Olympiad problems

2019 Saint Petersburg Mathematical Olympiad, 7

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?

1988 All Soviet Union Mathematical Olympiad, 477

Tags: inequalities , algebra , minimum

What is the minimal value of $\frac{b}{c + d} + \frac{c}{a + b}$ for positive real numbers $b$ and $c$ and non-negative real numbers $a$ and $d$ such that $b + c\ge a + d$?

2006 JBMO ShortLists, 15

Tags: geometry , prime number , integer , area , minimum

Let $A_1$ and $B_1$ be internal points lying on the sides $BC$ and $AC$ of the triangle $ABC$ respectively and segments $AA_1$ and $BB_1$ meet at $O$. The areas of the triangles $AOB_1,AOB$ and $BOA_1$ are distinct prime numbers and the area of the quadrilateral $A_1OB_1C$ is an integer. Find the least possible value of the area of the triangle $ABC$, and argue the existence of such a triangle.

1994 Tuymaada Olympiad, 5

Tags: trigonometry , inequalities , trigonometric inequality , minimum

Find the smallest natural number $n$ for which $sin \Big(\frac{1}{n+1934}\Big)<\frac{1}{1994}$ .

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.

2019 SAFEST Olympiad, 4

Tags: minimum , inequalities , algebra , sum

Let $a_1, a_2, . . . , a_{2019}$ be any positive real numbers such that $\frac{1}{a_1 + 2019}+\frac{1}{a_2 + 2019}+ ... +\frac{1}{a_{2019} + 2019}=\frac{1}{2019}$. Find the minimum value of $a_1a_2... a_{2019}$ and determine for which values of $a_1, a_2, . . . , a_{2019}$ this minimum occurs

1979 All Soviet Union Mathematical Olympiad, 275

Tags: minimum , combinatorics , chessboard

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] ?

2021 Regional Olympiad of Mexico Southeast, 3

Tags: minimum , inequalities , maximum

Let $a, b, c$ positive reals such that $a+b+c=1$. Prove that $$\min\{a(1-b),b(1-c),c(1-a)\}\leq \frac{1}{4}$$ $$\max\{a(1-b),b(1-c),c(1-a)\}\geq \frac{2}{9}$$

1963 All Russian Mathematical Olympiad, 032

Tags: geometry , equilateral , minimum

Given equilateral triangle with the side $l$. What is the minimal length $d$ of a brush (segment), that will paint all the triangle, if its ends are moving along the sides of the triangle.

2012 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: combinatorial geometry , rectangle , combinatorics , minimum

A rectangle is divided into $n^2$ smaller rectangle by $n - 1$ horizontal lines and $n - 1$ vertical lines. Between those rectangles there are exactly $5660$ which are not congruent. For what minimum value of $n$ is this possible?

1988 All Soviet Union Mathematical Olympiad, 481

Tags: minimum , cube , geometry , 3d geometry , broken line

A polygonal line connects two opposite vertices of a cube with side $2$. Each segment of the line has length $3$ and each vertex lies on the faces (or edges) of the cube. What is the smallest number of segments the line can have?

2014 Hanoi Open Mathematics Competitions, 4

Tags: perfect square , minimum , number theory

Find the smallest positive integer $n$ such that the number $2^n + 2^8 + 2^{11}$ is a perfect square. (A): $8$, (B): $9$, (C): $11$, (D): $12$, (E) None of the above.

1946 Moscow Mathematical Olympiad, 119

Tags: minimum , geometry , segment

On the legs of $\angle AOB$, the segments $OA$ and $OB$ lie, $OA > OB$. Points $M$ and $N$ on lines $OA$ and $OB$, respectively, are such that $AM = BN = x$. Find $x$ for which the length of $MN$ is minimal.

2022 Singapore MO Open, Q4

Tags: minimum , combinatorics , coloring , permutation

Let $n,k$, $1\le k\le n$ be fixed integers. Alice has $n$ cards in a row, where the card has position $i$ has the label $i+k$ (or $i+k-n$ if $i+k>n$). Alice starts by colouring each card either red or blue. Afterwards, she is allowed to make several moves, where each move consists of choosing two cards of different colours and swapping them. Find the minimum number of moves she has to make (given that she chooses the colouring optimally) to put the cards in order (i.e. card $i$ is at position $i$). NOTE: edited from original phrasing, which was ambiguous.

2010 Dutch IMO TST, 1

Tags: minimum , sequence , algebra

Consider sequences $a_1, a_2, a_3,...$ of positive integers. Determine the smallest possible value of $a_{2010}$ if (i) $a_n < a_{n+1}$ for all $n\ge 1$, (ii) $a_i + a_l > a_j + a_k$ for all quadruples $ (i, j, k, l)$ which satisfy $1 \le i < j \le k < l$.

2018 JBMO Shortlist, A4

Tags: algebra , minimum , product , system of equations

Let $k > 1, n > 2018$ be positive integers, and let $n$ be odd. The nonzero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and satisfy $$x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}$$ Find: a) the product $x_1 x_2 \ldots x_n$ as a function of $k$ and $n$ b) the least value of $k$, such that there exist $n,x_1,x_2,\ldots,x_n$ satisfying the given conditions.

1995 May Olympiad, 4

Tags: minimum , distance , 3d geometry , pyramid , geometry

Consider a pyramid whose base is an equilateral triangle $BCD$ and whose other faces are triangles isosceles, right at the common vertex $A$. An ant leaves the vertex $B$ arrives at a point $P$ of the $CD$ edge, from there goes to a point $Q$ of the edge $AC$ and returns to point $B$. If the path you made is minimal, how much is the angle $PQA$ ?

2013 India PRMO, 1

Tags: number theory , minimum

What is the smallest positive integer $k$ such that $k(3^3 + 4^3 + 5^3) = a^n$ for some positive integers $a$ and $n$, with $n > 1$?

1995 Tuymaada Olympiad, 8

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

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.

1974 All Soviet Union Mathematical Olympiad, 190

Tags: algebra , number theory , minimum

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.

1994 ITAMO, 5

Tags: geometry , maximum , cube , plane , area , minimum

Let $OP$ be a diagonal of a unit cube. Find the minimum and the maximum value of the area of the intersection of the cube with a plane through $OP$.

1983 Brazil National Olympiad, 5

Tags: positive integer , minimum , inequality , number theory

Show that $1 \le n^{1/n} \le 2$ for all positive integers $n$. Find the smallest $k$ such that $1 \le n ^{1/n} \le k$ for all positive integers $n$.

2009 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: minimum , algebra , inequalities

Find minimum of $x+y+z$ where $x$, $y$ and $z$ are real numbers such that $x \geq 4$, $y \geq 5$, $z \geq 6$ and $x^2+y^2+z^2 \geq 90$

1992 Nordic, 3

Tags: geometry , inradius , minimum

Prove that among all triangles with inradius $1$, the equilateral one has the smallest perimeter .

1987 Brazil National Olympiad, 4

Tags: minimum , combinatorial geometry , algebra , convex hull

Given points $A_1 (x_1, y_1, z_1), A_2 (x_2, y_2, z_2), .., A_n (x_n, y_n, z_n)$ let $P (x, y, z)$ be the point which minimizes $\Sigma ( |x - x_i| + |y -y_i| + |z -z_i| )$. Give an example (for each $n > 4$) of points $A_i $ for which the point $P$ lies outside the convex hull of the points $A_i$.