How many real solutions are there to the equation $ |x \minus{} |2x \plus{} 1\parallel{} \equal{} 3.$ (Here, $ |x|$ denotes the absolute value of $ x$: i.e., if $ x \geq 0,$ then $ |x| \equal{} |\minus{}x| \equal{} x.$) A. 0 B. 1 C. 2 D. 3 E. 4

Let $ x,y,z$ real numbers such that $ x \plus{} y \plus{} z \equal{} xy \plus{} yz \plus{} zx$. Find the minimum value of \[ {x \over x^2 \plus{} 1} \plus{} {y\over y^2 \plus{} 1} \plus{} {z\over z^2 \plus{} 1}\]

Find the number of 8-digit base-6 positive integers $(a_1a_2a_3a_4a_5a_6a_7a_8)_6$ (with leading zeros permitted) such that $(a_1a_2\ldots a_8)_6\mid(a_{i+1}a_{i+2}\ldots a_{i+8})_6$ for $i=1,2,\ldots,7$, where indices are taken modulo $8$ (so $a_9=a_1$, $a_{10}=a_2$, and so on). [i]Victor Wang[/i]

Let $ABC$ be a triangle and $D$, $E$, $F$ be the midpoints of arcs $BC$, $CA$, $AB$ on the circumcircle. Line $\ell_a$ passes through the feet of the perpendiculars from $A$ to $DB$ and $DC$. Line $m_a$ passes through the feet of the perpendiculars from $D$ to $AB$ and $AC$. Let $A_1$ denote the intersection of lines $\ell_a$ and $m_a$. Define points $B_1$ and $C_1$ similarly. Prove that triangle $DEF$ and $A_1B_1C_1$ are similar to each other.

Christopher and Robin are playing a game in which they take turns tossing a circular token of diameter 1 inch onto an infinite checkerboard whose squares have sides of 2 inches. If the token lands entirely in a square, the player who tossed the token gets 1 point; otherwise, the other player gets 1 point. A player wins as soon as he gets two more points than the other player. If Christopher tosses first, what is the probability that he will win? Express your answer as a fraction.

Let $A$ be a subset of the edges of an $n\times n $ table. Let $V(A)$ be the set of vertices from the table which are connected to at least on edge from $A$ and $j(A)$ be the number of the connected components of graph $G$ which it's vertices are the set $V(A)$ and it's edges are the set $A$. Prove that for every natural number $l$: $$\frac{l}{2}\leq min_{|A|\geq l}(|V(A)|-j(A)) \leq \frac{l}{2}+\sqrt{\frac{l}{2}}+1$$

$H$ is the orthocentre of $\triangle{ABC}$. $D$, $E$, $F$ are on the circumcircle of $\triangle{ABC}$ such that $AD \parallel BE \parallel CF$. $S$, $T$, $U$ are the semetrical points of $D$, $E$, $F$ with respect to $BC$, $CA$, $AB$. Show that $S, T, U, H$ lie on the same circle.

The point $P$ lies inside the triangle $ABC$. A line is drawn through $P$ parallel to each side of the triangle. The lines divide $AB$ into three parts length $c, c', c"$ (in that order), and $BC$ into three parts length $a, a', a"$ (in that order), and $CA$ into three parts length $b, b', b"$ (in that order). Show that $abc = a'b'c' = a"b"c"$.

A positive integer $n \ge 2$ is called [i]peculiar [/i] if the number $n \choose i$ + $n \choose j $ $-i-j$ is even for all integers $i$ and $j$ such that $0 \le i \le j \le n$. Determine all peculiar numbers.

Let $f : \mathbb R \to \mathbb R$ be a function such that $f(1)=1$ and \[f(x+y)=f(x)+f(y)\] And for all $x \in \mathbb R / \{0\}$ we have $f\left( \frac 1x \right) = \frac{1}{f(x)}.$ Find all such functions $f.$

Let $ L$ denote the set of all lattice points of the plane (points with integral coordinates). Show that for any three points $ A,B,C$ of $ L$ there is a fourth point $ D,$ different from $ A,B,C,$ such that the interiors of the segments $ AD,BD,CD$ contain no points of $ L.$ Is the statement true if one considers four points of $ L$ instead of three?

Find all polynomials $ P\in\mathbb{R}[x]$ such that for all $ r\in\mathbb{Q}$,there exist $ d\in\mathbb{Q}$ such that $ P(d)\equal{}r$

Let $ABC$ be an acute-angled triangle, $H$ the intersection point of its altitudes , and $AA'$ the diameter of the circumcircle of triangle $ABC$. Prove that the quadrilateral $HB A'C$ is a parallelogram.

Is it true that for all natural $n$, it is always possible to give each of the $n$ children a hat painted in one of $100$ colors so that if a girl is known to more than $2015$ boys, then not all of these boys have hats of the same color, and, if a boy is acquainted with more than $2015$ girls, don't all these girls have hats of the same color? [hide=original wording]Vai tiesa, ka visiem naturāliem n vienmēr iespējams katram no n bērniem iedot pa cepurei, kas nokrāsota vienā no 100 krāsām tā, ka, ja kāda meitene ir pazīstama ar vairāk nekā 2015 zēniem, tad ne visiem šiem zēniem cepures ir vienā krāsā, un, ja kāds zēns ir pazīštams ar vairāk nekā 2015 meitenēm, tad ne visām šīm meitenēm cepures ir vienā krāsā?[/hide]

Find all natural numbers $ x,y$ such that $ y| (x^{2}+1)$ and $ x^{2}| (y^{3}+1)$.

Alice throws two standard dice, with $A$ being the number on her first die and $B$ being the number on her second die. She then draws the line $Ax + By = 2013$. Boris also throws two standard dice, with $C$ being the number on his first die and $D$ being the number on his second die. He then draws the line $Cx + Dy = 2014$. Compute the probability that these two lines are parallel.

Alexandre and Herculano are at Campanha station waiting for the train. To entertain themselves, they decide to calculate the length of a freight train that passes through the station without changing its speed. When the front of the train passes them, Alexandre starts walking in the direction of the train's movement and Herculano starts walking in the opposite direction. The two walk at the same speed and each of them stops at the moment they cross the end of the train. Alexandre walked $45$ meters and Herculano $30$. How long is the train?

We have the following board of $2 \times 6$. [asy] unitsize(0.8 cm); int i; draw((0,0)--(6,0)); draw((0,1)--(6,1)); draw((0,2)--(6,2)); for (i = 0; i <= 6; ++i) { draw((i,0)--(i,2)); } dot("$A$", (0,2), NW); dot("$B$", (6,2), NE); dot("$C$", (3,0), S); [/asy] Find in how many ways you can go from point $A$ to point $B$, moving by the segments of the board, respecting the following rules: - You cannot pass through the same point twice. - You can only make three types of movements moving through the segments: To the right, up, down - You have to go through point $C$.

In $\triangle ABC$, points $P,Q$ satisfy $\angle PBC = \angle QBA$ and $\angle PCB = \angle QCA$, $D$ is a point on $BC$ such that $\angle PDB=\angle QDC$. Let $X,Y$ be the reflections of $A$ with respect to lines $BP$ and $CP$, respectively. Prove that $DX=DY$. [img]https://cdn.artofproblemsolving.com/attachments/a/7/f208f1651afc0fef9eef4c68ba36bf77556058.jpg[/img]

Evaluate $ \int_{\frac {\pi}{4}}^{\frac {3}{4}\pi } \cos \frac {1}{\sin \left(\frac {1}{\sin x}\right)}\cdot \cos \left(\frac {1}{\sin x}\right)\cdot \frac {\cos x}{\sin ^ 2 x\cdot \sin ^ 2 \left(\frac {1}{\sin x }\right)}\ dx$ Last Edited, Sorry kunny

On Lineland there are 2018 bus stations numbered 1 through 2018 from left to right. A self-driving bus that can carry at most $N$ passengers starts from station 1 and drives all the way to station 2018, while making a stop at each bus station. Each passenger that gets on the bus at station $i$ will get off at station $j$ for some $j>i$ (the value of $j$ may vary over different passengers). Call any group of four distinct stations $i_1, i_2, j_1, j_2$ with $i_u< j_v$ for all $u,v\in \{1,2\}$ a [i]good[/i] group. Suppose that in any good group $i_1, i_2, j_1, j_2$, there is a passenger who boards at station $i_1$ and de-boards at station $j_1$, or there is a passenger who boards at station $i_2$ and de-boards at station $j_2$, or both scenarios occur. Compute the minimum possible value of $N$. [i]Proposed by Yannick Yao[/i]

Let $a, b, c \in [0;1]$ be reals such that $ab+bc+ca=1$. Find the minimal and maximal value of $a^3+b^3+c^3$.

Let $A_1A_2...A_n$ be a regular polygon ($n > 4$), $T$ be the common point of $A_1A_2$ and $A_{n-1}A_n$ and $M$ be a point in the interior of the triangle $A_1A_nT$. Show that the equality $$\sum_{i=1}^{n-1} \frac{\sin^2 \left(\angle A_iMA_{i+1}\right)}{d(M,A_iA_{i+1}}=\frac{\sin^2 \left(\angle A_1MA_n\right)}{d(M,A_1A_n} $$ holds if and only if $M$ belongs to the circumcircle of the polygon.

A semicircle has diameter $XY$. A square $PQRS$ with side length 12 is inscribed in the semicircle with $P$ and $S$ on the diameter. Square $STUV$ has $T$ on $RS$, $U$ on the semicircle, and $V$ on $XY$. What is the area of $STUV$?

An infinite sequence of positive integers $a_1, a_2, \dots$ is called $good$ if (1) $a_1$ is a perfect square, and (2) for any integer $n \ge 2$, $a_n$ is the smallest positive integer such that $$na_1 + (n-1)a_2 + \dots + 2a_{n-1} + a_n$$ is a perfect square. Prove that for any good sequence $a_1, a_2, \dots$, there exists a positive integer $k$ such that $a_n=a_k$ for all integers $n \ge k$. [size=75](reposting because the other thread didn't get moved)[/size]