A teacher wants to divide the $2010$ questions she asked in the exams during the school year into three folders of $670$ questions and give each folder to a student who solved all $670$ questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations in which there are at most two students who did not solve any given question.

Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the following two conditions: \[ \left\{ \begin{array}{ll} \mbox{If } f(0) = 0, \mbox{ then } f(x) \neq 0 \mbox{ for any non-zero } x. \\ \\ f(x + y)f(y + z)f(z + x) = f(x + y + z)f(xy + yz + zx) - f(x)f(y)f(z) \quad \forall x, y, z \in \mathbb{R}. \end{array} \right. \]

The roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$ will be reciprocal if: $ \textbf{(A)}\ a \equal{} b \qquad\textbf{(B)}\ a \equal{} bc \qquad\textbf{(C)}\ c \equal{} a \qquad\textbf{(D)}\ c \equal{} b \qquad\textbf{(E)}\ c \equal{} ab$

a) The numbers $1, 2, . . . , 49$ are arranged in a square table as follows: [img]https://cdn.artofproblemsolving.com/attachments/5/0/c2e350a6ad0ebb8c728affe0ebb70783baf913.png[/img] Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $36$ numbers, etc., $7$ times. Find the sum of the numbers selected. b) The numbers $1, 2, . . . , k^2$ are arranged in a square table as follows: [img]https://cdn.artofproblemsolving.com/attachments/2/d/28d60518952c3acddc303e427483211c42cd4a.png[/img] Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $(k - 1)^2$ numbers, etc., $k$ times. Find the sum of the numbers selected.

In a convex $n$-gonal prism all sides are equal. For what $n$ is this prism right?

Let $a_1$ , $\ldots$ , $a_n$ be positive integers and $a$ a positive integer that is greater than $1$ and is divisible by the product $a_1a_2\ldots a_n$. Prove that $a^{n+1}+a-1$ is not divisible by the product $(a+a_1-1)(a+a_2-1)\ldots(a+a_n-1)$.

A museum has the shape of a (not necessarily convex) 3$n$-gon. Prove that $n$ custodians can be positioned so as to control all of the museum’s space.

Three congruent ellipses are mutually tangent. Their major axes are parallel. Two of the ellipses are tangent at the end points of their minor axes as shown. The distance between the centers of these two ellipses is $4$. The distances from those two centers to the center of the third ellipse are both $14$. There are positive integers m and n so that the area between these three ellipses is $\sqrt{n}-m \pi$. Find $m+n$. [asy] size(250); filldraw(ellipse((2.2,0),2,1),grey); filldraw(ellipse((0,-2),4,2),white); filldraw(ellipse((0,+2),4,2),white); filldraw(ellipse((6.94,0),4,2),white);[/asy]

Determine all pairs $(a,b)$ of integers with the property that the numbers $a^2+4b$ and $b^2+4a$ are both perfect squares.

Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

Let $T(a)$ be the sum of digits of $a$. For which positive integers $R$ does there exist a positive integer $n$ such that $\frac{T(n^2)}{T(n)}=R$?

Given is a cyclic quadrilateral $ABCD$. Points $K, L, M, N$ lying on sides $AB, BC, CD, DA$, respectively, satisfy $\angle ADK=\angle BCK$, $\angle BAL=\angle CDL$, $\angle CBM =\angle DAM$, $\angle DCN =\angle ABN$. Prove that lines $KM$ and $LN$ are perpendicular.

Let $a,b,c,d,e$ be non-negative real numbers such that $a+b+c+d+e>0$. What is the least real number $t$ such that $a+c=tb$, $b+d=tc$, $c+e=td$? $ \textbf{(A)}\ \frac{\sqrt 2}2 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \sqrt 2 \qquad\textbf{(D)}\ \frac32 \qquad\textbf{(E)}\ 2 $

If $x,y,z$ are positive real numbers with $x^2+y^2+z^2=3$, prove that $\frac32<\frac{1+y^2}{x+2}+\frac{1+z^2}{y+2}+\frac{1+x^2}{z+2}<3$

In a meeting there are 6 participants. It is known that among them there are seven pairs of friends and in any group of three persons there are at least two friends. Prove that: (a) there exists a person who has at least three friends; (b) there exists three persons who are friends with each other. [i]Valentin Vornicu[/i]

Let $ K$ be the figure bounded by the curve $ y\equal{}e^x$ and 3 lines $ x\equal{}0,\ x\equal{}1,\ y\equal{}0$ in the $ xy$ plane. (1) Find the volume of the solid formed by revolving $ K$ about the $ x$ axis. (2) Find the volume of the solid formed by revolving $ K$ about the $ y$ axis.

There are relatively prime positive integers $s$ and $t$ such that $$\sum_{n=2}^{100}\left(\frac{n}{n^2-1}- \frac{1}{n}\right)=\frac{s}{t}$$ Find $s + t$.

Determine the maximum value of the sum \[ \sum_{i < j} x_ix_j (x_i \plus{} x_j) \] over all $ n \minus{}$tuples $ (x_1, \ldots, x_n),$ satisfying $ x_i \geq 0$ and $ \sum^n_{i \equal{} 1} x_i \equal{} 1.$

Let $x_1,x_2,\cdots,x_n$ be iid rv. $S_n=\sum x_k$ (a) let $P(|x_1|\leq 1)=1$ , $E[x_1]=0$ , $E[x_1^2]=\sigma^2>0$ Prove that $\exists C>0$ , $\forall u\geq 2n\sigma^2$ $P(S_n\geq u)\leq e^{-C u \log(u/n\sigma^2)}$ (b) let $P(x_1=1)=P(x_1=-1)=\sigma^2/2$ , $P(x_1=0)=1-\sigma^2$ Prove that $\exists B_1<1,B_2>1,B_3>0$ , $\forall u\geq1, B_1 n\geq u\geq B_2 n\sigma^2$ $P(S_n\geq u)>e^{-B_3 u \log(u/n\sigma^2)}$

How many positive integers $a$ with $a\le 154$ are there such that the coefficient of $x^a$ in the expansion of \[(1+x^{7}+x^{14}+ \cdots +x^{77})(1+x^{11}+x^{22}+\cdots +x^{77})\] is zero? [i]Author: Ray Li[/i]

Find all integers $n\geq 3$ for which every convex equilateral $n$-gon of side length $1$ contains an equilateral triangle of side length $1$. (Here, polygons contain their boundaries.)

Find all positive real numbers $c$ such that the graph of $f\text{ : }\mathbb{R}\to\mathbb{R}$ given by $f(x)=x^3-cx$ has the property that the circle of curvature at any local extremum is centered at a point on the $x$-axis.

[b]Little Mario and the Cylindrical Beam[/b] Little Mario wishes to jump over a very long (practically infinite) cylindrical beam of radius $r$ whose axis is at a height $h$ from the ground. With what minimum initial speed must he launch himself if: [list=1] [*] Mario is allowed to touch the beam (neglect frictional effects)? [/*] [*] Mario is not allowed to touch the beam? [/*] [/list] Approximate Little Mario by a point particle for convenience. Acceleration due to gravity is $g$.

Consider a table with $n$ rows and $2n$ columns. we put some blocks in some of the cells. After putting blocks in the table we put a robot on a cell and it starts moving in one of the directions right, left, down or up. It can change the direction only when it reaches a block or border. Find the smallest number $m$ such that we can put $m$ blocks on the table and choose a starting point for the robot so it can visit all of the unblocked cells. (the robot can't enter the blocked cells.) Proposed by Seyed Mohammad Seyedjavadi and Alireza Tavakoli

Let $a,b,c$ be positive reals and sequences $\{a_{n}\},\{b_{n}\},\{c_{n}\}$ defined by $a_{k+1}=a_{k}+\frac{2}{b_{k}+c_{k}},b_{k+1}=b_{k}+\frac{2}{c_{k}+a_{k}},c_{k+1}=c_{k}+\frac{2}{a_{k}+b_{k}}$ for all $k=0,1,2,...$. Prove that $\lim_{k\to+\infty}a_{k}=\lim_{k\to+\infty}b_{k}=\lim_{k\to+\infty}c_{k}=+\infty$.