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]

Andile and Zandre play a game on a $2017 \times 2017$ board. At the beginning, Andile declares some of the squares [i]forbidden[/i], meaning the nothing may be placed on such a square. After that, they take turns to place coins on the board, with Zandre placing the first coin. It is not allowed to place a coin on a forbidden square or in the same row or column where another coin has already been placed. The player who places the last coin wins the game. What is the least number of squares Andile needs to declare as forbidden at the beginning to ensure a win? (Assume that both players use an optimal strategy.)

Let $n\geqslant 3$ be an integer and $a_1,a_2,\ldots,a_n$ be pairwise distinct positive real numbers with the property that there exists a permutation $b_1,b_2,\ldots,b_n$ of these numbers such that\[\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots=\frac{a_{n-1}}{b_{n-1}}\neq 1.\]Prove that there exist $a,b>0$ such that $\{a_1,a_2,\ldots,a_n\}=\{ab,ab^2,\ldots,ab^n\}.$ [i]Cristi Săvescu[/i]

Find all polynomials with real coefficients $P(x)$ such that $P(x^2+x+1)$ divides $P(x^3-1)$.

Find the number of permutations of the letters $ABCDE$ where the letters $A$ and $B$ are not adjacent and the letters $C$ and $D$ are not adjacent. For example, count the permutations $ACBDE$ and $DEBCA$ but not $ABCED$ or $EDCBA$.

Let $f(x)=|\log(x+1)|$ and let $a,b$ be two real numbers ($a<b$) satisfying the equations $f(a)=f\left(-\frac{b+1}{a+1}\right)$ and $f\left(10a+6b+21\right)=4\log 2$. Find $a,b$.

Let $ AH_1, BH_2, CH_3$ be the altitudes of an acute angled triangle $ ABC$. Its incircle touches the sides $ BC, AC$ and $ AB$ at $ T_1, T_2$ and $ T_3$ respectively. Consider the symmetric images of the lines $ H_1H_2, H_2H_3$ and $ H_3H_1$ with respect to the lines $ T_1T_2, T_2T_3$ and $ T_3T_1$. Prove that these images form a triangle whose vertices lie on the incircle of $ ABC$.

In a plane there are four fixed points $A, B, C, D$, no $3$ collinear. Construct a square with sides $a, b, c, d$ such that $A \in a$, $B \in b$, $C \in c$, $D \in d$.

In triangle $ABC$, $AB=\frac{20}{11} AC$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A triangle with integer sides is called Heronian if its area is an integer. Does there exist a Heronian triangle whose sides are the arithmetic, geometric and harmonic means of two positive integers?

Let $a,b,c$ be real numbers such that $a^3(b+c)+b^3(a+c)+c^3(a+b)=0$. Prove that $ab+bc+ca\leq0$.