Found problems: 127
1965 Swedish Mathematical Competition, 1
The feet of the altitudes in the triangle $ABC$ are $A', B', C'$. Find the angles of $A'B'C'$ in terms of the angles $A, B, C$. Show that the largest angle in $A'B'C'$ is at least as big as the largest angle in $ABC$. When is it equal?
2019 Dutch IMO TST, 4
There are $300$ participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of $n$ for which it is possible to satisfy the following conditions at the same time: each contestant plays at most $n$ games of chess, and for each $m$ with $1 \le m \le n$, there is a contestant playing exactly $m$ games of chess.
2007 Oral Moscow Geometry Olympiad, 2
Two circles intersect at points $P$ and $Q$. Point $A$ lies on the first circle, but outside the second. Lines $AP$ and $AQ$ intersect the second circle at points $B$ and $C$, respectively. Indicate the position of point $A$ at which triangle $ABC$ has the largest area.
(D. Prokopenko)
2023 Brazil Team Selection Test, 3
Show that for all positive real numbers $a, b, c$, we have that $$\frac{a+b+c}{3}-\sqrt[3]{abc} \leq \max\{(\sqrt{a}-\sqrt{b})^2, (\sqrt{b}-\sqrt{c})^2, (\sqrt{c}-\sqrt{a})^2\}$$
2021 New Zealand MO, 7
Let $a, b, c, d$ be integers such that $a > b > c > d \ge -2021$ and $$\frac{a + b}{b + c}=\frac{c + d}{d + a}$$ (and $b + c \ne 0 \ne d + a$). What is the maximum possible value of $ac$?
2005 Bosnia and Herzegovina Junior BMO TST, 1
Non-negative real numbers $x, y, z$ satisfy the following relations:
$3x + 5y + 7z = 10$ and $x + 2y + 5z = 6$.
Find the minimum and maximum of $w = 2x - 3y + 4z$.
2012 Czech-Polish-Slovak Junior Match, 1
There are a lot of different real numbers written on the board. It turned out that for each two numbers written, their product was also written. What is the largest possible number of numbers written on the board?
2002 Croatia Team Selection Test, 1
In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.
2013 Junior Balkan Team Selection Tests - Romania, 3
Find the minimum and the maximum value of the expression $\sqrt{4 -a^2} +\sqrt{4 -b^2} +\sqrt{4 -c^2}$
where $a,b, c$ are positive real numbers satisfying the condition $a^2 + b^2 + c^2=6$
2012 Mathcenter Contest + Longlist, 6 sl14
For a real number $a,b,c>0$ where $bc-ca-ab=1$ find the maximum value of $$P=\frac{4024}{1+a^2}-\frac{4024}{1+b^2}-\frac{2555}{1+c^2}$$ and find out when that holds .
[i](PP-nine)[/i]
1994 Italy TST, 1
Given a circle $\gamma$ and a point $P$ inside it, find the maximum and minimum value of the sum of the lengths of two perpendicular chords of $\gamma$ passing through $P$.
2002 Croatia Team Selection Test, 1
In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.
2021 Ukraine National Mathematical Olympiad, 8
There are $101$ not necessarily different weights, each of which weighs an integer number of grams from $1$ g to $2020$ g. It is known that at any division of these weights into two heaps, the total weight of at least one of the piles is no more than $2020$. What is the largest number of grams can weigh all $101$ weights?
(Bogdan Rublev)
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.
Cono Sur Shortlist - geometry, 1993.10
Let $\omega$ be the unit circle centered at the origin of $R^2$. Determine the largest possible value for the radius of the circle inscribed to the triangle $OAP$ where $ P$ lies the circle and $A$ is the projection of $P$ on the axis $OX$.
2003 Switzerland Team Selection Test, 4
Find the largest natural number $n$ that divides $a^{25} -a$ for all integers $a$.
2018 Junior Balkan Team Selection Tests - Romania, 2
Let $x, y,z$ be positive real numbers satisfying $2x^2+3y^2+6z^2+12(x+y+z) =108$. Find the maximum value of $x^3y^2z$.
Alexandru Gırban
1998 Rioplatense Mathematical Olympiad, Level 3, 2
Given an integer $n > 2$, consider all sequences $x_1,x_2,...,x_n$ of nonnegative real numbers such that $$x_1+ 2x_2 + ... + nx_n = 1.$$ Find the maximum value and the minimum value of $x_1^2+x_2^2+...+x_n^2$ and determine all the sequences $x_1,x_2,...,x_n$ for which these values are obtained.
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.
2009 Postal Coaching, 4
For positive integers $n \ge 3$ and $r \ge 1$, define $$P(n, r) = (n - 2)\frac{r^2}{2} - (n - 4) \frac{r}{2}$$
We call a triple $(a, b, c)$ of natural numbers, with $a \le b \le c$, an $n$-gonal Pythagorean triple if $P(n, a)+P(n, b) = P(n, c)$. (For $n = 4$, we get the usual Pythagorean triple.)
(a) Find an $n$-gonal Pythagorean triple for each $n \ge 3$.
(b) Consider all triangles $ABC$ whose sides are $n$-gonal Pythagorean triples for some $n \ge 3$. Find the maximum and the minimum possible values of angle $C$.
2015 Indonesia MO Shortlist, C5
A meeting was attended by $n$ people. They are welcome to occupy the $k$ table provided $\left( k \le \frac{n}{2} \right)$. Each table is occupied by at least two people. When the meeting begins, the moderator selects two people from each table as representatives for talk to. Suppose that $A$ is the number of ways to choose representatives to speak.
Determine the maximum value of $A$ that is possible.
1963 Swedish Mathematical Competition., 2
The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?
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.
2014 Estonia Team Selection Test, 3
Three line segments, all of length $1$, form a connected figure in the plane. Any two different line segments can intersect only at their endpoints. Find the maximum area of the convex hull of the figure.
2013 Saudi Arabia BMO TST, 3
Let $T$ be a real number satisfying the property:
For any nonnegative real numbers $a, b, c,d, e$ with their sum equal to $1$, it is possible to arrange them around a circle such that the products of any two neighboring numbers are no greater than $T$.
Determine the minimum value of $T$.