The function $f : \mathbb N \to \mathbb R$ satisfies $f(1) = 1, f(2) = 2$ and \[f (n+2) = f(n+2 - f(n+1) ) + f(n+1 - f(n) ).\] Show that $0 \leq f(n+1) - f(n) \leq 1$. Find all $n$ for which $f(n) = 1025$.

Let $E$ be the set of $1983^3$ points of the space $\mathbb R^3$ all three of whose coordinates are integers between $0$ and $1982$ (including $0$ and $1982$). A coloring of $E$ is a map from $E$ to the set {red, blue}. How many colorings of $E$ are there satisfying the following property: The number of red vertices among the $8$ vertices of any right-angled parallelepiped is a multiple of $4$ ?

In an acute triangle $\vartriangle ABC$, $D$ is the foot of altitude from $A$ to $BC$. Suppose that $AD = CD$, and define $N$ as the intersection of the median $CM$ and the line $AD$. Prove that $\vartriangle ABC$ is isosceles if and only if $CN = 2AM$.

Find the number of real solutions to the equation $(x^{2006}+1)(1+x^2+x^4+\ldots +x^{2004})=2006x^{2005}$

Let $a_1,a_2,\cdots, a_{31} ;b_1,b_2, \cdots, b_{31}$ be positive integers such that $a_1< a_2<\cdots< a_{31}\leq2015$ , $ b_1< b_2<\cdots<b_{31}\leq2015$ and $a_1+a_2+\cdots+a_{31}=b_1+b_2+\cdots+b_{31}.$ Find the maximum value of $S=|a_1-b_1|+|a_2-b_2|+\cdots+|a_{31}-b_{31}|.$

For a point $P$ on an ellipse, let $d$ be the distance from the center of the ellipse to the line tangent to the ellipse at $P.$ Prove that $(PF_1)(PF_2)d^2$ is constant as $P$ varies on the ellipse, where $PF_1$ and $PF_2$ are distances from $P$ to the foci $F_1$ and $F_2$ of the ellipse.

A set of identical square tiles with side length $1$ is placed on a (very large) floor. Every tile after the first shares an entire edge with at least one tile that has already been placed. - What is the largest possible perimeter for a figure made of $10$ tiles? - What is the smallest possible perimeter for a figure made of $10$ tiles? - What is the largest possible perimeter for a figure made of $2011$ tiles? - What is the smallest possible perimeter for a figure made of $2011$ tiles? Prove that your answers are correct.

For $a,b,c$ be real numbers greater than $1$, prove that \[\frac{a+b+c}{4} \geq \frac{\sqrt{ab-1}}{b+c}+\frac{\sqrt{bc-1}}{c+a}+\frac{\sqrt{ca-1}}{a+b}.\]

The sequence $(x_n)_{n\geqslant 0}$ is defined as such: $x_0=1, x_1=2$ and $x_{n+1}=4x_n-x_{n-1}$, for all $n\geqslant 1$. Determine all the terms of the sequence which are perfect squares. [i]George Stoica, Canada[/i]

Alice and Bob are running around a rectangular building measuring 100 by 200 meters. They start at the middle of a 200 meter side and run in the same direction, Alice running twice as fast as Bob. After Bob runs one lap around the building, what fraction of the time were Alice and Bob on the same side of the building?

Suppose that $a$ is real number. Sequence $a_1,a_2,a_3,....$ satisfies $$a_1=a, a_{n+1} = \begin{cases} a_n - \frac{1}{a_n}, & a_n\ne 0 \\ 0, & a_n=0 \end{cases} (n=1,2,3,..)$$ Find all possible values of $a$ such that $|a_n|<1$ for all positive integer $n$.

Let $n>1$ be an integer. Prove that there exist $m>n^n $ such that $\frac {n^m-m^n}{m+n} $ is a positive integer.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

Let $P,Q\in\mathbb{R}[x]$, such that $Q$ is a $2021$-degree polynomial and let $a_{1}, a_{2}, \ldots , a_{2022}, b_{1}, b_{2}, \ldots , b_{2022}$ be real numbers such that $a_{1}a_{2}\ldots a_{2022}\neq 0$. If for all real $x$ \[P(a_{1}Q(x) + b_{1}) + \ldots + P(a_{2021}Q(x) + b_{2021}) = P(a_{2022}Q(x) + b_{2022})\] prove that $P(x)$ has a real root.

Chris and Michael play a game on a board which is a rhombus of side length $n$ (a positive integer) consisting of two equilateral triangles, each of which has been divided into equilateral triangles of side length $ 1$. Each has a single token, initially on the leftmost and rightmost squares of the board, called the “home” squares (the illustration shows the case $n = 4$). [img]https://cdn.artofproblemsolving.com/attachments/e/b/8135203c22ce77c03c144850099ad1c575edb8.png[/img] A move consists of moving your token to an adjacent triangle (two triangles are adjacent only if they share a side). To win the game, you must either capture your opponent’s token (by moving to the triangle it occupies), or move on to your opponent’s home square. Supposing that Chris moves first, which, if any, player has a winning strategy?

Evaluate $\int_0^{2\pi} |x^2-\pi ^ 2 -\sin ^ 2 x|\ dx$.

It takes Mary $ 30$ minutes to walk uphill $ 1$ km from her home to school, but it takes her only $ 10$ minutes to walk from school to home along the same route. What is her average speed, in km/hr, for the round trip? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 3.125 \qquad \textbf{(C)}\ 3.5 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 4.5$

For which numbers $x$of the interval $0 < x <\pi$ holds: $$\frac{\tan 2x}{\tan x} -\frac{2 \cot 2x}{\cot x}=1$$

Present a family of subsets of the plane such that each one of its members is Lebesgue measurable, each one of its members intersects any circle, and the set of Lebesgue measures of all its members is the set of nonnegative real numbers.

Justin the robot is on a mission to rescue abandoned treasure from a minefield. To do this, he must travel from the point $(0, 0, 0)$ to $(4, 4, 4)$ in three-dimensional space, only taking one-unit steps in the positive $x, y,$ or $z$-directions. However, the evil David anticipated Justin's arrival, and so he has surreptitiously placed a mine at the point $(2,2,2)$. If at any point Justin is at most one unit away from this mine (in any direction), the mine detects his presence and explodes, thwarting Justin. How many paths can Justin take to reach his destination safely? [i]Proposed by Justin Stevens[/i]

In a non-isosceles triangle $ABC$ the centre of the incircle is denoted by $I$. The other intersection point of the angle bisector of $\angle BAC$ and the circumcircle of $\vartriangle ABC$ is $D$. The line through $I$ perpendicular to $AD$ intersects $BC$ in $F$. The midpoint of the circle arc $BC$ on which $A$ lies, is denoted by $M$. The other intersection point of the line $MI$ and the circle through $B, I$ and $C$, is denoted by $N$. Prove that $FN$ is tangent to the circle through $B, I$ and $C$.

Let $ P$, $ Q$, and $ R$ be the points on sides $ BC$, $ CA$, and $ AB$ of an acute triangle $ ABC$ such that triangle $ PQR$ is equilateral and has minimal area among all such equilateral triangles. Prove that the perpendiculars from $ A$ to line $ QR$, from $ B$ to line $ RP$, and from $ C$ to line $ PQ$ are concurrent.

An acute triangle $ABC$ has circumradius $R$, inradius $r$. $A$ is the biggest angle among $A,B,C$. Let $M$ be the midpoint of $BC$, and $X$ be the intersection of two lines that touches circumcircle of $ABC$ and goes through $B,C$ respectively. Prove the following inequality : $ \frac{r}{R} \geq \frac{AM}{AX}$.

$n > 10$ teams take part in a football tournament. Every team plays exactly once against each other. A win gives two points, a draw a point, and a defeat no point. After the tournament it turns out that each team gets exactly half their points in the games against the bottom $10$ teams has won (in particular, each of these $10$ teams has won the made half their points against the $9$ remaining). Determine all possible values ​​of $n$, and give an example of such a tournament for these values.

Let $f(x)=x^2+18$ have roots $r_1$ and $r_2$, and let $g(x)=x^2-8x+17$ have roots $r_3$ and $r_4$. If $h(x)=x^4+ax^3+bx^2+cx+d$ has roots $r_1+r_3$, $r_1+r_4$, $r_2+r_3$, and $r_2+r_4$, then find $h(4)$.