The volume of a pyramid whose base is an equilateral triangle of side length 6 and whose other edges are each of length $ \sqrt{15}$ is $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 9/2 \qquad \textbf{(C)}\ 27/2 \qquad \textbf{(D)}\ \frac{9\sqrt3}{2} \qquad \textbf{(E)}\ \text{none of these}$

An ant in the $xy$-plane is at the origin facing in the positive $x$-direction. The ant then begins a progression of moves, on the $n^{th}$ of which it first walks $\frac{1}{5^n}$ units in the direction it is facing and then turns $60^o$ degrees to the left. After a very large number of moves, the ant’s movements begins to converge to a certain point; what is the $y$-value of this point?

For each natural number $n$ we denote $a_n$ as the greatest perfect square less than or equal to $n$ and $b_n$ as the least perfect square greater than $n$. For example $a_9=3^2$, $b_9=4^2$ and $a_{20}=4^2$, $b_{20}=5^2$. Calculate: $$\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}$$

We’re playing a game with a sequence of $2008$ non-negative integers. A move consists of picking a integer $b$ from that sequence, of which the neighbours $a$ and $c$ are positive. We then replace $a, b$ and $c$ by $a - 1, b + 7$ and $c - 1$ respectively. It is not allowed to pick the first or the last integer in the sequence, since they only have one neighbour. If there is no integer left such that both of its neighbours are positive, then there is no move left, and the game ends. Prove that the game always ends, regardless of the sequence of integers we begin with, and regardless of the moves we make.

Prove that the equation $m!n! = k!$ has infinitely many solutions in which $m, n$ and $k$ are natural numbers greater than unity .

Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_0=1$ and $x_{n+1}=\sin(x_n)+\frac{\pi} {2}-1$ for all $n \geq 0$. Show that the sequence converges and find its limit.

Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation \[ \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}. \]

An ant is walking on a sidewalk and discovers $12$ sidewalk panels with leaves inscribed in them, as shown below. Find the number of ways in which the ant can traverse from point $A$ to point $B$ if it can only move [list] [*] up, down, or right (along the border of a sidewalk panel), or [*] up-right (along one of two margin halves of a leaf) [/list] and cannot visit any border or margin half more than once (an example path is highlighted in red). [asy] unitsize(1cm); int r = 4; int c = 5; for (int i = 0; i < r; ++i) { for (int j = 0; j < c; ++j) { pair A = (j,i); } } for (int i = 0; i < r; ++i) { for (int j = 0; j < c; ++j) { if (j != c-1) { draw((j,i)--(j+1,i)); } if (i != r-1) { draw((j,i)--(j,i+1)); } } } for (int i = 1; i < r+1; ++i) { for (int j = 0; j < c-2; ++j) { fill(arc((i,j),1,90,180)--cycle,deepgreen); fill(arc((i-1,j+1),1,270,360)--cycle,deepgreen); draw((i-1,j)--(i,j+1), heavygreen+linewidth(0.5)); draw((i-2/3,j+1/3)--(i-2/3,j+1/3+0.1), heavygreen); draw((i-1/3,j+2/3)--(i-1/3,j+2/3+0.1), heavygreen); draw((i-2/3,j+1/3)--(i-2/3+0.1,j+1/3), heavygreen); draw((i-1/3,j+2/3)--(i-1/3+0.1,j+2/3), heavygreen); draw(arc((i,j),1,90,180)); draw(arc((i-1,j+1),1,270,360)); } } draw((0,3)--(0,1), red+linewidth(1.5)); draw((0,3)--(0,1), red+linewidth(1.5)); draw(arc((1,1),1,90,180), red+linewidth(1.5)); draw((1,2)--(1,1)--(2,1), red+linewidth(1.5)); draw(arc((2,2),1,270,360), red+linewidth(1.5)); draw(arc((4,2),1,90,180), red+linewidth(1.5)); draw((4,3)--(4,0), red+linewidth(1.5)); dot((0,3)); dot((4,0)); label("$A$", (0,3), NW); label("$B$", (4,0), SE); [/asy]

Can we find a convex quadrilateral $ABCD$ with an interior point $P$ satisfying \[AB=AP, \quad BC=BP, \quad CD=CP, \quad \text{and} \quad DA=DP \quad ?\]

If $x > 0$ and $x^2 +\frac{1}{x^2}= 14$, find $x^5 +\frac{1}{x^5}$.

If we divide number $19250$ with one number, we get remainder $11$. If we divide number $20302$ with the same number, we get the reamainder $3$. Which number is that?

For a sale, a store owner reduces the price of a $10$ dollar scarf by $20\%$. Later the price is lowered again, this time by one-half the reduced price. The price is now $ \text{(A)}\ 2.00\text{ dollars}\qquad\text{(B)}\ 3.75\text{ dollars}\qquad\text{(C)}\ 4.00\text{ dollars}\qquad\text{(D)}\ 4.90\text{ dollars}\qquad\text{(E)}\ 6.40\text{ dollars} $

How many pairs of positive integers $ (a,b)$ are there such that $ \gcd(a,b) \equal{} 1$ and \[ \frac {a}{b} \plus{} \frac {14b}{9a} \]is an integer? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ \text{infinitely many}$

There are 7 numbers on a board. The product of any four of them is divisible by 2023. Prove that at least one of the numbers on the board is divisible by 119.

Three married couples arrange a party. They arrive at the party one at a time, the couples not necessarily arriving together. They all, upon arriving, shake the hand of everyone already there, except their own spouse. When everyone has arrived, someone asks all the others how many hands they shook upon arriving, and gets five different answers. How many hands did he himself shake upon arriving? A. 0 B. 1 C. 2 D. 3 E. 4

Let the numbers $1, 2, \ldots , n^2$ be written in the cells of an $n \times n$ square board so that the entries in each column are arranged increasingly. What are the smallest and greatest possible sums of the numbers in the $k^{th}$ row? ($k$ a positive integer, $1 \leq k \leq n$.)

Find the smallest possible number of divisors a positive integer $n$ may have, which satisfies the following conditions: 1. $24 \mid n+1$; 2. The sum of the squares of all divisors of $n$ is divisible by $48$ ($1$ and $n$ are included).

A regular tetrahedron has two vertices on the body diagonal of a cube with side length $12$. The other two vertices lie on one of the face diagonals not intersecting that body diagonal. Find the side length of the tetrahedron.

Let $a,b,c$ be positive integers which satisfy the condition: $16(a+b+c)\geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$. Prove that \[\sum_{cyc} \left( \frac{1}{a+b+\sqrt{2a+2c}} \right)^{3}\leq \frac{8}{9}\]

Let $E$ be an interior point of the convex quadrilateral $ABCD$. Construct triangles $\triangle ABF,\triangle BCG,\triangle CDH$ and $\triangle DAI$ on the outside of the quadrilateral such that the similarities $\triangle ABF\sim\triangle DCE,\triangle BCG\sim \triangle ADE,\triangle CDH\sim\triangle BAE$ and $ \triangle DAI\sim\triangle CBE$ hold. Let $P,Q,R$ and $S$ be the projections of $E$ on the lines $AB,BC,CD$ and $DA$, respectively. Prove that if the quadrilateral $PQRS$ is cyclic, then \[EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.\]

Given natural $n$. We shall call "universal" such a sequence of natural number $a_1, a_2, ... , a_k, k\ge n$, if we can obtain every transposition of the first $n$ natural numbers (i.e such a sequence of $n$ numbers, that every one is encountered only once) by deleting some its members. (Examples: ($1,2,3,1,2,1,3$) is universal for $n=3$, and ($1,2,3,2,1,3,1$) -- not, because you can't obtain ($3,1,2$) from it.) The goal is to estimate the length of the shortest universal sequence for given $n$. a) Give an example of the universal sequence of $n2$ members. b) Give an example of the universal sequence of $(n^2 - n + 1)$ members. c) Prove that every universal sequence contains not less than $n(n + 1)/2$ members d) Prove that the shortest universal sequence for $n=4$ contains 12 members e) Find as short universal sequence, as you can. The Organising Committee knows the method for $(n^2 - 2n +4) $ members.

[b]Form 1[/b] Consider the statements: $\textbf{(1)}\ p\text{ } \wedge\sim q\wedge r \qquad \textbf{(2)}\ \sim p\text{ } \wedge\sim q\wedge r\qquad \textbf{(3)}\ p\text{ } \wedge\sim q\text{ }\wedge \sim r \qquad \textbf{(4)}\ \sim p\text{ } \wedge q\text{ }\wedge r $, where $p,q,$ and $r$ are propositions. How many of these imply the truth of $(p\rightarrow q)\rightarrow r$? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$ [b]Form 2[/b] Consider the statements $(1)$ $p$ and $r$ are true and $q$ is false $(2)$ $r$ is true and $p$ and $q$ are false $(3)$ $p$ is true and $q$ and $r$ are false $(4)$ $q$ and $r$ are true and $p$ is false. How many of these imply the truth of the statement "$r$ is implied by the statement that $p$ implies $q$"? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Let $S$ is the set of permutation of {1,2,3,4,5,6,7,8} (1)For all $\sigma=\sigma_1\sigma_2...\sigma_8\in S$ Evaluate the sum of S=$\sigma_1\sigma_2+\sigma_3\sigma_4+\sigma_5\sigma_6+\sigma_7\sigma_8$. Then for all elements in $S$,what is the arithmetic mean of S? (Notice $S$ and S are different.) (2)In $S$, how many permutations are there which satisfies "For all $k=1,2,...,7$,the digit after k is [b]not[/b] (k+1)"?

[asy] real s=sqrt(3)/2; draw(box((0,0),(1,1))); draw((1+s,0.5)--(1,1)); draw((1+s,0.5)--(1,0)); draw((0,1)--(1+s,0.5)); label("$A$",(1,1),N); label("$B$",(1,0),S); label("$C$",(0,0),W); label("$D$",(0,1),W); label("$E$",(1+s,0.5),E); //Credit to TheMaskedMagician for the diagram [/asy] In the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle and point $E$ is outside square $ABCD$. What is the measure of $\measuredangle AED$ in degrees? $\textbf{(A) }10\qquad\textbf{(B) }12.5\qquad\textbf{(C) }15\qquad\textbf{(D) }20\qquad\textbf{(E) }25$

Two triangles $ABC,ABD$ (with the common side $c=AB$) are given in space. Triangle $ABC$ is right with hypotenuse $AB$, $ABD$ is equilateral. Denote $\varphi$ the dihedral angle between planes $ABC,ABD$. 1) Determine the length of $CD$ in terms of $a=BC,b=CA,c$ and $\varphi$. 2) Determine all values of $\varphi$ such that the tetrahedron $ABCD$ has four sides of the same length.