Prove that there exists an integer $n$, $n\geq 2002$, and $n$ distinct positive integers $a_1,a_2,\ldots,a_n$ such that the number $N= a_1^2a_2^2\cdots a_n^2 - 4(a_1^2+a_2^2+\cdots + a_n^2) $ is a perfect square.

Find all integers $N$ with the following property: for $10$ but not $11$ consecutive positive integers, each one is a divisor of $N$.

Let $X$ be the collection of all functions $f: \{0,1,\dots, 2016\} \rightarrow \{0,1,\dots, 2016\}$. Compute the number of functions $f \in X$ such that \[ \max_{g \in X} \left( \min_{0 \le i \le 2016} \big( \max (f(i), g(i)) \big) - \max_{0 \le i \le 2016} \big( \min (f(i),g(i)) \big) \right) = 2015. \]

$ A$ and $ B$ play the following game with a polynomial of degree at least 4: \[ x^{2n} \plus{} \_x^{2n \minus{} 1} \plus{} \_x^{2n \minus{} 2} \plus{} \ldots \plus{} \_x \plus{} 1 \equal{} 0 \] $ A$ and $ B$ take turns to fill in one of the blanks with a real number until all the blanks are filled up. If the resulting polynomial has no real roots, $ A$ wins. Otherwise, $ B$ wins. If $ A$ begins, which player has a winning strategy?

For real $\theta_i$, $i = 1, 2, \dots, 2011$, where $\theta_1 = \theta_{2012}$, find the maximum value of the expression \[ \sum_{i=1}^{2011} \sin^{2012} \theta_i \cos^{2012} \theta_{i+1}. \] [i]Proposed by Lewis Chen [/i]

Prove there do exist infinitely many positive integers $n$ such that if a prime $p$ divides $n(n+1)$ then $p^2$ also divides it (all primes dividing $n(n+1)$ bear exponent at least two). Exhibit (at least) two values, one even and one odd, for such numbers $n>8$. (Pál Erdös & Kurt Mahler)

p1. It is known that $a_1=2$ , $a_2=3$ . For $k > 2$, define $a_k=\frac{1}{2}a_{k-2}+\frac{1}{3}a_{k-1}$. Find the infinite sum of of $a_1+a_2+a_3+...$ p2. The [i]multiplied [/i] number is a natural number in two-digit form followed by the result time. For example, $7\times 8 = 56$, then $7856$ and $8756$ are multiplied numbers . $2\times 3 = 6$, then $236$ and $326$ are multiplied. $2\times 0 = 0$, then $200$ is the multiplied. For the record, the first digit of the number times can't be $0$. a. What is the difference between the largest and the smallest multiplied number? b. Find all the multiplied numbers that consist of three digits and each digit is square number. c. Given the following "boxes" that must be filled with multiple numbers. [img]https://cdn.artofproblemsolving.com/attachments/b/6/ac086a3d1a0549fae909c072224605430daf1d.png[/img] Determine the contents of the shaded box. Is this content the only one? d. Complete all the empty boxes above with multiplied numbers. p3. Look at the picture of the arrangement of three squares below. [img]https://cdn.artofproblemsolving.com/attachments/1/3/c0200abae77cc73260b083117bf4bafc707eea.png[/img]Prove that $\angle BAX + \angle CAX = 45^o$ p4. Prove that $(n-1)n (n^3 + 1)$ is always divisible by $6$ for all natural number $n$.

The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?

Let $A_1A_2A_3...A_{72}$ be a regurar $72$-gon with center $O$. Calculate an extenral angle of that polygon and the angles $\angle A_{45} OA_{46}$, $\angle A_{44} A_{45}A_{46}$. How many diagonals does this polygon have?

Let $A_1, A_2, ... , A_k$ be the subsets of $\left\{1,2,3,...,n\right\}$ such that for all $1\leq i,j\leq k$:$A_i\cap A_j \neq \varnothing$. Prove that there are $n$ distinct positive integers $x_1,x_2,...,x_n$ such that for each $1\leq j\leq k$: $$lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\}$$ [i]Proposed by Morteza Saghafian, Mahyar Sefidgaran[/i]

The cubic polynomials $p(x)$ and $q(x)$ satisfy • $p(1) = q(2)$ • $p(3) = q(4)$ • $p(5) = q(6)$ • $p(7) = q(8) + 13$. Find $p(9)-q(10)$.

Let $m, n > 1$ be coprime odd integers. Show that $$\big \lfloor \frac{m^{\phi (n)+1} + n^{\phi (m)+1}}{mn} \rfloor$$ is an even integer, where $\phi$ is Euler’s totient function.

Find all pairs $\left(m,n\right)$ of positive integers, with $m,n\geq2$, such that $a^n-1$ is divisible by $m$ for each $a\in \left\{1,2,3,\ldots,n\right\}$.

A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^r M,$ where $r$ and $M$ are positive integers and $M$ is not divisible by $3.$ What is $r?$ $\textbf{(A) }5 \qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Let $n \ge 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is [i]peaceful[/i] if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k \times k$ square which does not contain a rook on any of its $k^2$ unit squares.

Find the period of the repetend of the fraction $\frac{39}{1428}$ by using [i]binary[/i] numbers, i.e. its binary decimal representation. (Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2\cdots a_ka_1a_2\cdots a_ka_1a_2 \cdots a_k \cdots$. Here the repeated sequence $a_1a_2\cdots a_k$ is called the [i]repetend[/i] of the fraction, and the smallest length of the repetend, $k$, is called the [i]period[/i] of the decimal number.)

Determine all pairs $(p, q)$, $p, q \in \mathbb {N}$, such that $(p + 1)^{p - 1} + (p - 1)^{p + 1} = q^q$.

Prove that you can arrange arrows on the edges of a convex polyhedron such that each vertex contains at most three arrows.

Let $a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3$ be nine strictly positive real numbers. We set \[S_1 = a_1b_2c_3, \quad S_2 = a_2b_3c_1, \quad S_3 = a_3b_1c_2;\]\[T_1 = a_1b_3c_2, \quad T_2 = a_2b_1c_3, \quad T_3 = a_3b_2c_1.\] Suppose that the set $\{S1, S2, S3, T1, T2, T3\}$ has at most two elements. Prove that \[S_1 + S_2 + S_3 = T_1 + T_2 + T_3.\]

Prove that if the diagonals of a certain trapezoid are perpendicular, then the sum of the lengths of the bases of this trapezoid is not greater than the sum of the lengths of the sides of this trapezoid.

Prove that there exist infinitely many positive integers $k$ such that the sequence $\{x_n\}$ satisfying $$ x_1=1, x_2=k+2, x_{n+2}-(k+1)x_{n+1}+x_n=0(n \ge 0)$$ does not contain any prime number.

Let $ABCD$ be a quadrilateral with $AD = BC$ and $\angle DAB + \angle ABC = 120^\circ$. An equilateral triangle $DPC$ is erected in the exterior of the quadrilateral. Prove that the triangle $APB$ is also equilateral.

Triangle $ABC$ is equilateral with side length $\sqrt{3}$ and circumcenter at $O$. Point $P$ is in the plane such that $(AP)(BP)(CP) = 7$. Compute the difference between the maximum and minimum possible values of $OP$. [i]2015 CCA Math Bonanza Team Round #8[/i]

Let $R$ be a square region and $n \ge 4$ an integer. A point $X$ in the interior of $R$ is called [i]n-ray partitional[/i] if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional? $\textbf{(A)}\ 1500 \qquad \textbf{(B)}\ 1560 \qquad \textbf{(C)}\ 2320 \qquad \textbf{(D)}\ 2480 \qquad \textbf{(E)}\ 2500$

A game for two. One gives a digit and the second substitutes it instead of a star in the following difference: $$**** - **** = $$ Then the first gives the next digit, and so on $8$ times. The first wants to obtain the greatest possible difference, the second -- the least. Prove that: 1. The first can operate in such a way that the difference would be not less than $4000$, not depending on the second's behaviour. 2. The second can operate in such a way that the difference would be not greater than $4000$, not depending on the first's behaviour.