Given an isosceles triangle $ABC$ with $AB = AC$, suppose $D$ is the midpoint of the $AC$. The circumcircle of the $DBC$ triangle intersects the altitude from $A$ at point $E$ inside the triangle $ABC$, and the circumcircle of the triangle $AEB$ cuts the side $BD$ at point $F$. If $CF$ cuts $AE$ at point $G$, prove that $AE = EG$.

Compute all prime numbers $p$ such that $8p+1$ is a perfect square.

Let $p$ and $q$ be integers. If $k^2+pk+q>0$ for every integer $k$, show that $x^2+px+q>0$ for every real number $x$.

In the acute triangle $ ABC $, $ L $, $ H $ and $ M $ are the intersection points of bisectors, altitudes and medians, respectively, and $ O $ is the center of the circumscribed circle. Denote by $ X $, $ Y $ and $ Z $ the intersection points of $ AL $, $ BL $ and $ CL $ with a circle, respectively. Let $ N $ be a point on the line $ OL $ such that the lines $ MN $ and $ HL $ are parallel. Prove that $ N $ is the intersection point of the medians of $ XYZ $.

A man has $ \$10,000$ to invest. He invests $ \$4000$ at $ 5\%$ and $ \$3500$ at $ 4\%$. In order to have a yearly income of $ \$500$, he must invest the remainder at: $ \textbf{(A)}\ 6\% \qquad\textbf{(B)}\ 6.1\% \qquad\textbf{(C)}\ 6.2\% \qquad\textbf{(D)}\ 6.3\% \qquad\textbf{(E)}\ 6.4\%$

In the game of Flow, a path is drawn through a $3\times3$ grid of squares obeying the following rules: i A path is continuous with no breaks (it can be drawn without lifting a pencil). ii A path that spans multiple squares can only be drawn between colored squares that share a side. iii A path cannot go through a square more than once. Compute the number of ways to color a positive number of squares on the grid such that a valid path can be drawn. An example of one such coloring and a valid path is shown below. [Insert Diagram] [i]Proposed by Alex Li[/i]

$100$ numbers $1$, $1/2$, $1/3$, $...$, $1/100$ are written on the blackboard. One may delete two arbitrary numbers $a$ and $b$ among them and replace them by the number $a + b + ab$. After $99$ such operations only one number is left. What is this final number? (D. Fomin, Leningrad)

If ri are integers such that $0 \le r_i < 31$ and $r_i$ satisfies the polynomial $x^4 + x^3 + x^2 + x \equiv 30$ (mod $31$), find $$\sum^4_{i=1}(r^2_i + 1)^{-1} \,\,\,\, (mod \,\,\,\, 31)$$ where $x^{-1}$ is the modulo inverse of $x$, that is, it is the unique integer $y$ such that $0 < y < 31$ and $xy -1$ is divisible by $31$.

$S$ is a board containing all unit squares in the $xy$ plane whose vertices have integer coordinates and which lie entirely inside the circle $x^2 + y^2 = 1998^2$. In each square of $S$ is written $+1$. An allowed move is to change the sign of every square in $S$ in a given row, column or diagonal. Can we end up with exactly one $-1$ and $+1$ on the rest squares by a sequence of allowed moves?

Does there exist positive integer $n$ such that sum of all digits of number $n(4n+1)$ is equal to $2017$

If $n=\overline{a_1a_2\cdots a_{k-1}a_k}$, be $s(n)$ such that . If $k$ is even, $s(n)=\overline{a_1a_2}+\overline{a_3a_4}\cdots+\overline{a_{k-1}a_k}$ . If $k$ is odd, $s(n)=a_1+\overline{a_2a_3}\cdots+\overline{a_{k-1}a_k}$ For example $s(123)=1+23=24$ and $s(2021)=20+21=41$ Be $n$ is $digital$ if $s(n)$ is a divisor of $n$. Prove that among any 198 consecutive positive integers, all of them less than 2000021 there is one of them that is $digital$.

Is it possible to select $1000$ points in the plane so that $6000$ pairwise distances between them are equal?

Chandler the Octopus is at a tentacle party! At this party, there is $1$ creature with $2$ tentacles, $2$ creatures with $3$ tentacles, $3$ creatures with $4$ tentacles, all the way up to $14$ creatures with $15$ tentacles. Each tentacle is distinguishable from all other tentacles. For some $2\le m < n \le 15$, a creature with m tentacles “meets” a creature with n tentacles; “meeting” another creature consists of shaking exactly 1 tentacle with each other. Find the number of ways there are to pick distinct $m < n$ between $2$ and $15$, inclusive, and then to pick a creature with $m$ tentacles to “meet” a selected creature with $n$ tentacles. [i]Proposed by Armaan Tipirneni, Richard Chen, and Denise the Octopus[/i]

Let SABC be a regular triangular pyramid. Find the set of all points $ D (D! \equal{} S)$ in the space satisfing the equation $ |cos ASD \minus{} 2cosBSD \minus{} 2 cos CSD| \equal{} 3$.

Granny Smith has $\$63$. Elberta has $\$2$ more than Anjou and Anjou has one-third as much as Granny Smith. How many dollars does Elberta have? $ \text{(A)}\ 17\qquad\text{(B)}\ 18\qquad\text{(C)}\ 19\qquad\text{(D)}\ 21\qquad\text{(E)}\ 23 $

Find all real solutions of the system $$\begin{cases} x_1 +x_2 +...+x_{2000} = 2000 \\ x_1^4 +x_2^4 +...+x_{2000}^4= x_1^3 +x_2^3 +...+x_{2000}^3\end{cases}$$

In $\vartriangle ABC$, $AB = AC$ and $\angle BAC = 20^o$. A point $D$ lies on the side $AB$ and $AD = BC$. Find $\angle BCD$. (LF. Sharygin, Moscow)

If $x$ is a positive number such that $x^{x^{x^{x}}} = ((x^{x})^{x})^{x}$, find $(x^{x})^{(x^{x})}$.

Solve in integers the equation $$x^4 +4y^4 = 2(z^4 +4u^4)$$

Find all pairs $(x, y)$ of integers numbers such that $y^3+5=x(y^2+2)$

Let $M,N$ be the midpoints of the sides $AD,BC$ respectively of the convex quadrilateral $ABCD$, $K=AN \cap BM$, $L=CM \cap DN$. Find the smallest possible $c\in R$ such that $S(MKNL)<c \cdot S(ABCD)$ for any convex quadrilateral $ABCD$. I. Voronovich

Let $ABC$ be a triangle with $\angle A = 60^o$. The points $M,N,K$ lie on $BC,AC,AB$ respectively such that $BK = KM = MN = NC$. If $AN = 2AK$, find the values of $\angle B$ and $\angle C$. by Mahdi Etesami Fard

Two parallel lines $r,s$ and two points $P \in r$ and $Q \in s$ are given in a plane. Consider all pairs of circles $(C_P, C_Q)$ in that plane such that $C_P$ touches $r$ at $P$ and $C_Q$ touches $s$ at $Q$ and which touch each other externally at some point $T$. Find the locus of $T$.

Consider all binary sequences of length $n$. In a sequence that allows the interchange of positions of an arbitrary set of $k$ adjacent numbers, ($k < n$), two sequences are said to be [i]equivalent [/i] if they can be transformed from one sequence to another by a finite number of transitions as above. Find the number of sequences that are not equivalent.

Call an $n$-digit integer with distinct digits [i]mountainous [/i]if, for some integer $1 \le k \le n$, the first $k$ digits are in strictly ascending order and the following $n - k$ digits are in strictly descending order. How many $5$-digit mountainous integers with distinct digits are there?