Found problems: 43
Let $x,y,z$ be real numbers. Find the minimum value of the sum \begin{align*}|\cos(x)|+|\cos(y)|+|\cos(z)|+|\cos(x-y)|+|\cos(y-z)|+|\cos(z-x)|.\end{align*}
Let $k$ be a positive real number such that $$\frac{1}{k+a} + \frac{1}{k+b} + \frac{1}{k+c} \leq 1$$ for any positive positive real numbers $a$, $b$ and $c$ with $abc = 1$. Find the minimum value of $k$.
Given an equilateral triangle with sidelength $k$ cm. With lines parallel to it's sides, we split it into $k^2$ small equilateral triangles with sidelength $1$ cm. This way, a triangular grid is created. In every small triangle of sidelength $1$ cm, we place exactly one integer from $1$ to $k^2$ (included), such that there are no such triangles having the same numbers. With vertices the points of the grid, regular hexagons are defined of sidelengths $1$ cm. We shall name as [i]value [/i] of the hexagon, the sum of the numbers that lie on the $6$ small equilateral triangles that the hexagon consists of . Find (in terms of the integer $k>4$) the maximum and the minimum value of the sum of the values of all hexagons .
There are $2022$ students in winter school. Two arbitrary students are friend or enemy each other. Each turn, we choose a student $S$, make friends of $S$ enemies, and make enemies of $S$ friends. This continues until it satisfies the final condition.
[b]Final Condition[/b] : For any partition of students into two non-empty groups $A$, $B$, there exist two students $a$, $b$ such that $a\in A$, $b\in B$, and $a$, $b$ are friend each other.
Determine the minimum value of $n$ such that regardless of the initial condition, we can satisfy the final condition with no more than $n$ turns.
A sequence $\{z_n\}$ satisfies that for any positive integer $i,$ $z_i\in\{0,1,\cdots,9\}$ and $z_i\equiv i-1 \pmod {10}.$ Suppose there is $2021$ non-negative reals $x_1,x_2,\cdots,x_{2021}$ such that for $k=1,2,\cdots,2021,$ $$\sum_{i=1}^kx_i\geq\sum_{i=1}^kz_i,\sum_{i=1}^kx_i\leq\sum_{i=1}^kz_i+\sum_{j=1}^{10}\dfrac{10-j}{50}z_{k+j}.$$
Determine the least possible value of $\sum_{i=1}^{2021}x_i^2.$
In a country with $n$ towns, all the direct flights are of double destinations (back and forth). There are $r>2014$ rootes between different pairs of towns, that include no more than one intermediate stop (direction of each root matters). Find the minimum possible value of $n$ and the minimum possible $r$ for that value of $n$.
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.
Let $a,b,c$ be positive reals such that $abc=1$, $a+b+c=5$ and $$(ab+2a+2b-9)(bc+2b+2c-9)(ca+2c+2a-9)\geq 0$$.
Find the minimum value of $$\frac {1}{a}+ \frac {1}{b}+ \frac{1}{c}$$
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.
Let $a,b,c,d$ be positive real numbers. What is the minimum value of $$ \frac{(a^2+b^2+2c^2+3d^2)(2a^2+3b^2+6c^2+6d^2)}{(a+b)^2(c+d)^2}$$
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)}} $
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.
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 $\sqrt{x_1^2+x_2^2+\dots+x_n^2}\geq k\cdot\min(|x_1-x_2|,|x_2-x_3|,...,|x_{n-1}-x_n|,|x_n-x_1|)$
Determine the minimum value of the expression
$$\frac {a+1}{a(a+2)}+ \frac {b+1}{b(b+2)}+\frac {c+1}{c(c+2)}$$
for positive real numbers $a,b,c$ such that $a+b+c \leq 3$.
Determine the smallest integer $k$ for which there are three distinct positive integers $a$, $b$ and $c$, such that $$a^2 =bc \text{ and } k = 2b+3c-a.$$
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 $a$.
A set \(A\) of real numbers is framed when it is bounded and, for all \(a, b \in A\), not necessarily distinct, \((a-b)^{2} \in A\). What is the smallest real number that belongs to some framed set?
The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to rooms adjacent by side. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess rook (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 1$)?
The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to adjacent rooms. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess queen (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 2$)?
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.
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 $24 + \frac{36}{c}$ .
Today, Ivan the Confessor prefers continuous functions $f:[0,1]\to\mathbb{R}$ satisfying $f(x)+f(y)\geq |x-y|$ for all pairs $x,y\in [0,1]$. Find the minimum of $\int_0^1 f$ over all preferred functions.
(Proposed by Fedor Petrov, St. Petersburg State University)
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 $1 \leq n \leq 999$.
Let $m_n$ be the smallest value of the function ${{f}_{n}}\left( x \right)=\sum\limits_{k=0}^{2n}{{{x}^{k}}}$
Show that $m_n \to \frac{1}{2}$, as $n \to \infty.$
Two ships sail on the sea with constant speeds and fixed directions. It is known that at $9:00$ the distance between them was $20$ miles; at $9:35$, $15$ miles; and at $9:55$, $13$ miles. At what moment were the ships the smallest distance from each other, and what was that distance ?