Olympiad problems

2016 Ecuador Juniors, 2

Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$

2022 Moldova EGMO TST, 1

Tags: equation

Let $n$ be a positive integer. Solve the equation in $\mathbb{R}$ $$\sqrt[2n+1]{x}+\sqrt[2n+1]{x+1}+\sqrt[2n+1]{x+2}+\dots+\sqrt[2n+1]{x+n}=0.$$

1984 IMO Shortlist, 18

Tags: geometry , trigonometry , circles , radius

Inside triangle $ABC$ there are three circles $k_1, k_2, k_3$ each of which is tangent to two sides of the triangle and to its incircle $k$. The radii of $k_1, k_2, k_3$ are $1, 4$, and $9$. Determine the radius of $k.$

2018 Regional Olympiad of Mexico Center Zone, 5

Tags: diophantine equation , diophantine , number theory

Find all solutions of the equation $$p ^ 2 + q ^ 2 + 49r ^ 2 = 9k ^ 2-101$$ with $ p$, $q$ and $r$ positive prime numbers and $k$ a positive integer.

Kyiv City MO Juniors 2003+ geometry, 2016.8.5

Tags: geometry , angle bisector , angle

In the triangle $ABC$ the angle bisectors $AD$ and $BE$ are drawn. Prove that $\angle ACB = 60 {} ^ \circ$ if and only if $AE + BD = AB$. (Hilko Danilo)

2012 Belarus Team Selection Test, 3

Tags: trigonometry , triangle inequality , algebra , triangle

Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle. [i]Proposed by Canada[/i]

ICMC 4, 4

Tags: analysis , geometry

Let $\mathbb R^2$ denote the Euclidean plane. A continuous function $f : \mathbb R^2 \to \mathbb R^2$ maps circles to circles. (A point is not a circle.) Prove that it maps lines to lines. [i]Proposed by Tony Wang[/i]

2009 China Team Selection Test, 2

Tags: geometry , perpendicular bisector , angle bisector , law of sines

In convex quadrilateral $ ABCD$, $ CB,DA$ are external angle bisectors of $ \angle DCA,\angle CDB$, respectively. Points $ E,F$ lie on the rays $ AC,BD$ respectively such that $ CEFD$ is cyclic quadrilateral. Point $ P$ lie in the plane of quadrilateral $ ABCD$ such that $ DA,CB$ are external angle bisectors of $ \angle PDE,\angle PCF$ respectively. $ AD$ intersects $ BC$ at $ Q.$ Prove that $ P$ lies on $ AB$ if and only if $ Q$ lies on segment $ EF$.

2020 Peru EGMO TST, 3

Tags: geometry , incenter , circumcircle

Let $ABC$ be a triangle with $AB<AC$ and $I$ be your incenter. Let $M$ and $N$ be the midpoints of the sides $BC$ and $AC$, respectively. If the lines $AI$ and $IN$ are perpendicular, prove that the line $AI$ is tangent to the circumcircle of $\triangle IMC$.

1991 Federal Competition For Advanced Students, P2, 5

Tags: inequalities

For all positive integers $ n$ prove the inequality: $ \left( \frac{1\plus{}(n\plus{}1)^{n\plus{}1}}{n\plus{}2} \right)^{n\minus{}1}>\left( \frac{1\plus{}n^n}{n\plus{}1} \right)^n.$

2006 Germany Team Selection Test, 1

Tags: algebra , polynomial , number theory , composite number , prime number

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

2005 Germany Team Selection Test, 1

Tags: inequalities , number theory

Find the smallest positive integer $n$ with the following property: For any integer $m$ with $0 < m < 2004$, there exists an integer $k$ such that \[\frac{m}{2004}<\frac{k}{n}<\frac{m+1}{2005}.\]

1997 IMO Shortlist, 4

Tags: matrix , combinatorics , coloring , induction

An $ n \times n$ matrix whose entries come from the set $ S \equal{} \{1, 2, \ldots , 2n \minus{} 1\}$ is called a [i]silver matrix[/i] if, for each $ i \equal{} 1, 2, \ldots , n$, the $ i$-th row and the $ i$-th column together contain all elements of $ S$. Show that: (a) there is no silver matrix for $ n \equal{} 1997$; (b) silver matrices exist for infinitely many values of $ n$.

2018 PUMaC Live Round, 2.1

Tags:

Compute the period (i.e. length of the repeating part) of the decimal expansion of $\tfrac{1}{729}$.

2015 Danube Mathematical Competition, 4

Tags: number theory

Given an integer $n \ge 2$ ,determine the numbers that written in the form $a_1$$a_2$$+$$a_2$$a_3$$+$$...$$a_{k-1}$$a_k$ , where $k$ is an integer greater than or equal to 2, and $a_1$ ,... $a_k$ are positive integers with sum $n$.

2011 May Olympiad, 3

Tags: geometry , right triangle , isosceles , angle , perpendicular bisector

In a right triangle rectangle $ABC$ such that $AB = AC$, $M$ is the midpoint of $BC$. Let $P$ be a point on the perpendicular bisector of $AC$, lying in the semi-plane determined by $BC$ that does not contain $A$. Lines $CP$ and $AM$ intersect at $Q$. Calculate the angles that form the lines $AP$ and $BQ$.

1989 AMC 12/AHSME, 8

Tags: quadratic , algebra , quadratic formula

For how many integers $n$ between 1 and 100 does $x^2+x-n$ factor into the product of two linear factors with integer coefficients? $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 9 \qquad \text{(E)} \ 10$

2014 Germany Team Selection Test, 3

Tags: number theory

Let $a_1 \leq a_2 \leq \cdots$ be a non-decreasing sequence of positive integers. A positive integer $n$ is called [i]good[/i] if there is an index $i$ such that $n=\dfrac{i}{a_i}$. Prove that if $2013$ is [i]good[/i], then so is $20$.

1992 All Soviet Union Mathematical Olympiad, 577

Tags: number theory , digit

Find all integers $k > 1$ such that for some distinct positive integers $a, b$, the number $k^a + 1$ can be obtained from $k^b + 1$ by reversing the order of its (decimal) digits.

2017 Portugal MO, 5

Tags: geometry , quadrilateral

Let $[ABCD]$ be a convex quadrilateral with $AB = 2, BC = 3, CD = 7$ and $\angle B = 90^o$, for which there is a inscribed circle. Determine the radius of this circle. [img]https://1.bp.blogspot.com/-sDKOdmceJlY/X4KaJxi8AoI/AAAAAAAAMk8/7UkTzaWqQSkdqb0N_-r0CZZjD-OGZknSACLcBGAsYHQ/s260/2017%2Bportugal%2Bp5.png[/img]

1976 Chisinau City MO, 121

Tags: algebra , odd , polynomial , integer

Prove that the polynomial $P (x)$ with integer coefficients, taking odd values for $x = 0$ and $x= 1$, has no integer roots.

Kyiv City MO Seniors 2003+ geometry, 2007.10.3

Tags: geometry , circumcircle , incircle , angle bisector , locus

The points $ P, Q$ are given on the plane, which are the points of intersection of the angle bisector $AL$ of some triangle $ABC$ with an inscribed circle, and the point $W$ is the intersection of the angle bisector $AL$ with a circumscribed circle other than the vertex $A$. a) Find the geometric locus of the possible location of the vertex $A$ of the triangle $ABC$. b) Find the geometric locus of the possible location of the vertex $B$ of the triangle $ABC$.

2023 Junior Macedonian Mathematical Olympiad, 2

Tags: number theory , prime number

A positive integer is called [i]superprime[/i] if the difference between any two of its consecutive positive divisors is a prime number. Determine all superprime integers. [i]Authored by Nikola Velov[/i]

2001 All-Russian Olympiad, 1

Tags: combinatorics

Yura put $2001$ coins of $1$, $2$ or $3$ kopeykas in a row. It turned out that between any two $1$-kopeyka coins there is at least one coin; between any two $2$-kopeykas coins there are at least two coins; and between any two $3$-kopeykas coins there are at least $3$ coins. How many $3$-koyepkas coins could Yura put?

2008 Germany Team Selection Test, 3

Tags: number theory

Prove there is an integer $ k$ for which $ k^3 \minus{} 36 k^2 \plus{} 51 k \minus{} 97$ is a multiple of $ 3^{2008.}$