For integer $k \ge 1$, let $a_k =\frac{k}{4k^4 + 1}$. Find the least integer $n$ such that $a_1 + a_2 + a_3 + ... + a_n > \frac{505.45}{2022}$.

$(GBR 3)$ A smooth solid consists of a right circular cylinder of height $h$ and base-radius $r$, surmounted by a hemisphere of radius $r$ and center $O.$ The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point $P$ on the hemisphere such that $OP$ makes an angle $\alpha$ with the horizontal. Show that if $\alpha$ is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through $P$, show that it will cross the common circular section of the hemisphere and cylinder at a point $Q$ such that $\angle SOQ = \phi$, $S$ being where it initially crossed this section, and $\sin \phi = \frac{r \tan \alpha}{h}$.

Let $a,b,c$ be three postive real numbers such that $a+b+c=1$. Prove that $9abc\leq ab+ac+bc < 1/4 +3abc$.

Find all positive integers $n$ such that the sum of the squares of the five smallest divisors of $n$ is a square.

How many alphabetic sequences (that is, sequences containing only letters from $a\cdots z$) of length $2013$ have letters in alphabetic order?

Kolya and Vasya each have $8$ cards with numbers from $1$ to $8$ (each has all the numbers from $1$ to $8$). Kolya put $4$ cards on the table, and Vasya put a card with a larger number on each of them. Now Vasya puts his remaining $4$ cards on the table. a) Can Kolya always put his own card with a larger number on each of Vasya’s cards? b) Can Kolya always put on each of Vasya’s cards his own card with a number no less than on Vasya’s card?

Given $n \ge 2$ points on a circle, Alice and Bob play the following game. Initially, a tile is placed on one of the points and no segment is drawn. The players alternate in turns, with Alice to start. In a turn, a player moves the tile from its current position $P$ to one of the $n-1$ other points $Q$ and draws the segment $PQ$. This move is not allowed if the segment $PQ$ is already drawn. If a player cannot make a move, the game is over and the opponent wins. Determine, for each $n$, which of the two players has a winning strategy.

For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is [i]fibonatic[/i] when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not [i]fibonatic[/i] integers.

Let $ABC$ be an acute triangle. Let $M$ be the midpoint of the segment $BC$. Let $I, J, K$ be the incenters of triangles $ABC$, $ABM$, $ACM$, respectively. Let $P, Q$ be points on the lines $MK$, $MJ$, respectively, such that $\angle AJP=\angle ABC$ and $\angle AKQ=\angle BCA$. Let $R$ be the intersection of the lines $CP$ and $BQ$. Prove that the lines $IR$ and $BC$ are perpendicular.

If the rotational inertia of a sphere about an axis through the center of the sphere is $I$, what is the rotational inertia of another sphere that has the same density, but has twice the radius? $ \textbf{(A)}\ 2I \qquad\textbf{(B)}\ 4I \qquad\textbf{(C)}\ 8I\qquad\textbf{(D)}\ 16I\qquad\textbf{(E)}\ 32I $

Suppose $x, y, z$ are real numbers such that $x^2 + y^2 + z^2 + 2xyz = 1$. Prove that $8xyz \le 1$, with equality if and only if $(x, y,z)$ is one of the following: $$\left( \frac12, \frac12, \frac12 \right) , \left( -\frac12, -\frac12, \frac12 \right), \left(- \frac12, \frac12, -\frac12 \right), \left( \frac12,- \frac12, - \frac12 \right)$$

Which polygon inscribed in a given circle has the property that the sum of the squares of the lengths of its sides is maximum?

Bugs Bunny plays a game in the Euclidean plane. At the $n$-th minute $(n \geq 1)$, Bugs Bunny hops a distance of $F_n$ in the North, South, East, or West direction, where $F_n$ is the $n$-th Fibonacci number (defined by $F_1 = F_2 =1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$). If the first two hops were perpendicular, prove that Bugs Bunny can never return to where he started. [i]Proposed by Dylan Toh[/i]

The numbers $a, b, c, d$ and $e$ satisfy $$a + b < c + d < e + a < b + c < d + e .$$ Which of the numbers is the smallest, and which is the largest?

Three wooden equilateral triangles of side length $18$ inches are placed on axles as shown in the diagram to the right. Each axle is $30$ inches from the other two axles. A $144$-inch leather band is wrapped around the wooden triangles, and a dot at the top corner is painted as shown. The three triangles are then rotated at the same speed and the band rotates without slipping or stretching. Compute the length of the path that the dot travels before it returns to its initial position at the top corner. [asy] size(150); defaultpen(linewidth(0.8)+fontsize(10)); pair A=origin,B=(48,0),C=rotate(60,A)*B; path equi=(0,0)--(18,0)--(9,9*sqrt(3))--cycle,circ=circle(centroid(A,B,C)*18/48,1/3); picture a; fill(a,equi,grey); fill(a,circ,white); add(a); add(shift(15,15*sqrt(3))*a); add(shift(30,0)*a); draw(A--B--C--cycle,linewidth(1)); path top = circle(C,2/3); unfill(top); draw(top); real r=-5/2; draw((9,r+1)--(9,r-1)^^(9,r)--(39,r)^^(39,r-1)--(39,r+1)); label("$30$",(24,r),S); [/asy]

Let $ ABC$ be a triangle; $ \Gamma_A,\Gamma_B,\Gamma_C$ be three equal, disjoint circles inside $ ABC$ such that $ \Gamma_A$ touches $ AB$ and $ AC$; $ \Gamma_B$ touches $ AB$ and $ BC$; and $ \Gamma_C$ touches $ BC$ and $ CA$. Let $ \Gamma$ be a circle touching circles $ \Gamma_A, \Gamma_B, \Gamma_C$ externally. Prove that the line joining the circum-centre $ O$ and the in-centre $ I$ of triangle $ ABC$ passes through the centre of $ \Gamma$.

In the isosceles triangle $ABC$, with $AB=BC$, points $X$ and $Y$ are the midpoints of the sides $AB$ and $AC$, respectively. Point $Z$ is the foot of the perpendicular from $B$ to $CX$. Prove that the circumcenter of the triangle $XYZ$ is of the line $AC$.

Let $x\ne 1,y\ne 1$ and $x\ne y.$ Show that if \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y},\] then \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y}=x+y+z.\]

Prove that there exist $0 < \alpha< \beta<1$ numbers have the following properties. (i) for any sufficiently large n, n points can be specified in $\Bbb R^3$ , so that each point is equidistant from at least $n^\alpha$ other points. (ii) the above statement is no longer true with $n^\beta$ instead of $n^\alpha$

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$

Two natural numbers $d$ and $d'$, where $d'>d$, are both divisors of $n$. Prove that $d'>d+\frac{d^2}{n}$.

A game is played by two contestants A and B, each one having ten chips numbered from 1 to 10. The board of game consists of two numbered rows, from 1 to 1492 on the first row and from 1 to 1989 on the second. At the $n$-th turn, $n=1,2,\ldots,10$, A puts his chip numbered $n$ in any empty cell, and B puts his chip numbered $n$ in any empty cell on the row not containing the chip numbered $n$ from A. B wins the game if, after the 10th turn, both rows show the numbers of the chips in the same relative order. Otherwise, A wins. [list=a] [*] Which player has a winning strategy? [*] Suppose now both players has $k$ chips numbered 1 to $k$. Which player has a winning strategy? [*] Suppose further the rows are the set $\mathbb{Q}$ of rationals and the set $\mathbb{Z}$ of integers. Which player has a winning strategy? [/list]

Let $ f(x)\equal{}x^{2}\plus{}|x|$. Prove that $ \int_{0}^{\pi}f(\cos x)\ dx\equal{}2\int_{0}^{\frac{\pi}{2}}f(\sin x)\ dx$.

How many positive integers less than 4000 are not divisible by 2, not divisible by 3, not divisible by 5, and not divisible by 7?

Prove that every positive integer can be represented in the form \[3^{u_1} \ldots 2^{v_1} + 3^{u_2} \ldots 2^{v_2} + \ldots + 3^{u_k} \ldots 2^{v_k}\] with integers $u_1, u_2, \ldots , u_k, v_1, \ldots, v_k$ such that $u_1 > u_2 >\ldots > u_k\ge 0$ and $0 \le v_1 < v_2 <\ldots < v_k$.