Let $A$ be Abelian group of order $p^4$, where $p$ is a prime number, and which has a subgroup $N$ with order $p$ such that $A/N\approx\mathbb{Z}/p^3\mathbb{Z}$. Find all $A$ expect isomorphic.

Source: 1976 Euclid Part A Problem 9 ----- A circle has an inscribed triangle whose sides are $5\sqrt{3}$, $10\sqrt{3}$, and $15$. The measure of the angle subtended at the centre of the circle by the shortest side is $\textbf{(A) } 30 \qquad \textbf{(B) } 45 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } \text{none of these}$

The world famous infinite hotel with infinitely many floors (where the floors and the rooms on each floor are numbered with the positive integers) is full of guests: each room is occupied by exactly one guest. The manager of the hotel wants to carpet the corridor on each floor, and an infinite set of carpets of finite length (numbered with the positive integers) was obtained. Every guest marked an infinite number of carpets that they liked. Luckily, any two guests living on a different floor share only a finite number of carpets that they both like. Prove that the carpets can be distributed among the floors in a way that for every guest there are only finitely many carpets they like that are placed on floors different from the one where the guest is. [i]Proposed by András Imolay, Budapest[/i]

Let $n$ points be given inside a rectangle $R$ such that no two of them lie on a line parallel to one of the sides of $R$. The rectangle $R$ is to be dissected into smaller rectangles with sides parallel to the sides of $R$ in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect $R$ into at least $n + 1$ smaller rectangles. [i]Proposed by Serbia[/i]

Determine the last two digits of the number $2^5 + 2^{5^{2}} + 2^{5^{3}} +... + 2^{5^{1991}}$ , written in decimal notation.

Jesse and Tjeerd are playing a game. Jesse has access to $n\ge 2$ stones. There are two boxes: in the black box there is room for half of the stones (rounded down) and in the white box there is room for half of the stones (rounded up). Jesse and Tjeerd take turns, with Jesse starting. Jesse grabs in his turn, always one new stone, writes a positive real number on the stone and places put him in one of the boxes that isn't full yet. Tjeerd sees all these numbers on the stones in the boxes and on his turn may move any stone from one box to the other box if it is not yet full, but he may also choose to do nothing. The game stops when both boxes are full. If then the total value of the stones in the black box is greater than the total value of the stones in the white box, Jesse wins; otherwise win Tjeerd. For every $n \ge 2$, determine who can definitely win (and give a corresponding winning strategy).

The points $(0, 0)$, $(a, 5)$, and $(b, 11)$ are the vertices of an equilateral triangle. Find $ab$.

There are $4950$ ants. Assume that, for any three ants $A, B$ and $C$, if the ant $A$ is the boss of the ant $B$, and the ant $B$ is the boss of the ant $C$ then the ant $A$ is also the boss of the ant $C$. We want to divide the ants into $n$ groups so that in any group, either any two ants have the boss relationship or any two ants do not have the boss relationship. Find the smallest of $n$ we can always do in any case.

Prove that for all positive, entirely rational numbers $a$ and $b$ always holds $$\frac{a + b}{2} \ge \sqrt[a+b]{a^b \cdot b^a}.$$ When does the equal sign hold?

Find $\lim_{n\to\infty} n\int_0^{\frac{\pi}{2}} \frac{1}{(1+\cos x)^n}dx\ (n=1,\ 2,\ \cdots).$

Viorel participates in a mathematics competition with 50 problems. For each problem he answers correctly, he earns 4 points, and for each problem he answers incorrectly, he loses 1 point. If Viorel answered every problem and has 65 points, how many problems did he solve correctly?

Nine pawns forming a $3$ by $3$ square are placed in the lower left hand corner of an $8$ by $8$ chessboard. Any pawn may jump over another one standing next to it into a free square, i .e. may be reflected symmetrically with respect to a neighb our's centre (jumps may be horizontal , vertical or diagonal) . It is required to rearrange the nine pawns in another corner of the chessboard (in another $3$ by $3$ square) by means of such jumps. Can the pawns be thus re-arranged in the (a) upper left hand corner? (b) upper right hand corner? (J . E . Briskin)

A rectangular box $\mathcal{P}$ has distinct edge lengths $a, b,$ and $c$. The sum of the lengths of all $12$ edges of $\mathcal{P}$ is $13$, the sum of the areas of all $6$ faces of $\mathcal{P}$ is $\frac{11}{2}$, and the volume of $\mathcal{P}$ is $\frac{1}{2}$. What is the length of the longest interior diagonal connecting two vertices of $\mathcal{P}$? $\textbf{(A)}~2\qquad\textbf{(B)}~\frac{3}{8}\qquad\textbf{(C)}~\frac{9}{8}\qquad\textbf{(D)}~\frac{9}{4}\qquad\textbf{(E)}~\frac{3}{2}$

Consider all pairs of points $(a,b,c)$ and $(d,e, f )$ in the $3$-D coordinate system with $ad +be +c f = -2023$. What is the least positive integer that can be the distance between such a pair of points? [i]Proposed by William Hua[/i]

The sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=1, \; a_{n+1}=2a_{n}+\sqrt{3a_{n}^{2}+1}.\] Show that $a_{n}$ is an integer for every $n$.

Let $a\in\left[ \tfrac{1}{2},\ \tfrac{3}{2}\right]$ be a real number. Sequences $(u_n),\ (v_n)$ are defined as follows: $$u_n=\frac{3}{2^{n+1}}\cdot (-1)^{\lfloor2^{n+1}a\rfloor},\ v_n=\frac{3}{2^{n+1}}\cdot (-1)^{n+\lfloor 2^{n+1}a\rfloor}.$$ a. Prove that $${{({{u}_{0}}+{{u}_{1}}+\cdots +{{u}_{2018}})}^{2}}+{{({{v}_{0}}+{{v}_{1}}+\cdots +{{v}_{2018}})}^{2}}\le 72{{a}^{2}}-48a+10+\frac{2}{{{4}^{2019}}}.$$ b. Find all values of $a$ in the equality case.