Found problems: 85335
Let $ABC$ be an arbitrary triangle,$P$ is the intersection point of the altitude from $C$ and the tangent line from $A$ to the circumcircle. The bisector of angle $A$ intersects $BC$ at $D$ . $PD$ intersects $AB$ at $K$, if $H$ is the orthocenter then prove : $HK\perp AD$
The number $1234567890$ is written on the blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In one move, a player erases the number which is written on the blackboard, say, $m$, subtracts from $m$ any positive integer not exceeding the sum of the digits of $m$ and writes the obtained result instead of $m$. The first player who reduces the number written on the blackboard to $0$ wins. Determine which of the players has the winning strategy if the player $A$ makes the first move.
For each integer $n \ge 3$ solve in real numbers the system of equations:
$$\begin{cases} x_1^3 = x_2 + x_3 + 1 \\...\\x_{n-1}^3 = x_n+ x_1 + 1\\x_{n}^3 = x_1+ x_2 + 1 \end{cases}$$
Triangle $ABC$ has circumcenter $O$ and incenter $I$. The incircle is tangent to $AC, AB$ at $E, F$, respectively. $H$ is the orthocenter of $\triangle BIC$. $\odot(AEF)$ and $\odot(ABC)$ intersects again at $S$. $BC, AH$ intersects $OI$ again at $J, K$, respectively. Prove that $H, K, J, S$ are concyclic.
[i]Proposed by chengbilly[/i]
Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$.
- There is a real number $\xi$ with $P(\xi)=0$.
Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.
In each unit square of an infinite square grid a natural number is written. The polygons of area $n$ with sides going along the gridlines are called [i]admissible[/i], where $n > 2$ is a given natural number. The [i]value [/i] of an admissible polygon is defined as the sum of the numbers inside it. Prove that if the values of any two congruent admissible polygons are equal, then all the numbers written in the unit squares of the grid are equal. (We recall that a symmetric image of polygon $P$ is congruent to $P$.)
Let $a_1, a_2, ..., a_9$ be non-negative real numbers such that $a_1 = a_9 = 0$ and at least one of the remaining terms is different from $0$.
a) Prove that for some $i$ $(i = 2, ..., 8$) ,holds that $a_{i-1} + a_{i+1} < 2a_i.$
b) Will the previous statement be true, if we change the number $2$ for $1.9$ in the inequality?
We divide up the plane into disjoint regions using a circle, a rectangle and a triangle. What is the greatest number of regions that we can get?
Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying
\begin{align*}
f(x+f(y)) &= g(x) + h(y) \\
g(x+g(y)) &= h(x) + f(y) \\
h(x+h(y)) &= f(x) + g(y)
\end{align*}
for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.)
[i]Proposed by Evan Chen[/i]
Let $f:[0,1]\rightarrow\mathbb{R}$ be an integrable function such that:
\[0<\left\vert \int_{0}^{1}f(x)\, \text{d}x\right\vert\le 1.\]
Show that there exists $x_1\not= x_2, x_1,x_2\in [0,1]$, such that:
\[\int_{x_1}^{x_2}f(x)\, \text{d}x=(x_1-x_2)^{2002}\]
Consider $n$ segments in the plane which no two intersect and between their $2n$ endpoints no three are collinear. Is the following statement true?
Statement: There exists a simple $2n$-gon such that it's vertices are the $2n$ endpoints of the segments and each segment is either completely inside the polygon or an edge of the polygon.
Let be a natural number $ n, $ a permutation $ \sigma $ of order $ n, $ and $ n $ nonnegative real numbers $ a_1,a_2,\ldots , a_n. $ Prove the following inequality.
$$ \left( a_1^2+a_{\sigma (1)} \right)\left( a_2^2+a_{\sigma (2)} \right)\cdots \left( a_n^2+a_{\sigma (n)} \right)\ge \left( a_1^2+a_1 \right)\left( a_2^2+a_{2} \right)\cdots \left( a_n^2+a_n \right) $$
Let $n > 3$ be a positive integer. Find all integers $k$ such that $1 \le k \le n$ and for
which the following property holds:
If $x_1, . . . , x_n$ are $n$ real numbers such that $x_i + x_{i + 1} + ... + x_{i + k - 1} = 0$ for all integers $i > 1$ (indexes are taken modulo $n$), then $x_1 = . . . = x_n = 0$.
Proposed by [i]Vincent Jugé and Théo Lenoir, France[/i]
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$.
Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$.
[i]Ivan Chan Kai Chin, Malaysia[/i]
There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions ?
[i](i)[/i] Every route includes exactly three ports;
[i](ii)[/i] No two routes contain the same three ports;
[i](iii)[/i] The series offers exactly two routes to each tourist who desires to visit two different arbitrary ports.
Find all real solutions of the system of equations:
\[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$
For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.
Let $a_{1}$, $\cdots$, $a_{k}$ and $m_{1}$, $\cdots$, $m_{k}$ be integers with $2 \le m_{1}$ and $2m_{i}\le m_{i+1}$ for $1 \le i \le k-1$. Show that there are infinitely many integers $x$ which do not satisfy any of congruences \[x \equiv a_{1}\; \pmod{m_{1}}, x \equiv a_{2}\; \pmod{m_{2}}, \cdots, x \equiv a_{k}\; \pmod{m_{k}}.\]
How many squares must be painted at least on a $5 \times 5$ board such that in each row, in each column and in each $2 \times 2$ square is there at least one square painted?
On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.
Let $a, b$ and $c$ be integers such that $a + b + c = 0$.
(a) Show that $a^4 + b^4 + c^4$ is divisible by $a^2 + b^2 + c^2$.
(b) Show that $a^{100} + b^{100} + c^{100}$ is divisible by $a^2 + b^2 + c^2$.
.
A sequence $ (a_n)$ of positive integers satisfies: $ a_n\equal{}\sqrt{\frac{a_{n\minus{}1}^2\plus{}a_{n\plus{}1}^2}{2}}$ for all $ n \ge 1$. Prove that this sequence is constant.
The point ${D}$ inside the equilateral triangle ${\triangle ABC}$ satisfies ${\angle ADC = 150^o}$. Prove that a triangle with side lengths ${|AD|, |BD|, |CD|}$ is necessarily a right-angled triangle.
Given a sequence $ (c_n) $ of natural numbers defined recursively: $ c_1 = 2 $, $ c_{n+1} = \left[ \frac{3}{2}c_n\right] $. Prove that there are infinitely many even numbers and infinitely many odd numbers among the terms of this sequence.