Found problems: 99
Let $a$ and $b$ be positive real numbers with $a > b$. Find the smallest possible values of $$S = 2a +3 +\frac{32}{(a - b)(2b +3)^2}$$
The absolute value of the difference of the solutions of the equation $x^2 + px + q = 0$, with $p, q \in R$, is equal to $4$. Find the solutions of the equation if it is known that $(q + 1) p^2 + q^2$ takes the minimum value.
Diameter $AB$ is drawn to a circle with radius $1$. Two straight lines $s$ and $t$ touch the circle at points $A$ and $B$, respectively. Points $P$ and $Q$ are chosen on the lines $s$ and $t$, respectively, so that the line $PQ$ touches the circle. Find the smallest possible area of the quadrangle $APQB$.
Given a real number $c$ and an integer $m, m \ge 2$. Real numbers $x_1, x_2,... , x_m$ satisfy the conditions $x_1 + x_2 +...+ x_m = 0$ and $\frac{x^2_1 + x^2_2 + ...+ x^2_m}{m}= c$. Find max $(x_1, x_2,..., x_m)$ if it is known to be as small as possible.
Determine the maximal value of $k$ such that the inequality
$$\left(k +\frac{a}{b}\right) \left(k + \frac{b}{c}\right)\left(k + \frac{c}{a}\right)
\le \left( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\right) \left( \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}\right)$$
holds for all positive reals $a, b, c$.
Find the smallest possible positive integer n with the following property:
For all positive integers $x, y$ and $z$ with $x | y^3$ and $y | z^3$ and $z | x^3$ always to be true that $xyz| (x + y + z) ^n$.
(Gerhard J. Woeginger)
Find the minimum and the maximum value of the expression $\sqrt{4 -a^2} +\sqrt{4 -b^2} +\sqrt{4 -c^2}$
where $a,b, c$ are positive real numbers satisfying the condition $a^2 + b^2 + c^2=6$
On the midline of the isosceles trapezoid $ABCD$ ($BC \parallel AD$) find the point $K$, for which the sum of the angles $\angle DAK + \angle BCK$ will be the smallest.
Given positive numbers $a,b,c$, find the minimum of the function $f(x) = \sqrt{a^2 +x^2} +\sqrt{(b-x)^2 +c^2}$.
Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya .
[img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]
Two unit squares with parallel sides overlap by a rectangle of area $1/8$. Find the extreme values of the distance between the centers of the squares.
Find the smallest value of the expression $(x-1) (x -2) (x -3) (x - 4) + 10$.
We are allowed to put several brackets in the expression
$$\frac{29 : 28 : 27 : 26 :... : 17 : 16}{15 : 14 : 13 : 12 : ... : 3 : 2}$$
always in the same places below each other.
(a) Find the smallest possible integer value we can obtain in that way.
(b) Find all possible integer values that can be obtained.
Remark: in this problem, $$\frac{(29 : 28) : 27 : ... : 16}{(15 : 14) : 13 : ... : 2},$$ is valid position of parenthesis, on the other hand $$\frac{(29 : 28) : 27 : ... : 16}{15 : (14 : 13) : ... : 2}$$ is forbidden.
Given that $n> 2$ is a positive integer and a sequence of positive integers $a_1 <a_2 <...<a_n$. In
the subsets of the set $\{1,2,..., n\} $, there a subset $X$ such that $| \sum_{i \notin X} a_i -\sum_{i \in X} a_i |$ is the smallest .
Prove that there exists a sequence of positive integers $0<b_1 <b_2 <...<b_n$ such that $\sum_{i \notin X} b_i= \sum_{i \in X} b_i$.
In case this doesn't make sense, have a look at [url=https://drive.google.com/file/d/1xoBhJlG0xHwn6zAAA7AZDoaAqzZue-73/view]original wording in Vietnamese[/url].
Points $H_1, H_2, ... , H_n$ are arranged in the plane so that each distance $H_iH_j \le 1$. The point $P$ is chosen to minimise $\max (PH_i)$. Find the largest possible value of $\max (PH_i)$ for $n = 3$. Find the best upper bound you can for $n = 4$.
Let $n$ be a positive integer. Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $n$ distinct points.
Find the maximum and minimum of $x + y$ such that $x + y = \sqrt{2x-1}+\sqrt{4y+3}$
Given an integer $n > 2$, consider all sequences $x_1,x_2,...,x_n$ of nonnegative real numbers such that $$x_1+ 2x_2 + ... + nx_n = 1.$$ Find the maximum value and the minimum value of $x_1^2+x_2^2+...+x_n^2$ and determine all the sequences $x_1,x_2,...,x_n$ for which these values are obtained.
Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya .
[img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]
The lengths of the sides of triangle $ABC$ are $4,5,6$. For any point $D$ on one of the sides, draw the perpendiculars $DP, DQ$ on the other two sides. What is the minimum value of $PQ$?
Let $x,y,z$ be real numbers with $|x|,|y|,|z| > 2$. What is the smallest possible value of $|xyz+2(x+y+z)|$ ?
Let a triangle $ABC$ be given. For any point $X$ of the triangle denote $m(X)=\min\{XA,XB,XC\}.$ Find all points $X$ (of triangle $ABC$) such that $m(X)$ is maximal.
In a village, the maximum distance between two houses is $M$ and the minimum distance is $m$. Prove that if the village has $6$ houses, then $\frac{M}{m} \ge \sqrt3$.
You have an equilateral paper triangle of area $9$ and fold it in two, following a straight line that passes through the center of the triangle and does not contain any vertex of the triangle. Thus there remains a quadrilateral in which the two pieces overlap, and three triangles without overlaps. Determine the smallest possible value of the quadrilateral area of the overlay.
Determine the largest and smallest fractions $F = \frac{y-x}{x+4y}$
if the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$.