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?

Define $a \star b$ to be $2ab + a + b$. What is $((3 \star 4) \star 5) - (4 \star (5 \star 3))$ ?

a) $100$ numbers (some positive, some negative) are written in a row. All of the following three types of numbers are underlined: 1) every positive number, 2) every number whose sum with the number following it is positive, 3) every number whose sum with the two numbers following it is positive. Can the sum of all underlined numbers be (i) negative? (ii) equal to zero? b) $n$ numbers (some positive and some negative) are written in a row. Each positive number and each number whose sum with several of the numbers following it is positive is underlined. Prove that the sum of all underlined numbers is positive.