Let $a_n = 2^{n-1}$ for $n > 0$. Let \[ b_n = \sum\limits_{r+s \leq n} a_ra_s \] Find $b_n-b_{n-1}$, $b_n-2b_{n-1}$ and $b_n$.

On a circle $100$ points are chosen and for each point we wrote the multiple of its distances to the rest. Could the written numbers be $1,2,\dots, 100$ in some order?

Prove the following theorem: If the intersection of any plane that has more than one point in common with the surface $F$ is a circle, then $F$ is a sphere (surface).

Barry has a standard die containing the numbers 1-6 on its faces. He rolls the die continuously, keeping track of the sum of the numbers he has rolled so far, starting from 0. Let $E_n$ be the expected number of time he needs to until his recorded sum is at least $n$. It turns out that there exist positive reals $a, b$ such that $$\lim_{n \rightarrow \infty} E_n - (an + b) = 0$$ Find $(a,b)$. [i]Proposed by Dilhan Salgado[/i]

Does there exist a functional equation[sup]1[/sup] that has a solution and the range of any of its solutions is the set of integers? [sup]1[/sup][size=75]A [i]functional equation[/i] has the form $\mbox{\footnotesize \(E = 0\)}$, where $\mbox{\footnotesize \(E\)}$ is a function form. The set of function forms is the smallest set $\mbox{\footnotesize \(\mathcal{F}\)}$ which contains the variables $\mbox{\footnotesize \(x_1, x_2, \dots\)}$, the real numbers $\mbox{\footnotesize \(r \in \mathbb{R}\)}$, and for which $\mbox{\footnotesize \(E, E_1, E_2 \in \mathcal{F}\)}$ implies $\mbox{\footnotesize \(E_1+E_2 \in \mathcal{F}\)}$, $\mbox{\footnotesize \(E_1 \cdot E_2 \in \mathcal{F}\)}$, and $\mbox{\footnotesize \(f(E) \in \mathcal{F}\)}$, where $\mbox{\footnotesize \(f\)}$ is a fixed function symbol. The solution of the functional equation $\mbox{\footnotesize \(E = 0\)}$ is a function $\mbox{\footnotesize \(f: \mathbb{R} \to \mathbb{R}\)}$ such that $\mbox{\footnotesize \(E = 0\)}$ holds for all values of the variables. E.g. $\mbox{\footnotesize \(f\big(x_1 + f(\sqrt{2} \cdot x_2 \cdot x_2)\big) + (-\pi) + (-1) \cdot x_1 \cdot x_1 \cdot x_2 = 0\)}$ is a functional equation.[/size]

Solve into $N$: $$a^2 = 2^b +c^4$$

Find all pairs of real numbers $(x, y)$ that satisfy the equation $$\frac{(x+y)(2-\sin(x+y))}{4\sin^2(x+y)}=\frac{xy}{x+y}$$

Consider a sequence $P_{1}, P_{2}, \ldots$ of points in the plane such that $P_{1}, P_{2}, P_{3}$ are non-collinear and for every $n \geq 4, P_{n}$ is the midpoint of the line segment joining $P_{n-2}$ and $P_{n-3}$. Let $L$ denote the line segment joining $P_{1}$ and $P_{5}$. Prove the following: [list=a] [*] The area of the triangle formed by the points $P_{n}, P_{n-1}, P_{n-2}$ converges to zero as $n$ goes to infinity. [*] The point $P_{9}$ lies on $L$. [/list]

Let $ABCD$ be a trapezoid with $AB\parallel DC$. Let $M$ be the midpoint of $CD$. If $AD\perp CD, AC\perp BM,$ and $BC\perp BD$, find $\frac{AB}{CD}$. [i]Proposed by Nathan Ramesh

Let $ABC$ be a triangle inscribed in a circle $(O)$, and $M$ be some point on the perpendicular bisector of $BC$. Let $I_1, I_2$ be the incenters of triangles $MAB,MAC$. Prove that the incenters of triangles $A_II_1I_2$ are on a fixed line when $M$ varies on the perpendicular bisector. Trần Quang Hùng

Something related to this [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=845756#845756]problem[/url]: Prove that for a set $ S\subset\mathbb N$, there exists a sequence $ \{a_{i}\}_{i \equal{} 0}^{\infty}$ in $ S$ such that for each $ n$, $ \sum_{i \equal{} 0}^{n}a_{i}x^{i}$ is irreducible in $ \mathbb Z[x]$ if and only if $ |S|\geq2$. [i]By Omid Hatami[/i]

Let $\omega_a, \omega_b, \omega_c$ be the exscribed circles tangent to the sides $a, b, c$ of a triangle $ABC$, respectively, $ I_a, I_b, I_c$ be the centers of these circles, respectively, $T_a, T_b, T_c$ be the points of contact of these circles to the line $BC$, respectively. The lines $T_bI_c$ and $T_cI_b$ intersect at the point $Q$. Prove that the center of the circle inscribed in triangle $ABC$ lies on the line $T_aQ$.

Find all integer solutions of the equation \[\frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.\]

Let $p$ be a prime and $m \geq 2$ be an integer. Prove that the equation \[ \frac{ x^p + y^p } 2 = \left( \frac{ x+y } 2 \right)^m \] has a positive integer solution $(x, y) \neq (1, 1)$ if and only if $m = p$. [i]Romania[/i]

In a game similar to three card monte, the dealer places three cards on the table: the queen of spades and two red cards. The cards are placed in a row, and the queen starts in the center; the card configuration is thus RQR. The dealer proceeds to move. With each move, the dealer randomly switches the center card with one of the two edge cards (so the configuration after the first move is either RRQ or QRR). What is the probability that, after 2004 moves, the center card is the queen?

$P$ is any point of the tetrahedron $ABCD$ except $D$. Show that at least one of the three distances $DA$, $DB$, $DC$ exceeds at least one of the distances $PA$, $PB$ and $PC$.

(a) Prove that (p^2)-1 is divisible by 24 if p is a prime number greater than 3. (b) Prove that (p^2)-(q^2) is divisible by 24 if p and q are prime numbers greater than 3.

For every integer $n\geqslant2$ prove the inequality $$ \frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{2^{n-2}}{n!}\leqslant\frac{3}{2}, $$ where $k!=1\cdot2\cdot\ldots\cdot k$.

Let $n\geq 1$ be odd integer. There are $n$ arrows are arranged from left to right, such that each arrow points either to the left or to the right. Prove that there exists an arrow that is pointed to by exactly as many arrows as it is pointing to. Note: For example, for $n=5$ and the arrangement $\rightarrow\rightarrow\leftarrow\leftarrow\rightarrow$, the successive arrows (from the left) point respectively to $4$, $3$, $2$, $3$, $0$ arrows.

Let $S=\{1,2,3,...,12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product? [i]Proposed by Gabriel Wu[/i]

Suppse that $X=\{1,2, \ldots, 2^{1996}-1\}$, prove that there exist a subset $A$ that satisfies these conditions: a) $1\in A$ and $2^{1996}-1\in A$; b) Every element of $A$ except $1$ is equal to the sum of two (possibly equal) elements from $A$; c) The maximum number of elements of $A$ is $2012$. [i]Romania[/i]

A set of points of the plane is called [i] obtuse-angled[/i] if every three of it's points are not collinear and every triangle with vertices inside the set has one angle $ >91^o$. Is it correct that every finite [i] obtuse-angled[/i] set can be extended to an infinite [i]obtuse-angled[/i] set? (UK)

Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x,y$ taken from two different subsets, the number $x^2-xy+y^2$ belongs to the third subset.

Find the largest real constant $\lambda$ such that \[\frac{\lambda abc}{a+b+c}\leq (a+b)^2+(a+b+4c)^2\] For all positive real numbers $a,b,c.$

The area of the triangle $ABC$ shown in the figure is $1$ unit. Points $D$ and $E$ lie on sides $AC$ and $BC$ respectively, and also are its ''one third'' points closer to $C$. Let $F$ be that $AE$ and $G$ are the midpoints of segment $BD$. What is the area of the marked quadrilateral $ABGF$? [img]https://cdn.artofproblemsolving.com/attachments/4/e/305673f429c86bbc58a8d40272dd6c9a8f0ab2.png[/img]