Let $\triangle ABC$ be an acute triangle. Point $Z$ is on $A$ altitude and points $X$ and $Y$ are on the $B$ and $C$ altitudes out of the triangle respectively, such that: $\angle AYB=\angle BZC=\angle CXA=90$ Prove that $X$,$Y$ and $Z$ are collinear, if and only if the length of the tangent drawn from $A$ to the nine point circle of $\triangle ABC$ is equal with the sum of the lengths of the tangents drawn from $B$ and $C$ to the nine point circle of $\triangle ABC$.

We define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. What is \[ \left\lfloor \frac{5^{2017015}}{5^{2015}+7} \right\rfloor \mod 1000?\]

In some cells of a rectangular table $m\times n (m, n> 1)$ is one checker. $Baby$ cut along the lines of the grid this table so that it is split into two equal parts, with the number of pieces on each side were the same. $Carlson$ changed the arrangement of checkers on the board (and on each side of the cage is still worth no more than one pieces). Prove that the $Baby$ may again cut the board into two equal parts containing an equal number of pieces

Given an integer $n\geq 3$. For each $3\times3$ squares on the grid, call this $3\times3$ square isolated if the center unit square is white and other 8 squares are black, or the center unit square is black and other 8 squares are white. Now suppose one can paint an infinite grid by white or black, so that one can select an $a\times b$ rectangle which contains at least $n^2-n$ isolated $3\times 3$ square. Find the minimum of $a+b$ that such thing can happen. (Note that $a,b$ are positive reals, and selected $a\times b$ rectangle may have sides not parallel to grid line of the infinite grid.)

Let $R$ be an $20 \times 18$ grid of points such that adjacent points are $1$ unit apart. A fly starts at a point and jumps in straight lines to other points in $R$ in turn, such that each point in R is visited exactly once and no two jumps intersect at a point other than an endpoint of a jump, for a total of $359$ jumps. Call a jump small if it is of length $1$. What is the least number of small jumps? (The left configuration for a $4 \times 4$ grid has $9$ small jumps and $15$ total jumps, while the right configuration is invalid.)

Find all the pairs $a,b \in N$ such that $ab-1 |a^2 + 1$.

The diagram shows a circle with radius $24$ which contains two circles with radius $12$ tangent to each other and the larger circle. The smallest circle is tangent to the three other circles. What is the radius of the smallest circle? [asy] size(150); defaultpen(linewidth(0.8)); draw(unitcircle^^circle((0,0.5),0.5)^^circle((0,-0.5),0.5)^^circle((2/3,0),1/3));[/asy]

Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.

The inradius of triangle $ ABC$ is $ 1$ and the side lengths of $ ABC$ are all integers. Prove that triangle $ ABC$ is right-angled.

Triangle $ABC$ lies in the Cartesian Plane and has an area of 70. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20)$, respectively, and the coordinates of $A$ are $(p,q)$. The line containing the median to side $BC$ has slope $-5$. Find the largest possible value of $p+q$.

In a country named Fillip, there are three major cities called Alenda, Breda, Chenida. This country uses the unit of "FP". The distance between Alenda and Chenida is $100$ FP. Breda is $70$ FP from Alenda and $30$ FP from Chenida. Let us say that we take a road trip from Alenda to Chenida. After $2$ hours of driving, we are currently at $50$ FP away from Alenda and $50$ FP away from Chenida. How many FP are we away from Breda?

Let $a$ and $b$ be two positive integers such that $2ab$ divides $a^2+b^2-a$. Prove that $a$ is perfect square

For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set $$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$ where $i = 1, 2$. Determine the smallest positive integer $m$ such that $2f_1(m) - f_2(m) = 2017$.

We say that a function \( f: \mathbb{R} \to \mathbb{R} \) is morally odd if its graph is symmetric with respect to a point, that is, there exists \((x_0, y_0) \in \mathbb{R}^2\) such that if \((u, v) \in \{(x, f(x)) : x \in \mathbb{R}\}\), then \((2x_0 - u, 2y_0 - v) \in \{(x, f(x)) : x \in \mathbb{R}\}\). On the other hand, \( f \) is said to be morally even if its graph \(\{(x, f(x)) : x \in \mathbb{R}\}\) is symmetric with respect to some line (not necessarily vertical or horizontal). If \( f \) is morally even and morally odd, we say that \( f \) is parimpar. (a) Let \( S \subset \mathbb{R} \) be a bounded set and \( f: S \to \mathbb{R} \) be an arbitrary function. Prove that there exists \( g: \mathbb{R} \to \mathbb{R} \) that is parimpar such that \( g(x) = f(x) \) for all \( x \in S \). (b) Find all polynomials \( P \) with real coefficients such that the corresponding polynomial function \( P: \mathbb{R} \to \mathbb{R} \) is parimpar.

Janez wants to make an $m\times n$ grid (consisting of unit squares) using equal elements of the form $\llcorner$, where each leg of an element has the unit length. No two elements can overlap. For which values of $m$ and $n$ can Janez do the task?

Prove that there is a constant $c>0$ such that for any $n>3$ there exists a planar graph $G$ with $n$ vertices such that every straight-edged plane embedding of $G$ has a pair of edges with ratio of lengths at least $cn$.

Let $S$ be a set not empty of positive integers and $AB$ a segment with, initially, only points $A$ and $B$ colored by red. An operation consists of choosing two distinct points $X, Y$ colored already by red and $n\in S$ an integer, and painting in red the $n$ points $A_1, A_2,..., A_n$ of segment $XY$ such that $XA_1= A_1A_2= A_2A_3=...= A_{n-1}A_n= A_nY$ and $XA_1<XA_2<...<XA_n$. Find the least positive integer $m$ such exists a subset $S$ of $\{1,2,.., m\}$ such that, after a finite number of operations, we can paint in red the point $K$ in the segment $AB$ defined by $\frac{AK}{KB}= \frac{2709}{2022}$. Also, find the number of such subsets for such a value of $m$.

Justin chooses a number $n$ uniformly at random from the set of integers between $90$ and $99,$ inclusive. He then chooses a positive divisor $d$ of $n$ uniformly at random. Justin notices that $d$ and $n/d$ are relatively prime. If the probability that $n = 90$ can be expressed as $a/b$ for relatively prime positive integers, find $a + b.$

Each unit square of a $5 \times 5$ board is colored blue or yellow. Prove that there is a rectangle with sides parallel to the sides. axes of the board, such that its four corners are the same color.

For any positive integer $M$, the notation $M!$ denotes the product of the integers $1$ through $M$. What is the largest integer $n$ for which $5^n$ is a factor of the sum $98!+99!+100!$ ? $\textbf{(A) }23\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$: $$f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2$$ [i]Proposed by Ali Behrouz - Mojtaba Zare Bidaki[/i]

Let $f(x)$ be a differentiable function. Find the value of $x$ for which \[\{f(x)\}^2+(e+1)f(x)+1+e^2-2\int_0^x f(t)dt-2f(x)\int_0^x f(t)dt+2\left\{\int_0^x f(t)dt\right\}^2\] is minimized. [i]1978 Tokyo Medical College entrance exam[/i]

Prove that for every positive integer $n$ there exists a (not necessarily convex) polygon with no three collinear vertices, which admits exactly $n$ diffferent triangulations. (A [i]triangulation[/i] is a dissection of the polygon into triangles by interior diagonals which have no common interior points with each other nor with the sides of the polygon)

What is the minimum value of $ f(x) \equal{} |x \minus{} 1| \plus{} |2x \minus{} 1| \plus{} |3x \minus{} 1| \plus{} \cdots \plus{} |119x \minus{} 1|$? $ \textbf{(A)}\ 49 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 53$

Find all functions $f : Z^+ \to Z^+$ such that $n^3 - n^2 \le f(n) \cdot (f(f(n)))^2 \le n^3 + n^2$ for every $n$ in positive integers