Found problems: 85335
A square $ (n \minus{} 1) \times (n \minus{} 1)$ is divided into $ (n \minus{} 1)^2$ unit squares in the usual manner. Each of the $ n^2$ vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemse are regarded as different if at least one vertex is coloured differently in the two schemes.)
For $f(x)$ a positive , monotone decreasing function defined in $[0,1],$ prove that
$$ \int_{0}^{1} f(x) dx \cdot \int_{0}^{1} xf(x)^{2} dx \leq \int_{0}^{1} f(x)^{2} dx \cdot \int_{0}^{1} xf(x) dx.$$
Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1$, $A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths.
Prove that the lines $A_1B_2$, $B_1C_2$ and $C_1A_2$ are concurrent.
[i]Bogdan Enescu, Romania[/i]
Nadya has $2022$ cards, each with a number one or seven written on it. It is known that there are both cards.Nadya looked at all possible $2022$-digit numbers that can be composed from all these cards. What is the largest value that can take the greatest common divisor of all these numbers?
Prove that among any ten consecutive integers at least one is relatively prime to each of the others.
Given six real numbers $a$, $b$, $c$, $x$, $y$, $z$ such that $0 < b-c < a < b+c$ and $ax + by + cz = 0$.
What is the sign of the sum $ayz + bzx + cxy$ ?
\[\int_0^\infty \left(1-e^{-\pi/x^2}\right)^2\,\mathrm dx\]
[i]Proposed by Vlad Oleksenko[/i]
Let $ABC$ be a scalene and acute triangle with circumcenter $O$. Let $\omega$ be the circle with center $A$, tangent to $BC$ at $D$. Suppose there are two points $F$ and $G$ on $\omega$ such that $FG \perp AO$, $\angle BFD = \angle DGC$ and the couples of points $(B,F)$ and $(C,G)$ are in different halfplanes with respect to the line $AD$. Show that the tangents to $\omega$ at $F$ and $G$ meet on the circumcircle of $ABC$.
Let $K, T$ be the points of tangency of inscribed and exscribed circles to the side $BC$ triangle $ABC$, $M$ is the midpoint of the side $BC$. Using a compass and a ruler, construct triangle ABC given rays $AK$ and $AT$ (points $K, T$ are not marked on them) and point $M$.
In convex quadrilateral $ABCD$, let diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. Let the circumcircles of $ADE$ and $BCE$ intersect $\overline{AB}$ again at $P \neq A$ and $Q \neq B$, respectively. Let the circumcircle of $ACP$ intersect $\overline{AD}$ again at $R \neq A$, and let the circumcircle of $BDQ$ intersect $\overline{BC}$ again at $S \neq B$. Prove that $A$, $B$, $R$, and $S$ are concyclic.
[i]Tiger Zhang[/i]
If the parabola $ y \equal{} ax^2 \plus{} bx \plus{} c$ passes through the points $ ( \minus{} 1, 12), (0, 5),$ and $ (2, \minus{} 3),$ the value of $ a \plus{} b \plus{} c$ is:
$ \textbf{(A)}\ \minus{} 4 \qquad \textbf{(B)}\ \minus{} 2 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$
Let $u_1=1,u_2=1$ and for all $k \geq 1$'s
$$u_{k+2}=u_{k+1}+u_{k}$$
Prove that for all $m \geq 1$'s $5$ divides $u_{5m}$
Show that there are infinitely many prime numbers $p$ having the following property: there exists a natural number $n$, not dividing $p-1$, such that $p|n!+1$.
Given an integer $n \geq 3$, determine the maximum value of product of $n$ non-negative real numbers $x_1,x_2, \ldots , x_n$ when subjected to the condition
\begin{align*} \sum_{k=1}^n \frac{x_k}{1+x_k} =1 \end{align*}
Each of the $15$ coaches ranked the $50$ selected football players on the places from $1$ to $50$. For each football player, the highest and lowest obtained ranks differ by at most $5$. For each of the players, the sum of the ranks he obtained is computed, and the sums are denoted by $S_1\le S_2\le\ldots\le S_{50}$. Find the largest possible value of $S_1$.
In a graph with $n$ vertices every two vertices are connected by a unique path. For each two vertices $u$ and $v$, let $d(u, v)$ denote the distance
between $u$ and $v$, i.e. the number of edges in the path connecting these two vertices, and $\deg(u)$ denote the degree of a vertex $u$. Let $W$ be the sum of pairwise distances between the vertices, and $D$ the sum of weighted pairwise distances: $\sum_{\{u, v\}}(\deg(u)+\deg(v))d(u, v)$. Prove that $D=4W-n(n-1)$.
$BL $ is the bisector of an isosceles triangle $ABC $. A point $D $ is chosen on the Base $BC $ and a point $E $ is chosen on the lateral side $AB $ so that $AE=\frac {1}{2}AL=CD $. Prove that $LE=LD $.
Tuymaada 2017 Q5 Juniors
Compute the smallest positive integer that can be expressed as the product of four distinct integers.
[i]Proposed by Yannick Yao[/i]
What is the maximal number of solutions can the equation have $$\max \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0$$
where $a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10}$ are real numbers, all $a_i$ not equal to $0$.
How many ways are there to arrange the numbers $1$, $2$, $3$, $4$, $5$, $6$ on the vertices of a regular hexagon such that exactly 3 of the numbers are larger than both of their neighbors? Rotations and reflections are considered the same.
The $4$ code words
$$\square * \otimes \,\,\,\, \oplus \rhd \bullet \,\,\,\, * \square \bullet \,\,\,\, \otimes \oslash \oplus$$
they are in some order
$$AMO \,\,\,\, SUR \,\,\,\, REO \,\,\,\, MAS$$
Decrypt $$\otimes \oslash \square * \oplus \rhd \square \bullet \otimes $$
Suppose $H$ and $O$ are orthocenter and circumcenter of triangle $ABC$. $\omega$ is circumcircle of $ABC$. $AO$ intersects with $\omega$ at $A_1$. $A_1H$ intersects with $\omega$ at $A'$ and $A''$ is the intersection point of $\omega$ and $AH$. We define points $B',\ B'',\ C'$ and $C''$ similiarly. Prove that $A'A'',B'B''$ and $C'C''$ are concurrent in a point on the Euler line of triangle $ABC$.
Let $C$ and $D$ be points on the circle with center $O$ and diameter $[AB]$ where $C$ and $D$ are on different semicircles with diameter $[AB]$. Let $H$ be the foot perpendicular from $B$ to $[CD]$. If $|AO|=13$, $|AC|=24$, and $|HD|=12$, what is $\widehat{DCB}$ in degrees?
$
\textbf{(A)}\ 30
\qquad\textbf{(B)}\ 45
\qquad\textbf{(C)}\ 60
\qquad\textbf{(D)}\ 75
\qquad\textbf{(E)}\ 80
$
Let $\triangle ABC$ be an acute-angled triangle and $A_1, B_1$, and $C_1$ be the feet of the altitudes from $A, B$, and $C$, respectively. On the rays $AA_1, BB_1$, and $CC_1$, we have points $A_2, B_2$, and $C_2$ respectively, lying outside of $\triangle ABC$, such that
\[\frac{A_1A_2}{AA_1} = \frac{B_1B_2}{BB_1} = \frac{C_1C_2}{CC_1}.\]
If the intersections of $B_1C_2$ and $B_2C_1$, $C_1A_2$ and $C_2A_1$, and $A_1B_2$ and $A_2B_1$ are $A', B'$, and $C'$ respectively, prove that $AA', BB'$, and $CC'$ have a common point.
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i]
[b][i]Original Formulation[/i][/b]
Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f( \sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes.
[i]Proposed by Soviet Union. [/i]