Olympiad problems

2014 Polish MO Finals, 1

Denote the set of positive rational numbers by $\mathbb{Q}_{+}$. Find all functions $f: \mathbb{Q}_{+}\rightarrow \mathbb{Q}_{+}$ that satisfy $$\underbrace{f(f(f(\dots f(f}_{n}(q))\dots )))=f(nq)$$ for all integers $n\ge 1$ and rational numbers $q>0$.

2014 Polish MO Finals, 3

Tags: contests , geometry , circumcircle

In an acute triangle $ABC$ point $D$ is the point of intersection of altitude $h_a$ and side $BC$, and points $M, N$ are orthogonal projections of point $D$ on sides $AB$ and $AC$. Lines $MN$ and $AD$ cross the circumcircle of triangle $ABC$ at points $P, Q$ and $A, R$. Prove that point $D$ is the center of the incircle of $PQR$.

2021 Argentina National Olympiad, 6

Tags: combinatorics , contests

Milly chooses a positive integer $n$ and then Uriel colors each integer between $1$ and $n$ inclusive red or blue. Then Milly chooses four numbers $a, b, c, d$ of the same color (there may be repeated numbers). If $a+b+c= d$ then Milly wins. Determine the smallest $n$ Milly can choose to ensure victory, no matter how Uriel colors.

2021 Argentina National Olympiad Level 2, 1

Tags: contests , coprime numbers , number theory

You have two blackboards $A$ and $B$. You have to write on them some of the integers greater than or equal to $2$ and less than or equal to $20$ in such a way that each number on blackboard $A$ is co-prime with each number on blackboard $B.$ Determine the maximum possible value of multiplying the number of numbers written in $A$ by the number of numbers written in $B$.

2016 Bangladesh Mathematical Olympiad, 6

Tags: geometry , parallel , angle bisector , contests

$\triangle ABC$ is an isosceles triangle with $AC = BC$ and $\angle ACB < 60^{\circ}$. $I$ and $O$ are the incenter and circumcenter of $\triangle ABC$. The circumcircle of $\triangle BIO$ intersects $BC$ at $D \neq B$. (a) Do the lines $AC$ and $DI$ intersect? Give a proof. (b) What is the angle of intersection between the lines $OD$ and $IB$?

2015 District Olympiad, 2

Tags: integrals , real analysis , calculus , contests , integration

[b]a)[/b] Calculate $ \int_{0}^1 x\sin\left( \pi x^2\right) dx. $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} k\int_{\frac{k}{n}}^{\frac{k+1}{n}} \sin\left(\pi x^2\right) dx. $ [i]Florin Stănescu[/i]

2014 Contests, 1

Tags: contests , number theory , relatively prime

Let $k,n\ge 1$ be relatively prime integers. All positive integers not greater than $k+n$ are written in some order on the blackboard. We can swap two numbers that differ by $k$ or $n$ as many times as we want. Prove that it is possible to obtain the order $1,2,\dots,k+n-1, k+n$.

2015 Polish MO Finals, 1

Tags: contests , geometry

In triangle $ABC$ the angle $\angle A$ is the smallest. Points $D, E$ lie on sides $AB, AC$ so that $\angle CBE=\angle DCB=\angle BAC$. Prove that the midpoints of $AB, AC, BE, CD$ lie on one circle.

2021 Argentina National Olympiad Level 2, 6

Tags: algebra , contests

Decide if it is possible to choose $330$ points in the plane so that among all the distances that are formed between two of them there are at least $1700$ that are equal.

2014 Polish MO Finals, 2

Tags: contests , Diophantine equation , number theory

Find all pairs $(x,y)$ of positive integers that satisfy $$2^x+17=y^4$$.

2021 Argentina National Olympiad, 2

Tags: geometry , contests

In a semicircle with center $O$, let $C$ be a point on the diameter $AB$ different from $A, B$ and $O.$ Draw through $C$ two rays such that the angles that these rays form with the diameter $AB$ are equal and that they intersect at the semicircle at $D$ and at $E$. The line perpendicular to $CD$ through $D$ intersects the semicircle at $K.$ Prove that if $D\neq E,$ then $KE$ is parallel to $AB.$

2021 Argentina National Olympiad Level 2, 3

Tags: geometry , arcs , lengths , contests

A circle is divided into $2n$ equal arcs by $2n$ points. Find all $n>1$ such that these points can be joined in pairs using $n$ segments, all of different lengths and such that each point is the endpoint of exactly one segment.

2018 Bangladesh Mathematical Olympiad, 2

Tags: geometry , circles , Triangle , contests

BdMO National 2018 Higher Secondary P2 $AB$ is a diameter of a circle and $AD$ & $BC$ are two tangents of that circle.$AC$ & $BD$ intersect on a point of the circle.$AD=a$ & $BC=b$.If $a\neq b$ then $AB=?$

2016 Bangladesh Mathematical Olympiad, 4

Tags: number theory , algebra , Bdmo , contests

Consider the set of integers $ \left \{ 1, 2, \dots , 100 \right \} $. Let $ \left \{ x_1, x_2, \dots , x_{100} \right \}$ be some arbitrary arrangement of the integers $ \left \{ 1, 2, \dots , 100 \right \}$, where all of the $x_i$ are different. Find the smallest possible value of the sum $$S = \left | x_2 - x_1 \right | + \left | x_3 - x_2 \right | + \cdots+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | .$$

2021 Argentina National Olympiad, 6

Tags: algebra , contests

Decide if it is possible to choose $330$ points in the plane so that among all the distances that are formed between two of them there are at least $1700$ that are equal.

2015 Polish MO Finals, 3

Tags: contests , set theory

Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\{ 1,2,\dots, 1000\}$ there exist two sets, whose intersection contains at least $m$ numbers.

2017 ISI Entrance Examination, 7

Tags: number theory , counting , combinatorics , contests

Let $A=\{1,2,\ldots,n\}$. For a permutation $P=(P(1), P(2), \ldots, P(n))$ of the elements of $A$, let $P(1)$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $i,j \in A$: (a) if $i < j<P(1)$, then $j$ appears before $i$ in $P$; and (b) if $P(1)<i<j$, then $i$ appears before $j$ in $P$.

2018 Malaysia National Olympiad, A4

Tags: geometry , contests

Given a circle with diameter $AB$. Points $C$ and $D$ are selected on the circumference of the circle such that the chord $CD$ intersects $AB$ inside the circle, at point $P$. The ratio of the arc length $\overarc {AC}$ to the arc length $\overarc {BD}$ is $4 : 1$ , while the ratio of the arc length $\overarc{AD}$ to the arc length $\overarc {BC}$ is $3 : 2$ . Find $\angle{APC}$ , in degrees.

2021 Argentina National Olympiad, 1

Tags: Sequence , combinatorics , contests , Argentina

An infinite sequence of digits $1$ and $2$ is determined by the following two properties: i) The sequence is built by writing, in some order, blocks $12$ and blocks $112.$ ii) If each block $12$ is replaced by $1$ and each block $112$ by $2$, the same sequence is again obtained. In which position is the hundredth digit $1$? What is the thousandth digit of the sequence?

2021 Argentina National Olympiad, 5

Tags: combinatorics , contests

Mica wrote a list of numbers using the following procedure. The first number is $1$, and then, at each step, he wrote the result of adding the previous number plus $3$. The first numbers on Mica's list are $$1, 4, 7, 10, 13, 16,\dots.$$ Next, Facu underlined all the numbers in Mica's list that are greater than $10$ and less than $100000,$ and that have all their digits the same. What are the numbers that Facu underlined?

2021 Argentina National Olympiad, 6

Tags: number theory , contests , Tricubic numbers

We say that a positive integer $k$ is tricubic if there are three positive integers $a, b, c,$ not necessarily different, such that $k=a^3+b^3+c^3.$ a) Prove that there are infinitely many positive integers $n$ that satisfy the following condition: exactly one of the three numbers $n, n+2$ and $n+28$ is tricubic. b) Prove that there are infinitely many positive integers $n$ that satisfy the following condition: exactly two of the three numbers $n, n+2$ and $n+28$ are tricubic. c) Prove that there are infinitely many positive integers $n$ that satisfy the following condition: the three numbers $n, n+2$ and $n+28$ are tricubic.

2021 Argentina National Olympiad Level 2, 5

Tags: number theory , Perfect Squares , contests

Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.

2015 Polish MO Finals, 3

Tags: contests , number theory

Prove that for each positive integer $a$ there exists such an integer $b>a$, for which $1+2^a+3^a$ divides $1+2^b+3^b$.

2014 Polish MO Finals, 3

Tags: contests , geometry , 3D geometry , tetrahedron , sphere

A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.

2021 Argentina National Olympiad, 1

Tags: contests , coprime numbers , number theory

You have two blackboards $A$ and $B$. You have to write on them some of the integers greater than or equal to $2$ and less than or equal to $20$ in such a way that each number on blackboard $A$ is co-prime with each number on blackboard $B.$ Determine the maximum possible value of multiplying the number of numbers written in $A$ by the number of numbers written in $B$.