Olympiad problems

2019 MIG, 5

Tags:

How many distinct prime factors does the number $36$ have? $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }9\qquad\textbf{(E) }15$

A four-digit positive integer is called [i]doubly[/i] if its first two digits form some permutation of its last two digits. For example, $1331$ and $2121$ are both [i]doubly[/i]. How many four-digit [i]doubly[/i] positive integers are there?

2004 Balkan MO, 1

Tags: induction , algebra

The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation: \[ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n}) \] for all non-negative integers $m$ and $n$, $m \ge n$. If $a_1 = 3$ find $a_{2004}$.

2024 Assara - South Russian Girl's MO, 3

Tags: combinatorics , abstract algebra

In the cells of the $4\times N$ table, integers are written, modulo no more than $2024$ (i.e. numbers from the set $\{-2024, -2023,\dots , -2, -1, 0, 1, 2, 3,\dots , 2024\}$) so that in each of the four lines there are no two equal numbers. At what maximum $N$ could it turn out that in each column the sum of the numbers is equal to $2$? [i]G.M.Sharafetdinova[/i]

Kvant 2024, M2820

Tags: combinatorics

Let us name a move of the chess knight horizontal if it moves two cells horizontally and one vertically, and vertical otherwise. It is required to place the knight on a cell of a ${46} \times {46}$ board and alternate horizontal and vertical moves. Prove that if each cell is visited not more than once then the number of moves does not exceed 2024. Alexandr Gribalko

2010 Baltic Way, 7

Tags: combinatorics

There are some cities in a country; one of them is the capital. For any two cities $A$ and $B$ there is a direct flight from $A$ to $B$ and a direct flight from $B$ to $A$, both having the same price. Suppose that all round trips with exactly one landing in every city have the same total cost. Prove that all round trips that miss the capital and with exactly one landing in every remaining city cost the same.

2006 All-Russian Olympiad Regional Round, 8.1

Tags: number theory , digit

Find some nine-digit number $N$, consisting of different digits, such that among all the numbers obtained from $N$ by crossing out seven digits, there would be no more than one prime. Prove that the number found is correct. (If the number obtained by crossing out the digits starts at zero, then the zero is crossed out.)

2009 Junior Balkan Team Selection Test, 3

Tags: parallelogram , perpendicular bisector , geometry

Let $ ABCD$ be a convex quadrilateral, such that $ \angle CBD\equal{}2\cdot\angle ADB, \angle ABD\equal{}2\cdot\angle CDB$ and $ AB\equal{}CB$. Prove that quadrilateral $ ABCD$ is a kite.

2001 Tournament Of Towns, 3

Tags: geometric transformation , reflection , angle bisector , geometry

Point $A$ lies inside an angle with vertex $M$. A ray issuing from point $A$ is reflected in one side of the angle at point $B$, then in the other side at point $C$ and then returns back to point $A$ (the ordinary rule of reflection holds). Prove that the center of the circle circumscribed about triangle $\triangle BCM$ lies on line $AM$.

MOAA Team Rounds, 2018.1

Tags: geometry , team

In $\vartriangle ABC$, $AB = 3$, $BC = 5$, and $CA = 6$. Points $D$ and $E$ are chosen such that $ACDE$ is a square which does not overlap with $\vartriangle ABC$. The length of $BD$ can be expressed in the form $\sqrt{m + n\sqrt{p}}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of a prime. Determine the value of $m + n + p$.