Find the function $f(x)$ that satisfies the equation $$f'(x) + x^2f(x) = 0$$ knowing that $f(1) = e$. Graph this function and calculate the tangent of the curve at the point of abscissa $1$.

Let $A, B$ be two points on circle $\gamma$. At point $A$ and $B$ we construct tangents to $\gamma$, $AC$ and $BD$ respectively such that the tangents are both in the clockwise direction. Let the intersection between $AB$ and $CD$ be $P$ . If $AC = BD$, prove that $P$ bisects the line $CD$.

The sequences of natural numbers $p_n$ and $q_n$ are given such that $$p_1 = 1,\ q_1 = 1,\ p_{n + 1} = 2q_n^2-p_n^2,\ q_{n + 1} = 2q_n^2+p_n^2 $$ Prove that $p_n$ and $q_m$ are coprime for any m and n.

Let $A_M$ be the midpoint of side $BC$ of an acute-angled $\Delta ABC$, and $A_H$ be the foot of the altitude to this side. Points $B_M, B_H, C_M, C_H$ are defined similarly. Prove that one of the ratios $A_MA_H : A_HA, B_MB_H : B_HB, C_MC_H : C_HC$ is equal to the sum of two remaining ratios

(A.Zaslavsky, 9--10) Given four points $ A$, $ B$, $ C$, $ D$. Any two circles such that one of them contains $ A$ and $ B$, and the other one contains $ C$ and $ D$, meet. Prove that common chords of all these pairs of circles pass through a fixed point.

Prove that for every set of $2n$ lines in the plane, such that there are no two parallel lines, there are two lines that divide the plane into four quadrants such that in each quadrant the number of unbounded regions is equal to $n$. [asy] unitsize(1cm); pair[] A, B; pair P, Q, R, S; A[1] = (0,5.2); B[1] = (6.1,0); A[2] = (1.5,5.5); B[2] = (3.5,0); A[3] = (6.8,5.5); B[3] = (1,0); A[4] = (7,4.5); B[4] = (0,4); P = extension(A[2],B[2],A[4],B[4]); Q = extension(A[3],B[3],A[4],B[4]); R = extension(A[1],B[1],A[2],B[2]); S = extension(A[1],B[1],A[3],B[3]); fill(P--Q--S--R--cycle, palered); fill(A[4]--(7,0)--B[1]--S--Q--cycle, paleblue); draw(A[1]--B[1]); draw(A[2]--B[2]); draw(A[3]--B[3]); draw(A[4]--B[4]); label("Bounded region", (3.5,3.7), fontsize(8)); label("Unbounded region", (5.4,2.5), fontsize(8)); [/asy]

Tyler has an infinite geometric series with sum $10$. He increases the first term of his sequence by $4$ and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum $15$. Compute the common ratio of Tyler’s series.

Let $A=\{1,2,\cdots ,100\}$. Let $S$ be a subset of power set of $A$ such that any two elements of $S$ has nonzero intersection (Note that elements of $S$ are actually some subsets of $A$). Then the maximum possible cardinality of $S$ is [list=1] [*] $2^{99}$ [*] $2^{99}+1$ [*] $2^{99}+2^{98}$ [*] None of these [/list]

Find the number of partitions of the set $\{1,2,3,\cdots ,11,12\}$ into three nonempty subsets such that no subset has two elements which differ by $1$. [i]Proposed by Nathan Ramesh

This line graph represents the price of a trading card during the first $6$ months of $1993$. [asy] unitsize(18); for (int a = 0; a <= 6; ++a) { draw((4*a,0)--(4*a,10)); } for (int a = 0; a <= 5; ++a) { draw((0,2*a)--(24,2*a)); } draw((0,5)--(4,4)--(8,8)--(12,3)--(16,9)--(20,6)--(24,2),linewidth(1.5)); label("$Jan$",(2,0),S); label("$Feb$",(6,0),S); label("$Mar$",(10,0),S); label("$Apr$",(14,0),S); label("$May$",(18,0),S); label("$Jun$",(22,0),S); label("$\textbf{1993 PRICES FOR A TRADING CARD}$",(12,10),N); label("$\begin{tabular}{c}\textbf{P} \\ \textbf{R} \\ \textbf{I} \\ \textbf{C} \\ \textbf{E} \end{tabular}$",(-2,5),W); label("$1$",(0,2),W); label("$2$",(0,4),W); label("$3$",(0,6),W); label("$4$",(0,8),W); label("$5$",(0,10),W); [/asy] The greatest monthly drop in price occurred during $\text{(A)}\ \text{January} \qquad \text{(B)}\ \text{March} \qquad \text{(C)}\ \text{April} \qquad \text{(D)}\ \text{May} \qquad \text{(E)}\ \text{June}$

A circle with center $O$ passes through points $A$ and $C$ and intersects the sides $AB$ and $BC$ of triangle $ABC$ at points $K$ and $N$, respectively. The circumcircles of triangles $ABC$ and $KBN$ meet at distinct points $B$ and $M$. Prove that $\angle OMB = 90^o$.

Each student in a classroom has $0,1,2,3,4,5$ or $6$ pieces of candy. At each step the teacher chooses some of the students, and gives one piece of candy to each of them and also to any other student in the classroom who is friends with at least one of these students. A student who receives the seventh piece eats all $7$ pieces. Assume that for every pair of students in the classroom, there is at least one student who is friend swith exactly one of them. Show that no matter how many pieces each student has at the beginning, the teacher can make them to have any desired numbers of pieces after finitely many steps.

We are given a polynomial $f(x)=x^6-11x^4+36x^2-36$. Prove that for an arbitrary prime number $p$, $f(x)\equiv 0\pmod{p}$ has a solution.

The people in an infinitely long line are numbered $1,2,3,\dots$. Then, each person says either ``Karl" or ``Lark" independently and at random. Let $S$ be the set of all positive integers $i$ such that people $i$, $i+1$, and $i+2$ all say ``Karl," and define $X = \sum_{i \in S} 2^{-i}$. Then the expected value of $X^2$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$. [i]Proposed by Ankit Bisain[/i]

For what natural number $x$ will the value of the polynomial $x^3+7x^2+6x+1$ be the cube of a natural number?

Let $x_1,x_2,x_3, \cdots$ be a sequence defined by the following recurrence relation: \[ \begin{cases}x_{1}&= 4\\ x_{n+1}&= x_{1}x_{2}x_{3}\cdots x_{n}+5\text{ for }n\ge 1\end{cases} \] The first few terms of the sequence are $x_1=4,x_2=9,x_3=41 \cdots$ Find all pairs of positive integers $\{a,b\}$ such that $x_a x_b$ is a perfect square.