Natural nonzero numder, which consists of $ m$ digits, is called hiperprime, if its any segment, which consists $ 1,2,...,m$ digits is prime (for example $ 53$ is hiperprime, because numbers $ 53,3,5$ are prime). Find all hiperprime numbers.

Is the following statement true? For each positive integer $n$, we can find eight nonnegative integers $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$ such that $n=\frac{2^a-2^b}{2^c-2^d}\cdot\frac{2^e-2^f}{2^g-2^h}$.

Squence $(x_n)$ satisfies that $x_{n+1}=x_n-x_{n-1}(n\geq2)$. If $x_1=a,x_2=b$, $S_n=x_1+x_2+\cdots+x_n$. Wich one is correct? $\text{(A)}x_{100}=-a,S_{100}=2b-a$ $\text{(B)}x_{100}=-b,S_{100}=2b-a$ $\text{(C)}x_{100}=-a,S_{100}=b-a$ $\text{(D)}x_{100}=-b,S_{100}=b-a$

The incircle of a triangle $ A_1A_2A_3$ is centered at $ O$ and meets the segment $ OA_j$ at $ B_j$ , $ j \equal{} 1, 2, 3$. A circle with center $ B_j$ is tangent to the two sides of the triangle having $ A_j$ as an endpoint and intersects the segment $ OB_j$ at $ C_j$. Prove that \[ \frac{OC_1\plus{}OC_2\plus{}OC_3}{A_1A_2\plus{}A_2A_3\plus{}A_3A_1} \leq \frac{1}{4\sqrt{3}}\] and find the conditions for equality.

Let $\triangle ABC$ be a triangle such that $AB=5,AC=8,$ and $\angle BAC=60^{\circ}$. Let $\Gamma$ denote the circumcircle of $ABC$, and let $I$ and $O$ denote the incenter and circumcenter of $\triangle ABC$, respectively. Let $P$ be the intersection of ray $IO$ with $\Gamma$, and let $X$ be the intersection of ray $BI$ with $\Gamma$. If the area of quadrilateral $XICP$ can be expressed as $\frac{a\sqrt{b}+c\sqrt{d}}{e}$, where $a$ and $d$ are squarefree positive integers and $\gcd(a,c,e)=1$, compute $a+b+c+d+e$.

Which one of the following bar graphs could represent the data from the circle graph? [asy] unitsize(36); draw(circle((0,0),1),gray); fill((0,0)--arc((0,0),(0,-1),(1,0))--cycle,gray); fill((0,0)--arc((0,0),(1,0),(0,1))--cycle,black); [/asy] [asy] unitsize(4); fill((1,0)--(1,15)--(5,15)--(5,0)--cycle,gray); fill((6,0)--(6,15)--(10,15)--(10,0)--cycle,black); draw((11,0)--(11,20)--(15,20)--(15,0)); fill((26,0)--(26,15)--(30,15)--(30,0)--cycle,gray); fill((31,0)--(31,15)--(35,15)--(35,0)--cycle,black); draw((36,0)--(36,15)--(40,15)--(40,0)); fill((51,0)--(51,10)--(55,10)--(55,0)--cycle,gray); fill((56,0)--(56,10)--(60,10)--(60,0)--cycle,black); draw((61,0)--(61,20)--(65,20)--(65,0)); fill((76,0)--(76,10)--(80,10)--(80,0)--cycle,gray); fill((81,0)--(81,15)--(85,15)--(85,0)--cycle,black); draw((86,0)--(86,20)--(90,20)--(90,0)); fill((101,0)--(101,15)--(105,15)--(105,0)--cycle,gray); fill((106,0)--(106,10)--(110,10)--(110,0)--cycle,black); draw((111,0)--(111,20)--(115,20)--(115,0)); for(int a = 0; a < 5; ++a) { draw((25*a,21)--(25*a,0)--(25*a+16,0)); } label("(A)",(8,21),N); label("(B)",(33,21),N); label("(C)",(58,21),N); label("(D)",(83,21),N); label("(E)",(108,21),N); [/asy]

Consider the infinite sequence $\{a_i\}$ that extends the pattern \[1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, \dots\] Formally, $a_i = i-T(i)$ for all $i \geq 1$, where $T(i)$ represents the largest triangular number less than $i$ (triangle numbers are integers of the form $\frac{k(k+1)}2$ for some nonnegative integer $k$). Find the number of indices $i$ such that $a_i = a_{i + 2020}$. [i]Proposed by Gabriel Wu[/i]

For any $a>0$,set $\mathcal{S}(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}$. Show that there are no three positive reals $a,b,c$ such that \[ \mathcal{S}(a)\cap \mathcal{S}(b)=\mathcal{S}(b)\cap \mathcal{S}(c)=\mathcal{S}(c)\cap \mathcal{S}(a)=\emptyset \] \[ \mathcal{S}(a)\cup \mathcal{S}(b)\cup \mathcal{S}(c)=\mathbb{N} \]

Let $M$ be a positive integer and consider the set \[S=\{n \in \mathbb{N}\; \vert \; M^{2}\le n <(M+1)^{2}\}.\] Prove that the products of the form $ab$ with $a, b \in S$ are distinct.

Construct a right-angled triangle along its two medians, starting from the acute angles.

Sines of three acute angles form an arithmetic progression, while the cosines of these angles form a geometric progression. Prove that all three angles are equal.

Let $ABC$ be a triangle with $AB = AC$. Suppose that the bisector of $\angle ABC$ meets the side $AC$ at point $D$ such that $BC = BD+AD$. Find the measure of $\angle BAC$.

Let $n$ be a natural number. We define sequences $\langle a_i\rangle$ and $\langle b_i\rangle$ of integers as follows. We let $a_0=1$ and $b_0=n$. For $i>0$, we let $$\left( a_i,b_i\right)=\begin{cases} \left(2a_{i-1}+1,b_{i-1}-a_{i-1}-1\right) & \text{if } a_{i-1}<b_{i-1},\\ \left( a_{i-1}-b_{i-1}-1,2b_{i-1}+1\right) & \text{if } a_{i-1}>b_{i-1},\\ \left(a_{i-1},b_{i-1}\right) & \text{if } a_{i-1}=b_{i-1}.\end{cases}$$ Given that $a_k=b_k$ for some natural number $k$, prove that $n+3$ is a power of two.

LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid $A$ dollars and Bernardo had paid $B$ dollars, where $A < B$. How many dollars must LeRoy give to Bernardo so that they share the costs equally? $ \textbf{(A)}\ \frac{A+B}{2} \qquad \textbf{(B)}\ \frac{A-B}{2} \qquad \textbf{(C)}\ \frac{B-A}{2} \qquad \textbf{(D)}\ B-A \qquad \textbf{(E)}\ A+B $

The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller? $\textbf{(A) }\dfrac54\qquad\textbf{(B) }\dfrac32\qquad\textbf{(C) }\dfrac95\qquad\textbf{(D) }2\qquad\textbf{(E) }\dfrac52$

Find all positive integers $a,b$ for which $a^4+4b^4$ is a prime number.

Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$. Find the remainder when $d$ is divided by $2013$.

Let $a$ and $b$ be two sides of a triangle. How should the third side $c$ be chosen so that the points of contact of the incircle and the excircle with side $c$ divide that side into three equal segments? (The excircle corresponding to the side $c$ is the circle which is tangent to the side $c$ and to the extensions of the sides $a$ and $b$.) (Folklore)

Given a non-negative integer \( n \), determine the values of \( c \) for which the sequence of numbers \[ a_n = 4^n c + \frac{4^n - (-1)^n}{5} \] contains at least one perfect square.

Let $ABC$ be an acute, non isosceles with $I$ is its incenter. Denote $D, E$ as tangent points of $(I)$ on $AB,AC$, respectively. The median segments respect to vertex $A$ of triangles $ABE$ and $ACD$ meet$ (I)$ at$ P,Q,$ respectively. Take points $M, N$ on the line $DE$ such that $AM \parallel BE$ and $AN \parallel C D$ respectively. a) Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$. b) Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.

Is it possible to insert numbers $1, \ldots, 16$ into a table $4 \times 4$ (each cell should have a different number) so that every two adjacent cells (i.e. cells sharing a common side) have numbers $a$ and $b$ satisfying\\ (a) $|a-b| \geq 6$\\ (b) $|a-b| \geq 7$

For all non-negative real numbers $x,y,z$ with $x \geq y$, prove the inequality $$\frac{x^3-y^3+z^3+1}{6}\geq (x-y)\sqrt{xyz}.$$

Let $P$ on $AB$, $Q$ on $BC$, $R$ on $CD$ and $S$ on $AD$ be points on the sides of a convex quadrilateral $ABCD$. Show that the following are equivalent: (1) There is a choice of $P,Q,R,S$, for which all of them are interior points of their side, such that $PQRS$ has minimal perimeter. (2) $ABCD$ is a cyclic quadrilateral with circumcenter in its interior.

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant. [The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]

Given that $ x \plus{} \sin y \equal{} 2008$ and $ x \plus{} 2008 \cos y \equal{} 2007$, where $ 0 \leq y \leq \pi/2$, find the value of $ x \plus{} y$.