For each positive integer $n$, let $r_n$ be the smallest positive root of the equation $x^n = 7x - 4$. There are positive real numbers $a$, $b$, and $c$ such that \[\lim_{n \to \infty} a^n (r_n - b) = c.\] If $100a + 10b + c = \frac{p}{7}$ for some integer $p$, find $p$. [i]Proposed by Mehtaab Sawhney[/i]

If $S$ is a set which consists of $12$ elements, what is the maximum number of pairs $(a,b)$ such that $a, b\in S$ and $\frac{b}{a}$ is a prime number?

Equilateral triangle $ABC$ has area $1$. $A'$, $B'$, and $C'$ are the midpoints of $BC$, $CA$, and $AB$, respectively. $A''$, $B''$, $C''$ are the midpoints of $B'C'$, $C'A'$, and $A'B'$, respectively. The area of trapezoid $BB''C''C$ can be written as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$. Find $m+n$.

Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying \[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\] for all integers $a$ and $b$

We are given two mutually tangent circles in the plane, with radii $r_1, r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.

In a triangle scalene $ABC$, let $I$ be its incenter and $D$ the intersection of $AI$ and $BC$. Let $M$ and $N$ points where the incircle touches $AB$ and $AC$, respectively. Let $F$ be the second intersection of the circumcircle $(AMN)$ with the circumcircle $(ABC)$. Let $T$ the intersection of $AF$ and $BC$. Let $J$ be the intersection of $TI$ with the line parallel of $FI$ that passes through $D$. Prove that the line $AJ$ is perpendicular to $BC$.

By straightedge and compass, reconstruct a right triangle $ABC$ ($\angle C = 90^o$), given the vertices $A, C$ and a point on the bisector of angle $B$.

The radii of the circumcircle and the incircle of a right triangle are given. Cconstruct that triangle with compass and ruler, describe the construction and justify why it is correct.

If \[N=\frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}-\sqrt{3-2\sqrt{2}},\] then $N$ equals $\textbf{(A) }1\qquad\textbf{(B) }2\sqrt{2}-1\qquad\textbf{(C) }\frac{\sqrt{5}}{2}\qquad\textbf{(D) }\sqrt{\frac{5}{2}}\qquad \textbf{(E) }\text{none of these}$

A point $P$ lies on side $AB$ of a convex quadrilateral $ABCD$. Let $\omega$ be the inscribed circumference of triangle $CPD$ and $I$ the centre of $\omega$. It is known that $\omega$ is tangent to the inscribed circumferences of triangles $APD$ and $BPC$ at points $K$ and $L$ respectively. Let $E$ be the point where the lines $AC$ and $BD$ intersect, and $F$ the point where the lines $AK$ and $BL$ intersect. Prove that the points $E, I, F$ are collinear.

Let $f(x)=x^2+x+1$. Determine, with proof, all positive integers $n$ such that $f(k)$ divides $f(n)$ whenever $k$ is a positive divisor of $n$.

$ABC$ is a triangle. Take $a = BC$ etc as usual. Take points $T_1, T_2$ on the side $AB$ so that $AT_1 = T_1T_2 = T_2B$. Similarly, take points $T_3, T_4$ on the side BC so that $BT_3 = T_3T_4 = T_4C$, and points $T_5, T_6$ on the side $CA$ so that $CT_5 = T_5T_6 = T_6A$. Show that if $a' = BT_5, b' = CT_1, c'=AT_3$, then there is a triangle $A'B'C'$ with sides $a', b', c'$ ($a' = B'C$' etc). In the same way we take points $T_i'$ on the sides of $A'B'C' $ and put $a'' = B'T_6', b'' = C'T_2', c'' = A'T_4'$. Show that there is a triangle $A'' B'' C'' $ with sides $a'' b'' , c''$ and that it is similar to $ABC$. Find $a'' /a$.

$\vartriangle A_0B_0C_0$ has side lengths $A_0B_0 = 13$, $B_0C_0 = 14$, and $C_0A_0 = 15$. $\vartriangle A_1B_1C_1$ is inscribed in the incircle of $\vartriangle A_0B_0C_0$ such that it is similar to the first triangle. Beginning with $\vartriangle A_1B_1C_1$, the same steps are repeated to construct $\vartriangle A_2B_2C_2$, and so on infinitely many times. What is the value of $\sum_{i=0}^{\infty} A_iB_i$?

Triangle $ABC$ is inscribed in a circle with center $O'$. A circle with center $O$ is inscribed in triangle $ABC$. $AO$ is drawn, and extended to intersect the larger circle in $D$. Then, we must have: $\text{(A)}\ CD=BD=O'D \qquad \text{(B)}\ AO=CO=OD \qquad \text{(C)}\ CD=CO=BD \qquad\\ \text{(D)}\ CD=OD=BD \qquad \text{(E)}\ O'B=O'C=OD $ [asy] size(200); defaultpen(linewidth(0.8)+fontsize(12pt)); pair A=origin,B=(15,0),C=(5,9),O=incenter(A,B,C),Op=circumcenter(A,B,C); path incirc = incircle(A,B,C),circumcirc = circumcircle(A,B,C),line=A--3*O; pair D[]=intersectionpoints(circumcirc,line); draw(A--B--C--A--D[0]^^incirc^^circumcirc); dot(O^^Op,linewidth(4)); label("$A$",A,dir(185)); label("$B$",B,dir(355)); label("$C$",C,dir(95)); label("$D$",D[0],dir(O--D[0])); label("$O$",O,NW); label("$O'$",Op,E);[/asy]

The diagonals $AC$ and $BD$ of a convex cyclic quadrilateral $ABCD$ intersect at point $E$. Given that $AB = 39, AE = 45, AD = 60$ and $BC = 56$, determine the length of $CD.$

In triangle $ABC$ let $M$ and $N$ be midpoints of $BC$ and $AC,$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PBC = \angle PCA .$ Prove that if $\angle PNA = \angle AMB,$ then $ABC$ is isosceles triangle.

We consider the non-zero natural numbers $(m, n)$ such that the numbers $$\frac{m^2 + 2n}{n^2 - 2m} \,\,\,\, and \,\,\, \frac{n^2 + 2m}{m^2-2n}$$ are integers. a) Show that $|m - n| \le 2$: b) Find all the pairs $(m, n)$ with the property from hypothesis $a$.

Is there a real number $\alpha$ such that $\cos\alpha$ is irrational but $\cos 2\alpha$, $\cos 3\alpha$, $\cos 4\alpha$, $\cos 5\alpha$ are all rational? (Author: V. Senderov)

Let $n,k \in \mathbb{Z}^{+}$ with $k \leq n$ and let $S$ be a set containing $n$ distinct real numbers. Let $T$ be a set of all real numbers of the form $x_1 + x_2 + \ldots + x_k$ where $x_1, x_2, \ldots, x_k$ are distinct elements of $S.$ Prove that $T$ contains at least $k(n-k)+1$ distinct elements.

Find all functions $f:\mathbb{N}_0\to\mathbb{N}$ such that [b]1)[/b] \(f(a)\) divides \(a\) for every \(a\in\mathbb{N}_0\), and [b]2)[/b] for all \(a,b,k\in\mathbb{N}_0\) we have \[ f\bigl(f(a)+kb\bigr)\;=\;f\bigl(a + k\,f(b)\bigr). \]

Right isosceles triangle $T$ is placed in the first quadrant of the coordinate plane. Suppose that the projection of $T$ onto the $x$-axis has length $6$, while the projection of $T$ onto the $y$-axis has length $8$. What is the sum of all possible areas of the triangle $T$? [asy] import olympiad; size(120); defaultpen(linewidth(0.8)); pair A = (0.9,0.6), B = (1.7, 0.8), C = rotate(270, B)*A; pair PAx = (A.x,0), PBx = (B.x,0), PAy = (0, A.y), PCy = (0, C.y); draw(PAx--A--PAy^^PCy--C^^PBx--B, linetype("4 4")); draw(rightanglemark(A,B,C,3)); draw(A--B--C--cycle); draw((0,2)--(0,0)--(2,0),Arrows(size=8)); label("$6$",(PAx+PBx)/2,S); label("$8$",(PAy+PCy)/2,W); [/asy]

Let $ABC$ be a triangle and the points $M$ and $N$ on the sides $AB$ respectively $BC$, such that $2 \cdot \frac{CN}{BC} = \frac{AM}{AB}$. Let $P$ be a point on the line $AC$. Prove that the lines $MN$ and $NP$ are perpendicular if and only if $PN$ is the interior angle bisector of $\angle MPC$.

Let ${ABC}$ be an acute angled triangle, and ${H}$ a point in its interior. Let the reflections of ${H}$ through the sides ${AB}$ and ${AC}$ be called ${H_{c} }$ and ${H_{b} }$ , respectively, and let the reflections of H through the midpoints of these same sidesbe called ${H_{c}^{'} }$ and ${H_{b}^{'} }$, respectively. Show that the four points ${H_{b}, H_{b}^{'} , H_{c}}$, and ${H_{c}^{'} }$ are concyclic if and only if at least two of them coincide or ${H}$ lies on the altitude from ${A}$ in triangle ${ABC}$.

Let $p$ and $n$ be a natural numbers such that $p$ is a prime and $1+np$ is a perfect square. Prove that the number $n+1$ is sum of $p$ perfect squares.

Find the greatest integer $n$ such that $10^n$ divides $$\frac{2^{10^5} 5^{2^{10}}}{10^{5^2}}$$