Found problems: 85335
Let $p(x)$ be a nonconstant polynomial with real coefficients. For every positive integer~$n$, let $$q_n(x) = (x+1)^np(x)+x^n p(x+1) .$$
Prove that there are only finitely many numbers $n$ such that all roots of $q_n(x)$ are real.
Positive integers $ a$, $ b$, and $ 2009$, with $ a<b<2009$, form a geometric sequence with an integer ratio. What is $ a$?
$ \textbf{(A)}\ 7 \qquad
\textbf{(B)}\ 41 \qquad
\textbf{(C)}\ 49 \qquad
\textbf{(D)}\ 289 \qquad
\textbf{(E)}\ 2009$
Let $x_1$, ... , $x_{n+1} \in [0,1] $ and $x_1=x_{n+1} $. Prove that \[ \prod_{i=1}^{n} (1-x_ix_{i+1}+x_i^2)\ge 1. \]
A. Khrabrov, F. Petrov
A square $ABCD$ and point $E$ are drawn in a plane such that lengths $DE < BE$ and $\vartriangle ACE$ is equilateral. Compute $\angle BAE$.
Let $AD$ be an altitude of triangle $ABC$, and let $M$, $N$ and $P$ be midpoints of $AB$, $AD$ and $BC$, respectively. Furthermore let $K$ be a foot of perpendicular from point $D$ to line $AC$, and let $T$ be point on extension of line $KD$ (over point $D$) such that $\mid DT \mid = \mid MN \mid + \mid DK \mid$. If $\mid MP \mid = 2 \cdot \mid KN \mid$, prove that $\mid AT \mid = \mid MC \mid$.
Determine all positive real solutions of the following system of equations:
$$a =\ max \{ \frac{1}{b} , \frac{1}{c}\} \,\,\,\,\,\, b = \max \{ \frac{1}{c} , \frac{1}{d}\} \,\,\,\,\,\, c = \max \{ \frac{1}{d}, \frac{1}{e}\} $$
$$d = \max \{ \frac{1}{e} , \frac{1}{f }\} \,\,\,\,\,\, e = \max \{ \frac{1}{f} , \frac{1}{a}\} \,\,\,\,\,\, f = \max \{ \frac{1}{a} , \frac{1}{b}\}$$
Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$.
[i]Proposed by fattypiggy123[/i]
In the quadrilateral $ABCD$, we have $AB = AD$ and $\angle B = \angle D = 90 ^ \circ $. The points $P$ and $Q $ lie on $BC$ and $CD$, respectively, so that $AQ$ is perpendicular on $DP$. Prove that $AP$ is perpendicular to $BQ$.
The inscribed circle of triangle $ABC$, with centre $I$, touches sides $BC$, $CA$ and $AB$ at $D$, $E$ and
$F$, respectively. Let $P$ be a point, on the same side of $FE$ as $A$, for which $\angle PFE = \angle BCA$
and $\angle PEF = \angle ABC$. Prove that $P$, $I$ and $D$ lie on a straight line.
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function as
\[ f(x) = \begin{cases} \frac{1}{x-1}& (x > 1)\\ 1& (x=1)\\ \frac{x}{1-x} & (x<1) \end{cases} \]
Let $x_1$ be a positive irrational number which is a zero of a quadratic polynomial with integer coefficients. For every positive integer $n$, let $x_{n+1} = f(x_n)$. Prove that there exists different positive integers $k$ and $\ell$ such that $x_k = x_\ell$.
$(HUN 2)$ Prove that $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{5}{4}$
Determine all real $ x$ satisfying the equation \[ \sqrt[5]{x^3 \plus{} 2x} \equal{} \sqrt[3]{x^5\minus{}2x}.\] Odd roots for negative radicands shall be included in the discussion.
Let $m/n$, in lowest terms, be the probability that a randomly chosen positive divisor of $10^{99}$ is an integer multiple of $10^{88}$. Find $m + n$.
Let $D$ be the footpoint of the altitude from $B$ in the triangle $ABC$ , where $AB=1$ . The incircle of triangle $BCD$ coincides with the centroid of triangle $ABC$. Find the lengths of $AC$ and $BC$.
Find the smallest positive integer $N$ for which $N$ is divisible by $19$, and when the digits of $N$ are read in reverse order, the result (after removing any leading zeroes) is divisible by $36$.
[i]Proposed by Michael Tang[/i]
For each five distinct variables from the set $x_1,..., x_{10}$ there is a single card on which their product is written. Peter and Basil play the following game. At each move, each player chooses a card, starting with Peter. When all cards have been taken, Basil assigns values to the variables as he wants, so that $0 \le x_1 \le ... \le x_10$. Can Basil ensure that the sum of the products on his cards is greater than the sum of the products on Peter's cards?
(Ilya Bogdanov)
Higher Secondary P2
Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$. Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$.
Show that for any arbitrary triangle $ABC$, we have
\[\sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{abc}{(a+b)(b+c)(c+a)}.\]
Let $x,y,z$ be three positive integers with $\gcd(x,y,z)=1$. If
\[x\mid yz(x+y+z),\]
\[y\mid xz(x+y+z),\]
\[z\mid xy(x+y+z),\]
and
\[x+y+z\mid xyz,\]
show that $xyz(x+y+z)$ is a perfect square.
[i]Proposed by usjl[/i]
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
Let $f(n)$ be the number of ones that occur in the decimal representations of all the numbers from 1 to $n$. For example, this gives $f(8)=1$, $f(9)=1$, $f(10)=2$, $f(11)=4$, and $f(12)=5$. Determine the value of $f(10^{100})$.
An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers.
[b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$.
[b]b)[/b] Decide whether $a=2$ is friendly.
The points $A, B, C, D$ are marked on the straight line in this order. Circle $\omega_1$ passes through points $A$ and $C$, and the circle $\omega_2$ passes through points $B$ and $D$. On the circle $\omega_2$, the point $E$ is marked so that $AB = BE$, and on the circle $\omega_1$, the point $F$ is marked so that $CD = CF$. The line $AE$ intersects the circle $\omega_2$ a second time at point $X$, and the line $DF$ intersects the circle $\omega_1$ at point $Y$. Prove that the $XY$ lines and $AD$ is perpendicular.
[i]A.D.Tereshin[/i]
Let $BE$ and $CF$ be altitudes in triangle $ABC$. If $AE = 24$, $EC = 60$, and $BF = 31$, determine $AF$.
Find the number of all integer-sided [i]isosceles obtuse-angled[/i] triangles with perimeter $ 2008$.
[16 points out of 100 for the 6 problems]