Olympiad problems

2018 IOM, 4

Let $1 = d_0 < d_1 < \dots < d_m = 4k$ be all positive divisors of $4k$, where $k$ is a positive integer. Prove that there exists $i \in \{1, \dots, m\}$ such that $d_i - d_{i-1} = 2$. [i]Ivan Mitrofanov[/i]

2016 IOM, 5

Tags: IOM , algebra

Let $r(x)$ be a polynomial of odd degree with real coefficients. Prove that there exist only finitely many (or none at all) pairs of polynomials $p(x) $ and $q(x)$ with real coefficients satisfying the equation $(p(x))^3 + q(x^2) = r(x)$.

2018 IOM, 3

Tags: number theory , IOM

Let $k$ be a positive integer such that $p = 8k + 5$ is a prime number. The integers $r_1, r_2, \dots, r_{2k+1}$ are chosen so that the numbers $0, r_1^4, r_2^4, \dots, r_{2k+1}^4$ give pairwise different remainders modulo $p$. Prove that the product \[\prod_{1 \leqslant i < j \leqslant 2k+1} \big(r_i^4 + r_j^4\big)\] is congruent to $(-1)^{k(k+1)/2}$ modulo $p$. (Two integers are congruent modulo $p$ if $p$ divides their difference.) [i]Fedor Petrov[/i]

2018 IOM, 2

Tags: geometry , IOM

A convex quadrilateral $ABCD$ is circumscribed about a circle $\omega$. Let $PQ$ be the diameter of $\omega$ perpendicular to $AC$. Suppose lines $BP$ and $DQ$ intersect at point $X$, and lines $BQ$ and $DP$ intersect at point $Y$. Show that the points $X$ and $Y$ lie on the line $AC$. [i]Géza Kós[/i]

2018 IOM, 5

Tags: combinatorics , IOM

Ann and Max play a game on a $100 \times 100$ board. First, Ann writes an integer from 1 to 10 000 in each square of the board so that each number is used exactly once. Then Max chooses a square in the leftmost column and places a token on this square. He makes a number of moves in order to reach the rightmost column. In each move the token is moved to a square adjacent by side or vertex. For each visited square (including the starting one) Max pays Ann the number of coins equal to the number written in that square. Max wants to pay as little as possible, whereas Ann wants to write the numbers in such a way to maximise the amount she will receive. How much money will Max pay Ann if both players follow their best strategies? [i]Lev Shabanov[/i]

2016 IOM, 1

Tags: number theory , IOM

Find all positive integers $n$ such that there exist $n$ consecutive positive integers whose sum is a perfect square.

2018 IOM, 1

Tags: algebra , IOM

Solve the system of equations in real numbers: \[ \begin{cases*} (x - 1)(y - 1)(z - 1) = xyz - 1,\\ (x - 2)(y - 2)(z - 2) = xyz - 2.\\ \end{cases*} \] [i]Vladimir Bragin[/i]

2016 IOM, 3

Tags: IOM , algebra , geometry

Let $A_1A_2 . . . A_n$ be a cyclic convex polygon whose circumcenter is strictly in its interior. Let $B_1, B_2, ..., B_n$ be arbitrary points on the sides $A_1A_2, A_2A_3, ..., A_nA_1$, respectively, other than the vertices. Prove that $\frac{B_1B_2}{A_1A_3}+ \frac{B_2B_3}{A_2A_4}+...+\frac{B_nB_1}{A_nA_2}>1$.

2016 IOM, 6

Tags: combinatorics , IOM

In a country with $n$ cities, some pairs of cities are connected by one-way flights operated by one of two companies $A$ and $B$. Two cities can be connected by more than one flight in either direction. An $AB$-word $w$ is called implementable if there is a sequence of connected flights whose companies’ names form the word $w$. Given that every $AB$-word of length $ 2^n $ is implementable, prove that every finite $AB$-word is implementable. (An $AB$-word of length $k$ is an arbitrary sequence of $k$ letters $A $ or $B$; e.g. $ AABA $ is a word of length $4$.)

2016 IOM, 4

Tags: IOM , geometry

A convex quadrilateral $ABCD$ has right angles at $A$ and $C$. A point $E$ lies on the extension of the side $AD$ beyond $D$ so that$\angle ABE =\angle ADC$. The point $K$ is symmetric to the point $C$ with respect to point $A$. Prove that$\angle ADB =\angle AKE$ .