In a right triangle $ABC$ with a hypotenuse $AB$, the angle $A$ is greater than the angle $B$. Point $N$ lies on the hypotenuse $AB$ , such that $BN = AC$. Construct this triangle $ABC$ given the point $N$, point $F$ on the side $AC$ and a straight line $\ell$ containing the bisector of the angle $A$ of the triangle $ABC$. (Grigory Filippovsky)

A graph has $17$ points and each point has $4$ edges. Show that there are two points which are not joined and which are not both joined to the same point.

[b]9.[/b] Spin a regular coin repeatedly until the heads and tails turned up both reach the number $k$ ($k= 1, 2, \dots $); denote by $v_k$ the number of the necessary throws. Find the distribution of the random variable $v_k$ and the limit-distribution of the random variable $\frac {v_k -2k}{\sqrt {2k}}$ as $k \to \infty$. [b](P. 10)[/b]

There are distinct points $O, A, B, K_1, . . . , K_n, L_1, . . . , L_n$ on a plane such that no three points are collinear. The open line segments $K_1L_1, . . . , K_nL_n$ are coloured red, other points on the plane are left uncoloured. An allowed path from point $O$ to point $X$ is a polygonal chain with first and last vertices at points $O$ and $X$, containing no red points. For example, for $n = 1$, and $K_1 = (-1, 0)$, $L_1 = (1, 0)$, $O = (0,-1)$, and $X = (0,1)$, $OK_1X$ and $OL_1X$ are examples of allowed paths from $O$ to $X$, there are no shorter allowed paths. Find the least positive integer n such that it is possible that the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $A$ is closer to $B$ than to $A$, and the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $B$ is closer to $A$ than to $B$.

Determine all strictly increasing functions $f:\mathbb{N}_0\to\mathbb{N}_0$ which satisfy \[f(x)\cdot f(y)\mid (1+2x)\cdot f(y)+(1+2y)\cdot f(x)\]for all non-negative integers $x{}$ and $y{}$.

Let $S=\underbrace{111\dots 1}_{19}\underbrace{999\dots 9}_{19}$. Show that the $2S$-digit number \[\underbrace{111\dots 1}_{S}\underbrace{999\dots 9}_{S}\] is a multiple of $19$.

Let $A,B,C$ denote intersection points of diagonals $A_1A_4$ and $A_2A_5$, $A_1A_6$ and $A_2A_7$, $A_1A_9$ and $A_2A_{10}$ of the regular decagon $A_1A_2...A_{10}$ respectively Find the angles of the triangle $ABC$

Given $ a,b, c $ positive real numbers satisfying $ a+b+c=1 $. Prove that \[ \dfrac{1}{\sqrt{ab+bc+ca}}\ge \sqrt{\dfrac{2a}{3(b+c)}} +\sqrt{\dfrac{2b}{3(c+a)}} + \sqrt{\dfrac{2c}{3(a+b)}} \ge \sqrt{a} +\sqrt{b}+\sqrt{c} \]

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the equation $$f(x^{2019} + y^{2019}) = x(f(x))^{2018} + y(f(y))^{2018}$$ for all real numbers $x$ and $y$.

For each set $C$ of points in space, we designate by $P_C$ the set of planes containing at least three points of $C$. $\bullet$ Prove that there exists $C$ such that $\phi (P_C) = 1997$, where $\phi$ corresponds to the cardinality. $\bullet$ Determine the least number of points that $C$ must have so that the previous property can be fulfilled.

Given a triangle $ABC$ let $D$, $E$, $F$ be orthogonal projections from $A$, $B$, $C$ to the opposite sides respectively. Let $X$, $Y$, $Z$ denote midpoints of $AD$, $BE$, $CF$ respectively. Prove that perpendiculars from $D$ to $YZ$, from $E$ to $XZ$ and from $F$ to $XY$ are concurrent.

Let $p$ and $n$ be positive integers such that $p$ is prime and $1 + np$ is a perfect square. Prove that the number $n + 1$ can be expressed as the sum of $p$ perfect squares, where some of them can be equal.

Given a convex quadrilateral whose opposite sides are not parallel, and giving an internal point $P$. Find a parallelogram whose vertices are on the side lines of the rectangle and whose center is $P$. Give a method by which we can construct it (provided there is one). [img]https://1.bp.blogspot.com/-t4aCJza0LxI/X9j1qbSQE4I/AAAAAAAAMz4/V9pr7Cd22G4F320nyRLZMRnz18hMw9NHQCLcBGAsYHQ/s0/2012%2BDurer%2BC3.png[/img]

We are given a ruler with two marks at a distance 1. With its help we can do all possible constructions as with a ruler with no measurements, including one more: If there is a line $l$ and point $A$ on $l$, then we can construct points $P_1,P_2\in l$ for which $AP_1=AP_2=1$. By using this ruler, construct a perpendicular from a given point to a given line.

Two high school classes took the same test. One class of $ 20$ students made an average grade of $ 80\%$; the other class of $ 30$ students made an average grade of $ 70\%$. The average grade for all students in both classes is: $ \textbf{(A)}\ 75\% \qquad\textbf{(B)}\ 74\% \qquad\textbf{(C)}\ 72\% \qquad\textbf{(D)}\ 77\% \qquad\textbf{(E)}\ \text{none of these}$

Triangle $ ABC$ has $ AB\equal{}13$ and $ AC\equal{}15$, and the altitude to $ \overline{BC}$ has length $ 12$. What is the sum of the two possible values of $ BC$? $ \textbf{(A)}\ 15\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 19$

Find $\frac{area(CDF)}{area(CEF)}$ in the figure. [asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(5.75cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -2, xmax = 21, ymin = -2, ymax = 16; /* image dimensions */ /* draw figures */ draw((0,0)--(20,0)); draw((13.48,14.62)--(7,0)); draw((0,0)--(15.93,9.12)); draw((13.48,14.62)--(20,0)); draw((13.48,14.62)--(0,0)); label("6",(15.16,12.72),SE*labelscalefactor); label("10",(18.56,5.1),SE*labelscalefactor); label("7",(3.26,-0.6),SE*labelscalefactor); label("13",(13.18,-0.71),SE*labelscalefactor); label("20",(5.07,8.33),SE*labelscalefactor); /* dots and labels */ dot((0,0),dotstyle); label("$B$", (-1.23,-1.48), NE * labelscalefactor); dot((20,0),dotstyle); label("$C$", (19.71,-1.59), NE * labelscalefactor); dot((7,0),dotstyle); label("$D$", (6.77,-1.64), NE * labelscalefactor); dot((13.48,14.62),dotstyle); label("$A$", (12.36,14.91), NE * labelscalefactor); dot((15.93,9.12),dotstyle); label("$E$", (16.42,9.21), NE * labelscalefactor); dot((9.38,5.37),dotstyle); label("$F$", (9.68,4.5), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

Let $a,b\in \mathbb{R}$. Evaluate: \[\lim_{n\to \infty}\left(\sqrt{a^2n^2+bn}-an\right)\] Consider the sequence $(x_n)_{n\ge 1}$, defined by $x_n=\sqrt{n}-\lfloor \sqrt{n}\rfloor$. Denote by $A$ the set of the points $x\in \mathbb{R}$, for which there is a subsequence of $(x_n)_{n\ge 1}$ tending to $x$. a) Prove that $\mathbb{Q}\cap [0,1]\subset A$. b) Find $A$.

Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$: \[f(x-y+f(y))=f(x)+f(y).\]

Let $k$ be an integer. If the equation $(x-1)|x+1|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there?

A non self-intersecting polygon is given in a Cartesian coordinate system such that its perimeter contains no lattice points, and its vertices have no integer coordinates. A point is called semi-integer if exactly one of its coordinates is an integer. Let $P_1, P_2,\ldots, P_k$ denote the semi-integer points on the perimeter of the polygon. Let ni denote the floor of the non-integer coordinate of $P_i$. Prove that integers $n_1,n_2,\ldots ,n_k$ can be divided into two groups with the same sum. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

Let $C_1 , C_2$ be two circles in the plane intersecting at two distinct points. Let $P$ be the midpoint of a variable chord $AB$ of $C_2$ with the property that the circle on $AB$ as diameter meets $C_1$ at a point $T$ such that $P T$ is tangent to $C_1$ . Find the locus of $P$ .

Let $\Delta ABC$ be an acute scalene triangle with $AC<BC$, an orthocenter $H$ and altitudes $AE$, $BF$. The points $E'$ and $F'$ are symmetrical to $E$ and $F$ with respect to $A$ and $B$ respectively. Point $O$ is the center of the circumscribed circle of $ABC$ and $M$ is the midpoint of $AB$. Let $N$ be the midpoint of $OM$. Prove that the tangent through $H$ to the circumscribed circle of $\Delta E'HF'$ is perpendicular to line $CN$.

Each sidelength of a convex quadrilateral $ABCD$ is not less than $1$ and not greater than $2$. The diagonals of this quadrilateral meet at point $O$. Prove that $S_{AOB}+ S_{COD} \le 2(S_{AOD}+ S_{BOC})$.

$15.$ Let $n$ be the largest integer that is the product of exactly $3$ distinct prime numbers, $x,y,$ and $10x+y,$ where $x$ and $y$ are digits. What is the sum of digits of $n ?$