What is the largest possible value of the expression $$gcd \,\,\, (n^2 + 3, (n + 1)^2 + 3 )$$ for naturals $n$? [hide]original wording]Kāda ir izteiksmes LKD (n2 + 3, (n + 1)2 + 3) lielākā iespējamā vērtība naturāliem n? [/hide]

Let $ABC$ be a fixed scalene triangle. Suppose that $X, Y$ are variable points on segments $AB$, $AC$, respectively such that $BX = CY$ . Prove that the circumcircle of $\vartriangle AXY$ passes through a fixed point other than $A$.

Let $[ABCDEF]$ be a frustum of a regular pyramid. Let $G$ and $G'$ be the centroids of bases $ABC$ and $DEF$ respectively. It is known that $AB=36,DE=12$ and $GG'=35$. $a)$ Prove that the planes $(ABF),(BCD),(CAE)$ have a common point $P$, and the planes $(DEC),(EFA),(FDB)$ have a common point $P'$, both situated on $GG'$. $b)$ Find the length of the segment $[PP']$.

Suppose that $A=1,2,$ or $3$. Let $a$ and $b$ be relatively prime integers such that $a^{2}+Ab^2 =s^3$ for some integer $s$. Then, there are integers $u$ and $v$ such that $s=u^2 +Av^2$, $a =u^3 - 3Avu^2$, and $b=3u^{2}v -Av^3$.

A checkered polygon $A$ is drawn on the checkered plane. We call a cell of $A$ [i]internal[/i] if all $8$ of its adjacent cells belong to $A$. All other (non-internal) cells of $A$ we call [i]boundary[/i]. It is known that $1)$ each boundary cell has exactly two common sides with no boundary cells; and 2) the union of all boundary cells can be divided into isosceles trapezoid of area $2$ with vertices at the grid nodes (and acute angles of the trapezoids are equal $45^\circ$). Prove that the area of the polygon $A$ is congruent to $1$ modulo $4$.

In a room containing $N$ people, $N > 3$, at least one person has not shaken hands with everyone else in the room. What is the maximum number of people in the room that could have shaken hands with everyone else? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }N-1\qquad\textbf{(D) }N\qquad \textbf{(E) }\text{none of these}$

A sequence of integers $a_1, a_2, \ldots, a_n$ is said to be [i]sub-Fibonacci[/i] if $a_1=a_2=1$ and $a_i \le a_{i-1}+a_{i-2}$ for all $3 \le i \le n.$ How many sub-Fibonacci sequences are there with $10$ terms such that the last two terms are both $20$?

[b]p1.[/b] a) Given four points in the plane, no three of which lie on the same line, each subset of three points determines the vertices of a triangle. Can all these triangles have equal areas? If so, give an example of four points (in the plane) with this property, and then describe all arrangements of four joints (in the plane) which permit this. If no such arrangement exists, prove this. b) Repeat part a) with "five" replacing "four" throughout. [b]p2.[/b] Three people at the base of a long stairway begin a race up the stairs. Person A leaps five steps with each stride (landing on steps $5$, $10$, $15$, etc.). Person B leaps a little more slowly but covers six steps with each stride. Person C leaps seven steps with each stride. A picture taken near the end of the race shows all three landing simultaneously, with Person A twenty-one steps from the top, person B seven steps from the top, and Person C one step from the top. How many steps are there in the stairway? If you can find more than one answer, do so. Justify your answer. [b]p3. [/b]Let $S$ denote the sum of an infinite geometric series. Suppose the sum of the squares of the terms is $2S$, and that df the cubes is $64S/13$. Find the first three terms of the original series. [b]p4.[/b] $A$, $B$ and $C$ are three equally spaced points on a circular hoop. Prove that as the hoop rolls along the horizontal line $\ell$, the sum of the distances of the points $A, B$, and $C$ above line $\ell$ is constant. [img]https://cdn.artofproblemsolving.com/attachments/3/e/a1efd0975cf8ff3cf6acb1da56da1dce35d81e.png[/img] [b]p5.[/b] A set of $n$ numbers $x_1,x_2,x_3,...,x_n$ (where $n>1$) has the property that the $k^{th}$ number (that is, $x_k$ ) is removed from the set, the remaining $(n-1)$ numbers have a sum equal to $k$ (the subscript o $x_k$ ), and this is true for each $k = 1,2,3,...,n$. a) SoIve for these $n$ numbers b) Find whether at least one of these $n$ numbers can be an integer. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

It is given that there exist unique integers $m_1,\ldots, m_{100}$ such that \[0\leq m_1 < m_2 < \cdots < m_{100}\quad\text{and}\quad 2018 = \binom{m_1}1 + \binom{m_2}2 + \cdots + \binom{m_{100}}{100}.\] Find $m_1 + m_2 + \cdots + m_{100}$.

Let $ABC$ be a triangle right in $C$ and the points $D, E$ on the sides $BC$ and $CA$ respectively, such that $\frac{BD}{AC} =\frac{AE}{CD} = k$. Lines $BE$ and $AD$ intersect at $O$. Show that the angle $\angle BOD = 60^o$ if and only if $k =\sqrt3$.

Suppose that the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are distinct. Show that \[(x-a_{1})(x-a_{2}) \cdots (x-a_{n})-1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

Prove that the sum of the lengths of the diagonals of an arbitrary quadrilateral is less than the sum of the lengths of its sides.

The circuit diagram drawn (see figure ) contains a battery $B$, a lamp $L$ and five switches $S_1$ to $S_5$. The probability that switch $S_3$ is closed (makes contact) is $\frac23$, for the other four switches that probability is $\frac12$ (the probabilities are mutually independent). Calculate the probability that the light is on. [asy] unitsize (2 cm); draw((-1,1)--(-0.5,1)); draw((-0.25,1)--(1,1)--(1,0.25)); draw((1,-0.25)--(1,-1)--(0.05,-1)); draw((-0.05,-1)--(-1,-1)--(-1,0.25)); draw((-1,0.5)--(-1,1)); draw((-1,1)--(-0.5,0.5)); draw((-0.25,0.25)--(0,0)); draw((-1,0)--(-0.75,0)); draw((-0.5,0)--(0,0)); draw((0,1)--(0,0.75)); draw((0,0.5)--(0,0)); draw((-0.25,1)--(-0.5,1.25)); draw((-1,0.25)--(-1.25,0.5)); draw((-0.5,0.5)--(-0.25,0.5)); draw((0,0.75)--(0.25,0.5)); draw((-0.75,0)--(-0.5,-0.25)); draw(Circle((1,0),0.25)); draw(((1,0) + 0.25*dir(45))--((1,0) + 0.25*dir(225))); draw(((1,0) + 0.25*dir(135))--((1,0) + 0.25*dir(315))); draw((0.05,-0.9)--(0.05,-1.1)); draw((-0.05,-0.8)--(-0.05,-1.2)); label("$L$", (1.25,0), E); label("$B$", (-0.1,-1.1), SW); label("$S_1$", (-0.5,1.25), NE); label("$S_2$", (-1.25,0.5), SW); label("$S_3$", (-0.5,0.5), SW); label("$S_4$", (0.25,0.5), NE); label("$S_5$", (-0.5,-0.25), SW); [/asy]

Trapezoid $ABCD$ has sides $AB=92,BC=50,CD=19,AD=70$ $AB$ is parallel to $CD$ A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$.Given that $AP=\dfrac mn$ (Where $m,n$ are relatively prime).What is $m+n$?

Given the isosceles triangle $ABC$, where $AB = AC$. Let $D$ be a point in the segment $BC$ so that $BD = 2DC$. Suppose also that point $P$ lies on the segment $AD$ such that: $\angle BAC = \angle BP D$. Prove that $\angle BAC = 2\angle DP C$.

Given two circles of radii $r$ and $r'$ exterior to each other, construct a line parallel to a given line and intersecting the two circles in chords with the sum of lengths $\ell$.