Found problems: 594
The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.
$1+2+3+4+5+6=6+7+8$.
What is the smallest number $k$ greater than $6$ for which:
$1 + 2 +...+ k = k + (k+1) +...+ n$, with $n$ an integer greater than $k$ ?
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.
Find the number of ways to select three distinct numbers from ${1, 2, . . . , 3n}$ with a sum divisible by $3$.
Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$
Integers $1, 2,...,100$ are written on a circle, not necessarily in that order. Can it be that the absolute value of the di erence between any two adjacent integers is at least $30$ and at most $50$?
Two equal squares, one with red sides, another with blue ones, give an octagon in intersection. Prove that the sum of red octagon sides lengths is equal to the sum of blue octagon sides lengths.
Let $S$ be the set of integers that can be written in the form $50m + 3n$ where $m$ and $n$ are non-negative integers. For example $3, 50, 53$ are all in $S$. Find the sum of all positive integers not in $S$.
Denote $n$ as a natural number, and $m$ as the result of writing the digits of $n$ in reverse order. Determine, if they exist, the numbers of three digits which satisfy $2m + S = n$, $S$ being the sum of the digits of $n$.
For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.
The numbers from $1$ to $64$ must be written on the small squares of a chessboard, with a different number in each small square. Consider the $112$ numbers you can make by adding the numbers in two small squares which have a common edge. Is it possible to write the numbers in the squares so that these $112$ sums are all different?
Calculate the sum of all unsimplified fractions whose numerator and denominator are positive divisors of $1000$.
There are two points $A$ and $B$ in the plane.
a) Determine the set $M$ of all points $C$ in the plane for which $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$
b) Decide whether there is a point $C\in M$ such that $\angle ACB$ is maximal and if so, determine this angle.
Find a set of five different relatively prime natural numbers such, that the sum of an arbitrary subset is a composite number.
Prove that the rectangle $m\times n$ table can be filled with exact squares so, that the sums in the rows and the sums in the columns will be exact squares also.
Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.
Find positive real numbers $x,y,z$ that are solutions of the system
$x+y+z=xy+yz+zx$ and $xyz=1$ , and have the smallest possible sum.
Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.
* Given several numbers each of which is less than $1951$ and the least common multiple of any two of which is greater than $1951$. Prove that the sum of their reciprocals is less than $2$.
The numbers $1, 2, 3, ... , n$ are written on a blackboard (where $n \ge 3$). A move is to replace two numbers by their sum and non-negative difference. A series of moves makes all the numbers equal $k$. Find all possible $k$
Let $x, y, z$ be nonzero real numbers such that $ x + y + z = 0$ and $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}= 1 -xyz + \frac{1}{xyz}.$$ Determine the value of the expression '
$$\frac{x}{(1-xy) (1-xz)}+\frac{y}{(1- yx) (1- yz)}+\frac{z}{(1- zx) (1-zy)}.$$
Using each of the digits $1,2,3,\ldots ,8,9$ exactly once,we form nine,not necassarily distinct,nine-digit numbers.Their sum ends in $n$ zeroes,where $n$ is a non-negative integer.Determine the maximum possible value of $n$.
Find all points on the plane such that the sum of the distances of each of the four lines defined by the unit square of that plane is $4$.
Assume $a_1,a_2,...,a_{14}$ are positive integers such that $\sum_{i=1}^{14}3^{a_i} = 6558$.
Prove that the numbers $a_1,a_2,...,a_{14}$ consist of the numbers $1,...,7$, each taken twice.
In each cell of a table with $m$ rows and $n$ columns, where $m<n$, we put a non-negative real number such that each column contains at least one positive number.
Show that there is a cell with a positive number such that the sum of the numbers in its row is larger than the sum of the numbers in its column.