Let $r_1=2$ and $r_n = \prod^{n-1}_{k=1} r_i + 1$, $n \geq 2.$ Prove that among all sets of positive integers such that $\sum^{n}_{k=1} \frac{1}{a_i} < 1,$ the partial sequences $r_1,r_2, ... , r_n$ are the one that gets nearer to 1.

Find all real polynomials $P(x)$ such that for every four distinct natural numbers $a, b, c, d$ such that $a^2 + b^2 + c^2 = 2d^2$ with $gcd(a, b, c, d) = 1$ the following equality holds: $$2(P(d))^2 + 2P(ab + bc + ca) = (P(a + b + c))^2$$ .

Let an odd prime $p$ be a given number satisfying $2^h \neq 1 \pmod{p}$ for all $h < p-1, h \in \mathbb{N}^{*},$ and an even integer $a \in \left(\frac{p}{2},p \right).$ Let us consider the sequence $\{a_n\}^{\infty}_{n=0}$ defined by $a_0 = a$ and $a_{n+1} = p - b_n$ for $n = 0, 1, 2, \ldots$, where $b_n$ is the greatest odd divisor of $a_n.$ Show that $\{a_n\}$ is periodical and find its least positive period.

4. Find the biggest positive integer $n$, lesser thar $2012$, that has the following property: If $p$ is a prime divisor of $n$, then $p^2 - 1$ is a divisor of $n$.

Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?

Right triangle $\triangle{}ABC$ has $\angle{}C=90^{\circ{}}$. A fly is trapped inside $\triangle{}ABC$. It starts at point $D$, the foot of the altitude from $C$ to $\overline{AB}$, and then makes a (finite) sequence of moves. In each move, it flies in a direction parallel to either $\overline{AC}$ or $\overline{BC}$; upon reaching a leg of the triangle, it then flies to a point on $\overline{AB}$ in a direction parallel to $\overline{CD}$. For example, on its first move, the fly can move to either of the points $Y_1$ or $Y_2$, as shown. [asy] pair C = (0,0); pair A = (0,4); pair B = (5,0); draw(C--A); draw(C--B); draw(B--A); dot(A); dot(B); dot(C); label("$A$",A,NW); label("$C$",C,SW); label("$B$",B,SE); pair D = foot(C,A,B); draw(C--D,dotted); label("$D$",D,NE); dot(D); draw(rightanglemark(A,C,B)); pair B1 = foot(D,C,B); draw(D--B1,dotted); pair A1 = foot(D,A,C); draw(D--A1,dotted); pair Y1 = foot(A1,A,D); draw(A1--Y1,dotted); dot(Y1); label("$Y_1$",Y1,NE); pair Y2 = foot(B1,D,B); draw(B1--Y2,dotted); dot(Y2); label("$Y_2$",Y2,NE); draw(rightanglemark(C,A1,D)); draw(rightanglemark(C,B1,D)); draw(rightanglemark(B1,Y2,D)); draw(rightanglemark(A1,Y1,D)); draw(rightanglemark(C,D,A)); [/asy] Let $P$ and $Q$ be distinct points on $\overline{AB}$. Show that the fly can reach some point on $\overline{PQ}$.

Find all real functions $f$ defined on the real numbers except zero, satisfying $f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$

Given a triangle $ABC$. The circle $\omega_1$ passes through the vertex $B$ and touches the side $AC$ at the point $A$, and the circle $\omega_2$ passes through the vertex $C$ and touches the side $AB$ at the point $A$. The circles $\omega_1$ and $\omega_2$ intersect a second time at the point $D$. The line $AD$ intersects the circumcircle of the triangle $ABC$ at point $E$. Prove that $D$ is the midpoint of $AE$..

A positive integer is called [i]fancy[/i] if it can be expressed in the form $$2^{a_1}+2^{a_2}+ \cdots+ 2^{a_{100}},$$ where $a_1,a_2, \cdots, a_{100}$ are non-negative integers that are not necessarily distinct. Find the smallest positive integer $n$ such that no multiple of $n$ is a [i]fancy[/i] number. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

There are three polygons and the area of each one is $3$. They are drawn inside a square of area $6$. Find the greatest value of $m$ such that among those three polygons, we can always find two polygons so that the area of their overlap is not less than $m$.

Let points $A, B, C, A', B', C'$ be chosen in the plane such that no three of them are collinear, and let lines $AA'$, $BB'$ and $CC'$ be tangent to a given equilateral hyperbola at points $A$, $B$ and $C$, respectively. Assume that the circumcircle of $A'B'C'$ is the same as the nine-point circle of triangle $ABC$. Let $s(A')$ be the Simson line of point $A'$ with respect to the orthic triangle of $ABC$. Let $A^*$ be the intersection of line $B'C'$ and the perpendicular on $s(A')$ from the point $A$. Points $B^*$ and $C^*$ are defined in a similar manner. Prove that points $A^*$, $B^*$ and $C^*$ are collinear. [i]Submitted by Áron Bán-Szabó, Budapest[/i]

The base of the pyramid with vertex $S$ is a pentagon $ABCDE$ for which $BC>DE$ and $AB>CD$. If $AS$ is the longest edge of the pyramid prove that $BS>CS$. [i]Jordan Tabov[/i]

Call a number $a-b\sqrt2$ with $a$ and $b$ both positive integers $tiny$ if it is closer to zero than any number $c-d\sqrt2$ such that $c$ and $d$ are positive integers with $c<a$ and $d<b$. Three numbers which are tiny are $1-\sqrt2$, $3-2\sqrt2$, and $7-5\sqrt2$. Without using any calculator or computer, prove whether or not each of the following is tiny: \[(a)\ 58-41\sqrt2,\qquad\qquad (b)\ 99-70\sqrt2.\]

In the plane we are given two circles intersecting at $ X$ and $ Y$. Prove that there exist four points with the following property: (P) For every circle touching the two given circles at $ A$ and $ B$, and meeting the line $ XY$ at $ C$ and $ D$, each of the lines $ AC$, $ AD$, $ BC$, $ BD$ passes through one of these points.

grade IX P4, X P3 The bisectors of the angles $ A $ and $ C $ of the triangle $ ABC $ intersect the circumscirbed circle of this triangle at the points $ A_0 $ and $ C_0 $, respectively. The straight line passing through the center of the inscribed circle of triangle $ ABC $ parallel to the side of $ AC $, intersects with the line $ A_0C_0 $ at $ P $. Prove that the line $ PB $ is tangent to the circumcircle of the triangle $ ABC $. grade XI P4 The bisectors of the angles $ A $ and $ C $ of the triangle $ ABC $ intersect the sides at the points $ A_1 $ and $ C_1 $, and the circumcircle of this triangle at points $ A_0 $ and $ C_0 $ respectively. Straight lines $ A_1C_1 $ and $ A_0C_0 $ intersect at point $ P $. Prove that the segment connecting $ P $ with the center inscribed circles of triangle $ ABC $, parallel to $ AC $.

The residents of the local zoo are either rabbits or foxes. The ratio of foxes to rabbits in the zoo is $2:3$. After $10$ of the foxes move out of town and half the rabbits move to Rabbitretreat, the ratio of foxes to rabbits is $13:10$. How many animals are left in the zoo?

How many divisors of $30^{2003}$ are there which do not divide $20^{2000}$?

Determine the number of ordered pairs $(a,b)$ of integers such that $\log_a b + 6\log_b a=5$, $2 \leq a \leq 2005$, and $2 \leq b \leq 2005$.

Let $ABC$ be an acute scalene triangle such that $AB <AC$. The midpoints of sides $AB$ and $AC$ are $M$ and $N$, respectively. Let $P$ and $Q$ be points on the line $MN$ such that $\angle CBP = \angle ACB$ and $\angle QCB = \angle CBA$. The circumscribed circle of triangle $ABP$ intersects line $AC$ at $D$ ($D\ne A$) and the circumscribed circle of triangle $AQC$ intersects line $AB$ at $E$ ($E \ne A$). Show that lines $BC, DP,$ and $EQ$ are concurrent. Nicolás De la Hoz, Colombia

Find all primes $p$ for which the quotient \[ \dfrac{2^{p-1} - 1 }{p} \] is a square.

How many distinct triangles are there with prime side lengths and perimeter $100$? [i]Proposed by Muztaba Syed[/i] [hide=Solution][i]Solution.[/i] $\boxed{0}$ As the perimeter is even, $1$ of the sides must be $2$. Thus, the other $2$ sides are congruent by Triangle Inequality. Thus, for the perimeter to be $100$, both of the other sides must be $49$, but as $49$ is obviously composite, the answer is thus $\boxed{0}$.[/hide]

In convex pentagon $ ABCDE$, denote by $ AD\cap BE = F,BE\cap CA = G,CA\cap DB = H,DB\cap EC = I,EC\cap AD = J; AI\cap BE = A',BJ%Error. "capCA" is a bad command. = B',CF%Error. "capDB" is a bad command. = C',DG\cap EC = D',EH\cap AD = E'.$ Prove that $ \frac {AB'}{B'C}\cdot\frac {CD'}{D'E}\cdot\frac {EA'}{A'B}\cdot\frac {BC'}{C'D}\cdot\frac {DE'}{E'A} = 1$.

Given a regular convex $2m$- sided polygon $P$, show that there is a $2m$-sided polygon $\pi$ with the same vertices as $P$ (but in different order) such that $\pi$ has exactly one pair of parallel sides.

Let $ a$, $ b$, and $ c$ be positive real numbers such that $ ab + bc + ca = 3$. Prove the inequality \[ 3 + \sum_{\mathrm{\cyc}} (a - b)^2 \ge \frac {a + b^2c^2}{b + c} + \frac {b + c^2a^2}{c + a} + \frac {c + a^2b^2}{a + b} \ge 3. \]

Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\] [i]Proposed by Japan[/i]