Found problems: 76
1979 IMO Longlists, 68
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.
1976 IMO Longlists, 41
Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$
1983 IMO Shortlist, 17
Let $P_1, P_2, \dots , P_n$ be distinct points of the plane, $n \geq 2$. Prove that
\[ \max_{1\leq i<j\leq n} P_iP_j > \frac{\sqrt 3}{2}(\sqrt n -1) \min_{1\leq i<j\leq n} P_iP_j \]
1984 IMO Shortlist, 5
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.
2011 Indonesia TST, 3
Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define
\[
p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}.
\]
Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?
1966 IMO Longlists, 45
An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters.
1997 Nordic, 1
Let $A$ be a set of seven positive numbers. Determine the maximal number of triples $(x, y, z)$ of elements of $A$
satisfying $x < y$ and $x + y = z$.
1982 IMO Longlists, 17
[b](a)[/b] Find the rearrangement $\{a_1, \dots , a_n\}$ of $\{1, 2, \dots, n\}$ that maximizes
\[a_1a_2 + a_2a_3 + \cdots + a_na_1 = Q.\]
[b](b)[/b] Find the rearrangement that minimizes $Q.$
2018 Bosnia and Herzegovina EGMO TST, 4
It is given positive integer $n$. Let $a_1, a_2,..., a_n$ be positive integers with sum $2S$, $S \in \mathbb{N}$. Positive integer $k$ is called separator if you can pick $k$ different indices $i_1, i_2,...,i_k$ from set $\{1,2,...,n\}$ such that $a_{i_1}+a_{i_2}+...+a_{i_k}=S$. Find, in terms of $n$, maximum number of separators
1966 IMO Shortlist, 45
An alphabet consists of $n$ letters. What is the maximal length of a word if we know that any two consecutive letters $a,b$ of the word are different and that the word cannot be reduced to a word of the kind $abab$ with $a\neq b$ by removing letters.
2016 Iran Team Selection Test, 4
Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.
2002 Kazakhstan National Olympiad, 3
Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$.
1989 IMO Longlists, 84
Let $ n \in \mathbb{Z}^\plus{}$ and let $ a, b \in \mathbb{R}.$ Determine the range of $ x_0$ for which
\[ \sum^n_{i\equal{}0} x_i \equal{} a \text{ and } \sum^n_{i\equal{}0} x^2_i \equal{} b,\]
where $ x_0, x_1, \ldots , x_n$ are real variables.
1986 Traian Lălescu, 2.3
Among the spatial points $ A,B,C,D, $ at most two of are aparted at a distance greater than $ 1. $ Find the the maximum value of the expression:
$$ g(A,B,C,D) =AB+BC+ AD+CA+DB+DC. $$
1979 IMO Longlists, 15
Let $n \geq 2$ be an integer. Find the maximal cardinality of a set $M$ of pairs $(j, k)$ of integers, $1 \leq j < k \leq n$, with the following property: If $(j, k) \in M$, then $(k,m) \not \in M$ for any $m.$
2023 Indonesia MO, 8
Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. For example, $S(1, 2, 3) = \{ -1, -2, -1/2 \}$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$, the equation $2x^2+3x+1=0$ has roots $-1$ and $-1/2$, and for all the other permutations of $(1, 2, 3)$, the quadratic equations formed don't have any rational roots.
Determine the maximum number of elements in $S(a, b, c)$.
1972 IMO Shortlist, 3
The least number is $m$ and the greatest number is $M$ among $ a_1 ,a_2 ,\ldots,a_n$ satisfying $ a_1 \plus{}a_2 \plus{}...\plus{}a_n \equal{}0$. Prove that
\[ a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM\]
1979 IMO Longlists, 60
Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$
1966 IMO Shortlist, 44
What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$
1979 IMO Shortlist, 25
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.
1985 Austrian-Polish Competition, 7
Find an upper bound for the ratio
$$\frac{x_1x_2+2x_2x_3+x_3x_4}{x_1^2+x_2^2+x_3^2+x_4^2}$$
over all quadruples of real numbers $(x_1,x_2,x_3,x_4)\neq (0,0,0,0)$.
[i]Note.[/i] The smaller the bound, the better the solution.
2016 Taiwan TST Round 1, 2
Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.
1966 IMO Longlists, 44
What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$
1979 IMO, 1
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.
1995 IMO Shortlist, 2
Find the maximum value of $ x_{0}$ for which there exists a sequence $ x_{0},x_{1}\cdots ,x_{1995}$ of positive reals with $ x_{0} \equal{} x_{1995}$, such that
\[ x_{i \minus{} 1} \plus{} \frac {2}{x_{i \minus{} 1}} \equal{} 2x_{i} \plus{} \frac {1}{x_{i}},
\]
for all $ i \equal{} 1,\cdots ,1995$.