Let $p$ be an odd prime integer and $k$ a positive integer not divisible by $p$, $1\le k<2(p+1)$, and let $N=2kp+1$. Prove that the following statements are equivalent: (i) $N$ is a prime number (ii) there exists a positive integer $a$, $2\le a<n$, such that $a^{kp}+1$ is divisible by $N$ and $\gcd\left(a^k+1,N\right)=1$.

Given three positive integers $a,b,$ and $c$. Their greatest common divisor is $D$; their least common multiple is $m$. Then, which two of the following statements are true? $ \text{(1)}\ \text{the product MD cannot be less than abc} \qquad$ $\text{(2)}\ \text{the product MD cannot be greater than abc}\qquad$ $\text{(3)}\ \text{MD equals abc if and only if a,b,c are each prime}\qquad$ $\text{(4)}\ \text{MD equals abc if and only if a,b,c are each relatively prime in pairs}$ $\text{ (This means: no two have a common factor greater than 1.)}$ $ \textbf{(A)}\ 1,2 \qquad\textbf{(B)}\ 1,3\qquad\textbf{(C)}\ 1,4\qquad\textbf{(D)}\ 2,3\qquad\textbf{(E)}\ 2,4 $

For any permutation $p$ of set $\{1, 2, \ldots, n\}$, define $d(p) = |p(1) - 1| + |p(2) - 2| + \ldots + |p(n) - n|$. Denoted by $i(p)$ the number of integer pairs $(i, j)$ in permutation $p$ such that $1 \leqq < j \leq n$ and $p(i) > p(j)$. Find all the real numbers $c$, such that the inequality $i(p) \leq c \cdot d(p)$ holds for any positive integer $n$ and any permutation $p.$

The perimeter of a triangle is a natural number, its circumradius is equal to $\frac{65}{8}$, and the inradius is equal to $4$. Find the sides of the triangle.

Find all $4$-digit numbers $\overline{abba}$ that are equal to the product of some consecutive prime numbers.

Side $BC$ of triangle $ABC$ is equal to $a$ and the opposite angle is equal to $\theta$. A straight line passing through the midpoint of $BC$ and the center of the inscribed circle intersects lines $AB$ and $AC$ at points $M$ and $P$, respectively. Find the area of the quadrilateral (non-convex) $BMPC$.

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assume that no three points of $\mathcal S$ are collinear. A [i]windmill[/i] is a process that starts with a line $\ell$ going through a single point $P \in \mathcal S$. The line rotates clockwise about the [i]pivot[/i] $P$ until the first time that the line meets some other point belonging to $\mathcal S$. This point, $Q$, takes over as the new pivot, and the line now rotates clockwise about $Q$, until it next meets a point of $\mathcal S$. This process continues indefinitely. Show that we can choose a point $P$ in $\mathcal S$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $\mathcal S$ as a pivot infinitely many times. [i]Proposed by Geoffrey Smith, United Kingdom[/i]

In order to determine $\frac{1}{A}$ for $A>0$, one can use the iteration $X_{k+1}=X_{k}(2-AX_{k}),$ where $X_0$ is a selected starting value. Find the limitation, if any, on the starting value $X_0$ so that the above iteration converges to $\frac{1}{A}.$

Three of the $16$ squares from a $4 \times 4$ grid of squares are selected at random. The probability that at least one corner square of the grid is selected is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. $ \begin{tabular}{ | l | c | c | r| } \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline \end{tabular} $

Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.

Let $\{x\}$ denote the fractional part of $x$. Then $\lim \limits_{n\to \infty} \left\{ \left(1+\sqrt{2}\right)^{2n}\right\}$ equals [list=1] [*] 0 [*] 0.5 [*] 1 [*] Does not exists [/list]

Let $\mathbb{M}$ be the vector space of $m \times p$ real matrices. For a vector subspace $S\subseteq \mathbb{M}$, denote by $\delta(S)$ the dimension of the vector space generated by all columns of all matrices in $S$. Say that a vector subspace $T\subseteq \mathbb{M}$ is a $\emph{covering matrix space}$ if \[ \bigcup_{A\in T, A\ne \mathbf{0}} \ker A =\mathbb{R}^p \] Such a $T$ is minimal if it doesn't contain a proper vector subspace $S\subset T$ such that $S$ is also a covering matrix space. [list] (a) (8 points) Let $T$ be a minimal covering matrix space and let $n=\dim (T)$ Prove that \[ \delta(T)\le \dbinom{n}{2} \] (b) (2 points) Prove that for every integer $n$ we can find $m$ and $p$, and a minimal covering matrix space $T$ as above such that $\dim T=n$ and $\delta(T)=\dbinom{n}{2}$[/list]

Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$. (1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded? (2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$? Justify your answer.

Find the largest real number $\lambda$ with the following property: for any positive real numbers $p,q,r,s$ there exists a complex number $z=a+bi$($a,b\in \mathbb{R})$ such that $$ |b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0.$$

Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$, and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions: (1) For all $n\in \mathbb{Z}$, $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$ (2) For all $n\ge M$, $f(n)\ge 0$ (3) For all $n>m>M$, $n-m|f(n)-f(m)$ Show that $a$ is a perfect $r$-th power.

Find all $n \in \mathbb{N}$ such that $ \lfloor \sqrt{n}\rfloor$ divides $n$.

Given a positive integer $m$, let $d(m)$ be the number of positive divisors of $m$. Determine all positive integers $n$ such that $d(n) +d(n+ 1) = 5$.

For any three nonnegative reals $a$, $b$, $c$, prove that $\left|ca-ab\right|+\left|ab-bc\right|+\left|bc-ca\right|\leq\left|b^{2}-c^{2}\right|+\left|c^{2}-a^{2}\right|+\left|a^{2}-b^{2}\right|$. [i]Generalization.[/i] For any $n$ nonnegative reals $a_{1}$, $a_{2}$, ..., $a_{n}$, prove that $\sum_{i=1}^{n}\left|a_{i-1}a_{i}-a_{i}a_{i+1}\right|\leq\sum_{i=1}^{n}\left|a_{i}^{2}-a_{i+1}^{2}\right|$. Here, the indices are cyclic modulo $n$; this means that we set $a_{0}=a_{n}$ and $a_{n+1}=a_{1}$. darij

The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions: [b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$ [b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\] Prove that $ c \leq \frac{1}{4n}.$

The quadrilateral $ABCD$ is inscribed in a circle of radius $1$, so that $AB$ is a diameter of the circumference and $CD = 1$. A variable point $X$ moves along the semicircle determined by $AB$ that does not contain $C$ or $D$. Determine the position of $X$ for which the sum of the distances from $X$ to lines $BC, CD$ and $DA$ is maximum.

Let @ denote the "averaged with" operation: $a$ @ $b$ = $\frac{a+b}{2}$. Which of the following distributive laws hold for all numbers $x,y$ and $z$? I. x @ (y+z) = (x @ y) + (x @ z) II. x + (y @ z) = (x + y) @ (x + z) III. x @ (y @ z) = (x @ y) @ (x @ z) $ \textbf{(A)}\ \text{I only} \qquad \textbf{(B)}\ \text{II only} \qquad \textbf{(C)}\ \text{III only} \qquad \textbf{(D)}\ \text{I and III only} \qquad \textbf{(E)}\ \text{II and III only} $

Find the highest degree $ k$ of $ 1991$ for which $ 1991^k$ divides the number \[ 1990^{1991^{1992}} \plus{} 1992^{1991^{1990}}.\]

In an excavation in ancient Rome an unusual clock with $18$ divisions marked with Roman numerals (see figure). Unfortunately the watch was broken into $5$ pieces. The sum of the numbers on each piece was the same. Show how he could be broken the clock. [img]https://cdn.artofproblemsolving.com/attachments/7/a/6e83df1bb7adb13305239a152ac95a4a96f556.png[/img]

Triangles $ ABC$ and $ ADC$ are isosceles with $ AB \equal{} BC$ and $ AD \equal{} DC$. Point $ D$ is inside $ \triangle ABC$, $ \angle ABC \equal{} 40^\circ$, and $ \angle ADC \equal{} 140^\circ$. What is the degree measure of $ \angle BAD$? $ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 60$

For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4>m$.