Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation: \[a_{n+1}=10^n a_n^2\] (a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$. (b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.

Define alternate sum of a set of real numbers $A =\{a_1,a_2,...,a_k\}$ with $a_1 < a_2 <...< a_k$, the number $S(A) = a_k - a_{k-1} + a_{k-2} - ... + (-1)^{k-1}a_1$ (for example if $A = \{1,2,5, 7\}$ then $S(A) = 7 - 5 + 2 - 1$) Consider the alternate sums, of every subsets of $A = \{1, 2, 3, 4, 5, 6, 7, 8,9, 10\}$ and sum them. What is the last digit of the sum obtained?

Prove that the following statement is true for two natural nos. $m,n$ if and only $v(m) = v(n)$ where $v(k)$ is the highest power of $2$ dividing $k$. $\exists$ a set $A$ of positive integers such that $(i)$ $x,y \in \mathbb{N}, |x-y| = m \implies x \in A $ or $y \in A$ $(ii)$ $x,y \in \mathbb{N}, |x-y| = n \implies x \not\in A $ or $y \not\in A$

Let $a,b,c,d$ be positive real numbers such that $ 2(a+b+c+d)\ge abcd $. Prove that \[ a^2+b^2+c^2+d^2 \ge abcd .\]

For each vertex of a given tetrahedron, a sphere passing through that vertex and the midpoints of the edges outgoing from this vertex is constructed. Prove that these four spheres pass through a single point.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Let $ABC$ be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let $N = 2017$, and define the $A$-[i]ntipodes[/i] to be the points $A_1,\dots, A_N$ to be the points on segment $BC$ such that $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$, and similarly define the $B$, $C$-ntipodes. A line $\ell_A$ through $A$ is called a [i]qevian[/i] if it passes through an $A$-ntipode, and similarly we define qevians through $B$ and $C$. Compute the number of ordered triples $(\ell_A, \ell_B, \ell_C)$ of concurrent qevians through $A$, $B$, $C$, respectively. [i]Proposed by Brandon Wang[/i]

What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock? $ \text{(A)}\ 30\qquad\text{(B)}\ 45\qquad\text{(C)}\ 60\qquad\text{(D)}\ 75\qquad\text{(E)}\ 90 $

The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$?

Which of the following is true if $S = \dfrac 1{1^2} + \dfrac 1{2^2} + \dfrac 1{3^2} + \cdots + \dfrac 1{2001^2} + \dfrac 1{2002^2}$? $ \textbf{a)}\ 1\leq S < \dfrac 43 \qquad\textbf{b)}\ \dfrac 43 \leq S < 2 \qquad\textbf{c)}\ 2 \leq S < \dfrac 73$ $\textbf{d)}\ \dfrac 73 \leq S < \dfrac 52 \qquad\textbf{e)}\ \dfrac 52 \leq S < 3 $

Prove that the fraction $ \dfrac{21n \plus{} 4}{14n \plus{} 3}$ is irreducible for every natural number $ n$.

Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$. Alicia bikes towards $B$ at 18 miles per hour. Leaving at the same time, Beth bikes toward $A$ at 12 miles per hour. How many miles from City $A$ will they be when they meet? $\textbf{(A) }20\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$

Find all functions $\displaystyle f : \mathbb N^\ast \to \mathbb N^\ast$ ($\displaystyle N^\ast = \{ 1,2,3,\ldots \}$) with the property that, for all $\displaystyle n \geq 1$, \[ f(1) + f(2) + \ldots + f(n) \] is a perfect cube $\leq n^3$. [i]Dinu Teodorescu[/i]

Let $ N \equal{} \{1,2 \ldots, n\}, n \geq 2.$ A collection $ F \equal{} \{A_1, \ldots, A_t\}$ of subsets $ A_i \subseteq N,$ $ i \equal{} 1, \ldots, t,$ is said to be separating, if for every pair $ \{x,y\} \subseteq N,$ there is a set $ A_i \in F$ so that $ A_i \cap \{x,y\}$ contains just one element. $ F$ is said to be covering, if every element of $ N$ is contained in at least one set $ A_i \in F.$ What is the smallest value $ f(n)$ of $ t,$ so there is a set $ F \equal{} \{A_1, \ldots, A_t\}$ which is simultaneously separating and covering?

Let $ABCD$ be a convex quadrilateral such that $|AB|=10$, $|CD|=3\sqrt 6$, $m(\widehat{ABD})=60^\circ$, $m(\widehat{BDC})=45^\circ$, and $|BD|=13+3\sqrt 3$. What is $|AC|$ ? $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 12 $