Find the least positive integer $n$ such that for every prime number $p, p^2 + n$ is never prime.

For any $n\in\mathbb N$, denote by $a_n$ the sum $2+22+222+\cdots+22\ldots2$, where the last summand consists of $n$ digits of $2$. Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$.

Let $m, k$, and $c$ be positive integers with $k > c$, and let $\lambda$ be a positive, non-integer real root of the equation $\lambda^{m+1} - k \lambda^m - c = 0$. Let $f : Z^+ \to Z$ be defined by $f(n) = \lfloor \lambda n \rfloor$ for all $n \in Z^+$. Show that $f^{m+1}(n) \equiv cn - 1$ (mod $k$) for all $n \in Z^+$. (Here, $Z^+$ denotes the set of positive integers, $ \lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $f^{m+1}(n) = f(f(... f(n)...))$ where $f$ appears $m + 1$ times.)

A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$. The highest elevation is: $\textbf{(A)}\ 800 \qquad \textbf{(B)}\ 640\qquad \textbf{(C)}\ 400 \qquad \textbf{(D)}\ 320 \qquad \textbf{(E)}\ 160$

Let $n$ a positive integer and let $f\colon [0,1] \to \mathbb{R}$ an increasing function. Find the value of : \[ \max_{0\leq x_1\leq\cdots\leq x_n\leq 1}\sum_{k=1}^{n}f\left ( \left | x_k-\frac{2k-1}{2n} \right | \right )\]

A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is $ 1$ place to its right in the alphabet (assuming that the letter $ A$ is one place to the right of the letter $ Z$). The second time this same letter appears in the given message, it is replaced by the letter that is $ 1\plus{}2$ places to the right, the third time it is replaced by the letter that is $ 1 \plus{} 2 \plus{} 3$ places to the right, and so on. For example, with this code the word "banana" becomes "cbodqg". What letter will replace the last letter $ \text{s}$ in the message "Lee's sis is a Mississippi miss, Chriss!"? $ \textbf{(A)}\ \text{g}\qquad \textbf{(B)}\ \text{h}\qquad \textbf{(C)}\ \text{o}\qquad \textbf{(D)}\ \text{s}\qquad \textbf{(E)}\ \text{t}$

Prove that \[ a^2b^2(a^2+b^2-2) \geq (a+b)(ab-1) \] for all positive real numbers $a$ and $b.$

Given a $\Delta ABC$ where $\angle C = 90^{\circ}$, $D$ is a point in the interior of $\Delta ABC$ and lines $AD$ $,$ $BD$ and $CD$ intersect $BC$, $CA$ and $AB$ at points $P$ ,$Q$ and $R$ ,respectively. Let $M$ be the midpoint of $\overline{PQ}$. Prove that, if $\angle BRP$ $ =$ $ \angle PRC$ then $MR=MC$.

For a polynomial $P$ with integer coefficients, $P(5)$ is divisible by $2$ and $P(2)$ is divisible by $5$. Prove that $P(7)$ is divisible by $10$.

Let $ABC$ be a triangle and $H, O$ be its orthocenter and circumcenter, respectively. Construct a triangle by points $D_1, E_1, F_1,$ where $D_1$ lies on lines $BO$ and $AH$, $E_1$ lies on lines $CO$ and $BH$, and $F_1$ lies on lines $AO$ and $CH$. On the other hand, construct the other triangle $D_2E_2F_2$ that $D_2$ lies on $CO$ and $AH$, $E_2$ lies on $AO$ and $BH$, and $F_2$ lies on lines $BO$ and $CH$. Prove that triangles $D_1E_1F_1$ and $D_2E_2F_2$ are similar. [i]Proposed by Saintan Wu[/i]

It is given the equation $x^4-2ax^3+a(a+1)x^2-2ax+a^2=0$. a) Find the greatest value of $a$, such that this equation has at least one real root. b) Find all the values of $a$, such that the equation has at least one real root.

Paint the grid points of $L=\{0,1,\dots,n\}^2$ with red or green in such a way that every unit lattice square in $L$ has exactly two red vertices. How many such colorings are possible?

$n$ is a product of some two consecutive primes. $s(n)$ denotes the sum of the divisors of $n$ and $p(n)$ denotes the number of relatively prime positive integers not exceeding $n$. Express $s(n)p(n)$ as a polynomial of $n$.

5. Let $f : R \to R$ be a function that satisfies for all $x \in R$ (i) $| f(x)| \le 1$, and (ii) $f\left(x+\frac{13}{42}\right)+ f(x) = f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right)$ Prove that $f$ is a periodic function

$f : [-\sqrt{1995},\sqrt{1995}] \to\mathbb{R}$ is defined by $f(x) = x(1993+\sqrt{1995-x^{2}})$. Find its maximum and minimum values.

In the triangle $ABC$, the bisector of angle $A$ is equal to the half-sum of the height and median drawn from vertex $A$. Prove that if $\angle A$ is obtuse, then $AB = AC$.

Let $OABC$ be a tetrahedon with $\angle BOC=\alpha,\angle COA =\beta$ and $\angle AOB =\gamma$. The angle between the faces $OAB$ and $OAC$ is $\sigma$ and the angle between the faces $OAB$ and $OBC$ is $\rho$. Show that $\gamma > \beta \cos\sigma + \alpha \cos\rho$.

Let $ABC$ be a triangle with $AB = 3, AC = 4,$ and $BC = 5$, let $P$ be a point on $BC$, and let $Q$ be the point (other than $A$) where the line through $A$ and $P$ intersects the circumcircle of $ABC$. Prove that \[PQ\le \frac{25}{4\sqrt{6}}.\]

Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a,b,c$ by \[a=\root3\of n\kern1.5pt,\qquad b={1\over a-[a]}\kern1pt,\qquad c={1\over b-[b]}\kern1.5pt,\] where $[x]$ denotes the integer part of $x$. Prove that there are infinitely many such integers $n$ with the property that there exist integers $r,s,t$, not all zero, such that $ra+sb+tc=0$.

Let $a,b,c>0$ such that $a+b+c=1$. Prove: \[\frac{a^{2}}b+\frac{b^{2}}c+\frac{c^{2}}a \ge 3(a^{2}+b^{2}+c^{2}) \]

Put $x = \log_{10} 2$, $y = \log_{10} 3$. Then $15 < 16$ implies $1 - x + y < 4x$, so $1 + y < 5x$. Derive similar inequalities from $80 < 81$ and $243 < 250$. Hence show that \[ 0.47 < \log_{10} 3 < 0.482. \]

The bisector of angle $A$ in parallelogram $ABCD$ intersects side $BC$ at $M$ and the bisector of $\angle AMC$ passes through point $D$. Find angles of the parallelogram if it is known that $\angle MDC = 45^o$. [img]https://cdn.artofproblemsolving.com/attachments/e/7/7cfb22f0c26fe39aa3da3898e181ae013a0586.png[/img]

Let $P$ and $Q$ be distinct points in the plane of a triangle $ABC$ such that $AP : AQ = BP : BQ = CP : CQ$. Prove that the line $PQ$ passes through the circumcenter of the triangle.

Evaluate $ \int_{\minus{}\frac{1}{2}}^{\frac{1}{2}} \frac{x}{\{(2x\plus{}1)\sqrt{x^2\minus{}x\plus{}1}\plus{}(2x\minus{}1)\sqrt{x^2\plus{}x\plus{}1}\}\sqrt{x^4\plus{}x^2\plus{}1}}\ dx$.

Let ${\mathbb Q}^{+}$ be the set of positive rational numbers. Construct a function $f:{\mathbb Q}^{+}\rightarrow{\mathbb Q}^{+}$ such that \[f(xf(y)) = \frac{f(x)}{y}\] for all $x, y \in{\mathbb Q}^{+}$.