Let $p(x) = 3x^2 + 1$. Compute the largest prime divisor of $p(100) - p(3)$

A plane has no vertex of a regular dodecahedron on it,try to find out how many edges at most may the plane intersect the regular dodecahedron?

Prove the following theorem: If there are $n$ pairs of different points $P_i$, $i = 1, 2, ..., n$, $n > 2$ in three dimensions space, such that each of them is at a smaller distance from one and the same point $Q$ than any other $P_i$, then $n < 15$.

Prove: if $2^{2^n-1}-1$ is a prime, then $n$ is a prime.

Consider the set of all permutations, $\mathcal{P}$, of $\{1,2,\ldots,2022\}$. For permutation $P\in \mathcal{P}$, let $P_1$ denote the first element in $P$. Let $\text{sgn}(P)$ denote the sign of the permutation. Compute the following number modulo 1000: $$\displaystyle\sum_{P\in\mathcal{P}}\dfrac{P_1\cdot\text{sgn}(P)^{P_1}}{2020!}.$$ (The [i]sign[/i] of a permutation $P$ is $(-1)^k$, where $k$ is the minimum number of two-element swaps needed to reach that permutation). [i]Proposed by Nairit Sarkar[/i]

In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$. (a) Show that $AP$ and $BR$ are perpendicular. \\ (b) Show that $FM$ and $BM$ are perpendicular.

A real valued function $f$ is defined on the interval $(-1,2)$. A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$. Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$. Show that if $f'(1)>1$, then $f$ has a fixed point in the interval $(0,1)$.

Compute the radius of the sphere inscribed in the tetrahedron with coordinates $(2,0,0)$, $(4,0,0)$, $(0,1,0)$, and $(0,0,3)$.

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. [asy]unitsize(.3cm); defaultpen(linewidth(.8pt)); path c=Circle((0,2),1); filldraw(Circle((0,0),3),grey,black); filldraw(Circle((0,0),1),white,black); filldraw(c,white,black); filldraw(rotate(60)*c,white,black); filldraw(rotate(120)*c,white,black); filldraw(rotate(180)*c,white,black); filldraw(rotate(240)*c,white,black); filldraw(rotate(300)*c,white,black);[/asy]$ \textbf{(A)}\ \pi \qquad \textbf{(B)}\ 1.5\pi \qquad \textbf{(C)}\ 2\pi \qquad \textbf{(D)}\ 3\pi \qquad \textbf{(E)}\ 3.5\pi$

Arash and Babak play the following game, taking turns alternatively, on a $1400\times 1401$ table. Arash starts and in his turns he colors $k$, $L$-corners (any three cell of a square). Babak in his turn colors one $2\times 2$ square. Neither player is allowed to recolor any cell. Find all positive integers $k$ for which Arash has a winning strategy.

Two players, A and B, play the following game: they retire coins of a pile which contains initially 2006 coins. The players play removing alternatingly, in each move, from 1 to 7 coins, each player keeps the coins that retires. If a player wishes he can pass(he doesn't retire any coin), but to do that he must pay 7 coins from the ones he retired from the pile in past moves. These 7 coins are taken to a separated box and don't interfere in the game any more. The winner is the one who retires the last coin, and A starts the game. Determine which player can win for sure, it doesn't matter how the other one plays. Show the winning strategy and explain why it works.

Let $S$ be the solid in three-dimensional space consisting of all points $(x,y,z)$ satisfying the following six simultaneous conditions: $$ x,y,z \geq 0, \;\; x+y+z\leq 11, \;\; 2x+4y+3z \leq 36, \;\; 2x+3z \leq 44.$$ a) Determine the number $V$ of vertices of $S.$ b) Determine the number $E$ of edges of $S.$ c) Sketch in the $bc$-plane the set of points $(b, c)$ such that $(2,5,4)$ is one of the points $(x, y, z)$ at which the linear function $bx + cy + z$ assumes its maximum value on $S.$

The inscribed circle of the non-isosceles triangle $ABC$ touches sides $AB, BC$ and $AC$ at points $C_1, A_1$ and $B_1$, respectively. The circumscribed circle of the triangle $A_1BC_1$ intersects the lines $B_1A_1$ and $B_1C_1$ at the points $A_0$ and $C_0$, respectively. Prove that the orthocenter of triangle $A_0BC_0$, the center of the inscribed circle of triangle $ABC$ and the midpoint of the $AC$ lie on one straight line.

Let $a,b,c$ be the side lengths of any triangle. Prove that $$\frac{a}{\sqrt{2b^2+2c^2-a^2}}+\frac{b}{\sqrt{2c^2+2a^2-b^2 }}+\frac{c}{\sqrt{2a^2+2b^2-c^2}}\ge \sqrt{3}.$$ [i](Zhuge Liang)[/i]

Let $0 < u < 1$ and define \[u_1 = 1 + u, \quad u_2 = \frac{1}{u_1} + u, \quad \dots, \quad u_{n + 1} = \frac{1}{u_n} + u, \quad n \ge 1.\] Show that $u_n > 1$ for all values of $n = 1$, 2, 3, $\dots$.

The set of three-digit natural numbers formed from digits $1,2, 3, 4, 5, 6$ is called [i]nice [/i] if it satisfies the following condition: for any two different digits from $1, 2, 3, 4, 5, 6$ there exists a number from the set which contains both of them. For any nice set we calculate the sum of all its elements. Determine the smallest possible value of these sums. (E. Barabanov)

The trapezoid is inscribed in a circle. Prove that the sum of distances from any point of the circle to the midpoints of the lateral sides are not less than the diagonal of the trapezoid.

Evaluate $\int_{0}^{\pi/2}\frac{dx}{\left(\sqrt{\sin x}+\sqrt{\cos x}\right)^4}$.

a) Let $1, 2, 3, 5, 6, 7, 10, .., N$ be all the divisors of $N = 2\cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31$ (the product of primes $2$ to $31$) written in increasing order. Below this series of divisors, write the following series of $1$’s or $-1$’s: write $1$ below any number that factors into an even number of prime factors and below a $1$, write $-1$ below the remaining numbers. Prove that the sum of the series of $1$’s and $-1$’s is equal to $0$. b) Let $1, 2, 3, 5, 6, 7, 10, .., N$ be all the divisors of $N = 2\cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37$ (the product of primes $2$ to $37$) written in increasing order. Below this series of divisors, write the following series of $1$’s or $-1$’s: write $1$ below any number that factors into an even number of prime factors and below a $1$, write $-1$ below the remaining numbers. Prove that the sum of the series of $1$’s and $-1$’s is equal to $0$.

On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue. At the intersection of two walls there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet. After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls. Let $k(\mathfrak{L})$ be the largest number $k$ such that, no matter how Merlin paints the labyrinth $\mathfrak{L},$ Morgana can always place at least $k$ knights such that no two of them can ever meet. For each $n,$ what are all possible values for $k(\mathfrak{L}),$ where $\mathfrak{L}$ is a labyrinth with $n$ walls?

For integers $z$, let $\#(z)$ denote the number of integer ordered pairs $(x, y)$ that satisfy $x^2 - xy + y^2 = z$. How many integers $z$ between $0$ and $150$ inclusive satisfy $\#(z) \equiv 6$ (mod $12$)?

Prove that every multiple of $3$ can be written as a sum of four cubes (positive or negatives).

For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n\equiv1\pmod{2^n}$. Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n=a_{n+1}$.

Given a triangle $ABC$, let $P$ and $Q$ be points on segments $\overline{AB}$ and $\overline{AC}$, respectively, such that $AP=AQ$. Let $S$ and $R$ be distinct points on segment $\overline{BC}$ such that $S$ lies between $B$ and $R$, $\angle BPS=\angle PRS$, and $\angle CQR=\angle QSR$. Prove that $P,Q,R,S$ are concyclic (in other words, these four points lie on a circle).

A ticket for the tram costs 1 leva. On the queue in front of the ticket seller are standing $n$ people with a banknote of 1 leva and $m$ people with a banknote of 2 leva. The ticket seller has no money in his cash deck so he can only sell a ticket to a buyer with a banknote of 2 leva, if he has at least 1 banknote of 1 leva. Determine the probability that the ticket seller could sell tickets to all of the people standing in the queue.