Found problems: 76
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 $1$.
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$.
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?
1995 Czech and Slovak Match, 3
Consider all triangles $ABC$ in the cartesian plane whose vertices are at lattice points (i.e. with integer coordinates) and which contain exactly one lattice point (to be denoted $P$) in its interior. Let the line $AP$ meet $BC$ at $E$. Determine the maximum possible value of the ratio $\frac{AP}{PE}$.
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.
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 $$\sum_{i=1}^n\sin^2(x_i)=k.$$
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.
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.
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
\[ a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM\]
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 $1$.
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 Harvard-MIT Mathematics Tournament, 5
Let $a,b,c$ be positive real numbers such that $a+b+c=10$ and $ab+bc+ca=25$. Let $m=\min\{ab,bc,ca\}$. Find the largest possible value of $m$.
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 $m.$
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?
2023 Indonesia Regional, 3
Find the maximum value of an integer $B$ such that for every 9 distinct natural number with the sum of $2023$, there must exist a sum of 4 of the number that is greater than or equal to $B$
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.
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 $10 \times 10 \times 1 \ ?$
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.$
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.
1967 IMO Longlists, 19
The $n$ points $P_1,P_2, \ldots, P_n$ are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance $D_n$ between any two of these points has its largest possible value $D_n.$ Calculate $D_n$ for $n = 2$ to 7. and justify your answer.
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 $S(a, b, c)$.
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 $n_1 + \ldots + n_k = N$.
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:
$$ g(A,B,C,D) =AB+BC+ AD+CA+DB+DC. $$
1981 IMO Shortlist, 3
Find the minimum value of
\[\max(a + b + c, b + c + d, c + d + e, d + e + f, e + f + g)\]
subject to the constraints
(i) $a, b, c, d, e, f, g \geq 0,$
(ii)$ a + b + c + d + e + f + g = 1.$
1979 IMO Shortlist, 5
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 $m.$