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: 43

2010 Belarus Team Selection Test, 7.1
Tags: number theory , minimum value , minimum , absolute value
Find the smallest value of the expression $|3 \cdot 5^m - 11 \cdot 13^n|$ for all $m,n \in N$. (Folklore)

2021 European Mathematical Cup, 1
Tags: combinatorics , minimum value
Alice drew a regular $2021$-gon in the plane. Bob then labeled each vertex of the $2021$-gon with a real number, in such a way that the labels of consecutive vertices differ by at most $1$. Then, for every pair of non-consecutive vertices whose labels differ by at most $1$, Alice drew a diagonal connecting them. Let $d$ be the number of diagonals Alice drew. Find the least possible value that $d$ can obtain.

2014 Israel National Olympiad, 6
Tags: inequalities , algebra , minimum value
Let $n$ be a positive integer. Find the maximal real number $k$, such that the following holds: For any $n$ real numbers $x_1,x_2,...,x_n$, we have $\sqrt{x_1^2+x_2^2+\dots+x_n^2}\geq k\cdot\min(|x_1-x_2|,|x_2-x_3|,...,|x_{n-1}-x_n|,|x_n-x_1|)$

2018 VTRMC, 6
Tags: extremal , maximum , minimum value , sequence
For $n \in \mathbb{N}$, define $a_n = \frac{1 + 1/3 + 1/5 + \dots + 1/(2n-1)}{n+1}$ and $b_n = \frac{1/2 + 1/4 + 1/6 + \dots + 1/(2n)}{n}$. Find the maximum and minimum of $a_n - b_n$ for $1 \leq n \leq 999$.

1973 IMO, 3
Tags: algebra , polynomial , root , minimum value , coefficient
Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.

2018 Brazil Undergrad MO, 5
Tags: algebra , minimum value , minimization , set
Consider the set $A = \left\{\frac{j}{4}+\frac{100}{j}|j=1,2,3,..\right\} $ What is the smallest number that belongs to the $ A $ set?

2022 Turkey EGMO TST, 6
Tags: inequalities , minimum value , ratio
Let $x,y,z$ be positive real numbers satisfying the equations $$xyz=1\text{ and }\frac yz(y-x^2)+\frac zx(z-y^2)+\frac xy(x-z^2)=0$$ What is the minimum value of the ratio of the sum of the largest and smallest numbers among $x,y,z$ to the median of them.

2018 Israel National Olympiad, 3
Tags: absolute value , minimum value , maximum value , maximum and minimum , algebra , inequality
Determine the minimal and maximal values the expression $\frac{|a+b|+|b+c|+|c+a|}{|a|+|b|+|c|}$ can take, where $a,b,c$ are real numbers.

2024-IMOC, A2
Tags: algebra , minimum value , inequalities , absolute value
Given integer $n \geq 3$ and $x_1$, $x_2$, …, $x_n$ be $n$ real numbers satisfying $|x_1|+|x_2|+…+|x_n|=1$. Find the minimum of \[|x_1+x_2|+|x_2+x_3|+…+|x_{n-1}+x_n|+|x_n+x_1|.\] [i]Proposed by snap7822[/i]

1966 Vietnam National Olympiad, 1
Tags: algebra , inequalities , maximum value , minimum value
Let $x, y$ and $z$ be nonnegative real numbers satisfying the following conditions: (1) $x + cy \le 36$,(2) $2x+ 3z \le 72$, where $c$ is a given positive number. Prove that if $c \ge 3$ then the maximum of the sum $x + y + z$ is $36$, while if $c < 3$, the maximum of the sum is $24 + \frac{36}{c}$ .

2017 Azerbaijan Senior National Olympiad, G4
Tags: geometry , area of a triangle , area , convex polygon , minimum value
İn convex hexagon $ABCDEF$'s diagonals $AD,BE,CF$ intercepts each other at point $O$. If the area of triangles $AOB,COD,EOF$ are $4,6$ and $9$ respectively, find the minimum possible value of area of hexagon $ABCDEF$

2017 JBMO Shortlist, A2
Tags: algebra , minimum value
Let $a$ and $b$ be positive real numbers such that $3a^2 + 2b^2 = 3a + 2b$. Find the minimum value of $A =\sqrt{\frac{a}{b(3a+2)}} + \sqrt{\frac{b}{a(2b+3)}} $

1987 Nordic, 3
Tags: minimum value , increasing function , functional equation in n , algebra
Let $f$ be a strictly increasing function defined in the set of natural numbers satisfying the conditions $f(2) = a > 2$ and $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. Determine the smallest possible value of $a$.

1973 IMO Shortlist, 11
Tags: algebra , polynomial , root , minimum value , coefficient
Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.

2025 Romania National Olympiad, 1
Tags: combinatorics , minimum value , coloring
Let $N \geq 1$ be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer $k$ (not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are $x$ and $y$ respectively, we replace them with $x+k+1$ and $y+k^2+2$, which we color blue and red (in this order). We keep doing this procedure until the blue number is at least $N$. Determine the minimum value of the red number at the end of this procedure.

1975 Vietnam National Olympiad, 5
Tags: algebra , function , minimum value , maximum value , analysis
Show that the sum of the (local) maximum and minimum values of the function $\frac{tan(3x)}{tan^3x}$ on the interval $\big(0, \frac{\pi }{2}\big)$ is rational.

2010 Czech And Slovak Olympiad III A, 6
Tags: lcm , least common multiple , number theory , minimum value , minimum , inequalities
Find the minimum of the expression $\frac{a + b + c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c}$ where the variables $a, b, c$ are any integers greater than $1$ and $[x, y]$ denotes the least common multiple of numbers $x, y$.

2023 Turkey Team Selection Test, 6
Tags: minimum value , inequalities
Let $a,b,c,d$ be positive real numbers. What is the minimum value of $$ \frac{(a^2+b^2+2c^2+3d^2)(2a^2+3b^2+6c^2+6d^2)}{(a+b)^2(c+d)^2}$$