There are 8 distinct points on a plane, where no three are collinear. An ant starts at one of the points, then walks in a straight line to each one of the other points, visiting each point exactly once and stopping at the final point. This creates a trail of 7 line segments. What is the maximum number of times the ant can cross its own path as it walks? [i]Proposed by Gabriel Wu[/i]

Determine the smallest integer $n > 1$ with the property that $n^2(n - 1)$ is divisible by 2009.

Let $S$ be the set of sequences of length 2018 whose terms are in the set $\{1, 2, 3, 4, 5, 6, 10\}$ and sum to 3860. Prove that the cardinality of $S$ is at most \[2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.\]

Find the polynomials $ f(x)$ having the following properties: (i) $ f(0) \equal{} 1$, $ f'(0) \equal{} f''(0) \equal{} \cdots \equal{} f^{(n)}(0) \equal{} 0$ (ii) $ f(1) \equal{} f'(1) \equal{} f''(1) \equal{} \cdots \equal{} f^{(m)}(1) \equal{} 0$

For every positive integer $n$, let $p(n)$ denote the number of ways to express $n$ as a sum of positive integers. For instance, $p(4)=5$ because \[4=3+1=2+2=2+1+1=1+1+1.\] Also define $p(0)=1$. Prove that $p(n)-p(n-1)$ is the number of ways to express $n$ as a sum of integers each of which is strictly greater than 1. [i]Proposed by Fedor Duzhin, Nanyang Technological University.[/i]

If $x_1$ and $x_2$ are the solutions of the equation $x^2-(m+3)x+m+2=0$ Find all real values of $m$ such that the following inequations are valid $\frac {1}{x_1}+\frac {1}{x_2}>\frac{1}{2}$ and $x_1^2+x_2^2<5$

Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$. [i]Proposed by Estonia[/i]

Let $a, b, c, d, e$ be positive and [b]different [/b] divisors of $n$ where $n \in Z^{+}$. If $n=a^4+b^4+c^4+d^4+e^4$ let's call $n$ "marvelous" number. $a)$ Prove that all "marvelous" numbers are divisible by $5$. $b)$ Can count of "marvelous" numbers be infinity?

Consider all planes through the center of a $2\times2\times2$ cube that create cross sections that are regular polygons. The sum of the cross sections for each of these planes can be written in the form $a\sqrt b+c$, where $b$ is a square-free positive integer. Find $a+b+c$.

Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Find all positive integers $n$ satisfying $2n+7 \mid n! -1$.

Let $s_1, s_2, s_3$ be the respective sums of $n$, $2n$, $3n$ terms of the same arithmetic progression with $a$ as the first term and $d$ as the common difference. Let $R=s_3-s_2-s_1$. Then $R$ is dependent on: $ \textbf{(A)}\ a \text{ } \text{and} \text{ } d\qquad\textbf{(B)}\ d \text{ } \text{and} \text{ } n\qquad\textbf{(C)}\ a \text{ } \text{and} \text{ } n\qquad\textbf{(D)}\ a, d, \text{ } \text{and} \text{ } n\qquad$ $\textbf{(E)}\ \text{neither} \text{ } a \text{ } \text{nor} \text{ } d \text{ } \text{nor} \text{ } n $

If $x,y$ are real, then the $\textit{absolute value}$ of the complex number $z=x+yi$ is \[|z|=\sqrt{x^2+y^2}.\] Find the number of polynomials $f(t)=A_0+A_1t+A_2t^2+A_3t^3+t^4$ such that $A_0,\ldots,A_3$ are integers and all roots of $f$ in the complex plane have absolute value $\leq 1$.

Let $\mathbb{Z}^{+}$ be the set of positive integers. [b]a)[/b] Prove that there is only one function $f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$, strictly increasing, such that $f(f(n))=2n+1$ for every $n\in \mathbb{Z}^{+}$. [b]b)[/b] For the function in [b]a[/b]. Prove that for every $n\in \mathbb{Z}^{+}$ $\frac{4n+1}{3}\leq f(n)\leq \frac{3n+1}{2}$ [b]c) [/b] Prove that in each inequality side of [b]b[/b] the equality can reach by infinite positive integers $n$.

Let $a$ and $b$ be two positive reals such that the following inequality \[ ax^3 + by^2 \geq xy - 1 \] is satisfied for any positive reals $x, y \geq 1$. Determine the smallest possible value of $a^2 + b$. [i]Proposed by Fajar Yuliawan[/i]

Consider the sequence $p_{1},p_{2},...$ of primes such that for each $i\geq2$, either $p_{i}=2p_{i-1}-1$ or $p_{i}=2p_{i-1}+1$. Show that any such sequence has a finite number of terms.

Let $S$ be a set of integers with the following properties: [list] [*] $\{ 1, 2, \dots, 2025 \} \subseteq S$. [*] If $a, b \in S$ and $\gcd(a, b) = 1$, then $ab \in S$. [*] If for some $s \in S$, $s + 1$ is composite, then all positive divisors of $s + 1$ are in $S$. [/list] Prove that $S$ contains all positive integers.

In a geometric sequence of real numbers, the sum of the first two terms is 7, and the sum of the first 6 terms is 91. The sum of the first 4 terms is $\text{(A)}\ 28 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 49 \qquad \text{(E)}\ 84$

An $n\times n$ chessboard is given, where $n$ is an even positive integer. On every line, the unit squares are to be permuted, subject to the condition that the resulting table has to be symmetric with respect to its main diagonal (the diagonal from the top-left corner to the bottom-right corner). We say that a board is [i]alternative[/i] if it has at least one pair of complementary lines (two lines are complementary if the unit squares on them which lie on the same column have distinct colours). Otherwise, we call the board [i]nonalternative[/i]. For what values of $n$ do we always get from the $n\times n$ chessboard an alternative board?\\ \\ [i](Alexandru Petrescu and Andra Elena Mircea)[/i]

Georg has a circular game board with 100 squares labelled $1, 2, . . . , 100$. Georg chooses three numbers $a, b, c$ among the numbers $1, 2, . . . , 99$. The numbers need not be distinct. Initially there is a piece on the square labelled $100$. First, Georg moves the piece $a$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Then he moves the piece $b$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Finally, he moves the piece $c$ squares forward $33$ times and puts a caramel on each of the squares the piece lands on. Thus he puts a total of $99$ caramels on the board. Georg wins all the caramels on square number $1$. How many caramels can Georg win, at most? [img]https://cdn.artofproblemsolving.com/attachments/d/c/af438e5feadca5b1bfc98ae427f6fc24655e29.png[/img]

A complex number $z$ whose real and imaginary parts are integers satisfies $\left(Re(z) \right)^4 +\left(Re(z^2)\right)^2 + |z|^4 =(2018)(81)$, where $Re(w)$ and $Im(w)$ are the real and imaginary parts of $w$, respectively. Find $\left(Im(z) \right)^2$ .

Show that \[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\] holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.

Let $a_1,...,a_n$ be $n$ real numbers. If for each odd positive integer $k\leqslant n$ we have $a_1^k+a_2^k+\ldots+a_n^k=0$, then for each odd positive integer $k$ we have $a_1^k+a_2^k+\ldots+a_n^k=0$. [i]Proposed by M. Didin[/i]

The golden ratio $\phi = \frac{1+\sqrt5}{2}$ satisfies the property $\phi^2 =\phi + 1$. Point $P$ lies inside equilateral triangle $\vartriangle ABC$ such that $PA = \phi$, $PB = 2$, and angle $\angle APC$ measures $150$ degrees. What is the measure of $\angle BPC$ in degrees?

In a triangle, both the sides and the angles form arithmetic sequences. Determine the angles of the triangle.