Found problems: 127
1964 Swedish Mathematical Competition, 5
$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$. We seek estimates of $a_1$. If $n = 2$, find the smallest and largest possible values of $a_1$. Find corresponding estimates for other values of $n$.
1999 Ukraine Team Selection Test, 3
Let $m,n$ be positive integers with $m \le n$, and let $F$ be a family of $m$-element subsets of $\{1,2,...,n\}$ satisfying $A \cap B \ne \varnothing$ for all $A,B \in F$. Determine the maximum possible number of elements in $F$.
Kyiv City MO 1984-93 - geometry, 1986.10.5
Let $E$ be a point on the side $AD$ of the square $ABCD$. Find such points $M$ and $K$ on the sides $AB$ and $BC$ respectively, such that the segments $MK$ and $EC$ are parallel, and the quadrilateral $MKCE$ has the largest area.
2008 Postal Coaching, 3
Let $ABC$ be a triangle. For any point $X$ on $BC$, let $AX$ meet the circumcircle of $ABC$ in $X'$. Prove or disprove: $XX'$ has maximum length if and only if $AX$ lies between the median and the internal angle bisector from $A$.
1999 Kazakhstan National Olympiad, 8
Let $ {{a} _ {1}}, {{a} _ {2}}, \ldots, {{a} _ {n}} $ be permutation of numbers $ 1,2, \ldots, n $, where $ n \geq 2 $.
Find the maximum value of the sum $$ S (n) = | {{a} _ {1}} - {{a} _ {2}} | + | {{a} _ {2}} - {{a} _ {3}} | + \cdots + | {{a} _ {n-1}} - {{a} _ {n}} |. $$
2017 Puerto Rico Team Selection Test, 5
Let $a, b$ be two real numbers that satisfy $a^3 + b^3 = 8-6ab$.
Find the maximum value and the minimum value that $a + b$ can take.
2000 Switzerland Team Selection Test, 9
Two given circles $k_1$ and $k_2$ intersect at points $P$ and $Q$.
Construct a segment $AB$ through $P$ with the endpoints at $k_1$ and $k_2$ for which $AP \cdot PB$ is maximal.
2006 MOP Homework, 5
Let $a_1, a_2,...,a_{2005}, b_1, b_2,...,b_{2005}$ be real numbers such that $(a_ix - b_i)^2 \ge \sum_{j\ne i,j=1}^{2005} (a_jx - b_j)$ for all real numbers x and every integer $i$ with $1 \le i \le 2005$. What is maximal number of positive $a_i$'s and $b_i$'s?
2009 Ukraine Team Selection Test, 11
Suppose that integers are given $m <n $. Consider a spreadsheet of size $n \times n $, whose cells arbitrarily record all integers from $1 $ to ${{n} ^ {2}} $. Each row of the table is colored in yellow $m$ the largest elements. Similarly, the blue colors the $m$ of the largest elements in each column. Find the smallest number of cells that are colored yellow and blue at a time
1997 Poland - Second Round, 6
Let eight points be given in a unit cube. Prove that two of these points are on a distance not greater than $1$.
1983 Swedish Mathematical Competition, 4
$C$, $C'$ are concentric circles with radii $R$, $R'$. A rectangle has two adjacent vertices on $C$ and the other two vertices on $C'$. Find its sides if its area is as large as possible.
2015 Grand Duchy of Lithuania, 4
We denote by gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd$(4, 6, 8)=2$ and gcd $(12, 15)=3$.) Suppose that positive integers $a, b, c$ satisfy the following four conditions:
$\bullet$ gcd $(a, b, c)=1$,
$\bullet$ gcd $(a, b + c)>1$,
$\bullet$ gcd $(b, c + a)>1$,
$\bullet$ gcd $(c, a + b)>1$.
a) Is it possible that $a + b + c = 2015$?
b) Determine the minimum possible value that the sum $a+ b+ c$ can take.
1981 Czech and Slovak Olympiad III A, 5
Let $n$ be a positive integer. Determine the maximum of the sum $x_1+\cdots+x_n$ where $x_1,\ldots,x_n$ are non-negative integers satisfying the condition \[x_1^3+\cdots+x_n^3\le7n.\]
2018 Estonia Team Selection Test, 3
Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.
2002 Estonia National Olympiad, 4
Let $a_1, ... ,a_5$ be real numbers such that at least $N$ of the sums $a_i+a_j$ ($i < j$) are integers. Find the greatest value of $N$ for which it is possible that not all of the sums $a_i+a_j$ are integers.
2018 Estonia Team Selection Test, 3
Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.
1974 Czech and Slovak Olympiad III A, 2
Let a triangle $ABC$ be given. For any point $X$ of the triangle denote $m(X)=\min\{XA,XB,XC\}.$ Find all points $X$ (of triangle $ABC$) such that $m(X)$ is maximal.
2016 Czech-Polish-Slovak Junior Match, 1
Let $AB$ be a given segment and $M$ be its midpoint. We consider the set of right-angled triangles $ABC$ with hypotenuses $AB$. Denote by $D$ the foot of the altitude from $C$. Let $K$ and $L$ be feet of perpendiculars from $D$ to the legs $BC$ and $AC$, respectively. Determine the largest possible area of the quadrilateral $MKCL$.
Czech Republic
2012 Czech And Slovak Olympiad IIIA, 4
Inside the parallelogram $ABCD$ is a point $X$. Make a line that passes through point $X$ and divides the parallelogram into two parts whose areas differ from each other the most.
2001 Estonia National Olympiad, 2
Find the maximum value of $k$ for which one can choose $k$ integers out of $1,2... ,2n$ so that none of them divides another one.
2022 Saudi Arabia JBMO TST, 2
Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$ . Find the maximum value of the following expression $$P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c$$
2012 Estonia Team Selection Test, 5
Let $x, y, z$ be positive real numbers whose sum is $2012$. Find the maximum value of $$ \frac{(x^2 + y^2 + z^2)(x^3 + y^3 + z^3)}{(x^4 + y^4 + z^4)}$$
1973 Chisinau City MO, 67
The product of $10$ natural numbers is equal to $10^{10}$. What is the largest possible sum of these numbers?
2018 Malaysia National Olympiad, A2
The product of $10$ integers is $1024$. What is the greatest possible sum of these $10$ integers?
2013 Saudi Arabia Pre-TST, 2.2
The quadratic equation $ax^2 + bx + c = 0$ has its roots in the interval $[0, 1]$. Find the maximum of $\frac{(a - b)(2a - b)}{a(a - b + c)}$.