , contest problem

Two ants are moving along the edges of a convex polyhedron. The route of every ant ends in its starting point, so that one ant does not pass through the same point twice along its way. On every face $F$ of the polyhedron are written the number of edges of $F$ belonging to the route of the first ant and the number of edges of $F$ belonging to the route of the second ant. Is there a polyhedron and a pair of routes described as above, such that only one face contains a pair of distinct numbers? [i]Proposed by Nikolai Beluhov[/i]

2017 Turkey EGMO TST, 3

Tags: Turkey , EGMO , TST , algebra , inequalities , contest problem

For all positive real numbers $x,y,z$ satisfying the inequality $$\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}\leq 3,$$ prove that $$\frac{x^2}{y^3}+\frac{y^2}{z^3}+\frac{z^2}{x^3}\geq \frac{x}{y}+\frac{y}{z}+\frac{z}{x}.$$

2017 Turkey EGMO TST, 1

Tags: Turkey , EGMO , TST , number theory , contest problem

Let $m,k,n$ be positive integers. Determine all triples $(m,k,n)$ satisfying the following equation: $3^m5^k=n^3+125$

2017 Turkey EGMO TST, 5

Tags: Turkey , EGMO , TST , combinatorics , contest problem

In a $12\times 12$ square table some stones are placed in the cells with at most one stone per cell. If the number of stones on each line, column, and diagonal is even, what is the maximum number of the stones? [b]Note[/b]. Each diagonal is parallel to one of two main diagonals of the table and consists of $1,2\ldots,11$ or $12$ cells.

2019 Romanian Master of Mathematics Shortlist, original P4

Tags: geometry , Equilateral Triangle , circumcircle , contest problem

Let there be an equilateral triangle $ABC$ and a point $P$ in its plane such that $AP<BP<CP.$ Suppose that the lengths of segments $AP,BP$ and $CP$ uniquely determine the side of $ABC$. Prove that $P$ lies on the circumcircle of triangle $ABC.$

2017 Turkey EGMO TST, 4

Tags: Turkey , EGMO , TST , geometry , contest problem

Russian TST 2019, P3

Tags: polynomial , algebra , contest problem , multivariate polynomial

Let $P(x)$ be a nonconstant complex coefficient polynomial and let $Q(x,y)=P(x)-P(y).$ Suppose that polynomial $Q(x,y)$ has exactly $k$ linear factors unproportional two by tow (without counting repetitons). Let $R(x,y)$ be factor of $Q(x,y)$ of degree strictly smaller than $k$. Prove that $R(x,y)$ is a product of linear polynomials. [b]Note: [/b] The [i]degree[/i] of nontrivial polynomial $\sum_{m}\sum_{n}c_{m,n}x^{m}y^{n}$ is the maximum of $m+n$ along all nonzero coefficients $c_{m,n}.$ Two polynomials are [i]proportional[/i] if one of them is the other times a complex constant. [i]Proposed by Navid Safaie[/i]

2016 Bangladesh Mathematical Olympiad, 9

Tags: combinatorics , calculus , contest problem , definite integrals , Integers

Consider the integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$. [b](a)[/b] Show that the integral $Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}$, where $j$ is not a function of $x$, is $Z(j)=e^{j^{2}/4a} Z(0)$. [b](b)[/b] Show that $$\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n},$$ where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times\cdots\times3\times 1$. [b](c)[/b] What is the number of ways to form $n$ pairs from $2n$ distinct objects? Interpret the previous part of the problem in term of this answer.

2016 Bangladesh Mathematical Olympiad, 1

Tags: Bdmo , contests , contest problem , number theory , modular arithmetic

(a) Show that $n(n + 1)(n + 2)$ is divisible by $6$. (b) Show that $1^{2015} + 2^{2015} + 3^{2015} + 4^{2015} + 5^{2015} + 6^{2015}$ is divisible by $7$.

Russian TST 2019, P2

Tags: combinatorics , polyhedron , edge , contest problem

Two ants are moving along the edges of a convex polyhedron. The route of every ant ends in its starting point, so that one ant does not pass through the same point twice along its way. On every face $F$ of the polyhedron are written the number of edges of $F$ belonging to the route of the first ant and the number of edges of $F$ belonging to the route of the second ant. Is there a polyhedron and a pair of routes described as above, such that only one face contains a pair of distinct numbers? [i]Proposed by Nikolai Beluhov[/i]

2015 Bangladesh Mathematical Olympiad, 2

Tags: number theory , Bdmo , contest problem , Contest Preparation , Integer , solutions

[b][u]BdMO National Higher Secondary Problem 3[/u][/b] Let $N$ be the number if pairs of integers $(m,n)$ that satisfies the equation $m^2+n^2=m^3$ Is $N$ finite or infinite?If $N$ is finite,what is its value?

Kvant 2019, M2573

Tags: combinatorics , polyhedron , edge , contest problem

Two ants are moving along the edges of a convex polyhedron. The route of every ant ends in its starting point, so that one ant does not pass through the same point twice along its way. On every face $F$ of the polyhedron are written the number of edges of $F$ belonging to the route of the first ant and the number of edges of $F$ belonging to the route of the second ant. Is there a polyhedron and a pair of routes described as above, such that only one face contains a pair of distinct numbers? [i]Proposed by Nikolai Beluhov[/i]

2019 Romanian Master of Mathematics Shortlist, original P6

Tags: polynomial , algebra , contest problem , multivariate polynomial

Let $P(x)$ be a nonconstant complex coefficient polynomial and let $Q(x,y)=P(x)-P(y).$ Suppose that polynomial $Q(x,y)$ has exactly $k$ linear factors unproportional two by tow (without counting repetitons). Let $R(x,y)$ be factor of $Q(x,y)$ of degree strictly smaller than $k$. Prove that $R(x,y)$ is a product of linear polynomials. [b]Note: [/b] The [i]degree[/i] of nontrivial polynomial $\sum_{m}\sum_{n}c_{m,n}x^{m}y^{n}$ is the maximum of $m+n$ along all nonzero coefficients $c_{m,n}.$ Two polynomials are [i]proportional[/i] if one of them is the other times a complex constant. [i]Proposed by Navid Safaie[/i]