Found problems: 155
For a positive integer $s$, denote with $v_2(s)$ the maximum power of $2$ that divides $s$.
Prove that for any positive integer $m$ that: $$v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.$$
(FYROM)
Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$.
Find the greatest value of $M = abc$
Let $a_1,a_2,...,a_9 \ge - 1$ and $a^3_1+a^3_2+...+a^3_9= 0$.
Determine the maximal value of $M = a_1 + a_2 + ... + a_9$.
Consider the set of all rectangles with a given area $S$.
Find the largest value o $ M = \frac{16-p}{p^2+2p}$ where $p$ is the perimeter of the rectangle.
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 $1 \leq n \leq 999$.
A unit square has a circle of radius $r$ with center at it's midpoint. The four quarter circles are centered on the vertices of the square and are tangent to the central circle (see figure). Find the maximum and minimum possible value of the area of the striped figure in the figure and the corresponding values of $r$ such these, the maximum and minimum are achieved.
[img]https://2.bp.blogspot.com/-DOT4_B5Mx-8/XnmsTlWYfyI/AAAAAAAALgs/TVYkrhqHYGAeG8eFuqFxGDCTnogVbQFUwCK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs1.4.png[/img]
In each cell of $5 \times 5$ table there is one number from $1$ to $5$ such that every number occurs exactly once in every row and in every column. Number in one column is [i]good positioned[/i] if following holds:
- In every row, every number which is left from [i]good positoned[/i] number is smaller than him, and every number which is right to him is greater than him, or vice versa.
- In every column, every number which is above from [i]good positoned[/i] number is smaller than him, and every number which is below to him is greater than him, or vice versa.
What is maximal number of good positioned numbers that can occur in this table?
Show that the maximum number of spheres of radius $1$ that can be placed touching a fixed sphere of radius $1$ so that no pair of spheres has an interior point in common is between $12$ and $14$.
Board with dimesions $2018 \times 2018$ is divided in unit cells $1 \times 1$. In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain black chips. If $W$ is number of remaining white chips, and $B$ number of remaining black chips on board and $A=min\{W,B\}$, determine maximum of $A$
Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$.
Find the minimum value and the maximum value of the product $abcd$.
The world and the european champion are determined in the same tournament carried in one round. There are $20$ teams and $k$ of them are european. The european champion is determined according to the results of the games only between those $k$ teams. What is the greatest $k$ such that the situation, when the single european champion is the single world outsider, is possible if:
a) it is hockey (draws allowed)?
b) it is volleyball (no draws)?
There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly $1200$. What is the maximum possible number of cups in the kitchen?
What maximal area can have a triangle if its sides $a,b,c$ satisfy inequality $0\le a\le 1\le b\le 2\le c\le 3$ ?
For a positive integer $n$, we de ne $D_n$ as the largest integer that is a divisor of $a^n + (a + 1)^n + (a + 2)^n$ for all positive integers $a$.
1. Show that for all positive integers $n$, the number $D_n$ is of the form $3^k$ with $k \ge 0$ an integer.
2. Show that for all integers $k \ge 0$ there exists a positive integer n such that $D_n = 3^k$.
16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.
The real numbers $x_1, x_2, ... , x_{1991}$ satisfy $$|x_1 - x_2| + |x_2 - x_3| + ... + |x_{1990} - x_{1991}| = 1991$$ What is the maximum possible value of $$|s_1 - s_2| + |s_2 - s_3| + ... + |s_{1990} - s_{1991}|$$ where $$s_n = \frac{x_1 + x_2 + ... + x_n}{n}?$$
A book contains $30$ stories. Each story has a different number of pages under $31$. The first story starts on page $1$ and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?
In a football tournament there are n teams, with ${n \ge 4}$, and each pair of teams meets exactly once. Suppose that, at the end of the tournament, the final scores form an arithmetic sequence where each team scores ${1}$ more point than the following team on the scoreboard. Determine the maximum possible score of the lowest scoring team, assuming usual scoring for football games (where the winner of a game gets ${3}$ points, the loser ${0}$ points, and if there is a tie both teams get ${1}$ point).
Define the maximal value of the ratio of a three-digit number to the sum of its digits.
Let $a, b$ be two real numbers that satisfy $a^3 + b^3 = 8-6ab$.
Find the maximum value and the minimum value that $a + b$ can take.
Given regular $1976$-gon. The midpoints of all the sides and diagonals are marked. What is the greatest number of the marked points lying on one circumference?
What is the maximal possible length of the segment, being cut out by the sides of the triangle on the tangent to the inscribed circle, being drawn parallel to the base, if the triangle's perimeter equals $2p$?
What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers?
A number is written in each square of a chessboard, so that each number not on the border is the mean of the $4$ neighboring numbers. Show that if the largest number is $N$, then there is a number equal to $N$ in the border squares.
Among all orthogonal projections of a regular tetrahedron to all possible planes, find the projection of the greatest area.