Found problems: 76
1967 IMO Shortlist, 1
Find whether among all quadrilaterals, whose interiors lie inside a semi-circle of radius $r$, there exist one (or more) with maximum area. If so, determine their shape and area.
2006 China Team Selection Test, 2
Given positive integer $n$, find the biggest real number $C$ which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than $1$ is less than $C$, then we can divide the set of numbers into no more than $n$ groups so that the sum of reciprocals of every group is less than $1$.
1996 IMO Shortlist, 5
Let $ P(x)$ be the real polynomial function, $ P(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d.$ Prove that if $ |P(x)| \leq 1$ for all $ x$ such that $ |x| \leq 1,$ then
\[ |a| \plus{} |b| \plus{} |c| \plus{} |d| \leq 7.\]
1977 IMO Longlists, 57
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
1991 IMO Shortlist, 27
Determine the maximum value of the sum
\[ \sum_{i < j} x_ix_j (x_i \plus{} x_j)
\]
over all $ n \minus{}$tuples $ (x_1, \ldots, x_n),$ satisfying $ x_i \geq 0$ and $ \sum^n_{i \equal{} 1} x_i \equal{} 1.$
1989 IMO Shortlist, 26
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.
1979 IMO Shortlist, 11
Given real numbers $x_1, x_2, \dots , x_n \ (n \geq 2)$, with $x_i \geq \frac 1n \ (i = 1, 2, \dots, n)$ and with $x_1^2+x_2^2+\cdots+x_n^2 = 1$ , find whether the product $P = x_1x_2x_3 \cdots x_n$ has a greatest and/or least value and if so, give these values.
1983 Czech and Slovak Olympiad III A, 1
Let $n$ be a positive integer and $k\in[0,n]$ be a fixed real constant. Find the maximum value of $$\left|\sum_{i=1}^n\sin(2x_i)\right|$$ where $x_1,\ldots,x_n$ are real numbers satisfying $$\sum_{i=1}^n\sin^2(x_i)=k.$$
1989 IMO Shortlist, 24
For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)
2005 Bosnia and Herzegovina Team Selection Test, 3
Let $n$ be a positive integer such that $n \geq 2$. Let $x_1, x_2,..., x_n$ be $n$ distinct positive integers and $S_i$ sum of all numbers between them except $x_i$ for $i=1,2,...,n$. Let $f(x_1,x_2,...,x_n)=\frac{GCD(x_1,S_1)+GCD(x_2,S_2)+...+GCD(x_n,S_n)}{x_1+x_2+...+x_n}.$
Determine maximal value of $f(x_1,x_2,...,x_n)$, while $(x_1,x_2,...,x_n)$ is an element of set which consists from all $n$-tuples of distinct positive integers.
1972 IMO Longlists, 11
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
\[ a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM\]
1969 IMO Shortlist, 48
$(NET 3)$ Let $x_1, x_2, x_3, x_4,$ and $x_5$ be positive integers satisfying
\[x_1 +x_2 +x_3 +x_4 +x_5 = 1000,\]
\[x_1 -x_2 +x_3 -x_4 +x_5 > 0,\]
\[x_1 +x_2 -x_3 +x_4 -x_5 > 0,\]
\[-x_1 +x_2 +x_3 -x_4 +x_5 > 0,\]
\[x_1 -x_2 +x_3 +x_4 -x_5 > 0,\]
\[-x_1 +x_2 -x_3 +x_4 +x_5 > 0\]
$(a)$ Find the maximum of $(x_1 + x_3)^{x_2+x_4}$
$(b)$ In how many different ways can we choose $x_1, . . . , x_5$ to obtain the desired maximum?
1995 IMO, 4
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$.
1982 IMO Longlists, 4
[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 $Q.$
2016 Ukraine Team Selection Test, 6
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 $i = 1, \cdots , 2n$.
2015 Romania Team Selection Test, 5
Given an integer $N \geq 4$, determine the largest value the sum
$$\sum_{i=1}^{\left \lfloor{\frac{k}{2}}\right \rfloor+1}\left( \left \lfloor{\frac{n_i}{2}}\right \rfloor+1\right)$$
may achieve, where $k, n_1, \ldots, n_k$ run through the integers subject to $k \geq 3$, $n_1 \geq \ldots\geq n_k\geq 1$ and $n_1 + \ldots + n_k = N$.
1966 IMO Shortlist, 46
Let $a,b,c$ be reals and
\[f(a, b, c) = \left| \frac{ |b-a|}{|ab|} +\frac{b+a}{ab} -\frac 2c \right| +\frac{ |b-a|}{|ab|} +\frac{b+a}{ab} +\frac 2c\]
Prove that $f(a, b, c) = 4 \max \{\frac 1a, \frac 1b,\frac 1c \}.$
2015 IMO Shortlist, A3
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 $i = 1, \cdots , 2n$.
1977 IMO, 2
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
2006 China Team Selection Test, 2
Given positive integer $n$, find the biggest real number $C$ which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than $1$ is less than $C$, then we can divide the set of numbers into no more than $n$ groups so that the sum of reciprocals of every group is less than $1$.
2024 Indonesia TST, C
Given a sequence of integers $A_1,A_2,\cdots A_{99}$ such that for every sub-sequence that contains $m$ consecutive elements, there exist not more than $max\{ \frac{m}{3} ,1\}$ odd integers. Let $S=\{ (i,j) \ | i<j \}$ such that $A_i$ is even and $A_j$ is odd. Find $max\{ |S|\}$.
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 $i = 1, \cdots , 2n$.
1999 IMO, 2
Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality
\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]
holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.
1976 IMO, 1
Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$
1983 IMO Longlists, 35
Let $P_1, P_2, \dots , P_n$ be distinct points of the plane, $n \geq 2$. Prove that
\[ \max_{1\leq i<j\leq n} P_iP_j > \frac{\sqrt 3}{2}(\sqrt n -1) \min_{1\leq i<j\leq n} P_iP_j \]