Olympiad problems

2022 May Olympiad, 4

Ana and Bruno have an $8 \times 8$ checkered board. Ana paints each of the $64$ squares with some color. Then Bruno chooses two rows and two columns on the board and looks at the $4$ squares where they intersect. Bruno's goal is for these $4$ squares to be the same color. How many colors, at least, must Ana use so that Bruno can't fulfill his objective? Show how you can paint the board with this amount of colors and explain because if you use less colors then Bruno can always fulfill his goal.

2007 Argentina National Olympiad, 1

Tags: number theory , diophantine equation , diophantine

Find all the prime numbers $p$ and $q$ such that $ p^2+q=37q^2+p $. Clarification: $1$ is not a prime number.

2011 Paraguay Mathematical Olympiad, 3

Tags: number theory

If number $\overline{aaaa}$ is divided by $\overline{bb}$, the quotient is a number between $140$ and $160$ inclusively, and the remainder is equal to $\overline{(a-b)(a-b)}$. Find all pairs of positive integers $(a,b)$ that satisfy this.

1954 Miklós Schweitzer, 4

Tags: function

[b]4.[/b] Find all functions of two variables defined over the entire plane that satisfy the relations $f(x+u,y+u)=f(x,y)+u$ and $f(xv,yv)= f(x,y) v$ for any real numbers $x,y,u,v$. [b](R.12)[/b]

2012 Gheorghe Vranceanu, 1

Tags: number theory

Find the natural numbers $ n $ which have the property that $ \log_2 \left( 1+2^n \right) $ is rational. [i]Cornel Berceanu[/i]

2019 Estonia Team Selection Test, 4

Tags: algebra , polynomial

Let us call a real number $r$ [i]interesting[/i], if $r = a + b\sqrt2$ for some integers a and b. Let $A(x)$ and $B(x)$ be polynomial functions with interesting coefficients for which the constant term of $B(x)$ is $1$, and $Q(x)$ be a polynomial function with real coefficients such that $A(x) = B(x) \cdot Q(x)$. Prove that the coefficients of $Q(x)$ are interesting.

2007 IMC, 2

Tags: modular arithmetic , algebra , polynomial

Let $ x$, $ y$ and $ z$ be integers such that $ S = x^{4}+y^{4}+z^{4}$ is divisible by 29. Show that $ S$ is divisible by $ 29^{4}$.

2005 ITAMO, 2

Tags: modular arithmetic , euler , function , number theory , totient function , algebra

Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and \[a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}\] For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$?

1999 IMC, 6

Tags: function , real analysis

(a) Let $p>1$ a real number. Find a real constant $c_p$ for which the following statement holds: If $f: [-1,1]\rightarrow\mathbb{R}$ is a continuously differentiable function with $f(1)>f(-1)$ and $|f'(y)|\le1 \forall y\in[-1,1]$, then $\exists x\in[-1,1]: f'(x)>0$ so that $\forall y\in[-1,1]: |f(y)-f(x)|\le c_p\sqrt[p]{f'(x)}|y-x|$. (b) What if $p=1$?

1999 Vietnam Team Selection Test, 3

Tags: vector , parallelogram , rotation , geometry

Let a convex polygon $H$ be given. Show that for every real number $a \in (0, 1)$ there exist 6 distinct points on the sides of $H$, denoted by $A_1, A_2, \ldots, A_6$ clockwise, satisfying the conditions: [b]I.[/b] $(A_1A_2) = (A_5A_4) = a \cdot (A_6A_3)$. [b]II.[/b] Lines $A_1A_2, A_5A_4$ are equidistant from $A_6A_3$. (By $(AB)$ we denote vector $AB$)