Let $S(N)$ be the number of 1's in the binary representation of an integer $N$, and let $D(N) = S(N + 1) - S(N)$. Compute the sum of $D(N)$ over all $N$ such that $1 \le N \le 2017$ and $D(N) < 0$.

$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$

Can we say that two triangles are congruent if the radii of the inscribed circles, the radii of the circumscribed circles, and the areas of these triangles are equal?

Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.

Let $ n$ be a composite. Prove that there exists positive integer $ m$ satisfying $ m|n, m\le\sqrt {n},$ and $ d(n)\le d^3(m).$ Where $ d(k)$ denotes the number of positive divisors of positive integer $ k.$

For an odd positive integer $n>1$ define $S_n$ to be the set of the residues of the powers of two, modulo $n{}$. Do there exist distinct $n{}$ and $m{}$ whose corresponding sets $S_n$ and $S_m$ coincide? [i]Proposed by D. Kuznetsov[/i]

Suppose that $ s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences \[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\] are both arithmetic progressions. Prove that the sequence $ s_1, s_2, s_3, \ldots$ is itself an arithmetic progression. [i]Proposed by Gabriel Carroll, USA[/i]

The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$ in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.

Consider the statements: I: $ (\sqrt { \minus{} 4})(\sqrt { \minus{} 16}) \equal{} \sqrt {( \minus{} 4)( \minus{} 16)}$, II: $ \sqrt {( \minus{} 4)( \minus{} 16)} \equal{} \sqrt {64}$, and $ \sqrt {64} \equal{} 8$. Of these the following are [u]incorrect[/u]. $ \textbf{(A)}\ \text{none} \qquad \textbf{(B)}\ \text{I only} \qquad \textbf{(C)}\ \text{II only} \qquad \textbf{(D)}\ \text{III only} \qquad \textbf{(E)}\ \text{I and III only}$

Let ${\omega}$ be the circumcircle of an acute-angled triangle ${ABC}$. Point ${D}$ lies on the arc ${BC}$ of ${\omega}$ not containing point ${A}$. Point ${E}$ lies in the interior of the triangle ${ABC}$, does not lie on the line ${AD}$, and satisfies ${\angle DBE =\angle ACB}$ and ${\angle DCE = \angle ABC}$. Let ${F}$ be a point on the line ${AD}$ such that lines ${EF}$ and ${BC}$ are parallel, and let ${G}$ be a point on ${\omega}$ different from ${A}$ such that ${AF = FG}$. Prove that points ${D,E, F,G}$ lie on one circle. (Slovakia)

Determine all polynomials $p(x)$ with real coefficients such that $p((x+1)^3)=(p(x)+1)^3$ and $p(0)=0$.

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$, \[xf(x+f(y))=(y-x)f(f(x)).\] [i]Proposed by Nikola Velov, Macedonia[/i]

Let the [i]subbishop[/i] (a bishop is the figure moving only by a diagonal) be a figure moving only by diagonal but only in the next cells (squares) of the chessboard. Find the maximal count of subbishops over a chessboard $n\times n$, no two of which are not attacking. [i]V. Chukanov[/i]

Problems 17, 18, and 19 refer to the following: At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 1.5 cups flour, 2 eggs, 3 tablespoons butter, 3/4 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes. They learn that a big concert is scheduled for the same night and attendance will be down $25\%$. How many recipes of cookies should they make for their smaller party? $ \text{(A)}\ 6\qquad\text{(B)}\ 8\qquad\text{(C)}\ 9\qquad\text{(D)}\ 10\qquad\text{(E)}\ 11 $

Set $X$ has $56$ elements. Determine the least positive integer $n$ such that for any 15 subsets of $X$, if the union of any $7$ of the subsets has at least $n$ elements, then 3 of the subsets have nonempty intersection.

If $(a,~b,~c)$ is a triple of real numbers, define [list] [*] $g(a,~b,~c)=(a+b,~b+c,~a+c)$, and [*] $g^n(a,~b,~c)=g(g^{n-1}(a,~b,~c))$ for $n\ge 2$[/list] Suppose that there exists a positive integer $n$ so that $g^n(a,~b,~c)=(a,~b,~c)$ for some $(a,~b,~c)\neq (0,~0,~0)$. Prove that $g^6(a,~b,~c)=(a,~b,~c)$

Let $A_1, A_2, A_3$ be nonempty sets of integers such that for $\{i, j, k\} = \{1, 2, 3\}$ holds $$(x \in A_i, y\in A_j) \Rightarrow (x + y \in A_k, x - y \in A_k).$$ Prove that at least two of the sets $A_1, A_2, A_3$ are equal. Can any of these sets be disjoint?

Show that every rational number $r$ can be written as the sum of numbers in the form $\frac{a}{p^k}$ where $p$ is prime, $a$ is an integer and $k$ is natural.

The sequence $a_0,a_1,a_2,...$ is defined such that $a_0 = 2+ \sqrt3$, $a_1 =\sqrt{5-2\sqrt5}$, and $$a_n a_{n-1}a_{n-2} - a_n + a_{n-1} + a_{n-2} = 0.$$ Find the least positive integer $n$ such that $a_n = 1$.

Consider a triangle $ ABC $. On the line $ AC $ take a point $ B_1 $ such that $ AB = AB_1 $ and in addition, $ B_1 $ and $ C $ are located on the same side of the line with respect to the point $ A $. The bisector of the angle $ A $ intersects the side $ BC $ at a point that we will denote as $ A_1 $. Let $ P $ and $ R $ be the circumscribed circles of the triangles $ ABC $ and $ A_1B_1C $ respectively. They intersect at points $ C $ and $ Q $. Prove that the tangent to the circle $ R $ at the point $ Q $ is parallel to the line $ AC $.

Prove that if $ a $, $ b $, $ c $, $ d $ are the sides of a quadrilateral in which a circle can be circumscribed and a circle can be inscribed in it, then the area $ S $ of the quadrilateral is given by $$S = \sqrt{abcd}.$$

A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle? [asy] unitsize(0.25cm); pair A, B, C, O; A = (-8, 0); B = (8, 0); C = (0, 15); O = (0, 0); draw(arc(O, 120/17, 0, 180)); draw(A--B--C--cycle); [/asy] $\textbf{(A) }4 \sqrt{3}\qquad\textbf{(B) } \dfrac{120}{17}\qquad\textbf{(C) }10\qquad\textbf{(D) }\dfrac{17\sqrt{2}}{2}\qquad \textbf{(E) }\dfrac{17\sqrt{3}}{2}$

Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition \[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 \] for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$ [i]Author: Nikolai Nikolov, Bulgaria[/i]

Define $H_n = 1+\frac{1}{2}+\cdots+\frac{1}{n}$. Let the sum of all $H_n$ that are terminating in base 10 be $S$. If $S = m/n$ where m and n are relatively prime positive integers, find $100m+n$. [i]Proposed by Lewis Chen[/i]

The orthocenter of triangle $ABC$ lies on its circumcircle. One of the angles of $ABC$ must equal: (The orthocenter of a triangle is the point where all three altitudes intersect.) $$ \mathrm a. ~ 30^\circ\qquad \mathrm b.~60^\circ\qquad \mathrm c. ~90^\circ \qquad \mathrm d. ~120^\circ \qquad \mathrm e. ~\text{It cannot be deduced from the given information.} $$