This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 76

2002 Kazakhstan National Olympiad, 3
Tags: modular arithmetic , combinatorics , extremal combinatorics , maximization
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 $m$.

1979 IMO Shortlist, 20
Tags: algebra , maximization , optimization , lagrange , calculus
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 $a.$

1984 IMO Longlists, 20
Tags: inequality , three variable inequality , calculus , maximization
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

1976 IMO Shortlist, 10
Tags: number theory , product , maximization
Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$

2024 Indonesia TST, C
Tags: combinatorics , maximization
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 $max\{ |S|\}$.

1985 Austrian-Polish Competition, 7
Tags: inequalities , maximization , ratio , four variables
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.

2024 Indonesia TST, C
Tags: combinatorics , maximization
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 $max\{ |S|\}$.

1972 IMO Longlists, 11
Tags: inequality , optimization , maximization
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 \[ a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM\]

1982 IMO Longlists, 17
Tags: sequence , maximization , minimization , rearrangement , algebra
[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 $Q.$

1967 IMO Shortlist, 1
Tags: quadrilateral , area , maximization , geometry
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.

1983 IMO Shortlist, 17
Tags: geometry , geometric inequality , optimization , minimization , maximization , point set
Let $P_1, P_2, \dots , P_n$ be distinct points of the plane, $n \geq 2$. Prove that \[ \max_{1\leq i<j\leq n} P_iP_j > \frac{\sqrt 3}{2}(\sqrt n -1) \min_{1\leq i<j\leq n} P_iP_j \]

1989 IMO Shortlist, 24
Tags: geometry , 3d geometry , sphere , combinatorics , maximization , geometric inequality
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.)

1979 IMO, 1
Tags: trigonometry , geometry , 3d geometry , maximization
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 Longlists, 84
Tags: maximization , minimization , optimization , algebra , calculus
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.

2018 Bosnia and Herzegovina EGMO TST, 4
Tags: combinatorics , maximization , extremal combinatorics
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 Longlists, 44
Tags: combinatorics , packing , 3d geometry , maximization
What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$

1984 IMO Shortlist, 5
Tags: inequality , three variable inequality , calculus , maximization
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

1967 IMO Shortlist, 3
Tags: geometry , point set , euclidean distance , maximization
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.

1999 IMO Shortlist, 1
Tags: inequalities , algebra , maximization
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.

2015 Romania Team Selection Tests, 5
Tags: maximization , algebra , inequalities , sum
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, 45
Tags: combinatorics , maximization , extremal combinatorics , combinatorics of words
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
Tags: combinatorics , maximization , triplets , extremal combinatorics
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 $x + y = z$.

1966 IMO Longlists, 46
Tags: algebra , maximization , optimization , function
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 Harvard-MIT Mathematics Tournament, 6
Tags: algebra , number theory , maximization
Let $a,b,c,d,e$ be nonnegative integers such that $625a+250b+100c+40d+16e=15^3$. What is the maximum possible value of $a+b+c+d+e$?

1995 Nordic, 3
Tags: inequalities , algebra , maximization
Let $n \ge 2$ and let $x_1, x_2, ..., x_n$ be real numbers satisfying $x_1 +x_2 +...+x_n \ge 0$ and $x_1^2+x_2^2+...+x_n^2=1$. Let $M = max \{x_1, x_2,... , x_n\}$. Show that $M \ge \frac{1}{\sqrt{n(n-1)}}$ (1) .When does equality hold in (1)?