Found problems: 85335
You are given a checkered square, the side of which is $n – 1$ long and contains $n \geq 10$ nodes. A non-return path is a path along edges, the intersection of which with any horizontal or vertical line is a segment, point or empty set, and which does not pass along any edge more than once. What is the smallest number of non-return paths that can cover all the edges?
(An edge is a unit segment between adjacent nodes.)
Sabine rolls a fair $14$-sided die numbered $1$ to $14$ and gets a value of $x$. She then draws $x$ cards uniformly at random (without replacement) from a deck of $14$ cards, each of which labeled a different integer from $1$ to $14$. She finally sums up the value of her die roll and the value on each card she drew to get a score of $S$. Let $A$ be the set of all obtainable scores. Compute the probability that $S$ is greater than or equal to the median of $A$.
Let $ABC$ be a triangle and $I$ is your incenter, let $P$ be a point in $AC$ such that $PI$ is perpendicular to $AC$, and let $D$ be the reflection of $B$ wrt circumcenter of $\triangle ABC$. The line $DI$ intersects again the circumcircle of $\triangle ABC$ in the point $Q$. Prove that $QP$ is the angle bisector of the angle $\angle AQC$.
Let $ABCD$ be the circumscribed quadrilateral with the incircle $(I)$. The circle $(I)$ touches $AB, BC, C D, DA$ at $M, N, P,Q$ respectively. Let $K$ and $L$ be the circumcenters of the triangles $AMN$ and $APQ$ respectively. The line $KL$ cuts the line $BD$ at $R$. The line $AI$ cuts the line $MQ$ at $J$. Prove that $RA = RJ$.
All vertices of a regular 100-gon are colored in 10 colors. Prove that there exist 4 vertices of the given 100-gon which are the vertices of a rectangle and which are colored in at most 2 colors.
There are $100$ cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from $1$ to $100$. In a move, Madina is allowed to choose any $3$ cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?
[i]Shubin Yakov, Russia[/i]
In each square of a $10\times 10$ board, there is an arrow pointing upwards, downwards, left, or right. Prove that it is possible to remove $50$ arrows from the board, such that no two remaining arrows point at each other.
The beaver is chess piece that move to $2$ cells by horizontal or vertical. Every cell of $100 \times 100$ chessboard colored in some color,such that we can not get from one cell to another with same color with one move of beaver or knight. What minimal color do we need?
Standing in a circle are $99$ girls, each with a candy. In each move, each girl gives her candy to either neighbour. If a girl receives two candies in the same move, she eats one of them. What is the minimum number of moves after which only one candy remains?
The triangle $ABC$ is given. Inside its sides $AB$ and $AC$, the points $X$ and $Y$ are respectively selected Let $Z$ be the intersection of the lines $BY$ and $CX$. Prove the inequality $$[BZX] + [CZY]> 2 [XY Z]$$, where $[DEF]$ denotes the content of the triangle $DEF$.
(David Hruska, Josef Tkadlec)
During the class interval, $n$ children sit in a circle and play the game described below. The teacher goes around the children clockwisely and hands out candies to them according to the following regulations: Select a child, give him a candy; and give the child next to the first child a candy too; then skip over one child and give next child a candy; then skip over two children; give the next child a candy; then skip over three children; give the next child a candy;...
Find the value of $n$ for which the teacher can ensure that every child get at least one candy eventually (maybe after many circles).
Let $ABC$ be a triangle with circumcircle $\omega$, and let $X$ be the reflection of $A$ in $B$. Line $CX$ meets $\omega$ again at $D$. Lines $BD$ and $AC$ meet at $E$, and lines $AD$ and $BC$ meet at $F$. Let $M$ and $N$ denote the midpoints of $AB$ and $ AC$.
Can line $EF$ share a point with the circumcircle of triangle $AMN?$
[i]Proposed by Sayandeep Shee[/i]
Let $(x+1)^p(x-3)^q=x^n+a_1x^{n-1}+\ldots+a_{n-1}x+a_n$, where $p,q$ are positive integers.
(a) Prove that if $a_1=a_2$, then $3n$ is a perfect square.
(b) Prove that there exists infinitely many pairs $(p,q)$ for which $a_1=a_2$.
Given an isosceles trapezoid and draw one of its diagonals (so we get $2$ triangles). On each triangle we get, there is an ant walking along the edge. There speed are equal and unchanged. These $2$ ants also didn't change their walking directions and their directions are opposite.
Prove that: for all initial locations of the ants, they will meet each other at some time.
Find all integers $m$ and $n$ such that $2m^2 +n^2 = 2mn+3n$
The captain distributed $4000$ gold coins among $40$ pirates. A group of $5$ pirates is called poor if those $5$ pirates received, together, $500$ coins or less. The captain made the distribution so that there were the minimum possible number of poor groups of $5$ pirates. Determine how many poor $5$ pirate groups there are.
Clarification: Two groups of $5$ pirates are considered different if there is at least one pirate in one of them who is not in the other.
A $2N\times2N$ board is covered by non-overlapping dominos of $1\times2$ size. A lame rook (which can only move one cell at a time, horizontally or vertically) has visited each cell once on its route across the board. Call a move by the rook longitudinal if it is a move from one cell of a domino to another cell of the same domino. What is:
[list=a]
[*]the maximum possible number of longitudinal moves?
[*]the minimum possible number of longitudinal moves?
[/list]
A square $ABCD$ is given. Two circles are inscribed into angles $A$ and $B$, and the sum of their diameters is equal to the sidelength of the square. Prove that one of their common tangents passes through the midpoint of $AB$.
Given a triangle with sides $a, b, c$ that satisfy $\frac{a}{b+c}=\frac{c}{a+b}$. Determine the angles of this triangle, if you know that one of them is equal to $120^0$.
Call a polynomial positive reducible if it can be written as a product of two nonconstant polynomials with positive real coefficients. Let $ f(x)$ be a polynomial with $ f(0)\not\equal{}0$ such that $ f(x^n)$ is positive reducible for some natural number $ n$. Prove that $ f(x)$ itself is positive reducible. [L. Redei]
Some cells of a $10\times10$ board are marked so that each cell has an even number of neighboring (i.e. sharing a side) marked cells. Find the maximum possible number of marked cells.
Consider the line (E): $5x-10y+3=0$ . Prove that:
a) Line $(E)$ doesn't pass through points with integer coordinates.
b) There is no point $A(a_1,a_2)$ with $ a_1,a_2 \in \mathbb{Z}$ with distance from $(E)$ less then $\frac{\sqrt3}{20}$.
Let $5$ and $13$ be lengths of two sides of a right triangle. Compute the sum of all possible lengths of the third side.
For each positive integer $n$ , define $a_n = 20 + n^2$ and $d_n = gcd(a_n, a_{n+1})$. Find the set of all values that are taken by $d_n$ and show by examples that each of these values is attained.
Let a, b, and c be nonzero integers. Show that there exists an integer k such that
$$gcd\left(a+kb, c\right) = gcd\left(a, b, c\right)$$