Let $a \in \mathbb R$ and let $z_1, z_2, \ldots, z_n$ be complex numbers of modulus $1$ satisfying the relation \[\sum_{k=1}^n z_k^3=4(a+(a-n)i)- 3 \sum_{k=1}^n \overline{z_k}\] Prove that $a \in \{0, 1,\ldots, n \}$ and $z_k \in \{1, i \}$ for all $k.$

Suppose that $ \{a_n\}$ is an arithmetic sequence with \[a_1 \plus{} a_2 \plus{} \cdots \plus{} a_{100} \equal{} 100\quad\text{and}\quad a_{101} \plus{} a_{102} \plus{} \cdots \plus{} a_{200} \equal{} 200.\] What is the value of $ a_2 \minus{} a_1$? $ \textbf{(A)}\ 0.0001 \qquad \textbf{(B)}\ 0.001 \qquad \textbf{(C)}\ 0.01 \qquad \textbf{(D)}\ 0.1 \qquad \textbf{(E)}\ 1$

For a polynomial $ P$ of degree 2000 with distinct real coefficients let $ M(P)$ be the set of all polynomials that can be produced from $ P$ by permutation of its coefficients. A polynomial $ P$ will be called [b]$ n$-independent[/b] if $ P(n) \equal{} 0$ and we can get from any $ Q \in M(P)$ a polynomial $ Q_1$ such that $ Q_1(n) \equal{} 0$ by interchanging at most one pair of coefficients of $ Q.$ Find all integers $ n$ for which $ n$-independent polynomials exist.

There are $n$ distinct points in the plane, no three of which are collinear. Suppose that $A$ and $B$ are two of these points. We say that segment $AB$ is independent if there is a straight line such that points $A$ and $B$ are on one side of the line, and the other $n-2$ points are on the other side. What is the maximum possible number of independent segments?

Let be two fixed points $ B,C. $ Find the locus of the spatial points $ A $ such that $ ABC $ is a nondegenerate triangle and the expression $$ R^2 (A)\cdot\sin \left( 2\angle ABC\right)\cdot\sin \left( 2\angle BCA\right) $$ has the greatest value possible, where $ R(A) $ denotes the radius of the excirlce of $ ABC. $

This problem consists of four parts. 1. Show that for any nonzero integers $m,n,$ and prime $p$, we have $v_p(mn)=v_p(m)+v_p(n).$ 2. Show that if an off prime $p$, a positive integer $k$ and integers $a,b$ satisfy $p \nmid ~^\text{'}~p|a-b$ and $p\nmid k$, then $v_p(a^k-b^k)=v_p(a-b).$ 3. Show that if $p$ is an off prime with $p|a-b$ and $p\nmid a,b$, then $v_p(a^p-b^p)=v_p(a-b)+1)$. 4. Show that if an odd prime $p$, a positive integer $k$ and integers $a,b$ satisfy $p\nmid a,b ~^\text{'}~ p|a-b$, then $v_p(a^k-b^k)=v_p(a-b)$. Proposed by LTE.

Consider the $9\times 9$ grid of lattice points $\{(x,y) | 0 \le x, y \le 8\}$. How many rectangles with nonzero area and sides parallel to the $x, y$ axes are there such that each corner is one of the lattice points and the point $(4, 4)$ is not contained within the interior of the rectangle? ($(4,4)$ is allowed to lie on the boundary of the rectangle).

The diagram below shows an isosceles triangle with base $21$ and height $28$. Inscribed in the triangle is a square. Find the area of the shaded region inside the triangle and outside of the square. [asy] size(170); defaultpen(linewidth(0.8)); draw((0,0)--(1,1)); pair A=(5,0),B=(-5,0),C=(0,14), invis[]={(1,2),(-1,2)}; pair intsquare[]={extension(origin,invis[0],A,C),extension(origin,invis[1],B,C)}; path triangle=A--B--C--cycle,square=(intsquare[0]--intsquare[1]--(intsquare[1].x,0)--(intsquare[0].x,0)--cycle); fill(triangle,gray); unfill(square); draw(triangle^^square); [/asy]

Let $ x$ be chosen at random from the interval $ (0,1)$. What is the probability that \[ \lfloor\log_{10}4x\rfloor \minus{} \lfloor\log_{10}x\rfloor \equal{} 0? \]Here $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$. $ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 3{20} \qquad \textbf{(C) } \frac 16 \qquad \textbf{(D) } \frac 15 \qquad \textbf{(E) } \frac 14$

A point $ P$ in the interior of triangle $ ABC$ satisfies \[ \angle BPC \minus{} \angle BAC \equal{} \angle CPA \minus{} \angle CBA \equal{} \angle APB \minus{} \angle ACB.\] Prove that \[ \bar{PA} \cdot \bar{BC} \equal{} \bar{PB} \cdot \bar{AC} \equal{} \bar{PC} \cdot \bar{AB}.\]

Let $\{a_{n}\}$ be a sequence which satisfy $a_{1}=5$ and $a_{n=}\sqrt[n]{a_{n-1}^{n-1}+2^{n-1}+2.3^{n-1}} \qquad \forall n\geq2$ [b](a)[/b] Find the general fomular for $a_{n}$ [b](b)[/b] Prove that $\{a_{n}\}$ is decreasing sequences

The sequence $ \{a_{n}\}_{n \ge 1}$ is defined by \[ a_{1}=1, \; a_{2}=2, \; a_{3}=24, \; a_{n}=\frac{ 6a_{n-1}^{2}a_{n-3}-8a_{n-1}a_{n-2}^{2}}{a_{n-2}a_{n-3}}\ \ \ \ (n\ge4).\] Show that $ a_{n}$ is an integer for all $ n$, and show that $ n|a_{n}$ for every $ n\in\mathbb{N}$.

Let $m, n$ be positive integers. Starting with all positive integers written in a line, we can form a list of numbers in two ways: $(1)$ Erasing every $m$-th and then, in the obtained list, erasing every $n$-th number; $(2)$ Erasing every $n$-th number and then, in the obtained list, erasing every $m$-th number. A pair $(m,n)$ is called [i]good[/i] if, whenever some positive integer $k$ occurs in both these lists, then it occurs in both lists on the same position. (a) Show that the pair $(2, n)$ is good for any $n\in \mathbb{N}$. (b) Is there a good pair $(m, n)$ with $2<m<n$?

If $\frac{b}{a} = 2$ and $\frac{c}{b} = 3$, what is the ratio of $a + b$ to $b + c$? $ \textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{3}{8}\qquad\textbf{(C)}\ \frac{3}{5}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{3}{4} $

Find all functions $f : \mathbb{Z}_{+} \rightarrow \mathbb{Z}_{+}$ that satisfy $f(mf(n)) = n+f(2015m)$ for all $m,n \in \mathbb{Z}_{+}$.

At the theater children get in for half price. The price for $5$ adult tickets and $4$ child tickets is $\$24.50$. How much would $8$ adult tickets and $6$ child tickets cost? $\textbf{(A) }\$35\qquad \textbf{(B) }\$38.50\qquad \textbf{(C) }\$40\qquad \textbf{(D) }\$42\qquad \textbf{(E) }\$42.50$

Let $ABCD$ be a cyclic quadrilateral, and let $E$, $F$, $G$, and $H$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. Let $W$, $X$, $Y$ and $Z$ be the orthocenters of triangles $AHE$, $BEF$, $CFG$ and $DGH$, respectively. Prove that the quadrilaterals $ABCD$ and $WXYZ$ have the same area.

Show that for every natural number $n$ there are $n$ natural numbers $ x_1 < x_2 < ... < x_n $ such that $$\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}-\frac{1}{x_1x_2...x_n}\in \mathbb{N}\cup {0}$$ (15 points )

Let $n$ be a positive integer. Two players, Alice and Bob, are playing the following game: - Alice chooses $n$ real numbers; not necessarily distinct. - Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are $\frac{n(n-1)}{2}$ such sums; not necessarily distinct.) - Bob wins if he finds correctly the initial $n$ numbers chosen by Alice with only one guess. Can Bob be sure to win for the following cases? a. $n=5$ b. $n=6$ c. $n=8$ Justify your answer(s). [For example, when $n=4$, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]

Santiago, Daniel and Fátima practice for the Math Olympics. Santiago thinks of a regular polygon and Daniel of another, without telling Fatima what the polygons are. They just tell you that one of the polygons has $3$ more sides than the other and that an angle of one of the polygons measures $10$ degrees more than one angle of the other. From this, and knowing that each interior angle of a regular polygon of $n$ sides measures $\frac{180(n-2)}{n}$ degrees, Fatima identifies what the polygons are. How many sides do the polygons that James and Daniel chose, have?

[u]Set 4[/u] [b]4.1[/b] Quadrilateral $ABCD$ (with $A, B, C$ not collinear and $A, D, C$ not collinear) has $AB = 4$, $BC = 7$, $CD = 10$, and $DA = 5$. Compute the number of possible integer lengths $AC$. [img]https://cdn.artofproblemsolving.com/attachments/1/6/4f43873a64bc00a0e6173002ccd80e8f1529a9.png[/img] [b]4.2[/b] Let $T$ be the answer from the previous part. $2T$ congruent isosceles triangles with base length $b$ and leg length $\ell$ are arranged to form a parallelogram as shown below (not necessarily the correct number of triangles). If the total length of all drawn line segments (not double counting overlapping sides) is exactly three times the perimeter of the parallelogram, find $\frac{\ell}{b}$. [img]https://cdn.artofproblemsolving.com/attachments/5/c/744f503ed822bc43acafe2633e6108022f2c88.png[/img] [b]4.3[/b] Let $T$ be the answer from the previous part. Rectangle $R$ has length $T$ times its width. $R$ is inscribed in a square $S$ such that the diagonals of $ S$ are parallel to the sides of $R$. What proportion of the area of $S$ is contained within $R$? [img]https://cdn.artofproblemsolving.com/attachments/a/1/0928dd1ffbeb4d7dee9b697fdb7696cc70c03d.png[/img] PS. You should use hide for answers.

In Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and her points-per-game average for the $10$ games was also an integer. What is the product of the number of points she scored in the ninth and tenth games? $\textbf{(A)}\ 35\qquad \textbf{(B)}\ 40\qquad \textbf{(C)}\ 48\qquad \textbf{(D)}\ 56\qquad \textbf{(E)}\ 72$

Let $S = \{1,2,3,\ldots,n\}$. Consider a function $f\colon S\to S$. A subset $D$ of $S$ is said to be invariant if for all $x\in D$ we have $f(x)\in D$. The empty set and $S$ are also considered as invariant subsets. By $\deg (f)$ we define the number of invariant subsets $D$ of $S$ for the function $f$. [b]i)[/b] Show that there exists a function $f\colon S\to S$ such that $\deg (f)=2$. [b]ii)[/b] Show that for every $1\leq k\leq n$ there exists a function $f\colon S\to S$ such that $\deg (f)=2^{k}$.

For every positive integer $n$, let $\operatorname{mod_5}(n)$ be the remainder obtained when $n$ is divided by $5$. Define a function $f : \{0, 1, 2, 3, \dots\} \times \{0, 1, 2, 3, 4\} \to \{0, 1, 2, 3, 4\}$ recursively as follows: \[f(i, j) = \begin{cases} \operatorname{mod_5}(j+1) & \text{if }i=0\text{ and }0\leq j\leq 4 \\ f(i-1, 1) & \text{if }i\geq 1\text{ and }j=0 \text{, and}\\ f(i-1, f(i, j-1)) & \text{if }i\geq 1\text{ and }1\leq j\leq 4 \end{cases}\] What is $f(2015, 2)$? $\textbf{(A) }0 \qquad\textbf{(B) }1 \qquad\textbf{(C) }2 \qquad\textbf{(D) }3 \qquad\textbf{(E) }4$

Find the largest real number $k$, such that for any positive real numbers $a,b$, $$(a+b)(ab+1)(b+1)\geq kab^2$$