Let P be a set of 4n points in the plane such that none of the three points are collinear. Prove that if n is large enough, then the following two statements are equivalent. (i) P can be divided into n four-element subsets such that each subset forms the vertices of a convex quadrilateral. (ii) P can not be split into two sets A and B, each with an odd number of elements, so that each convex quadrilateral whose vertices are in P has an even number of vertices in A and B.

Find the positive integers $n$ with $n \geq 4$ such that $[\sqrt{n}]+1$ divides $n-1$ and $[\sqrt{n}]-1$ divides $n+1$. [hide="Remark"]This problem can be solved in a similar way with the one given at [url=http://www.mathlinks.ro/Forum/resources.php?c=1&cid=97&year=2006]Cono Sur Olympiad 2006[/url], problem 5.[/hide]

28. The numbers $1-10$ are written in a circle randomly. Find the expected number of numbers which are at least $2$ larger than an adjacent number. 29. We want to design a new chess piece, the American, with the property that (i) the American can never attack itself, and (ii) if an American $A_1$ attacks another American $A_2$, then $A_2$ also attacks $A_1$. Let $m$ be the number of squares that an American attacks when placed in the top left corner of an 8 by 8 chessboard. Let $n$ be the maximal number of Americans that can be placed on the $8$ by $8$ chessboard such that no Americans attack each other, if one American must be in the top left corner. Find the largest possible value of $mn$. 30. On the blackboard, Amy writes $2017$ in base-$a$ to get $133201_a$. Betsy notices she can erase a digit from Amy’s number and change the base to base-$b$ such that the value of the the number remains the same. Catherine then notices she can erase a digit from Betsy’s number and change the base to base-$c$ such that the value still remains the same. Compute, in decimal, $a + b + c$.

Find all primes $p$ and positive integers $n$ satisfying \[n \cdot 5^{n-n/p} = p! (p^2+1) + n.\]

Let $ABC$ be an acute triangle such that $\angle B > \angle C$. Let $D$ and $E$ be the points on the segments $BC$ and $CA$, respectively, such that $AD$ bisects $\angle A$ and $BE\perp AC$. Finally, let $M$ be the midpoint of the side $BC$. Suppose that the circumcircle of $\triangle CDE$ intersects $AD$ again at a point $X$ different from $D$. Prove that $\angle XME = 90^{\circ} - \angle BAC$.

In the table shown in the figure, Zosia replaced eight numbers with their negatives. It turned out that each row and each column contained exactly two negative numbers. Prove that after this change, the sum of all sixteen numbers in the table is equal to $0$. [center] [img] https://wiki-images.artofproblemsolving.com//2/2e/17-3-5.png [/img] [/center]

Suppose $ABCD$ and $A_1B_1C_1D_1$ be quadrilaterals with corresponding angles equal. Also $AB=A_1B_1$, $AC=A_1C_1$, $BD=B_1D_1$. Are the quadrilaterals necessarily congruent?

Let $n$ be a fixed positive odd integer. Take $m+2$ [b]distinct[/b] points $P_0,P_1,\ldots ,P_{m+1}$ (where $m$ is a non-negative integer) on the coordinate plane in such a way that the following three conditions are satisfied: 1) $P_0=(0,1),P_{m+1}=(n+1,n)$, and for each integer $i,1\le i\le m$, both $x$- and $y$- coordinates of $P_i$ are integers lying in between $1$ and $n$ ($1$ and $n$ inclusive). 2) For each integer $i,0\le i\le m$, $P_iP_{i+1}$ is parallel to the $x$-axis if $i$ is even, and is parallel to the $y$-axis if $i$ is odd. 3) For each pair $i,j$ with $0\le i<j\le m$, line segments $P_iP_{i+1}$ and $P_jP_{j+1}$ share at most $1$ point. Determine the maximum possible value that $m$ can take.

You are given $n \geq 3$ distinct real numbers. Prove that one can choose either $3$ numbers with positive sum, or $2$ numbers with negative sum. [i]Proposed by Mykhailo Shtandenko[/i]

[b]7.1[/b] Prove that a natural number with an odd number of divisors is a perfect square. [b]7.2[/b] In a triangle $ABC$ with area $S$, medians $AK$ and $BE$ are drawn, intersecting at the point $O$. Find the area of the quadrilateral $CKOE$. [img]https://cdn.artofproblemsolving.com/attachments/0/f/9cd32bef4f4459dc2f8f736f7cc9ca07e57d05.png[/img] [b]7.3 .[/b] The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers. When you need to swap tires so that the car can travel the longest possible distance with the same tires? [b]7.4 [/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it? [b]7.5 / 8.4[/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ . [b]7.6[/b] Black paint was sprayed onto a white surface. Prove that there are two points of the same color, the distance between which is $1965$ meters. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].

The sequence of functions $f_0,f_1,f_2,...$ is given by the conditions: $f_0(x) = |x|$ for all $x \in R$ $f_{n+1}(x) = |f_n(x)-2|$ for $n=0,1,2,...$ and all $x \in R$. For each positive integer $n$, solve the equation $f_n(x)=1$.

Solve in real numbers $\frac{(x+2)^4}{x^3}-\frac{(x+2)^2}{2x}\ge - \frac{x}{16}$

In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2+10$, $10+2$, and $3+2+2+5$ are three such ways.

Show that for any $8$ distinct positive real numbers, one can choose a quadraple of them $(a,b,c,d)$ , all distinct such that $$(ac+bd)^2 \ge \frac{2+\sqrt3}{4}\left(a^2+b^2 \right)\left(c^2+d^2 \right)$$ [i]Proposed by Evan Chang (squareman), USA[/i]

Given $x,y,z$, three different integers. Prove that $$(x-y)^5+(y-z)^5+(z-x)^5$$ is divisible by $$5(x-y)(y-z)(z-x)$$

Find the smallest natural number $n$ which has the following properties: a) Its decimal representation has a 6 as the last digit. b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.

An approximate construction of a regular pentagon goes as follows. Inscribe an arbitrary convex pentagon $P_1P_2P_3P_4P_5$ in a circle. Now choose an arror bound $\epsilon > 0$ and apply the following procedure. (a) Denote $P_0 = P_5$ and $P_6 = P_1$ and construct the midpoint $Q_i$ of the circular arc $P_{i-1}P_{i+1}$ containing $P_i$. (b) Rename the vertices $Q_1,...,Q_5$ as $P_1,...,P_5$. (c) Repeat this procedure until the difference between the lengths of the longest and the shortest among the arcs $P_iP_{i+1}$ is less than $\epsilon$. Prove this procedure must end in a finite time for any choice of $\epsilon$ and the points $P_i$.

How many ordered triples of integers $(x,y,z)$ satisfy \[36x^2+100y^2+225z^2=12600?\] [i]Proposed by Bill Fei and Mahith Gottipati [/i]

[b]p1.[/b] Let the series an be defined as $a_1 = 1$ and $a_n =\sum^{n-1}_{i=1} a_ia_{n-i}$ for all positive integers $n$. Evaluate $\sum^{\infty}_{i=1} \left(\frac14\right)^ia_i$. [b]p2.[/b] $a, b, c$ and $d$ are distinct real numbers such that $$a + \frac{1}{b}= b +\frac{1}{c}= c +\frac{1}{d}= d +\frac{1}{a}= x$$ Find |x|. [b]p3.[/b] Find all ordered tuples $(w, x, y, z)$ of complex numbers satisfying $$x + y + z + xy + yz + zx + xyz = -w$$ $$y + z + w + yz + zw + wy + yzw = -x$$ $$z + w + x + zw + wx + xz + zwx = -y$$ $$w + x + y + wx + xy + yw + wxy = -z$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $(a, b)$, such that $\frac{a^2}{2ab^2-b^3+1}$ is also a positive integer.

Construct outside the acute-angled triangle $ABC$ the isosceles triangles $ABA_B, ABB_A , ACA_C,ACC_A ,BCB_C$ and $BCC_B$, so that $$AB = AB_A = BA_B, AC = AC_A=CA_C, BC = BC_B = CB_C$$ and $$\angle BAB_A = \angle ABA_B =\angle CAC_A=\angle ACA_C= \angle BCB_C =\angle CBC_B = a < 90^o$$. Prove that the perpendiculars from $A$ to $B_AC_A$, from $B$ to $A_BC_B$ and from $C$ to $A_CB_C$ are concurrent

Show that any triangle has two sides whose lengths $a$ and $b$ satisfy $\frac{\sqrt{5}-1}{2}<\frac{a}{b}<\frac{\sqrt{5}+1}{2}$.

Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that \[ P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \mbox{for all real $x,y,z,u$}, \] and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$.

Anita plays the following single-player game: She is given a circle in the plane. The center of this circle and some point on the circle are designated “known points”. Now she makes a series of moves, each of which takes one of the following forms: (i) She draws a line (infinite in both directions) between two “known points”; or (ii) She draws a circle whose center is a “known point” and which intersects another “known point”. Once she makes a move, all intersections between her new line/circle and existing lines/circles become “known points”, unless the new/line circle is identical to an existing one. In other words, Anita is making a ruler-and-compass construction, starting from a circle. What is the smallest number of moves that Anita can use to construct a drawing containing an equilateral triangle inscribed in the original circle?

Let $ABCD$ be a tetrahedron and $O$ its incenter, and let the line $OD$ be perpendicular to $AD$. Find the angle between the planes $DOB$ and $DOC.$