In the convex quadrilateral $ABCD$ we have $\angle B = \angle C$ and $\angle D = 90^{\circ}.$ Suppose that $|AB| = 2|CD|.$ Prove that the angle bisector of $\angle ACB$ is perpendicular to $CD.$

Let $k$ be a given positive integer. Define $x_{1}= 1$ and, for each $n > 1$, set $x_{n+1}$ to be the smallest positive integer not belonging to the set $\{x_{i}, x_{i}+ik | i = 1, . . . , n\}$. Prove that there is a real number $a$ such that $x_{n}= [an]$ for all $n \in\mathbb{ N}$.

After a typist has written ten letters and had addressed the ten corresponding envelopes, a careless mailing clerk inserted the letters in the envelopes at random, one letter per envelope. What is the probability that [b]exactly[/b] nine letters were inserted in the proper envelopes?

Find all integer solutions to the equation $x^2=y^2(x+y^4+2y^2)$ .

The equation $2000x^6+100x^5+10x^3+x-2=0$ has exactly two real roots, one of which is $\frac{m+\sqrt{n}}r,$ where $m, n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r>0.$ Find $m+n+r.$

On hyperbola $y=\frac{1}{x}$ points $A_1,\ldots,A_{10}$ are chosen such that $(A_i)_x=2^{i-1}a$, where $a$ is some positive constant. Find the area of $A_1A_2 \ldots A_{10}$

Let $a,b,c \in \mathbb{R}^{+}$ such that $abc=1$. Prove that: $$a^2b + b^2c + c^2a \ge \sqrt{(a+b+c)(ab + bc +ca)}$$

Find all real solutions of the system of equations $$x^2-y^2=2(xz+yz+x+y),$$$$y^2-z^2=2(yx+zx+y+z),$$$$z^2-x^2=2(zy+xy+z+x).$$

Let $a,b,b',c,m,q$ be positive integers, where $m>1,q>1,|b-b'|\ge a$. It is given that there exist a positive integer $M$ such that $$S_q(an+b)\equiv S_q(an+b')+c\pmod{m}$$ holds for all integers $n\ge M$. Prove that the above equation is true for all positive integers $n$. (Here $S_q(x)$ is the sum of digits of $x$ taken in base $q$).

In the training of a state, the coach proposes a game. The coach writes four real numbers on the board in order from least to greatest: $a < b < c < d$. Each Olympian draws the figure on the right in her notebook and arranges the numbers inside the corner shapes, however she wants, putting a number on each one. Once arranged, on each segment write the square of the difference of the numbers at its ends. Then, add the $4$ numbers obtained. [img]https://cdn.artofproblemsolving.com/attachments/9/a/ea348c637ae266c908e0b97e64605808b3b1d2.png[/img] For example, if Vania arranges them as in the figure on the right, then the result would be $$ (c - b)^2 + (b- a)^2 + (a - d)^2 + (d - c)^2.$$ [img]https://cdn.artofproblemsolving.com/attachments/8/b/9c5375d66a4a6344b2bce333534fa7fac2ad6c.png[/img] The Olympians with the lowest result win. In what ways can you arrange the numbers to win? Give all the possible solutions.

The positive real numbers $x$ and $y$ satisfy the relation $x + y = 3 \sqrt{xy}$. Find the value of the numerical expression $$E=\left| \frac{x-y}{x+y}+\frac{x^2-y^2}{x^2+y^2}+\frac{x^3-y^3}{x^3+y^3}\right|.$$

The vertices $M$, $N$, $K$ of rectangle $KLMN$ lie on the sides $AB$, $BC$, $CA$ respectively of a regular triangle $ABC$ in such a way that $AM = 2$, $KC = 1$. The vertex $L$ lies outside the triangle. Find the value of $\angle KMN$.

Let $O$ be the circumcenter and $I$ be the incenter of an acute triangle $ABC$ with $m(\widehat{B}) \neq m(\widehat{C})$. Let $D$, $E$, $F$ be the midpoints of the sides $[BC]$, $[CA]$, $[AB]$, respectively. Let $T$ be the foot of perpendicular from $I$ to $[AB]$. Let $P$ be the circumcenter of the triangle $DEF$ and $Q$ be the midpoint of $[OI]$. If $A$, $P$, $Q$ are collinear, prove that \[\dfrac{|AO|}{|OD|}-\dfrac{|BC|}{|AT|}=4.\]

Let $ x_{1} ,x_{2} ,\ldots ,x_{9}$ be real numbers on $ \left[ \minus{} 1,1\right]$. If $ \sum _{i \equal{} 1}^{9}x_{i}^{3} \equal{} 0$, then what is the largest possible value of $ \sum _{i \equal{} 1}^{9}x_{i}$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac {3}{2} \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ \frac {9}{2} \qquad\textbf{(E)}\ \text{None}$

Let $ k $ be a natural number. A function $ f:S:=\left\{ x_1,x_2,...,x_k\right\}\longrightarrow\mathbb{R} $ is said to be [i]additive[/i] if, whenever $ n_1x_1+n_2x_2+\cdots +n_kx_k=0, $ it holds that $ n_1f\left( x_1\right)+n_2f\left( x_2\right)+\cdots +n_kf\left( x_k\right)=0, $ for all natural numbers $ n_1,n_2,...,n_k. $ Show that for every additive function and for every finite set of real numbers $ T, $ there exists a second function, which is a real additive function defined on $ S\cup T $ and which is equal to the former on the restriction $ S. $

Serge and Tanya want to show Masha a magic trick. Serge leaves the room. Masha writes down a sequence $(a_1, a_2, \ldots , a_n)$, where all $a_k$ equal $0$ or $1$. After that Tanya writes down a sequence $(b_1, b_2, \ldots , b_n)$, where all $b_k$ also equal $0$ or $1$. Then Masha either does nothing or says “Mutabor” and replaces both sequences: her own sequence by $(a_n, a_{n-1}, \ldots , a_1)$, and Tanya’s sequence by $(1 - b_n, 1 - b_{n-1}, \ldots , 1 - b_1)$. Masha’s sequence is covered by a napkin, and Serge is invited to the room. Serge should look at Tanya’s sequence and tell the sequence covered by the napkin. For what $n$ Serge and Tanya can prepare and show such a trick? Serge does not have to determine whether the word “Mutabor” has been pronounced.

For a positive integer $n$, $S(n)$ denotes the sum of its digits and $U(n)$ its unit digit. Determine all positive integers $n$ with the property that \[n = S(n) + U(n)^2.\]

Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.

For nonzero integers $n$, let $f(n)$ be the sum of all positive integers $b$ for which all solutions $x$ to $x^2 +bx+n = 0$ are integers, and let $g(n)$ be the sum of all positive integers $c$ for which all solutions $x$ to $cx + n = 0$ are integers. Compute $\sum^{2020}_{n=1} (f(n) - g(n))$.

On the inner surface of a fixed circle, rolls a wheel half the radius of the circle, without slipping. We marked a point red on the wheel. Prove that while the wheel makes a turn, the point moves on a line. [img]https://1.bp.blogspot.com/-PhgUWk0eU2c/X9j1gNJ7w3I/AAAAAAAAMzo/gP13TIZq7YsvNDBGVISkMQSdjwCgk_zwQCLcBGAsYHQ/s0/2014%2BDurer%2BD2.png[/img]

Represent an arbitrary positive integer as an expression involving only $3$ twos and any mathematical signs. (P. Dirac)

Let $ABC$ be a right triangle with a right angle in $C$ and a circumcenter $U$. On the sides $AC$ and $BC$, the points $D$ and $E$ lie in such a way that $\angle EUD = 90 ^o$. Let $F$ and $G$ be the projection of $D$ and $E$ on $AB$, respectively. Prove that $FG$ is half as long as $AB$. (Walther Janous)

Is it true that in any $2$-connected graph with a countably infinite number of vertices it's always possible to find a trail that is infinite in one direction? [i]Submitted by Balázs Bursics and Anett Kocsis, Budapest[/i]

Alice and Bob are independently trying to figure out a secret password to Cathy’s bitcoin wallet. Both of them have already figured out that: $\bullet$ it is a $4$-digit number whose first digit is $5$. $\bullet$ it is a multiple of $9$; $\bullet$ The larger number is more likely to be a password than a smaller number. Moreover, Alice figured out the second and the third digits of the password and Bob figured out the third and the fourth digits. They told this information to each other but not actual digits. After that the conversation followed: Alice: ”I have no idea what the number is.” Bob: ”I have no idea too.” After that both of them knew which number they should try first. Identify this number

The distance between towns $A$ and $B$ is $999$ km. At every kilometer of the road that connects $A$ and $B$ a sign shows the distances to $A$ and $B$ as follows: $\fbox{0-999}$ , $\fbox{1-998}$ ,$\fbox{2-997}$ , $ . . . $ , $\fbox{998-1}$ , $\fbox{999-0}$ How many signs are there, with both distances written with the help of only two distinct digits?