Found problems: 99
Mummy-trolley huts are located on a straight line at points with coordinates $x_1, x_2,...., x_n$. In this village are going to build $3$ stores $A, B$ and $C$, of which will be brought every day to all Moomin-trolls chocolates, bread and water. For the delivery of chocolate, the store takes the distance from the store to the hut, raised to the square; for bread delivery , take the distance from the store to the hut; for water delivery take distance $1$, if the distance is greater than $1$ km, but do not take anything otherwise.
a) Where to build each of the stores so that the total cost of all Moomin-trolls for delivery wasthe smallest?
b) Where to place the TV tower, if the fee for each Moomin-troll is the maximum distance from the TV tower to the farthest hut from it?
c) How will the answer change if the Moomin-troll huts are not located in a straight line, and on the plane?
[hide=original wording]
На прямiй розташованi хатинки Мумi-тролей в точках з координатами x1, x2, . . . , xn. В цьому селi бираються побудувати 3 магазина A, B та C, з яких будуть кожен день привозити всiм Мумi-тролям шоколадки, хлiб та воду. За доставку шоколадки мага- зин бере вiдстань вiд магазину до хатинки, пiднесену до квадрату; за доставку хлiба – вiдстань вiд магазину до хатинки; за доставку води беруть 1, якщо вiдстань бiльша 1 км, та нiчого не беруть в супротивному випадку.
1. Де побудувати кожний з магазинiв, щоб загальнi витрати всiх Мумi-тролей на доставку були найменшими?
2. Де розташувати телевежу, якщо плата для кожного Мумi-троля – максимальна вiдстань вiд телевежi до самої вiддаленої вiд неї хатинки?
3. Як змiниться вiдповiдь, якщо хатинки Мумi-тролей розташованi не на прямiй, а на площинi?[/hide]
For a natural number $n$, let $w(n)$ denote the number of (positive) prime divisors of $n$.
Find the smallest positive integer $k$ such that $2^{w(n)} \le k \sqrt[4]{ n}$ for each $n \in N$.
In triangle $ABC$, the lengths of all sides are integers, $\angle B=2 \angle A$ and $\angle C> 90^o$. Find the smallest possible perimeter of this triangle.
Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$.
What is the smallest possible radius?
(a) Find the point $M$ in the plane of triangle $ABC$ for which the sum $MA + MB+ MC$ is minimal.
(b) Given a parallelogram $ABCD$ whose angles do not exceed $120^o$, determine $min \{MA+ MB+NC+ND+ MN | M,N$ are in the plane $ABCD\}$ in terms of the sides and angles of the parallelogram.
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]
Let $x, y, z$ be reals summing to $1$ which minimizes $2x^2 + 3y^2 + 4z^2$. Find $x$.
Find the largest possible value in the real numbers of the term $$\frac{3x^2 + 16xy + 15y^2}{x^2 + y^2}$$ with $x^2 + y^2 \ne 0$.
Find the maximum and the minimum values of $S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2)$ for real numbers $x_1, x_2, y_1,y_2$ with $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.
For which positive $x$ does the expression $x^{1000}+x^{900}+x^{90}+x^6+\frac{1996}{x}$ attain the smallest value?
Find the smallest possible value of the expression $$\frac{(a+b) (b + c)}{a + 2b+c}$$ where $a, b, c$ are arbitrary numbers from the interval $[1,2]$.
For an integer $n \ge 2$, find all real numbers $x$ for which the polynomial $f(x) = (x-1)^4 +(x-2)^4 +...+(x-n)^4$ takes its minimum value.
Consider the cube with the vertices at the points $(\pm 1, \pm 1, \pm 1)$. Let $V_1,...,V_{95}$ be arbitrary points within this cube. Denote $v_i = \overrightarrow{OV_i}$, where $O = (0,0,0)$ is the origin. Consider the $2^{95}$ vectors of the form $s_1v_1 + s_2v_2 +...+ s_{95}v_{95}$, where $s_i = \pm 1$.
(a) If $d = 48$, prove that among these vectors there is a vector $w = (a, b, c)$ such that $a^2 + b^2 + c^2 \le 48$.
(b) Find a smaller $d$ (the smaller, the better) with the same property.
Find the maximum and minimum values of $x^2 + 2y^2 + 3z^2$ for real $x, y, z$ satisfying $x^2 + y^2 + z^2 = 1$.
Find the side lengths of the triangle $ABC$ with area $S$ and $\angle BAC = x$ such that the side $BC$ is as short as possible.
If $x, y, z, w$ are nonnegative real numbers satisfying \[\left\{ \begin{array}{l}y = x - 2003 \\ z = 2y - 2003 \\ w = 3z - 2003 \\
\end{array} \right.
\] find the smallest possible value of $x$ and the values of $y, z, w$ corresponding to it.
(a) Let $x$ be a real number with $x \ge 1$. Prove that $x^3 - 5x^2 + 8x - 4 \ge 0$.
(b) Let $a, b \ge 1$ real numbers. Find the minimum value of the expression $ab(a + b - 10) + 8(a + b)$. Determine
also the real number pairs $(a, b)$ that make this expression equal to this minimum value.
The positive differences $a_i-a_j$ of five different positive integers $a_1, a_2, a_3, a_4, a_5$ are all different (there are altogether $10$ such differences). Find the least possible value of the largest number among the $a_i$.
For positive integers $n \ge 3$ and $r \ge 1$, define $$P(n, r) = (n - 2)\frac{r^2}{2} - (n - 4) \frac{r}{2}$$
We call a triple $(a, b, c)$ of natural numbers, with $a \le b \le c$, an $n$-gonal Pythagorean triple if $P(n, a)+P(n, b) = P(n, c)$. (For $n = 4$, we get the usual Pythagorean triple.)
(a) Find an $n$-gonal Pythagorean triple for each $n \ge 3$.
(b) Consider all triangles $ABC$ whose sides are $n$-gonal Pythagorean triples for some $n \ge 3$. Find the maximum and the minimum possible values of angle $C$.
In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$.
[img]https://1.bp.blogspot.com/-WD8WXIE6DK4/XzcC9GYsa6I/AAAAAAAAMXg/vl2OrbAdChEYrRpemYmj6DiOrdOSqj_IgCLcBGAsYHQ/s178/2001%2BMohr%2Bp3.png[/img]
A circle is inscribed in an angle with vertex $O$, touching its sides at points $M$ and $N$. On an arc $MN$ nearest to point $O$, an arbitrary point $P$ is selected. At point $P$, a tangent is drawn to the circle $P$, intersecting the sides of the angle at points $A$ and $B$. Prove that that the length of the segment $AB$ is the smallest when $P$ is its midpoint.
Define $g(x)$ as the largest value of$ |y^2 - xy|$ for $y$ in $[0, 1]$. Find the minimum value of $g$ (for real $x$).
Given a circle $\gamma$ and a point $P$ inside it, find the maximum and minimum value of the sum of the lengths of two perpendicular chords of $\gamma$ passing through $P$.
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
$a_1, a_2, ... , a_n$ are constants such that $f(x) = 1 + a_1 cos x + a_2 cos 2x + ...+ a_n cos nx \ge 0$ for all $x$. We seek estimates of $a_1$. If $n = 2$, find the smallest and largest possible values of $a_1$. Find corresponding estimates for other values of $n$.