Investigate the singular points on the curve $y=x^3$ in the projective plane.

Consider a right -angle triangle $ABC$ with $A=90^{o}$, $AB=c$ and $AC=b$. Let $P\in AC$ and $Q\in AB$ such that $\angle APQ=\angle ABC$ and $\angle AQP = \angle ACB$. Calculate $PQ+PE+QF$, where $E$ and $F$ are the projections of $B$ and $Q$ onto $BC$, respectively.

Throw $ n$ balls in to $ 2n$ boxes.　Suppose each ball comes into each box with equal probability of entering in any boxes. Let $ p_n$ be the probability such that any box has ball less than or equal to one. Find the limit $ \lim_{n\to\infty} \frac{\ln p_n}{n}$

A sequence $(a_1, a_2,...,a_k)$ consisting of pairwise different cells of an $n\times n$ board is called a cycle if $k \ge 4$ and cell ai shares a side with cell $a_{i+1}$ for every $i = 1,2,..., k$, where $a_{k+1} = a_1$. We will say that a subset $X$ of the set of cells of a board is [i]malicious [/i] if every cycle on the board contains at least one cell belonging to $X$. Determine all real numbers $C$ with the following property: for every integer $n \ge 2$ on an $n\times n$ board there exists a malicious set containing at most $Cn^2$ cells.

Four lines are given in a plane so that any three of them determine a triangle. One of these lines is parallel to a median in the triangle determined by the other three lines. Prove that each of the other three lines also has this property.

Here is a problem we (me and my colleagues) suggested and was given at the competition this year. The problem statement is very natural and short. However, we have not seen such a problem before. A real $2024 \times 2024$ matrix $A$ is called nice if $(Av, v) = 1$ for every vector $v\in \mathbb{R}^{2024}$ with unit norm. a) Prove that the only nice matrix such that all of its eigenvalues are real is the identity matrix. b) Find an example of a nice non-identity matrix

Bus tickets from a transportation company are numbered with six digits, ranging from 000000 to 999999. A ticket is considered "lucky" if the sum of the first three digits equals the sum of the last three digits. For example, ticket 721055 is lucky, whereas 003101 is not. Determine how many consecutive tickets a person must buy to guarantee obtaining at least one lucky ticket, regardless of the starting ticket number.

Using the digits: $ 1,2,3,4,5,$ players $ A$ and $ B$ compose a $ 2005$-digit number $ N$ by selecting one digit at a time: $ A$ selects the first digit, $ B$ the second, $ A$ the third and so on. Player $ A$ wins if and only if $ N$ is divisible by $ 9$. Who will win if both players play as well as possible?

Let $ k$ be a positive integer, $ z$ a complex number, and $ \varepsilon <\frac12$ a positive number. Prove that the following inequality holds for infinitely many positive integers $ n$: \[ \mid \sum_{0\leq l \leq \frac{n}{k+1}} \binom{n-kl}{l}z^l \mid \geq (\frac 12-\varepsilon)^n.\] [i]P. Turan[/i]

Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)

Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$. [i]Proposed by fattypiggy123[/i]

Call a positive number $n$ [i]fine[/i], if there exists a prime number $p$ such that $p|n$ and $p^2\nmid n$. Prove that at least 99% of numbers $1, 2, 3, \ldots, 10^{12}$ are fine numbers.

What is angle $B$ of triangle$ ABC$, if it is known that the altitudes drawn from $A$ and $C$ intersect inside the triangle and one of them is divided by of intersection point into equal parts, and the other one in the ratio of $2: 1$, counting from the vertex?

Let $n$ be a positive integer. Morgane has coloured the integers $1,2,\ldots,n$. Each of them is coloured in exactly one colour. It turned out that for all positive integers $a$ and $b$ such that $a<b$ and $a+b \leqslant n$, at least two of the integers among $a$, $b$ and $a+b$ are of the same colour. Prove that there exists a colour that has been used for at least $2n/5$ integers. \\ \\ (Vincent Jugé)

Show that there is no positive integer $k$ for which the equation \[(n-1)!+1=n^{k}\] is true when $n$ is greater than $5$.

A large flat plate of glass is suspended $\sqrt{2/3}$ units above a large flat plate of wood. (The glass is infinitely thin and causes no funny refractive effects.) A point source of light is suspended $\sqrt{6}$ units above the glass plate. An object rests on the glass plate of the following description. Its base is an isosceles trapezoid $ABCD$ with $AB \parallel DC$, $AB = AD = BC = 1$, and $DC = 2$. The point source of light is directly above the midpoint of $CD$. The object’s upper face is a triangle $EF G$ with $EF = 2$, $EG = F G =\sqrt3$. $G$ and $AB$ lie on opposite sides of the rectangle $EF CD$. The other sides of the object are $EA = ED = 1$, $F B = F C = 1$, and $GD = GC = 2$. Compute the area of the shadow that the object casts on the wood plate.

The sequence $a_n$ for $n \in \mathbb{N}$ is defined as follows: \[ a_0 = 6, \quad a_1 = 7, \quad a_{n+2} = 3a_{n+1} - 2a_n \] Find all values of $n$ such that $n^2 = a_n$.

Eight points are equally spaced around a circle of radius $r$. If we draw a circle of radius $1$ centered at each of the eight points, then each of these circles will be tangent to two of the other eight circles that are next to it. IF $r^2=a+b\sqrt{2}$, where $a$ and $b$ are integers, then what is $a+b$? $\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) }6\qquad\text{(E) }7$

Let $ a,b$ be two complex numbers. Prove that roots of $ z^{4}\plus{}az^{2}\plus{}b$ form a rhombus with origin as center, if and only if $ \frac{a^{2}}{b}$ is a non-positive real number.

If $a$, $b$, $c$ are rational, show that \[ \frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}+\frac{1}{(a-b)^2} \] is the square of a rational.

$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$

A pair of positive integer numbers \((a, b)\) is given. It turns out that for every positive integer number \(n\), for which the numbers \((n - a)(n + b)\) and \(n^2 - ab\) are positive, they have the same number of divisors. Is it necessarily true that \(a = b\)? [i]Proposed by Oleksii Masalitin[/i]

Find the number of integers $x$ which satisfy the equation $(x^2-5x+5)^{x+5}=1$.

Find the integer that is closest to $ 1000 \sum_{n=3}^{10000}\frac{1}{n^{2}-4}.$

Find all polynomials $P(x)$ with integer coefficients such that $P(P(n) + n)$ is a prime number for infinitely many integers $n$.