Let $a_1, a_2, a_3, a_4, a_5$ be non-negative real numbers satisfied \[\sum_{k = 1}^{5} a_k = 20 \ \ \ \ \text{and} \ \ \ \ \sum_{k=1}^{5} a_k^2 = 100\] Find the minimum and maximum of $\text{max} \{a_1, a_2, a_3, a_4, a_5\}$

Let $ABC$ be an isosceles triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD=2DC$. Let $P$ be a point on the segment $AD$ such that $\angle BAC=\angle BPD$. Prove that $\angle BAC=2\angle DPC$.

Prove that the application $ \mathbb{R}\ni x\mapsto 2x+ \{ x\} $ and its inverse are bijective and continuous.

If $ m,n\in Z$, then $ m^{2} \plus{} 3mn \minus{} 4n^{2}$ cannot be $\textbf{(A)}\ 69 \qquad\textbf{(B)}\ 76 \qquad\textbf{(C)}\ 91 \qquad\textbf{(D)}\ 94 \qquad\textbf{(E)}\ \text{None}$

Let $a,b,c$ be real numbers with $0 < a< 1$, $0 < b < 1$, $0 < c < 1$, and $a+b + c = 2$. Prove that $\dfrac{a}{1-a} \cdot \dfrac{b}{1-b} \cdot \dfrac{c}{1-c} \geq 8$.

Let $D$ and $E$ be points on the sides $BC$ and $AC$ of the triangle $\triangle ABC$, respectively. The circumcircle of $\triangle ADC$ meets the circumcircle of $\triangle BCE$ for the second time at $F$. The line $FE$ meets the line $AD$ at $G$, while the line $FD$ meets the line $BE$ at $H$. Prove that the lines $CF$, $AH$ and $BG$ pass through the same point. [i]Authored by Petar Filipovski[/i]

Prove that every convex $n$-gon of area $1$ contains a quadrilateral of area at least $\frac12 $. .

Two segment $AB,CD$ of the same length are given in plane such that lines $AB,CD$ are not parallel. Consider a point $S$ with the following property: the image of segment $AB$ under point reflection with respect to $S$ is identical to the mirror-image of segment $CD$ with respect to some axis. Find the locus of all such points $S.$

Consider $111$ distinct points which lie on or in the internal of a circle with radius 1.Prove that there are at least $1998$ segments formed by these points with length $\leq \sqrt{3}$

Find all functions $ f : Z\rightarrow Z$ for which we have $ f (0) \equal{} 1$ and $ f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n$, for every natural number $ n$.

Let $p{}$ be a fixed prime number. Juku and Miku play the following game. One of the players chooses a natural number $a$ such that $a>1$ and $a$ is not divisible by $p{}$, his opponent chooses any natural number $n{}$ such that $n>1$. Miku wins if the natural number written as $n{}$ "$1$"s in the positional numeral system with base $a$ is divisible by $p{}$, otherwise Juku wins. Which player has a winning strategy if: (a) Juku chooses the number $a$, tells it to Miku and then Miku chooses the number $n{}$; (b) Juku chooses the number $n{}$, tells it to Miku and then Miku chooses the number $a$?

[u]Set 3[/u] [b]p7.[/b] On unit square $ABCD$, a point $P$ is selected on segment $CD$ such that $DP =\frac14$ . The segment $BP$ is drawn and its intersection with diagonal $AC$ is marked as $E$. What is the area of triangle $AEP$? [b]p8.[/b] Five distinct points are arranged on a plane, creating ten pairs of distinct points. Seven pairs of points are distance $1$ apart, two pairs of points are distance $\sqrt3$ apart, and one pair of points is distance $2$ apart. Draw a line segment from one of these points to the midpoint of a pair of these points. What is the longest this line segment can be? [b]p9.[/b] The inhabitants of Mars use a base $8$ system. Mandrew Mellon is competing in the annual Martian College Interesting Competition of Math (MCICM). The first question asks to compute the product of the base $8$ numerals $1245415_8$, $7563265_8$, and $ 6321473_8$. Mandrew correctly computed the product in his scratch work, but when he looked back he realized he smudged the middle digit. He knows that the product is $1014133027\blacksquare 27662041138$. What is the missing digit? PS. You should use hide for answers.

Consider the sequence $ (x_n)_{n\ge 0}$ where $ x_n\equal{}2^{n}\minus{}1\ ,\ n\in \mathbb{N}$. Determine all the natural numbers $ p$ for which: \[ s_p\equal{}x_0\plus{}x_1\plus{}x_2\plus{}...\plus{}x_p\] is a power with natural exponent of $ 2$.

Four masts stand on a flat horizontal piece of land at the vertices of a square $ABCD$. The height of the mast on $A$ is $7$ meters, of the mast on $B$ $13$ meters, and of the mast on $C$ $15$ meters. Within the square there is a point $P$ on the ground equidistant from each of the tops of these three masts. (a) What length must the sides of the square be at least for this to be possible? (b) The distance from $P$ to the top of the mast on $D$ is equal to the distance from$ P$ to each of the tops of the three other masts. Calculate the height of the mast at $D$.

On the side $AB$ of the cyclic quadrilateral $ABCD$ there is a point $X$ such that diagonal $AC$ bisects the segment $DX$, and the diagonal $BD$ bisects the segment $CX$. What is the smallest possible ratio $|AB | : |CD|$ in such a quadrilateral ?

In the figure on the right, $O$ is the center of the circle, $OK$ and $OA$ are perpendicular to one another, $M$ is the midpoint of $OK$, $BN$ is parallel to $OK$, and $\angle AMN=\angle NMO$. Determine the measure of $\angle A B N$ in degrees. [asy] defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, A=dir(90), K=dir(180), M=0.5*dir(180), N=2/5*dir(90), B=dir(degrees((2/5, sqrt(21/25)))+90); draw(K--O--A--M--N--B--A^^Circle(origin,1)); label("$A$", A, dir(O--A)); label("$K$", K, dir(O--K)); label("$B$", B, dir(O--B)); label("$N$", N, E); label("$M$", M, S); label("$O$", O, SE);[/asy]

An odd prime $p$ is called a prime of the year $2022$ if there is a positive integer $n$ such that $p^{2022}$ divides $n^{2022}+2022$. Show that there are infinitely many primes of the year $2022$.

A regular hexagon is inscribed in a circle of radius $ 10$ inches. Its area is: $ \textbf{(A)}\ 150\sqrt{3} \text{ sq. in.} \qquad \textbf{(B)}\ \text{150 sq. in.} \qquad \textbf{(C)}\ 25\sqrt{3} \text{ sq. in.} \qquad \textbf{(D)}\ \text{600 sq. in.} \qquad \textbf{(E)}\ 300\sqrt{3} \text{ sq. in.}$

Let $E$ be a finite set of points in space such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that $E$ contains the vertices of a tetrahedron $T = ABCD$ such that $T \cap E = \{A,B,C,D\}$ (including interior points of $T$ ) and such that the projection of $A$ onto the plane $BCD$ is inside a triangle that is similar to the triangle $BCD$ and whose sides have midpoints $B,C,D.$

Determine the gcd of all numbers of the form $p^8-1$, with p a prime above 5.

Let $P$ be a polynomial satisfying $P(x + 1) + P(x - 1) = x^3$ for all real numbers $x$. Find the value of $P(12)$.

A finite set $\mathcal{L}$ of coplanar lines, no three of which are concurrent, is called [i]odd[/i] if, for every line $\ell$ in $\mathcal{L}$ the total number of lines in $\mathcal{L}$ crossed by $\ell$ is odd. [list=a] [*]Prove that every finite set of coplanar lines, no three of which are concurrent, extends to an odd set of coplanar lines. [*]Given a positive integer $n$ determine the smallest nonnegative integer $k$ satisfying the following condition: Every set of $n$ coplanar lines, no three of which are concurrent, extends to an odd set of $n+k$ coplanar lines. [/list]

Prove that the following inequality holds for all positive $\{a_i\}$: $$\frac{1}{a_1} + \frac{2}{a_1+a_2} + ... +\frac{ n}{a_1+...+a_n} < 4\left(\frac{1}{a_1} + ... + \frac{1}{a_n}\right)$$

A circle passing through $B$ and $C$ meets the side $[AB]$ of $\triangle ABC$ at $D$, and $[AC]$ at $E$. The circumcircle of $\triangle ACD$ intersects with $BE$ at a point $F$ outside $[BE]$. If $|AD| = 4, |BD|= 8$, then what is $|AF|$? $\textbf{(A)}\ \sqrt3 \qquad\textbf{(B)}\ 2\sqrt6 \qquad\textbf{(C)}\ 4\sqrt6 \qquad\textbf{(D)}\ \sqrt6 \qquad\textbf{(E)}\ \text{None}$

Let $r,n$ be positive integers. For a set $A$, let ${A \choose r}$ denote the family of all $r$-element subsets of $A$. Prove that if $A$ is infinite and $f : {A \choose r} \to {1,2,...,n}$ is any function, then there exists an infinite subset $B$ of $A$ such that $f(X) = f(Y)$ for all $X,Y \in {B \choose r}$.