Let $ABCD$ be a convex quadrilateral with $AB = BD = 8$ and $CD = DA = 6$. Let $P$ be a point on side $AB$ such that $DP$ is bisector of angle $\angle ADB$ and let $Q$ be a point on side $BC$ such that $DQ$ is bisector of angle $\angle CDB$. Calculate the radius of the circumcircle of triangle $DPQ$. Note: The circumcircle of a triangle is the circle that passes through its three vertices.

$\frac{1000^2}{252^2 - 248^2}$ equals $\textbf{(A) }62,500\qquad \textbf{(B) }1000\qquad\textbf{(C) }500\qquad\textbf{(D) }250\qquad\textbf{(E) } \frac{1}{2}$

A point $(x,y)$ is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line $x+y = 2$. Then $x$ is A. $\sqrt{2} - 1$ B. $\frac{1}{2}$ C. $2 - \sqrt{2}$ D. 1 E. Not uniquely determined

Let $a_1=2$ and, for every positive integer $n$, let $a_{n+1}$ be the smallest integer strictly greater than $a_n$ that has more positive divisors than $a_n$. Prove that $2a_{n+1}=3a_n$ only for finitely many indicies $n$. [i] Proposed by Ilija Jovčevski, North Macedonia[/i]

Find all functions $f:\mathbb N\to\mathbb N$ such that $$f(x)+f(y)\mid x^2-y^2$$holds for all $x,y\in\mathbb N$.

$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.

Let $f(z)=\frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $|a|=1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $|b|$? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \sqrt{2}-1 \qquad \textbf{(C)}\ \sqrt{3}-1 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

Let be a ring $ A. $ Denote with $ N(A) $ the subset of all nilpotent elements of $ A, $ with $ Z(A) $ the center of $ A, $ and with $ U(A) $ the units of $ A. $ Prove: [b]a)[/b] $ Z(A)=A\implies N(A)+U(A)=U(A) . $ [b]b)[/b] $ \text{card} (A)\in\mathbb{N}\wedge a+U(A)\subset U(A)\implies a\in N(A) . $

Consider $A = \{1, 2, ..., 16\}$. A partition of $A$ into nonempty sets $A_1, A_2,..., A_n$ is said to be good if none of the Ai contains elements $a, b, c$ (not necessarily distinct) such that $a = b + c$. (a) Find a good partition $\{A_1, A_2, A_3, A_4\}$ of $A$. (b) Prove that no partition $\{A_1, A_2, A_3\}$ of $A$ is good

For any integer $n \ge 1$, show that $$\sum_{k=1}^{n} \frac{2^k}{\sqrt{k+0.5}} \le 2^{n+1}\sqrt{n+1}-\frac{4n^{3/2}}{3}$$

Consider a triangle $ABC$ with circumcenter $O$ and incenter $I$. Incircle touches sides $BC,CA$ and $AB$ at $D, E$ and $F$. $K$ is a point such that $KF$ is tangent to circumcircle of $BFD$ and $KE$ is tangent to circumcircle of $CED$. Prove that $BC,OI$ and $AK$ are concurrent.

Convex equiangular hexagon $ABCDEF$ has $AB = CD = EF = \sqrt 3$ and $BC = DE = FA = 2.$ Points $X, Y,$ and $Z$ are situated outside the hexagon such that $AEX, ECY,$ and $CAZ$ are all equilateral triangles. Compute the area of the region bounded by lines $XF, YD, $ and $ZB.$

The cells of a $100 \times 100$ table contain non-zero numbers. It turned out that all $100$ hundred-digit numbers written horizontally are divisible by 11. Could it be that exactly $99$ hundred-digit numbers written vertically are also divisible by $11$?

Let two natural number $n > 1$ and $m$ be given. Find the least positive integer $k$ which has the following property: Among $k$ arbitrary integers $a_1, a_2, \ldots, a_k$ satisfying the condition $a_i - a_j$ ( $1 \leq i < j \leq k$) is not divided by $n$, there exist two numbers $a_p, a_s$ ($p \neq s$) such that $m + a_p - a_s$ is divided by $n$.

A wooden block (mass $M$) is hung from a peg by a massless rope. A speeding bullet (with mass $m$ and initial speed $v_\text{0}$) collides with the block at time $t = \text{0}$ and embeds in it. Let $S$ be the system consisting of the block and bullet. Which quantities are conserved between $t = -\text{10 s}$ and $ t = \text{+10 s}$? [asy] // Code by riben draw(circle((0,0),0.3),linewidth(2)); filldraw(circle((0,0),0.3),gray); draw((0,-0.8)--(0,-15.5),linewidth(2)); draw((5,-15.5)--(-5,-15.5)--(-5,-20.5)--(5,-20.5)--cycle,linewidth(2)); filldraw((5,-15.5)--(-5,-15.5)--(-5,-20.5)--(5,-20.5)--cycle,gray); draw((-15,-18)--(-16,-17)--(-18,-17)--(-18,-19)--(-16,-19)--cycle,linewidth(2)); filldraw((-15,-18)--(-16,-17)--(-18,-17)--(-18,-19)--(-16,-19)--cycle,gray); [/asy] (A) The total linear momentum of $S$. (B) The horizontal component of the linear momentum of $S$. (C) The mechanical energy of $S$. (D) The angular momentum of $S$ as measured about a perpendicular axis through the peg. (E) None of the above are conserved.

Quadrilateral $ABCD$ with side lengths $AB=7, BC = 24, CD = 20, DA = 15$ is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form $\frac{a\pi - b}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ have no common prime factor. What is $a+b+c$? $\textbf{(A) } 260 \qquad \textbf{(B) } 855 \qquad \textbf{(C) } 1235 \qquad \textbf{(D) } 1565 \qquad \textbf{(E) } 1997$

An integer $N \ge 2$ is given. A collection of $N(N + 1)$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $N(N - 1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions hold: ($1$) no one stands between the two tallest players, ($2$) no one stands between the third and fourth tallest players, $\;\;\vdots$ ($N$) no one stands between the two shortest players. Show that this is always possible. [i]Proposed by Grigory Chelnokov, Russia[/i]

Let $M>1$ be a natural number. Tom and Jerry play a game. Jerry wins if he can produce a function $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying [list] [*]$f(M) \ne M$ [/*] [*] $f(k)<2k$ for all $k \in \mathbb{N}$[/*] [*] $f^{f(n)}(n)=n$ for all $n \in \mathbb{N}$. For each $\ell>0$ we define $f^{\ell}(n)=f\left(f^{\ell-1}(n)\right)$ and $f^0(n)=n$[/*] [/list] Tom wins otherwise. Prove that for infinitely many $M$, Tom wins, and for infinitely many $M$, Jerry wins. [i]Proposed by Anant Mudgal[/i]

A point $P$ was chosen on the smaller arc $BC$ of the circumcircle of the acute-angled triangle $ABC$. Points $R$ and $S$ on the sides$ AB$ and $AC$ are respectively selected so that $CPRS$ is a parallelogram. Point $T$ on the arc $AC$ of the circumscribed circle of $\vartriangle ABC$ such that $BT \parallel CP$. Prove that $\angle TSC = \angle BAC$. (Anton Trygub)

Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]

For real numbers $a, b, c, d \in [0, 1]$, find the largest possible value of the following expression: $$a^2+b^2+c^2+d^2-ab-bc-cd-da$$ [i]Proposed by Mykhailo Shtandenko[/i]

Given two circles $(O_1)$ and $(O_2)$ intersect at $A$ and $B$. Let $d_1$ and $d_2$ be two lines through $A$ and be symmetric with respect to $AB$. The line $d_1$ cuts $(O_1)$ and $(O_2)$ at $G, E$ ($\ne A$), respectively, the line $d_2$ cuts $(O_1)$ and $(O_2)$ at $F, H$ ($\ne A$), respectively, such that $E$ is between $A, G$ and $F$ is between $A, H$. Let $J$ be the intersection of $EH$ and $FG$. The line $BJ$ cuts $(O_1), (O_2)$ at $K, L$ ($\ne B$), respectively. Let $N$ be the intersection of $O_1K$ and $O_2L$. Prove that the circle $(NLK)$ is tangent to $AB$.

An ant starts at the bottom left corner of a $5\times5$ grid of dots and walks to the top right corner. It can walk from one dot to any dot that is horizontally or vertically adjacent to it. If it never walks between the same pair of dots twice, what is the length of the longest path the ant can take? $\text{(A) }30\qquad\text{(B) }31\qquad\text{(C) }32\qquad\text{(D) }33\qquad\text{(E) }34$

Let $\mathcal{S}$ be a set of integers. Given a real positive $r$, we say that $\mathcal{S}$ is a $r$-discerning, if for any pair $m, n > 1$ of distinct integers such that $\left| \frac{m-n}{m+n} \right|  < r$, there exists $a \in \mathcal{S}$ and $k \geq 1$ such that $a^k$ divides $m$ but not $n$, or $a^k$ divides $n$ but not $m$ 1. Show that for every $r > 0$ every $r$-discerning set contains an infinite number of primes. 2. For every $r > 0$ determine the maximal possible cardinality of $\mathcal{P} \backslash \mathcal{S}$ where $\mathcal{P}$ is the set of primes and $\mathcal{S} \subseteq \mathcal{P}$ is a $r$-discerning set.

Let $ABCD$ be a parallelogram, and let $P$ be a point on the side $AB$. Let the line through $P$ parallel to $BC$ intersect the diagonal $AC$ at point $Q$. Prove that $$|DAQ|^2 = |PAQ| \times |BCD| ,$$ where $|XY Z|$ denotes the area of triangle $XY Z$.