Found problems: 85335
Find the coefficient of $x^2$ in the following polynomial
$$(1 -x)^2(1 + 2x)^2(1 - 3x)^2... (1 -11x)^2.$$
The number $2^{1997}$ has $m$ decimal digits, while the number $5^{1997}$ has $n$ digits. Evaluate $m+n$.
Several small villages are situated on the banks of a straight river. On one side, there are 20 villages in a row, and on the other there are 15 villages in a row. We would like to build bridges, each of which connects a village on the one side with a village on the other side. The bridges must not cross, and it should be possible to get from any village to any other village using only those bridges (and not any roads that might exist between villages on the same side of the river). How many different ways are there to build the bridges.
$6$ musicians gathered at a chamber music festival. At each scheduled concert, some of the musicians played while the others listened as members of the audience. What is the least number of such concerts which would need to be scheduled so that every $2$ musicians each must play for the other in some concert?
Let $\Delta ABC$ be an acute triangle and $\omega$ be its circumcircle. Perpendicular from $A$ to $BC$ intersects $BC$ at $D$ and $\omega$ at $K$. Circle through $A$, $D$ and tangent to $BC$ at $D$ intersect $\omega$ at $E$. $AE$ intersects $BC$ at $T$. $TK$ intersects $\omega$ at $S$. Assume, $SD$ intersects $\omega$ at $X$. Prove that $X$ is the reflection of $A$ with respect to the perpendicular bisector of $BC$.
For every $3$-digit natural number $n$ (leading digit of $n$ is nonzero), we consider the number $n_0$ obtained from $n$ eliminating all possible digits that are zero. For example, if $n = 207$, then $n_0 = 27$. Determine the number of three-digit positive integers $n$, for which $n_0$ is a divisor of $n$ different from $n$.
A parallelepiped with the side lengths $ a$, $ b$, $ c$ is cut by a plane through its intersection of diagonals which is perpendicular to one of these diagonals. Calculate the area of the intersection of the plane and the parallelepiped.
Prove that, for any integer $x$, $x^2 +5x+16$ is not divisible by $169$.
The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$.
[b]a.)[/b] Prove that $0\le b_n<2$.
[b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.
Evaluate $\int_1^3 \frac{\ln (x+1)}{x^2}dx$.
[i]2010 Hirosaki University entrance exam[/i]
Let $p>2$ be a prime and let
\[\mathcal{A}=\{n \in \mathbb{N}: 2p \mid n \text{ and } p^2\nmid n \text{ and } n \mid 3^n-1\}.\]
Prove that
\[\limsup_{k \to \infty} \frac{\vert \mathcal{A} \cap [1,k]\vert}{k} \le \frac{2\log 3}{p\log p}.\]
At a point on the real line sits a greyhound. On one of the sides a hare runs, away from the hound. The only thing known is that the (maximal) speed of the hare is strictly less than the (maximal) speed of the greyhound (but not their precise ratio). Does the greyhound have a strategy for catching the hare in a finite amount of time?
([i]Dan Schwarz[/i])
How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?
$ \textbf{(A)}\ 96 \qquad \textbf{(B)}\ 104 \qquad \textbf{(C)}\ 112 \qquad \textbf{(D)}\ 120 \qquad \textbf{(E)}\ 256$
Does there exist a finite set $M$ of nonzero real numbers, such that for any natural number $n$ a polynomial of degree no less than $n$ with coeficients in $M$, all of whose roots are real and belong to $M$?
[i]E. Malinnikova[/i]
Circles $C$ and $C'$ intersect at $O$ and $X$. A circle center $O$ meets $C$ at $Q$ and $R$ and meets $C'$ at $P$ and $S$. $PR$ and $QS$ meet at $Y$ distinct from $X$. Show that $\angle YXO = 90^o$.
A candle and a man are placed in a dihedral mirror angle. How many reflections can the man see ?
$a, b, c$ are real positive numbers for which $a+b+c=3$. Prove that $a^{12}+b^{12}+c^{12}+8(ab+bc+ca) \geq 27$
Regular hexagon $P_1P_2P_3P_4P_5P_6$ has side length $2$. For $1 \le i \le 6$, let $C_i$ be a unit circle centered at $P_i$ and $\ell_i$ be one of the internal common tangents of $C_i$ and $C_{i+2}$, where $C_7 = C_1$ and $C_8 = C_2$. Assume that the lines $\{\ell_1, \ell_2, \ell_3, \ell_4, \ell_5,\ell_6\}$ bound a regular hexagon. The area of this hexagon can be expressed as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.
Find all pairs of distinct primes $(p,q)$ such that $p$ and $q$ are both prime factors of $p^3+q^2+1$, and are the only such prime factors.
[i]Proposed by Takeda Shigenori[/i]
Consider a finite field $ K. $
[b]a)[/b] Prove that there is an element $ k $ in $ K $ having the property that the polynom $ X^3+k $ is irreducible in $ K[X], $ if $ \text{ord} (K)\equiv 1\pmod {12}. $
[b]b)[/b] Is [b]a)[/b] still true if, intead, $ \text{ord} (K) \equiv -1\pmod{12} ? $
[i]Dorel Miheț[/i]
Define sequence $\{a_n\}$ as following: $a_0=0$, $a_1=1$, and $a_{i}=2a_{i-1}-a_{i-2}+2$ for all $i\geq 2$. Determine the value of $a_{1000}.$
[i] Proposed by Yannick Yao [/i]
Define the sequences $a_n$ and $b_n$ as follows: $a_1 = 2017$ and $b_1 = 1$.
For $n > 1$, if there is a greatest integer $k > 1$ such that $a_n$ is a perfect $k$th power, then $a_{n+1} =\sqrt[k]{a_n}$, otherwise $a_{n+1} = a_n + b_n$. If $a_{n+1} \ge a_n$ then $b_{n+1} = b_n$, otherwise $b_{n+1} = b_n + 1$. Find $a_{2017}$.
Let $\{x_{n}\}_{n\ge0}$ and $\{y_{n}\}_{n\ge0}$ be two sequences defined recursively as follows \[x_{0}=1, \; x_{1}=4, \; x_{n+2}=3 x_{n+1}-x_{n},\] \[y_{0}=1, \; y_{1}=2, \; y_{n+2}=3 y_{n+1}-y_{n}.\] [list=a][*] Prove that ${x_{n}}^{2}-5{y_{n}}^{2}+4=0$ for all non-negative integers. [*] Suppose that $a$, $b$ are two positive integers such that $a^{2}-5b^{2}+4=0$. Prove that there exists a non-negative integer $k$ such that $a=x_{k}$ and $b=y_{k}$.[/list]
Let $ABC$ be a triangle and let $P$ be a point on $BC$. Points $M$ and $N$ lie on $AB$ and $AC$, respectively such that $MN$ is not parallel to $BC$ and $AMP N$ is a parallelogram. Line $MN$ meets the circumcircle of $ABC$ at $R$ and $S$. Prove that the circumcircle of triangle $RP S$ is tangent to $BC$.
The sequence $\{x_n\}_{n=0}^\infty$ is defined by the following equations:
$x_n=\sqrt{x_{n-1} x_{n-2}+\frac{n}{2}}$ ,$\forall$ $n\geq 2$, $x_0=x_1=1$.
Prove that there exist a real number $a$, such that $an<x_n<an+1$ for each natural number $n$.