Found problems: 85335
Given is a $n \times n$ chessboard. With the same probability, we put six pawns on its six cells. Let $p_n$ denotes the probability that there exists a row or a column containing at least two pawns. Find $\lim_{n \to \infty} np_n$.
Points $A_1, A_2, ..., A_{10}$ are marked on a circle clockwise. It is known that these points can be divided into pairs of points symmetric with respect to the centre of the circle. Initially at each marked point there was a grasshopper. Every minute one of the grasshoppers jumps over its neighbour along the circle so that the resulting distance between them doesn't change. It is not allowed to jump over any other grasshopper and to land at a point already occupied. It occurred that at some moment nine grasshoppers were found at points $A_1, A_2, ... , A_9$ and the tenth grasshopper was on arc $A_9A_{10}A_1$. Is it necessarily true that this grasshopper was exactly at point $A_{10}$?
Find the sum of the first 5 positive integers $n$ such that $n^2 - 1$ is the product of 3 distinct primes.
One side of a given triangle is $ 18$ inches. Inside the triangle a line segment is drawn parallel to this side forming a trapezoid whose area is one-third of that of the triangle. The length of this segment, in inches, is:
$ \textbf{(A)}\ 6\sqrt {6} \qquad \textbf{(B)}\ 9\sqrt {2} \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 6\sqrt {3} \qquad \textbf{(E)}\ 9$
It's known that there is always a prime between $n$ and $2n-7$ for all $n \ge 10$. Prove that, with the exception of $1$, $4$, and $6$, every natural number can be written as the sum of distinct primes.
Given a triangle $ABC$. Suppose I is its incentre, and $X, Y, Z$ are the incentres of triangles $AIB, BIC$ and $AIC$ respectively. The incentre of triangle $XYZ$ coincides with $I$. Is it necessarily true that triangle $ABC$ is regular?
For positive integer $n$, prove that at least one of the numbers $$A=2n-1 , B=5n-1, C=13n-1$$ is not perfect square
$n$ distinct integers are given, all of which are bigger than $-a$, where $a$ is a positive integer. It turned out that the amount of odd numbers among them is equal to the biggest even number, and the amount of even numbers is equal to the biggest odd numbers
a) Find the least possible value of $n$ for all $a$
b) For each $a \geq 2$ find the maximum possible value of $n$
Using $3$ colors, red, blue and yellow, how many different ways can you color a cube (modulo rigid rotations)?
The expression $ 1 \minus{} \frac {1}{1 \plus{} \sqrt {3}} \plus{} \frac {1}{1 \minus{} \sqrt {3}}$ equals:
$ \textbf{(A)}\ 1 \minus{} \sqrt {3} \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \minus{} \sqrt {3} \qquad\textbf{(D)}\ \sqrt {3} \qquad\textbf{(E)}\ 1 \plus{} \sqrt {3}$
Call a positive integer [i]almost square[/i] if it is not a perfect square, but all of its digits are perfect squares. For example, both $149$ and $904$ are almost square, but $144$ and $936$ are not. Find the number of positive integers less than $1000$ that are not almost square.
Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$
Find all pairs of integers $(p,q)$ for which all roots of the trinomials $x^2+px+q$ and $x^2+qx+p$ are integers.
Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$?
Suppose that $a,b,c$ are positive numbers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ is an integer. Show that $abc$ is a perfect cube.
Given $ \triangle ABC$ with point $ D$ inside. Let $ A_0\equal{}AD\cap BC$, $ B_0\equal{}BD\cap AC$, $ C_0 \equal{}CD\cap AB$ and $ A_1$, $ B_1$, $ C_1$, $ A_2$, $ B_2$, $ C_2$ are midpoints of $ BC$, $ AC$, $ AB$, $ AD$, $ BD$, $ CD$ respectively. Two lines parallel to $ A_1A_2$ and $ C_1C_2$ and passes through point $ B_0$ intersects $ B_1B_2$ in points $ A_3$ and $ C_3$respectively. Prove that $ \frac{A_3B_1}{A_3B_2}\equal{}\frac{C_3B_1}{C_3B_2}$.
Prove that there exist infinitely many non prime positive integers $n$ such that $7^{n-1} - 3^{n-1}$ is divisible by $n$.
Lê Anh Vinh
Polynomial $P$ with non-negative real coefficients and function $f:\mathbb{R}^+\to \mathbb{R}^+$ are given such that for all $x, y\in \mathbb{R}^+$ we have
$$f(x+P(x)f(y)) = (y+1)f(x)$$
(a) Prove that $P$ has degree at most 1.
(b) Find all function $f$ and non-constant polynomials $P$ satisfying the equality.
On account of the annual fluctuation of temperature the ground at the town of Ν freezes to a depth of 2 metres. To what depth would it freeze on account of a daily fluctuation of the same amplitude?
Some cities of a country consisting of $n$ cities are connected by round trip flights so that there are at least $k$ flights from any city and any city is reachable from any city. Prove that for any such flight organization these flights can be distributed among $n-k$ air companies so that one can reach any city from any city by using of at most one flight of each air company.
Let $I$ be the incenter of a triangle $ABC$; $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ be the orthocenters of the triangles $BIC$, $AIC$, $AIB$; $M_{a}$, $M_{b}$, $M_{c}$ be the midpoints of $BC$, $CA$, $AB$, and $S_{a}$, $S_{b}$, $S_{c}$ be the midpoints of $AA^{\prime}$, $BB^{\prime}$, $CC^{\prime}$. Prove that $M_{a}S_{a}$, $M_{b}S_{b}$, $M_{c}S_{c}$ concur.
Proposed by: S Kuznetsov
Let $p$ be an odd prime number and $\mathbb{Z}_{>0}$ be the set of positive integers. Suppose that a function $f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}$ satisfies the following properties:
[list]
[*] $f(1,1)=0$.
[*] $f(a,b)+f(b,a)=1$ for any pair of relatively prime positive integers $(a,b)$ not both equal to 1;
[*] $f(a+b,b)=f(a,b)$ for any pair of relatively prime positive integers $(a,b)$.
[/list]
Prove that
$$\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.$$
Circles ω[size=35]1[/size] and ω[size=35]2[/size] have no common point. Where is outerior tangents a and b, interior tangent c. Lines a, b and c touches circle
ω[size=35]1[/size] respectively on points A[size=35]1[/size], B[size=35]1[/size] and C[size=35]1[/size], and circle ω[size=35]2[/size] – respectively
on points A[size=35]2[/size], B[size=35]2[/size] and C[size=35]2[/size]. Prove that triangles A[size=35]1[/size]B[size=35]1[/size]C[size=35]1[/size] and A[size=35]2[/size]B[size=35]2[/size]C[size=35]2[/size]
area ratio is the same as ratio of ω[size=35]1[/size] and ω[size=35]2[/size] radii.
Show that given any $4$ points in the plane we can find two whose distance apart is not an odd integer.
Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$.
Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set.
[i]Dan Schwarz, Romania[/i]