Find two consecutive integers with the property that the sums of their digits are each divisible by $2006$.

Let $H$ be an orhocenter of an acute triangle $ABC$ and $M$ midpoint of side $BC$. If $D$ and $E$ are foots of perpendicular of $H$ on internal and external angle bisector of angle $\angle BAC$, prove that $M$, $D$ and $E$ are collinear

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

Let $n$ be a natural number divisible by $3$. We have a $n \times n$ table and each square is colored either black or white. Suppose that for all $m \times m$ sub-tables from the table ($m > 1$), the number of black squares is not more than white squares. Find the maximum number of black squares.

Let $N{}$ be the number of positive integers with $10$ digits $\overline{d_9d_8\cdots d_0}$ in base $10$ (where $0\le d_i\le9$ for all $i$ and $d_9>0$) such that the polynomial \[d_9x^9+d_8x^8+\cdots+d_1x+d_0\] is irreducible in $\Bbb Q$. Prove that $N$ is even. (A polynomial is irreducible in $\Bbb Q$ if it cannot be factored into two non-constant polynomials with rational coefficients.)

4. Given is an integer $n\geq 1$. Find out the number of possible values of products $k \cdot m$, where $k,m$ are integers satisfying $n^{2}\leq k \leq m \leq (n+1)^{2}$.

Let $A'B'C'$ denote the reflection of scalene and acute triangle $ABC$ across its Euler-line. Let $P$ be an arbitrary point of the nine-point circle of $ABC$. For every point $X$, let $p(X)$ denote the reflection of $X$ across $P$. [b]a)[/b] Let $e_{AB}$ denote the line connecting the orthogonal projection of $A$ to line $BB'$ and the orthogonal projection of $B$ to line $AA'$. Lines $e_{BC}$ and $e_{CA}$ are defined analogously. Prove that these three lines are concurrent (and denote their intersection by $K$). [b]b)[/b] Prove that there are two choices of $P$ such that lines $Ap(A')$, $Bp(B')$ and $Cp(C')$ are concurrent, and the four points $p(A)p(A')\cap BC$, $p(B)p(B')\cap CA$, $p(C)p(C')\cap AB$, and $K$ are collinear. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

Solve the system $\begin {cases} 16x^3 + 4x = 16y + 5 \\ 16y^3 + 4y = 16x + 5 \end{cases}$

If $x_{1}, x_{2}, . . . , x_{n}$ are positive numbers, prove the inequality $\frac{x_{1}^{3}}{x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}}+\frac{x_{2}^{3}}{x_{2}^{2}+x_{2}x_{3}+x_{3}^{2}}+...+\frac{x_{n}^{3}}{x_{n}^{2}+x_{n}x_{1}+x_{1}^{2}}\geq\frac{x_{1}+x_{2}+...+x_{n}}{3}$.

Prove that for every positive integer $m$ we can find a finite set $S$ of points in the plane, such that given any point $A$ of $S$, there are exactly $m$ points in $S$ at unit distance from $A$.

Inside an acute triangle $ABC$ is a point $P$ that is not the circumcenter. Prove that among the segments $AP$, $BP$ and $CP$, at least one is longer and at least one is shorter than the circumradius of $ABC$.

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

The number of $ x$-intercepts on the graph of $ y \equal{} \sin(1/x)$ in the interval $ (0.0001,0.001)$ is closest to $ \textbf{(A)}\ 2900 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 3100 \qquad \textbf{(D)}\ 3200 \qquad \textbf{(E)}\ 3300$

Prove that any triangle can be cut by at most into $3$ parts, from which an isosceles triangle is formed.

Let $a,\ b$ are real numbers such that $a+b=1$. Find the minimum value of the following integral. \[\int_{0}^{\pi}(a\sin x+b\sin 2x)^{2}\ dx \]

Let $n$ be a positive integer, let $f_1(x),\ldots,f_n(x)$ be $n$ bounded real functions, and let $a_1,\ldots,a_n$ be $n$ distinct reals. Show that there exists a real number $x$ such that $\sum^n_{i=1}f_i(x)-\sum^n_{i=1}f_i(x-a_i)<1$.

In the sequence $ (a_n)$ with general term $ a_n \equal{} n^3 \minus{} (2n \plus{} 1)^2$, does there exist a term that is divisible by 2006?

Let $p$ be a prime. Let $f(x)$ be the number of ordered pairs $(a, b)$ of positive integers less than $p$, such that $a^b \equiv x \pmod p$. Suppose that there do not exist positive integers $x$ and $y$, both less than $p$, such that $f(x) = 2f(y)$, and that the maximum value of $f$ is greater than $2018$. Find the smallest possible value of $p$.

For a positive integer $n$, there is a school with $n$ people. For a set $X$ of students in this school, if any two students in $X$ know each other, we call $X$ [i]well-formed[/i]. If the maximum number of students in a well-formed set is $k$, show that the maximum number of well-formed sets is not greater than $3^{(n+k)/3}$. Here, an empty set and a set with one student is regarded as well-formed as well.

The points $A_1,B_1,C_1,D_1$ and $A_2,B_2,C_2,D_2$ are orthogonal projections of the $ABCD$ tetrahedron vertices on two planes. Prove that it is possible to move one of the planes to provide the parallelness of lines $(A_1A_2), (B_1B_2), (C_1C_2)$ and $(D_1D_2)$ .

A sequence \( a_1, a_2, \ldots \) of real numbers satisfies the following condition: for every positive integer \( k \geq 2 \), there exists a positive integer \( i < k \) such that \( a_i + a_k = k \). It is known that for some \( j \), the fractional parts of the numbers \( a_j \) and \( a_{j+1} \) are equal. Prove that for some positive integers \( x \neq y \), the equality \[ a_x - a_y = x - y \] holds. [i]The fractional part of a real number \( a \) is defined as the number \( \{a\} \in [0, 1) \), which satisfies the condition \( a = n + \{a\} \), where \( n \) is an integer. For example, \( \{-3\} = 0 \), \( \{3.14\} = 0.14 \), and \( \{-3.14\} = 0.86 \).[/i] [i]Proposed by Mykhailo Shtandenko[/i]

Arthur has a regular 11-gon. He labels the vertices with the letters in $CORONAVIRUS$ in consecutive order. Every non-ordered set of 3 letters that forms an isosceles triangle is a member of a set $S$, i.e. $\{C, O, R\}$ is in $S$. How many elements are in $S$? [i]Proposed by Sammy Chareny[/i]

Let $S$ be the locus of all points $(x,y)$ in the first quadrant such that $\dfrac{x}{t}+\dfrac{y}{1-t}=1$ for some $t$ with $0<t<1$. Find the area of $S$.

A digital watch displays hours and minutes with $ \text c{AM}$ and $ \text c{PM}$. What is the largest possible sum of the digits in the display? $ \textbf{(A) } 17\qquad \textbf{(B) } 19\qquad \textbf{(C) } 21\qquad \textbf{(D) } 22\qquad \textbf{(E) } 23$

The sequence $(a_n)$ is defined by $a_1 =\sqrt2$ and $a_{n+1} =\sqrt{2-\sqrt{4-a_n^2}}$. Let $b_n =2^{n+1}a_n$. Prove that $b_n \le 7$ and $b_n < b_{n+1}$ for all $n$.