Found problems: 85335
For every integer $n\geqslant2$ prove the inequality
$$
\frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{2^{n-2}}{n!}\leqslant\frac{3}{2},
$$
where $k!=1\cdot2\cdot\ldots\cdot k$.
The square $ABCD$ is divided into $8$ equal right triangles and the square $KLMN$, as shown in the figure. Find the area of the square $ABCD$ if $KL = 5, PS = 8$.
[img]https://1.bp.blogspot.com/-B2QIHvPcIx0/X4BhUTMDhSI/AAAAAAAAMj4/4h0_q1P6drskc5zSvtfTZUskarJjRp5LgCLcBGAsYHQ/s0/Yasinsky%2B2020%2Bp1.png[/img]
For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?
$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }9 \qquad \textbf{(D) }14 \qquad \textbf{(E) }16$
Which of the following cannot be equal to $x^2 + \dfrac 1{4x}$ where $x$ is a positive real number?
$
\textbf{a)}\ \sqrt 3 -1
\qquad\textbf{b)}\ 2\sqrt 2 - 2
\qquad\textbf{c)}\ \sqrt 5 - 1
\qquad\textbf{d)}\ 1
\qquad\textbf{e)}\ \text{None of above}
$
Bishops on a chessboard move along the diagonals ( that is , on lines parallel to the two main diagonals ) . Prove that the maximum number of non-attacking bishops on an $n*n$ chessboard is $2n-2$. (Two bishops are said to be attacking if they are on a common diagonal).
Consider an integer $n \geq 1$, $a_1,a_2, \ldots , a_n$ real numbers in $[-1,1]$ satisfying
\begin{align*}a_1+a_2+\ldots +a_n=0 \end{align*}
and a function $f: [-1,1] \mapsto \mathbb{R}$ such
\begin{align*} \mid f(x)-f(y) \mid \le \mid x-y \mid \end{align*}
for every $x,y \in [-1,1]$. Prove
\begin{align*} \left| f(x) - \frac{f(a_1) +f(a_2) + \ldots + f(a_n)}{n} \right| \le 1 \end{align*}
for every $x$ $\in [-1,1]$. For a given sequence $a_1,a_2, \ldots ,a_n$, Find $f$ and $x$ so hat the equality holds.
What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?
Determine all strictly increasing functions $f: R \to R$ satisfying $f (f(x) + y) = f(x + y) + f (0)$ for all $x,y \in R$.
Is there an integer coefficients polynomial $P(x)$ satisfying \[ \begin{cases} P(1+\sqrt[3]{2})=1+\sqrt[3]{2}\\ P(1+\sqrt{5})=2+3\sqrt{5}\end{cases} \]
Sasha wrote a non-zero number on the board and added it to it on the right, one non-zero digit at a time, until he writes out a million digits. Prove that an exact square has been written on the board no more than $100$ times.
Find all real-valued functions $f$ satisfying $f(2x + f(y)) + f(f(y)) = 4x + 8y$ for all real numbers $x$ and $y$.
$x=\frac{1}{\log_{\frac{1}{2}} \frac{1}{3}}+\frac{1}{\log_{\frac{1}{5}} \frac{1}{3}}$, then
$\text{(A)}x\in(-2,-1)\qquad\text{(B)}x\in(1,2)\qquad\text{(C)}x\in(-3,-2)\qquad\text{(D)}x\in(2,3)$
Find all $a , b , c \in \mathbb{N}$ for which \[1997^a+15^b=2012^c\]
Let $ABCD$ be a square. Let $E, Z$ be points on the sides $AB, CD$ of the square respectively, such that $DE\parallel BZ$. Assume that the triangles $\triangle EAD, \triangle ZCB$ and the parallelogram $BEDZ$ have the same area.
If the distance between the parallel lines $DE$ and $BZ$ is equal to $1$, determine the area of the square.
On the bisector of angle $A$ of triangle $ABC$, points $D$ and $F$ are taken inside the triangle so that $\angle DBC = \angle FBA$. Prove that:
a) $\angle DCB = \angle FCA$
b) a circle passing through $B$ and $F$ and tangent to the segment $BC$ is tangle to the circumscribed circle of the triangle $ABC$.
The incircle of a right triangle $ABC$ touches the hypotenuse $AB$ at a point $D$. Show that the area of $\triangle ABC$ equals $AD\cdot DB$.
A sequence $a_0, a_1, \dots , a_n, \dots$ of positive integers is constructed as follows:
[list]
[*] If the last digit of $a_n$ is less than or equal to $5$, then this digit is deleted and $a_{n+1}$ is the number consisting of the remaining digits. (If $a_{n+1}$ contains no digits, the process stops.)
[*] Otherwise, $a_{n+1}= 9a_n$.
[/list]
Can one choose $a_0$ so that this sequence is infinite?
Let $\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\triangle{ABC}$. If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$, then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$, $b$, and $c$ are positive integers. Find $a+b+c$.
[color=#00f]Note that this problem is null because a diagram is impossible.[/color]
[i]Proposed by Andy Xu[/i]
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(2f(x)) = f(x - f(y)) + f(x) + y$$ for all $x, y \in \mathbb{R}$.
Let $ABC$ be an acute, non-isosceles triangle with orthocenter $H$. Let $D, E, F$ be the reflections of $H$ over $BC, CA, AB$, respectively, and let $A', B', C'$ be the reflections of $A, B, C$ over $BC, CA, AB$, respectively. Let $S$ be the circumcenter of triangle $A'B'C'$, and let $H'$ be the orthocenter of triangle $DEF$. Define $J$ as the center of the circle passing through the three projections of $H$ onto the lines $B'C', C'A', A'B'$. Prove that $HJ$ is parallel to $H'S$.
Let $N$ be an odd number, $N\geq 3$. $N$ tennis players take part in a championship. Before starting the championship, a commission puts the players in a row depending on how good they think the players are. During the championship, every player plays with every other player exactly once, and each match has a winner. A match is called [i]suprising[/i] if the winner was rated lower by the commission. At the end of the tournament, players are arranged in a line based on the number of victories they have achieved. In the event of a tie, the commission's initial order is used to decide which player will be higher.
It turns out that the final order is exactly the same as the commission's initial order. What is the maximal number of suprising matches that could have happened.
Express $\sum_{k=1}^\infty\frac{6^k}{(3^{k+1}-2^{k+1})(3^k-2^k)}$ as a rational number.
Let $ p,q,r $ be prime numbers. Solve the equation $ p^{2q}+q^{2p}=r $
Find all possibilities: how many acute angles can there be in a convex polygon?
2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.