$C_1:\frac{x^2}{a^2}+y^2=1(a>0),C_2:y^2=2(x+m)$, one intersection of $C_1$ and $C_2$ is $P$, and $P$ is above the $x$-axis. [b](a)[/b] Find the range value of $m$ (express with $a$). [b](b)[/b] $O(0,0),A(-a,0)$. If $0<a<\frac{1}{2}$, find the maximum value of $S_{\triangle OAP}$.

How many $8$-digit numbers in base $4$ formed of the digits $1,2, 3$ are divisible by $3$?

Let $\phi(n)$ denote $\textit{Euler's phi function}$, the number of integers $1\leq i\leq n$ that are relatively prime to $n$. (For example, $\phi(6)=2$ and $\phi(10)=4$.) Let \[S=\sum_{d|2008}\phi(d),\] in which $d$ ranges through all positive divisors of $2008$, including $1$ and $2008$. Find the remainder when $S$ is divided by $1000$.

Given equilateral triangle $ABC$. Some line, parallel to $[AC]$ crosses $[AB]$ and $[BC]$ in $M$ and $P$ points respectively. Let $D$ be the centre of $PMB$ triangle, $E$ be the midpoint of the $[AP]$ segment. Find the angles of triangle $DEC$ .

Consider an infinite strictly increasing sequence of positive integers $a_1$, $a_2$,$...$ where for any real number $C$, there exists an integer $N$ where $a_k >Ck$ for any $k >N$. Do there necessarily exist inifinite many indices $k$ where $2a_k <a_{k-1}+a_{k+1}$ for any $0<i<k$?

Consider $ 2011 $ positive real numbers $ a_1,a_2,\ldots ,a_{2011} . $ If they are in geometric progression, show that there exists a real number $ \lambda $ such that any $ i\in\{ 1,2,\ldots , 1005 \} $ implies $ \lambda =a_i\cdot a_{2012-i} . $ Disprove the converse. [i]Teodor Radu[/i]

Let $P = \{(x, y) | x, y \in \{0, 1, 2,... , 2015\}\}$ be a set of points on the plane. Straight wires of unit length are placed to connect points in $P$ so that each piece of wire connects exactly two points in $P$, and each point in $P$ is an endpoint of exactly one wire. Prove that no matter how the wires are placed, it is always possible to draw a straight line parallel to either the horizontal or vertical axis passing through midpoints of at least $506$ pieces of wire.

In $\vartriangle ABC$ on the sides $BC, CA, AB$ mark feet of altitudes $H_1, H_2, H_3$ and the midpoint of sides $M_1, M_3, M_3$. Let $H$ be orthocenter $\vartriangle ABC$. Suppose that $X_2, X_3$ are points symmetric to $H_1$ wrt $BH_2$ and $CH_3$. Lines $M_3X_2$ and $M_2X_3$ intersect at point $X$. Similarly, $Y_3,Y_1$ are points symmetric to $H_2$ wrt $C_3H$ and $AH_1$.Lines $M_1Y_3$ and $M_3Y_1$ intersect at point $Y.$ Finally, $Z_1,Z_2$ are points symmetric to $H_3$ wrt $AH_1$ and $BH_2$. Lines $M_1Z_2$ and $M_2Z_1$ intersect at the point $Z$ Prove that $H$ is the incenter $\vartriangle XYZ$ .

Let $n>1$ be an integer and $S_n$ the set of all permutations $\pi : \{1,2,\cdots,n \} \to \{1,2,\cdots,n \}$ where $\pi$ is bijective function. For every permutation $\pi \in S_n$ we define: \[ F(\pi)= \sum_{k=1}^n |k-\pi(k)| \ \ \text{and} \ \ M_{n}=\frac{1}{n!}\sum_{\pi \in S_n} F(\pi) \] where $M_n$ is taken with all permutations $\pi \in S_n$. Calculate the sum $M_n$.

Let $n$ and $r$ two positive integers. It is wanted to make $r$ subsets $A_1,\ A_2,\dots,A_r$ from the set $\{0,1,\cdots,n-1\}$ such that all those subsets contain exactly $k$ elements and such that, for all integer $x$ with $0\leq{x}\leq{n-1}$ there exist $x_1\in{}A_1,\ x_2\in{}A_2 \dots,x_r\in{}A_r$ (an element of each set) with $x=x_1+x_2+\cdots+x_r$. Find the minimum value of $k$ in terms of $n$ and $r$.

Let $a, b, c$ be positive real numbers. Prove that $$\frac {3(ab + bc + ca)}{2(a^2b^2+b^2c^2+c^2a^2)}\leq \frac1{a^2 + bc} + \frac1{b^2 + ca} + \frac1{c^2 + ab}\leq\frac{a+b+c}{2abc}.$$

The nodes of a three dimensional unit cube lattice with all three coordinates even are coloured red and blue otherwise. A convex polyhedron with all vertices red is given. Assuming the number of red points on its border is $n$. How many blue vertices can be on its border?

Let $p$ be a prime number. All natural numbers from $1$ to $p$ are written in a row in ascending order. Find all $p$ such that this sequence can be split into several blocks of consecutive numbers, such that every block has the same sum. [i]A. Khrabov[/i]

In a triangle $A B C$, let $E$ be the midpoint of $A C$ and $F$ be the midpoint of $A B$. The medians $B E$ and $C F$ intersect at $G$. Let $Y$ and $Z$ be the midpoints of $B E$ and $C F$ respectively. If the area of triangle $A B C$ is 480 , find the area of triangle $G Y Z$.

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Positive integers $a,b,c$ satisfy the equations $a^2=b^3+ab$ and $c^3=a+b+c$. Prove that $a=bc$.

Let $AH_A$, $BH_B$, $CH_C$ be the altitudes of the acute-angled $\Delta ABC$. Let $X$ be an arbitrary point of segment $CH_C$, and $P$ be the common point of circles with diameters $H_CX$ and BC, distinct from $H_C$. The lines $CP$ and $AH_A$ meet at point $Q$, and the lines $XP$ and $AB$ meet at point $R$. Prove that $A, P, Q, R, H_B$ are concyclic.

Let $A$ be an infinite set of points in the plane such that inside each circle there are only a finite number of points of $A$, with the following properties: $\bullet$ $(0, 0)$ belongs to $A$. $\bullet$ If $(a, b)$ and $(c, d)$ belong to $A$, then $(a-c, b-d)$ belongs to $A$. $\bullet$ There is a value of $\alpha$ such that by rotating the set $A$ with center at $(0, 0)$ and angle $\alpha$, the set $A$ is obtained again. Prove that $\alpha$ must be equal to $\pm 60^{\circ}$ or $\pm 90^{\circ}$ or $\pm 120^{\circ}$ or $\pm 180^{\circ}$.

Given a straight line with points $A, B, C$ and $D$. Construct using $AB$ and $CD$ regular triangles (in the same half-plane). Let $E,F$ be the third vertex of the two triangles (as in the figure) . The circumscribed circles of triangles $AEC$ and $BFD$ intersect in $G$ ($G$ is is in the half plane of triangles). Prove that the angle $AGD$ is $120^o$ [img]https://1.bp.blogspot.com/-66akc83KSs0/X9j2BBOwacI/AAAAAAAAM0M/4Op-hrlZ-VQRCrU8Z3Kc3UCO7iTjv5ZQACLcBGAsYHQ/s0/2011%2BDurer%2BC5.png[/img]

In how many ways can Eve fill each of the six squares of a $2 \times 3$ grid with either a $0$ or a $1$, such that Anne can then divide the grid into three congruent rectangles: one containing two $0$s, one containing two $1$s, and one containing a $0$ and a $1$? [i]Proposed by Michael Tang[/i]

For a given value $t$, we consider number sequences $a_1, a_2, a_3,...$ such that $a_{n+1} =\frac{a_n + t}{a_n + 1}$ for all $n \ge 1$. (a) Suppose that $t = 2$. Determine all starting values $a_1 > 0$ such that $\frac43 \le a_n \le \frac32$ holds for all $n \ge 2$. (b) Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.

Sofiya and Marquis are playing a game. Sofiya announces to Marquis that she's thinking of a polynomial of the form $f(x)=x^3+px+q$ with three integer roots that are not necessarily distinct. She also explains that all of the integer roots have absolute value less than (and not equal to) $N$, where $N$ is some fixed number which she tells Marquis. As a "move" in this game, Marquis can ask Sofiya about any number $x$ and Sofiya will tell him whether $f(x)$ is positive negative, or zero. Marquis's goal is to figure out Sofiya's polynomial. If $N=3\cdot 2^k$ for some positive integer $k$, prove that there is a strategy which allows Marquis to identify the polynomial after making at most $2k+1$ "moves".

Diagonals of an isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ are perpendicular. Let $DE$ be the perpendicular from $D$ to $AB$, and let $CF$ be the perpendicular from $C$ to $DE$. Prove that angle $DBF$ is equal to half of angle $FCD$.

Let $\ast$ be the symbol denoting the binary operation on the set $S$ of all non-zero real numbers as follows: For any two numbers $a$ and $b$ of $S$, $a\ast b=2ab$. Then the one of the following statements which is not true, is $\textbf{(A) }\ast\text{ is commutative over }S \qquad\textbf{(B) }\ast\text{ is associative over }S\qquad$ $\textbf{(C) }\frac{1}{2}\text{ is an identity element for }\ast\text{ in }S\qquad\textbf{(D) }\text{Every element of }S\text{ has an inverse for }\ast\qquad$ $\textbf{(E) }\dfrac{1}{2a}\text{ is an inverse for }\ast\text{ of the element }a\text{ of }S$

Let $f(x):\mathbb {Q} \rightarrow \mathbb {Q}$ be a function satisfying $f(x+2y)+f(2x-y)=5f(x)+5f(y)$ Find all such functions.