The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations \[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.

Parabola $y^2=2px$, two fixed points $A(a,b),B(-a,0)(ab\neq0,b^2\neq 2pa)$. $M$ is a point on the parabola, $AM$ intersects the parabola at $M_1$, $BM$ intersects the parabola at $M_2$. Prove: When $M$ changes, line $M_1M_2$ passes a fixed point, and find the fixed point.

Let $n$ be a positive integer. Define a sequence by setting $a_{1}= n$ and, for each $k > 1$, letting $a_{k}$ be the unique integer in the range $0\leq a_{k}\leq k-1$ for which $a_{1}+a_{2}+...+a_{k}$ is divisible by $k$. For instance, when $n = 9$ the obtained sequence is $9,1,2,0,3,3,3,...$. Prove that for any $n$ the sequence $a_{1},a_{2},...$ eventually becomes constant.

Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.

Consider the function $y(x)$ satisfying the differential equation $y'' = -(1+\sqrt{x})y$ with $y(0)=1$ and $y'(0)=0.$ Prove that $y(x)$ vanishes exactly once on the interval $0< x< \pi \slash 2,$ and find a positive lower bound for the zero.

Let $f:\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ be a continuous function such that $f(0)=0$ and $$f(x)+f(f(x))+f(f(f(x)))=3x$$ for all $x>0$. Show that $f(x)=x$ for all $x>0$.

Let $AC$ and $BD$ be two chords of a circle with center $O$ such that they intersect at right angles inside the circle at the point $M$. Suppose $K$ and $L$ are midpoints of the chords $AB$ and $CD$ respectively. Prove that $OKML$ is a parallelogram.

A cross-section of a river is a trapezoid with bases $10$ and $16$ and slanted sides of length $5$. At this section the water is flowing at $\pi$ mph. A little ways downstream is a dam where the water flows through $4$ identical circular holes at $16$ mph. What is the radius of the holes?

Let $ ABC$ be an acute-angled triangle with altitude $ AK$. Let $ H$ be its ortho-centre and $ O$ be its circum-centre. Suppose $ KOH$ is an acute-angled triangle and $ P$ its circum-centre. Let $ Q$ be the reflection of $ P$ in the line $ HO$. Show that $ Q$ lies on the line joining the mid-points of $ AB$ and $ AC$.

At the vertices of a regular hexagon are written six nonnegative integers whose sum is $2003^{2003}$. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.

Let $ a_1,a_2, \cdots ,a_{2015} $ be $2015$-tuples of positive integers (not necessary distinct) and let $ k $ be a positive integers. Denote $\displaystyle f(i)=a_i+\frac{a_1a_2 \cdots a_{2015}}{a_i} $. a) Prove that if $ k=2015^{2015} $, there exist $ a_1, a_2, \cdots , a_{2015} $ such that $ f(i)= k $ for all $1\le i\le 2015 $.\\ b) Find the maximum $k_0$ so that for $k\le k_0$, there are no $k$ such that there are at least $ 2 $ different $2015$-tuples which fulfill the above condition.

Point $P$ lies inside the triangle $ABC$. Points $K, L, M$ are symmetrics of point $P$ wrt the midpoints of the sides $BC, CA, AB$. Prove that the straight $AK, BL, CM$ intersect at one point.

Let $(a,b,c,d)$ be a solution to the system \begin{align*}a+b&=15,\\ab+c+d&=78,\\ad+bc&=160,\\cd&=96.\end{align*} Find the greatest possible value of $a^2+b^2+c^2+d^2$.

Given a positive integer \(n\), let \(\phi(n)\) denote the number of positive integers less than or equal to \(n\) that are relatively prime to \(n\). Find all possible positive integers \(k\) for which there exist positive integers \(1 \leq a_1 < a_2 < \dots < a_k\) such that: \[ \left\lfloor \frac{\phi(a_1)}{a_1} + \frac{\phi(a_2)}{a_2} + \dots + \frac{\phi(a_k)}{a_k} \right\rfloor = 2024 \]

Let $A_0A_1A_2$ be a scalene triangle. Find the locus of the centres of the equilateral triangles $X_0X_1X_2$ , such that $A_k$ lies on the line $X_{k+1}X_{k+2}$ for each $k=0,1,2$ (with indices taken modulo $3$).

Let $\{a_{n}\}$ be a strictly increasing positive integers sequence such that $\gcd(a_{i}, a_{j})=1$ and $a_{i+2}-a_{i+1}>a_{i+1}-a_{i}$. Show that the infinite series \[\sum^{\infty}_{i=1}\frac{1}{a_{i}}\] converges.

An $8\times 8$ chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper.

In an attempt to copy down from the board a sequence of six positive integers in arithmetic progression, a student wrote down the five numbers, \[ 113,137,149,155,173, \] accidentally omitting one. He later discovered that he also miscopied one of them. Can you help him and recover the original sequence?

Find all derivable functions that have real domain and codomain, and are equal to their second functional power.

A strip consists of $n$ squares which are numerated in their order by integers $1,2,3,..., n$. In the beginning, one square is empty while each remaining square contains one piece. Whenever a square contains a piece and its some neighbouring square contains another piece while the square immediately following the neighbouring square is empty, one may raise the first piece over the second one to the empty square, removing the second piece from the strip. Find all possibilites which square can be initially empty, if it is possible to reach a state where the strip contains only one piece and a) $n = 2008$, b) $n = 2009$.

$\angle 1 + \angle 2 = 180^\circ $ $\angle 3 = \angle 4$ Find $\angle 4.$ [asy]pair H,I,J,K,L; H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0); draw(H--I--J--cycle); draw(K--L--J); draw(arc((0,0),dir(70),(1,0),CW)); label("$70^\circ$",dir(35),NE); draw(arc(I,I+dir(250),I+dir(290),CCW)); label("$40^\circ$",I+1.25*dir(270),S); label("$1$",J+0.25*dir(162.5),NW); label("$2$",J+0.25*dir(17.5),NE); label("$3$",L+dir(162.5),WNW); label("$4$",K+dir(-52.5),SE); [/asy] $\textbf{(A)}\ 20^\circ \qquad \textbf{(B)}\ 25^\circ \qquad \textbf{(C)}\ 30^\circ \qquad \textbf{(D)}\ 35^\circ \qquad \textbf{(E)}\ 40^\circ$

After swimming around the ocean with some snorkling gear, Joshua walks back to the beach where Alexis works on a mural in the sand beside where they drew out symbol lists. Joshua walks directly over the mural without paying any attention. "You're a square, Josh." "No, $\textit{you're}$ a square," retorts Joshua. "In fact, you're a $\textit{cube}$, which is $50\%$ freakier than a square by dimension. And before you tell me I'm a hypercube, I'll remind you that mom and dad confirmed that they could not have given birth to a four dimension being." "Okay, you're a cubist caricature of male immaturity," asserts Alexis. Knowing nothing about cubism, Joshua decides to ignore Alexis and walk to where he stashed his belongings by a beach umbrella. He starts thinking about cubes and computes some sums of cubes, and some cubes of sums: \begin{align*}1^3+1^3+1^3&=3,\\1^3+1^3+2^3&=10,\\1^3+2^3+2^3&=17,\\2^3+2^3+2^3&=24,\\1^3+1^3+3^3&=29,\\1^3+2^3+3^3&=36,\\(1+1+1)^3&=27,\\(1+1+2)^3&=64,\\(1+2+2)^3&=125,\\(2+2+2)^3&=216,\\(1+1+3)^3&=125,\\(1+2+3)^3&=216.\end{align*} Josh recognizes that the cubes of the sums are always larger than the sum of cubes of positive integers. For instance, \begin{align*}(1+2+4)^3&=1^3+2^3+4^3+3(1^2\cdot 2+1^2\cdot 4+2^2\cdot 1+2^2\cdot 4+4^2\cdot 1+4^2\cdot 2)+6(1\cdot 2\cdot 4)\\&>1^3+2^3+4^3.\end{align*} Josh begins to wonder if there is a smallest value of $n$ such that \[(a+b+c)^3\leq n(a^3+b^3+c^3)\] for all natural numbers $a$, $b$, and $c$. Joshua thinks he has an answer, but doesn't know how to prove it, so he takes it to Michael who confirms Joshua's answer with a proof. What is the correct value of $n$ that Joshua found?

Find all triples $a,b,c$ of positive integers such that \[ ( 1 + \frac{1}{a} ) ( 1 + \frac{1}{b}) ( 1 + \frac{1}{c} ) = 3. \]

Suppose that $a,b$ are two odd positive integers such that $2ab+1 \mid a^2 + b^2 + 1$. Prove that $a=b$. (15 points)

Determine all real solutions of the following system of equations: $$x+y^2=y^3$$ $$y+x^2=x^3$$