Found problems: 127
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 $n = 4$.
Find inside the triangle $ABC$, points $G$ and $H$ for which, respectively, the geometric mean and the harmonic mean of the distances to the sides of the triangle acquire maximum values. In which cases is the segment $GH$ parallel to one of the sides of the triangle? Find the length of such a segment $GH$.
Let an even positive integer $2k$ be given. Find such relatively prime positive integers $x, y$ that maximize the product $xy$.
Find the maximum and minimum of $x + y$ such that $x + y = \sqrt{2x-1}+\sqrt{4y+3}$
For a real number $a,b,c>0$ where $bc-ca-ab=1$ find the maximum value of $$P=\frac{4024}{1+a^2}-\frac{4024}{1+b^2}-\frac{2555}{1+c^2}$$ and find out when that holds .
[i](PP-nine)[/i]
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.
Let $180^o < \theta_1 < \theta_2 <...< \theta_n = 360^o$. For $i = 1,2,..., n$, $P_i = (\cos \theta_i^o, \sin \theta_i^o)$ is a point on the circle $C$ with centre $(0,0)$ and radius $1$. Let $P$ be any point on the upper half of $C$. Find the coordinates of $P$ such that the sum of areas $[PP_1P_2] + [PP_2P_3] + ...+ [PP_{n-1}P_n]$ attains its maximum.
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.
Let $P$ be a point in the interior of a triangle $ABC$ and let $d_a,d_b,d_c$ be its distances to $BC,CA,AB$ respectively. Prove that max $(AP, BP, CP) \ge \sqrt{d_a^2+d_b^2+d_c^2}$
Let $a, b, c, d > 0$ satisfy $36a + 4b + 4c + 3d = 25$. What is the maximum possible value of $ab^{1/2}c^{1/3}d^{1/4}$ ?
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 $\frac{M}{m} \ge \sqrt3$.
Let $\vartriangle ABC$ be a triangle with $\angle ABC > \angle BCA \ge 30^o$ . The angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $CA$ and $AB$ at $D$ and $E$ respectively, and $BD$ and $CE$ intersect at $P$. Suppose that $P D = P E$ and the incircle of $\vartriangle ABC$ has unit radius. What is the maximum possible length of $BC$?
Some distinct positive integers were written on a blackboard such that the sum of any two integers is a power of $2$. What is the maximal possible number written on the blackboard?
Determine the largest and smallest fractions $F = \frac{y-x}{x+4y}$
if the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$.
$C$, $C'$ are concentric circles with radii $R$, $R'$. A rectangle has two adjacent vertices on $C$ and the other two vertices on $C'$. Find its sides if its area is as large as possible.
An $n \times m$ table ( $n \le m$ ) is filled in accordance with the rules of the game "Minesweeper": mines are placed at some cells (not more than one mine at the cell) and in the remaining cells one writes the number of the mines in the neighboring (by side or by vertex) cells. Then the sum of allnumbers in the table is computed (this sum is equal to $9$ for the picture). What is the largest possible value of this sum?
(V. Lebed)
[img]https://cdn.artofproblemsolving.com/attachments/2/9/726ccdbc57807788a5f6e88a5acb42b10a6cc0.png[/img]
Let $S$ be the set of all tetrahedra which satisfy:
(1) the base has area $1$,
(2) the total face area is $4$, and
(3) the angles between the base and the other three faces are all equal.
Find the element of $S$ which has the largest volume.
Let $n \ge 4$ be an integer. Find the maximum value of the area of a $n$-gon which is inscribed in the circle of radius $1$ and has two perpendicular diagonals.
Let $N$ be the set of positive integers.
Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers
Let $a, b, c, d$ be distinct non-zero real numbers satisfying the following two conditions:
$ac = bd$ and $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 4$.
Determine the largest possible value of the expression $\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}$.
A square is dissected into $n$ congruent non-convex polygons whose sides are parallel to the sides of the square, and no two of these polygons are parallel translates of each other. What is the maximum value of $n$? (4)
The squares of a chessboard have side $4$. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?
Let $a_1, ... ,a_5$ be real numbers such that at least $N$ of the sums $a_i+a_j$ ($i < j$) are integers. Find the greatest value of $N$ for which it is possible that not all of the sums $a_i+a_j$ are integers.
There are a lot of different real numbers written on the board. It turned out that for each two numbers written, their product was also written. What is the largest possible number of numbers written on the board?
Given a circle of radius $R$. Find the ratio of the largest area of the circumscribed quadrilateral to the smallest area of the inscribed one.