Found problems: 85335
There are non-constant polynom $P(x)$ with integral coefficients and natural number $n$. Suppose that $a_0=n$, $a_k=P(a_{k-1})$ for any natural $k$. Finally, for every natural $b$ there is number in sequence $a_0, a_1, a_2, \ldots$ that is $b$-th power of some natural number that is more than 1. Prove that $P(x)$ is linear polynom.
The fox and pinocchio have grown a tree on the field of miracles with $8$ golden coins. It is known that exactly $3$ of them are counterfeit. All the real coins weigh the same, the counterfeit coins also weigh the same but are lighter. The fox and pinocchio have collected the coins and wish to divide them. The fox is going to give 3 coins to pinocchio, but pinocchio wants to check whether they all are real. Can he check this using $2$ weighings on a balance scale with no weights?
Find all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that for any real numbers \(x\) and \(y\), the following inequality holds:
\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq f\left((x+y)^2\right)
\]
[i]Proposed by Anton Trygub[/i]
Two lines intersect inside a unit square, splitting it into four regions. Find the maximum product of the areas of the four regions.
[i]Proposed by Nathan Ramesh
On a examination of $n$ questions a student answers correctly $15$ of the first $20$. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is $50\%$, how many different values of $n$ can there be?
$ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ \text{the problem cannot be solved} $
A small point-like object is thrown horizontally off of a $50.0$-$\text{m}$ high building with an initial speed of $10.0 \text{ m/s}$. At any point along the trajectory there is an acceleration component tangential to the trajectory and an acceleration component perpendicular to the trajectory. How many seconds after the object is thrown is the tangential component of the acceleration of the object equal to twice the perpendicular component of the acceleration of the object? Ignore air resistance.
$ \textbf{(A)}\ 2.00\text{ s}$
$\textbf{(B)}\ 1.50\text{ s}$
$\textbf{(C)}\ 1.00\text{ s}$
$\textbf{(D)}\ 0.50\text{ s}$
$\textbf{(E)}\ \text{The building is not high enough for this to occur.} $
Let $x, y, z \in R^+$. Prove that
$$\frac{x}{x +\sqrt{(x + y)(x + z)}}+\frac{y}{y +\sqrt{(y + z)(y + x)}}+\frac{z}{z +\sqrt{(x + z)(z + y)}} \le 1$$
Solve the recurrence $R_0=1, R_n=nR_{n-1}+2^n\cdot n!$.
Find all prime numbers whose base $b$ representations (for some $b$) contain each of the digits $0,1,\ldots,b-1$ exactly once. (Digit $0$ may appear as the first digit.)
Let $n$ be a positive integer such that $1 \leq n \leq 1000$. Let $M_n$ be the number of integers in the set $X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}$. Let
$$
a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. }
$$
Find $a-b$.
Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta=\angle PAB=\angle QPC=\angle RQB=\cdots$ will produce in a length that is $120$ meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.)
[asy]
import olympiad;
draw((-50,15)--(50,15));
draw((50,15)--(50,-15));
draw((50,-15)--(-50,-15));
draw((-50,-15)--(-50,15));
draw((-50,-15)--(-22.5,15));
draw((-22.5,15)--(5,-15));
draw((5,-15)--(32.5,15));
draw((32.5,15)--(50,-4.090909090909));
label("$\theta$", (-41.5,-10.5));
label("$\theta$", (-13,10.5));
label("$\theta$", (15.5,-10.5));
label("$\theta$", (43,10.5));
dot((-50,15));
dot((-50,-15));
dot((50,15));
dot((50,-15));
dot((50,-4.09090909090909));
label("$D$",(-58,15));
label("$A$",(-58,-15));
label("$C$",(58,15));
label("$B$",(58,-15));
label("$S$",(58,-4.0909090909));
dot((-22.5,15));
dot((5,-15));
dot((32.5,15));
label("$P$",(-22.5,23));
label("$Q$",(5,-23));
label("$R$",(32.5,23));
[/asy]
$\textbf{(A)}~\arccos\frac{5}{6}\qquad\textbf{(B)}~\arccos\frac{4}{5}\qquad\textbf{(C)}~\arccos\frac{3}{10}\qquad\textbf{(D)}~\arcsin\frac{4}{5}\qquad\textbf{(E)}~\arcsin\frac{5}{6}$
Given are $8$ weights weighing $1, 2, . . . , 8$ grams, but it is not known which one how much does it weigh. Baron Munchausen claims that he remembers which of the weights weighs how much, and to prove that he is right he is ready to conduct one weighing, as a result of which the weight of at least one of the weights will be unambiguously established. Is he cheating?
Determine if there exist positive real numbers $x, \alpha$, so that for any non-empty finite set of positive integers $S$, the inequality
\[\left|x-\sum_{s\in S}\frac{1}{s}\right|>\frac{1}{\max(S)^\alpha}\]
holds, where $\max(S)$ is defined as the maximum element of the finite set $S$.
Let $ I $ be the incentre of $ ABC $ and $ D,E,F $ be the feet of the perpendiculars from $ I $ to $ BC,CA,AB, $ respectively. Show that
$$ \frac{AB}{DE} +\frac{BC}{EF} +\frac{CA}{FD}\ge 6. $$
Doug and Dave shared a pizza with $ 8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $ \$8$, and there was an additional cost of $ \$2$ for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?
$ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$
Determine all $f: \mathbb{R}\rightarrow\mathbb{R}$ for which
\[ 2\cdot f(x)-g(x)=f(y)-y \textrm{ and } f(x)\cdot g(x) \geq x+1. \]
Triangle $ABC$ has side lengths $AB=7, BC=8, CA=9.$ Define $M,N,P$ to be the midpoints of sides $BC,CA,AB,$ respectively. The circumcircles of $\triangle APN$ and $\triangle ABM$ intersect at another point $K.$ Find $NK.$
[i]Individual #15[/i]
A $4 \times 4$ board has all its squares painted white. An allowed operation is to choose a rectangle that contains $3$ squares and paint each of the boxes as follows:
a) If the box is white then it is painted black.
b) If the box is black then it is painted white.
Prove that by applying the allowed operation several times, it is impossible get the entire board painted black.
Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate
\[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \][i]Proposed by Sammy Luo[/i]
Let $c_1,c_2,\ldots ,c_n,b_1,b_2,\ldots ,b_n$ $(n\geq 2)$ be positive real numbers. Prove that the equation
\[ \sum_{i=1}^nc_i\sqrt{x_i-b_i}=\frac{1}{2}\sum_{i=1}^nx_i\]
has a unique solution $(x_1,\ldots ,x_n)$ if and only if $\sum_{i=1}^nc_i^2=\sum_{i=1}^nb_i$.
Let $ABC$ be a triangle with two isogonal points $ P$ and $Q$ . Let $D, E$ be the projection of $P$ on $AB$, $AC$. $G$ is the projection of $Q$ on $BC$. $U$ is the projection of $G$ on $DE$, $ L$ is the projection of $P$ on $AQ$, $K$ is the symmetric of $L$ wrt $UG$.
Prove that $UK$ passes through a fixed point when $P$ and $Q$ vary.
Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$. Find the angle $\angle MXK $.
In acute-angled $\bigtriangleup ABC$, $H$ is the orthocenter, $O$ is the circumcenter and $I$ is the incenter. Given that $\angle C > \angle B > \angle A$, prove that $I$ lies within $\bigtriangleup BOH$.
In a five term arithmetic sequence, the first term is $2020$ and the last term is $4040.$ Find the second term of the sequence.
[i]Proposed by Ada Tsui[/i]
Let $n,m,r$ be positive integers such that $n>m$ and both $n^2+r, m^2+r$ are powers of $2$. Show that $n>\frac{2m^2}{r}$.