Olympiad problems

2016 Auckland Mathematical Olympiad, 4

Find the smallest positive value of $36^k - 5^m$, where $k$ and $m$ are positive integers.

1974 All Soviet Union Mathematical Olympiad, 190

Tags: minimum , algebra , number theory

Among all the numbers representable as $36^k - 5^l$ ($k$ and $l$ are natural numbers) find the smallest. Prove that it is really the smallest.

2017 Thailand Mathematical Olympiad, 4

Tags: combinatorics , minimum

In a math competition, $14$ schools participate, each sending $14$ students. The students are separated into $14$ groups of $14$ so that no two students from the same school are in the same group. The tournament organizers noted that, from the competitors, exactly $15$ have participated in the competition before. The organizers want to select two representatives, with the conditions that they must be former participants, must come from different schools, and must also be in different groups. It turns out that there are $ n$ ways to do this. What is the minimum possible value of $n$?

2019 Saint Petersburg Mathematical Olympiad, 7

Tags: combinatorics , game , minimum , game strategy

In a circle there are $2019$ plates, on each lies one cake. Petya and Vasya are playing a game. In one move, Petya points at a cake and calls number from $1$ to $16$, and Vasya moves the specified cake to the specified number of check clockwise or counterclockwise (Vasya chooses the direction each time). Petya wants at least some $k$ pastries to accumulate on one of the plates and Vasya wants to stop him. What is the largest $k$ Petya can succeed?

1995 Tuymaada Olympiad, 8

Tags: geometry , geometric inequality , minimum , sum , distance

Inside the triangle $ABC$ a point $M$ is given . Find the points $P,Q$ and $R$ lying on the sides $AB,BC$ and $AC$ respectively and such so that the sum $MP+PQ+QR+RM$ is the smallest.

2012 Oral Moscow Geometry Olympiad, 5

Tags: minimum , sum , distance , geometry , circles

Inside the circle with center $O$, points $A$ and $B$ are marked so that $OA = OB$. Draw a point $M$ on the circle from which the sum of the distances to points $A$ and $B$ is the smallest among all possible.

2014 Sharygin Geometry Olympiad, 8

Tags: geometry , convex polygon , minimum

A convex polygon $P$ lies on a flat wooden table. You are allowed to drive some nails into the table. The nails must not go through $P$, but they may touch its boundary. We say that a set of nails blocks $P$ if the nails make it impossible to move $P$ without lifting it off the table. What is the minimum number of nails that suffices to block any convex polygon $P$? (N. Beluhov, S. Gerdgikov)

1979 All Soviet Union Mathematical Olympiad, 275

Tags: chessboard , minimum , combinatorics

What is the least possible number of the checkers being required a) for the $8\times 8$ chess-board, b) for the $n\times n$ chess-board, to provide the property: [i]Every line (of the chess-board fields) parallel to the side or diagonal is occupied by at least one checker[/i] ?

2005 Sharygin Geometry Olympiad, 15

Tags: geometry , lattice points , minimum , radius , circles

Given a circle centered at the origin. Prove that there is a circle of smaller radius that has no less points with integer coordinates.

1991 All Soviet Union Mathematical Olympiad, 546

Tags: geometry , combinatorics , combinatorial geometry , minimum

The figure below is cut along the lines into polygons (which need not be convex). No polygon contains a $2 \times 2$ square. What is the smallest possible number of polygons? [missing figure]

1999 Spain Mathematical Olympiad, 6

Tags: combinatorial geometry , parallel lines , minimum , plane , combinatorics

A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?

2015 Bulgaria National Olympiad, 2

Tags: rectangle , coloring , combinatorics , color problem , minimum

One hundred and one of the squares of an $n\times n$ table are colored blue. It is known that there exists a unique way to cut the table to rectangles along boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$.