Let $ABCD$ be a convex quadrilateral with $AC \perp BD$, and let $P$ be the intersection of $AC$ and $BD$. Suppose that the distance from $P$ to $AB$ is $99$, the distance from $P$ to $BC$ is $63$, and the distance from $P$ to $CD$ is $77$. What is the distance from $P$ to $AD$?

Find all real numbers $x$ such that $$x^3 = \{(x + 1)^3\}$$ where $\{y\}$ denotes the fractional part of $y$, i.e. the difference between $y$ and the largest integer less than or equal to $y$.

Find the positive real numbers $x,y$ for which $\frac{x+y}{2},\sqrt{xy},\frac{2xy}{x+y},\sqrt{\frac{x^2 +y^2}{2}}$ are integers whose sum is $66$.

Cerena, Faith, Edna, and Veronica each have a cube. Aarnő knows that the side lengths of each of their cubes are distinct integers greater than $1$, and he is trying to guess their exact values. Each girl fully paints the surface of her cube in Carolina blue before splitting the entire cube into $1\times1\times1$ cubes. Then, [list=disc] [*] Cerena reveals how many of her $1\times1\times1$ cubes have exactly $0$ blue faces. [*] Faith reveals how many of her $1\times1\times1$ cubes have exactly $1$ blue faces. [*] Edna reveals how many of her $1\times1\times1$ cubes have exactly $2$ blue faces. [*] Veronica reveals how many of her $1\times1\times1$ cubes have exactly $3$ blue faces. [/list] Whose side lengths can Aarnő deduce from these statements? [i]Jason Lee[/i]

A [i]snake of length $k$[/i] is an animal which occupies an ordered $k$-tuple $(s_1, \dots, s_k)$ of cells in a $n \times n$ grid of square unit cells. These cells must be pairwise distinct, and $s_i$ and $s_{i+1}$ must share a side for $i = 1, \dots, k-1$. If the snake is currently occupying $(s_1, \dots, s_k)$ and $s$ is an unoccupied cell sharing a side with $s_1$, the snake can [i]move[/i] to occupy $(s, s_1, \dots, s_{k-1})$ instead. The snake has [i]turned around[/i] if it occupied $(s_1, s_2, \dots, s_k)$ at the beginning, but after a finite number of moves occupies $(s_k, s_{k-1}, \dots, s_1)$ instead. Determine whether there exists an integer $n > 1$ such that: one can place some snake of length $0.9n^2$ in an $n \times n$ grid which can turn around. [i]Nikolai Beluhov[/i]

Solve in the set of $ 2\times 2 $ integer matrices the equation $$ X^2-4\cdot X+4\cdot\left(\begin{matrix}1\quad 0\\0\quad 1\end{matrix}\right) =\left(\begin{matrix}7\quad 8\\12\quad 31\end{matrix}\right) . $$

Let $ABC$ be an equilateral triangle, with sidelength equal to $a$. Let $P$ be a point in the interior of triangle $ABC$, and let $D,E$ and $F$ be the feet of the altitudes from $P$ on $AB, BC$ and $CA$, respectively. Prove that $\frac{|PD|+|PE|+|PF|}{3a}=\frac{\sqrt{3}}{6}$

Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.

Points $ A$ and $ B$ lie on the circle with center $ O.$ Let point $ C$ lies outside the circle; let $ CS$ and $ CT$ be tangents to the circle. $ M$ be the midpoint of minor arc $ AB$ of $ (O).$ $ MS,\,MT$ intersect $ AB$ at points $ E,\,F$ respectively. The lines passing through $ E,\,F$ perpendicular to $ AB$ cut $ OS,\,OT$ at $ X$ and $ Y$ respectively. A line passed through $ C$ intersect the circle $ (O)$ at $ P,\,Q$ ($ P$ lies on segment $ CQ$). Let $ R$ be the intersection of $ MP$ and $ AB,$ and let $ Z$ be the circumcentre of triangle $ PQR.$ Prove that: $ X,\,Y,\,Z$ are collinear.

Let $n \geqslant 3$ be a positive integer and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2. Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of \[ \left|1-\sum_{i \in X} a_{i}\right| \] is minimised. Prove that there exists a strictly increasing sequence of $n$ positive real numbers $\left(b_{1}, b_{2}, \ldots, b_{n}\right)$ with sum equal to 2 such that \[ \sum_{i \in X} b_{i}=1. \]

$N$ in natural. There are natural numbers from $N^3$ to $N^3+N$ on the board. $a$ numbers was colored in red, $b$ numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that $b|a$

Prove that for positive reals $a$,$b$,$c$ so that $a+b+c+abc=4$, \[\left (1+\dfrac{a}{b}+ca \right )\left (1+\dfrac{b}{c}+ab \right)\left (1+\dfrac{c}{a}+bc \right) \ge 27\] holds.

An [i]$n$-type triangle[/i] where $n\geqslant 2$ is formed by the cells of a $(2n+1)\times(2n+1)$ board, situated under both main diagonals. For instance, a $3$-type triangle looks like this:[img]https://i.ibb.co/k4fmwWY/Screenshot-2024-07-31-153932.png[/img]Determine the maximal length of a sequence with pairwise distinct cells in an $n$-type triangle, such that, beggining with the second one, any cell of the sequence has a common side with the previous one. [i]Cristi Săvescu[/i]

Find the number of real solutions $x$, in radians, to $$\sin(x)=\frac{x}{1000}.$$

A sequence $a_1,a_2,\ldots$ of positive integers satisfies the following properties.[list][*]$a_1 = 1$ [*]$a_{3n+1} = 2a_n + 1$ [*]$a_{n+1}\ge a_n$ [*]$a_{2001} = 200$[/list]Find the value of $a_{1000}$. [i]Note[/i]. In the original statement of the problem, there was an extra condition:[list][*]every positive integer appears at least once in the sequence.[/list]However, with this extra condition, there is no solution, i.e., no such sequence exists. (Try to prove it.) The problem as written above does have a solution.

Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that $$f(x + f(y)) = f(x) + f(y)$$ for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.

Let $p$ be a prime. A [i]complete residue class modulo $p$[/i] is a set containing at least one element equivalent to $k \pmod{p}$ for all $k$. (a) Show that there exists an $n$ such that the $n$th row of Pascal's triangle forms a complete residue class modulo $p$. (b) Show that there exists an $n \le p^2$ such that the $n$th row of Pascal's triangle forms a complete residue class modulo $p$.

Consider the matrices $$A = \left(\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right) \\ \mbox{ and } \\ B = \left(\begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right).$$ Let $k\geq 1$ an integer. Prove that for any nonzero $i_1,i_2,\dots,i_{k-1},j_1,j_2,\dots,j_k$ and any integers $i_0,i_k$ it holds that $$A^{i_0}B^{j_1}A^{i_1}B^{j_2}\cdots A^{i_{k-1}}B^{i_k}A^{i_k} \not = I.$$

Let $p,q$ be positive reals with sum 1. Show that for any $n$-tuple of reals $(y_1,y_2,...,y_n)$, there exists an $n$-tuple of reals $(x_1,x_2,...,x_n)$ satisfying $$p\cdot \max\{x_i,x_{i+1}\} + q\cdot \min\{x_i,x_{i+1}\} = y_i$$ for all $i=1,2,...,2017$, where $x_{2018}=x_1$.

When $10^{93}-93$ is expressed as a single whole number, the sum of the digits is $\text{(A)}\ 10 \qquad \text{(B)}\ 93 \qquad \text{(C)}\ 819 \qquad \text{(D)}\ 826 \qquad \text{(E)}\ 833$

On the sides of triangle $ ABC$ we consider points $ C_1,C_2 \in (AB), B_1,B_2 \in (AC), A_1,A_2 \in (BC)$ such that triangles $ A_1,B_1,C_1$ and $ A_2B_2C_2$ have a common centroid. Prove that sets $ [A_1,B_1]\cap [A_2B_2], [B_1C_1]\cap[B_2C_2], [C_1A_1]\cap [C_2A_2]$ are not empty.

Let $P(X) = a_n X^n + a_{n-1} X^{n-1} + \cdots + a_1 X + a_0$ be a polynomial with real coefficients such that $0 \leqslant a_i \leqslant a_0$ for $i = 1, 2, \ldots, n$. Prove that, if $P(X)^2 = b_{2n} X^{2n} + b_{2n-1} X^{2n-1} + \cdots + b_{n+1} X^{n+1} + \cdots + b_1 X + b_0$, then $4 b_{n+1} \leqslant P(1)^2$.

Let $ABC$ be a right triangle and isosceles, with $\angle C = 90^o$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint of $AC$. Let $ P$ be such that $MNP$ is an equilateral triangle with $ P$ inside the quadrilateral $MBCN$. Calculate the measure of $\angle CAP$

Let $I$ denote the center of the circle inscribed in the right triangle $ABC$ with right angle at the vertex $A$. Next, denote by $M$ and $N$ the midpoints of the lines $AB$ and $BI$. Prove that the line $CI$ is tangent to the circumscribed circle of triangle $BMN$. (Patrik Bak, Josef Tkadlec)

Find all quadruples $(a,b,c,d)$ of positive integers satisfying that $a^2+b^2=c^2+d^2$ and such that $ac+bd$ divides $a^2+b^2$.