For which values of N is it possible to write numbers from 1 to N in some order so that for any group of two or more consecutive numbers, the arithmetic mean of these numbers is not whole?

Let $N$ be a positive integer. Consider the sequence $a_1, a_2, ..., a_N$ of positive integers, none of which is a multiple of $2^{N+1}$. For $n \ge N +1$, the number $a_n$ is defined as follows: choose $k$ to be the number among $1, 2, ..., n - 1$ for which the remainder obtained when $a_k$ is divided by $2^n$ is the smallest, and define $a_n = 2a_k$ (if there are more than one such $k$, choose the largest such $k$). Prove that there exist $M$ for which $a_n = a_M$ holds for every $n \ge M$.

The centers of the three circles A, B, and C are collinear with the center of circle B lying between the centers of circles A and C. Circles A and C are both externally tangent to circle B, and the three circles share a common tangent line. Given that circle A has radius $12$ and circle B has radius $42,$ find the radius of circle C.

Consider a flat field on which there exist a valley in the form of an infinite strip with arbitrary width $\omega$. There exist a polyhedron of diameter $d$(Diameter in a polyhedron is the maximum distance from the points on the polyhedron) is in one side and a pit of diameter $d$ on the other side of the valley. We want to roll the polyhedron and put it into the pit such that the polyhedron and the field always meet each other in one point at least while rolling (If the polyhedron and the field meet each other in one point at least then the polyhedron would not fall into the valley). For crossing over the bridge, we have built a rectangular bridge with a width of $\frac{d}{10}$ over the bridge. Prove that we can always put the polyhedron into the pit considering the mentioned conditions. (You will earn a good score if you prove the decision for $\omega = 0$).

Letters $A,B,C,$ and $D$ represent four different digits from 0,1,2,3...9. If $(A+B)/(C+D)$ is an integer that is as large as possible, what is the value of $A+B$? $\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17$

sorry if this has been posted before . given a fixed circle $(O)$ and two fixed point $B,C$ on it.point A varies on circle $(O)$. let $I$ be the midpoint of $BC$ and $H$ be the orthocenter of $\triangle ABC$. ray $IH$ meet $(O)$ at $K$ ,$AH$ meet $BC$ at $D$ ,$KD$ meet $(O)$ at $M$ .a line pass $M$ and perpendicular to $BC$ meet $AI$ at $N$. a) prove that $N$ varies on a fixed circle. b) a circle pass $N$ and tangent to $AK$ at $A$ cut $AB,AC$ at $P,Q$. let $J$ be the midpoint of $PQ$ .prove that $AJ$ pass through a fixed point.

A plane through the center of a torus is tangent to the torus. Prove that the intersection of the plane and the torus consists of two circles.

For positive reals $a,b,c$ with $\sqrt{a}+\sqrt{b}+\sqrt{c}\ge 3$ prove that $$\frac{a^3}{a^2+b}+\frac{b^3}{b^2+c}+\frac{c^3}{c^2+a}\ge \frac{3}{2}$$

A big $ L$ is formed as shown. What is its area? [asy]unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(12pt)); draw((0,0)--(5,0)--(5,2)--(2,2)--(2,8)--(0,8)--cycle); label("5",(2.5,0),S); label("2",(5,1),E); label("2",(1,8),N); label("8",(0,4),W);[/asy]$ \textbf{(A)}\ 22 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 30$

For how many pairs of primes $(p, q)$, is $p^2 + 2pq^2 + 1$ also a prime?

Say that a rational number is [i]special[/i] if its decimal expansion is of the form $0.\overline{abcdef}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are digits (possibly equal) that include each of the digits $2$, $0$, $1$, and $5$ at least once (in some order). How many special rational numbers are there?

Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that the following two conditions hold: [b]1. [/b] For all real numbers $a$ and $b$ satisfying $a^2 + b^2 = 1$, We have $f(x) + f(y) \geq f(ax + by)$ for all real numbers $x, y$. [b]2.[/b] For all real numbers $x$ and $y$, there exist real numbers $a$ and $b$, such that $a^2 + b^2 = 1$ and $f(x) + f(y) = f(ax + by)$. [i]Proposed by Zaza Melikidze, Georgia[/i]

Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$

Given two non-parallel segments $AB$ and $CD$ on the plane, find the locus of points $P$ on the plane such that the area of triangle $ABP$ is equal to the area of triangle $CDP$.

Give the answer about the following propositions $(p),\ (q)$ whether they are true or not. If the answer is true, then give the proof and if the answer is false, then give the proof by giving the counter example. $(p)$ If we can form a triangle such that one of inner angles of the triangle is $60^\circ$ by choosing 3 points from the vertices of a regular $n$-polygon, then $n$ is a multiple of 3. $(q)$ In $\triangle{ABC},\ \triangle{A'B'C'}$, if $AB=A'B',\ BC=B'C',\ \angle{A}=\angle{A'}$, then these triangles are congruent. 30 points

Mr. DoBa has a bag of markers. There are 2 blue, 3 red, 4 green, and 5 yellow markers. Mr. DoBa randomly takes out two markers from the bag. The probability that these two markers are different colors can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Raina Yang[/i]