Does the equation $$z(y-x)(x+y)=x^3$$ have finitely many solutions in the set of positive integers? [i]Proposed by Nikola Velov[/i]

Let $a$ and $b$ be positive integers such that $10< \gcd(a,b) < 20$ and $220 < \text{lcm}(a,b) < 230$. Find the difference between the smallest and largest possible values of $ab$.

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that its derivative $f'$ is a continuous function. Moreover, assume that for all $x\in\mathbb{R}$, $$0\leqslant \vert f'(x)\vert\leqslant \frac{1}{2}$$ Define a sequence of real numbers $\{a_n\}_{n\in\mathbb{N}}$ by :$$a_1=1~~\text{and}~~a_{n+1}=f(a_n)~\text{for all}~n\in\mathbb{N}$$ Prove that there exists a positive real number $M$ such that for all $n\in\mathbb{N}$, $$\vert a_n\vert \leqslant M$$

The three sides of a right triangle have integral lengths which form an arithmetic progression. One of the sides could have length $\text{(A)}\ 22 \qquad \text{(B)}\ 58 \qquad \text{(C)}\ 81 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 361$

Determine all real solutions $(x,y)$ of the following system of equations: \begin{align*} x&=3x^2y-y^3,\\ y &= x^3-3xy^2 \end{align*}

How many 4-digit numbers $ABCD$ are there with the property that $|A-B|= |B-C|= |C-D|$? Note that the first digit $A$ of a four-digit number cannot be zero.

Let $P_1P_2\ldots P_n$ be a convex polygon in the plane. We assume that for any arbitrary choice of vertices $P_i,P_j$ there exists a vertex in the polygon $P_k$ distinct from $P_i,P_j$ such that $\angle P_iP_kP_j=60^{\circ}$. Show that $n=3$. [i]Radu Todor[/i]

The extension of the bisector of angle $A$ of triangle $ABC$ intersects with the circumscribed circle of this triangle at point $W$. A straight line is drawn through $W$, which is parallel to side $AB$ and intersects sides $BC$ and $AC$ , at points $N$ and $K$, respectively. Prove that the line $AW$ is tangent to the circumscribed circle of $\vartriangle CNW$. (Sergey Yakovlev)

A particle moves on a circle with center $O$, starting from rest at a point $P$ and coming to rest again at a point $Q$, without coming to rest at any intermediate point. Prove that the acceleration vector of the particle does not vanish at any point between $P$ and $ Q$ and that, at some point $R$ between $P$ and $Q$, the acceleration vector points in along the radius $RO.$

Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.

For geometric series $(a_n)$ with all items real, if $S_{10}=10,S_{30}=70$, then $S_{40}=$ $\text{(A)}150\qquad\text{(B)}-200\qquad\text{(C)}150\text{ or }-200\qquad\text{(D)}-50\text{ or }400$ Note: $S_n=\sum_{i=1}^{n}a_i$.

Let $n$ be a positive integer. Solve the equation in $\mathbb{R}$ $$\sqrt[2n+1]{x}+\sqrt[2n+1]{x+1}+\sqrt[2n+1]{x+2}+\dots+\sqrt[2n+1]{x+n}=0.$$

Let $p$ be a real number and $f(x)=x^p-x+p$. Prove that: (a) Every root $\alpha$ of $f(x)$ satisfies $|\alpha|<p^{\frac1{p-1}}$; (b) If $p$ is a prime number, then $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.

If $ a,b,c $ are all in the interval $ (0,1) $ or all in the interval $ \left( 1,\infty \right), $ then $$ 1\le\sum_{\text{cyc}} \frac{\log_a^7 b\cdot \log_b^3c}{\log_c a +2\log_a b} . $$ [i]Gheorghe Duță[/i]

Find the coefficient of $x$ in the expansion of $(1 + x)(1 - 2x)(1 + 3x)(1 - 4x) ...(1 - 2008x)$.

The Planar National Park is a subset of the Euclidean plane consisting of several trails which meet at junctions. Every trail has its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints). Finally, no trails begin and end at the same two junctions. (An example of one possible layout of the park is shown to the left below, in which there are six junctions and nine trails.) [center] [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZS9mLzc1YmNjN2YxMWZhZTNhMTVkZTQ4NWE1ZDIyMDNhN2I5NzY0NTBlLnBuZw==&rn=Z3JhcGguUE5H[/img] [/center] A visitor walks through the park as follows: she begins at a junction and starts walking along a trail. At the end of that first trail, she enters a junction and turns left. On the next junction she turns right, and so on, alternating left and right turns at each junction. She does this until she gets back to the junction where she started. What is the largest possible number of times she could have entered any junction during her walk, over all possible layouts of the park?

Let $f(x)=\frac{\sqrt{2+\sqrt2}x+\sqrt{2-\sqrt2}}{-\sqrt{2-\sqrt2}x+\sqrt{2+\sqrt2}}$. Find $\underbrace{f(f(\cdots f}_{1987\text{ times}}(x)\cdots))$.

Let $f$ be a function on a set $X$. Prove that $$f(X-f(X))=f(X)-f(f(X)),$$ where for a set $S$, the notation $f(S)$ means $\{f(a) | a \in S\}$.

In triangle $ABC$, $\measuredangle C=\theta$ and $\measuredangle B=2\theta$, where $0^{\circ} <\theta < 60^{\circ}$. The circle with center $A$ and radius $AB$ intersects $AC$ at $D$ and intersects $BC$, extended if necessary, at $B$ and at $E$ ($E$ may coincide with $B$). Then $EC=AD$ $ \textbf{(A)}\ \text{for no values of}\ \theta \qquad\textbf{(B)}\ \text{only if}\ \theta=45^{\circ} \qquad\textbf{(C)}\ \text{only if}\ 0^{\circ} < \theta \le 45^{\circ} \\ \qquad\textbf{(D)}\ \text{only if}\ 45^{\circ} \le \theta < 60^{\circ} \qquad\textbf{(E)}\ \text{for all}\ \theta \ \text{such that}\ 0^{\circ} <\theta < 60^{\circ} $

Figure 1 is called a "stack map." The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes, and Fig. 3 shows the view of the stacked cubes as seen from the front. Which of the following is the front view for the stack map in Fig. 4? [asy] unitsize(24); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((0,1)--(2,1)); draw((5,0)--(7,0)--(7,1)--(20/3,4/3)--(20/3,13/3)--(19/3,14/3)--(16/3,14/3)--(16/3,11/3)--(13/3,11/3)--(13/3,2/3)--cycle); draw((20/3,13/3)--(17/3,13/3)--(17/3,10/3)--(14/3,10/3)--(14/3,1/3)); draw((20/3,10/3)--(17/3,10/3)--(17/3,7/3)--(20/3,7/3)); draw((17/3,7/3)--(14/3,7/3)); draw((7,1)--(6,1)--(6,2)--(5,2)--(5,0)); draw((5,1)--(6,1)--(6,0)); draw((20/3,4/3)--(6,4/3)); draw((17/3,13/3)--(16/3,14/3)); draw((17/3,10/3)--(16/3,11/3)); draw((14/3,10/3)--(13/3,11/3)); draw((5,2)--(13/3,8/3)); draw((5,1)--(13/3,5/3)); draw((6,2)--(17/3,7/3)); draw((9,0)--(11,0)--(11,4)--(10,4)--(10,3)--(9,3)--cycle); draw((11,3)--(10,3)--(10,0)); draw((11,2)--(9,2)); draw((11,1)--(9,1)); draw((13,0)--(16,0)--(16,2)--(13,2)--cycle); draw((13,1)--(16,1)); draw((14,0)--(14,2)); draw((15,0)--(15,2)); label("Figure 1",(1,0),S); label("Figure 2",(17/3,0),S); label("Figure 3",(10,0),S); label("Figure 4",(14.5,0),S); label("$1$",(1.5,.2),N); label("$2$",(.5,.2),N); label("$3$",(.5,1.2),N); label("$4$",(1.5,1.2),N); label("$1$",(13.5,.2),N); label("$3$",(14.5,.2),N); label("$1$",(15.5,.2),N); label("$2$",(13.5,1.2),N); label("$2$",(14.5,1.2),N); label("$4$",(15.5,1.2),N);[/asy] [asy] unitsize(18); draw((0,0)--(3,0)--(3,2)--(1,2)--(1,4)--(0,4)--cycle); draw((0,3)--(1,3)); draw((0,2)--(1,2)--(1,0)); draw((0,1)--(3,1)); draw((2,0)--(2,2)); draw((5,0)--(8,0)--(8,4)--(7,4)--(7,3)--(6,3)--(6,2)--(5,2)--cycle); draw((8,3)--(7,3)--(7,0)); draw((8,2)--(6,2)--(6,0)); draw((8,1)--(5,1)); draw((10,0)--(12,0)--(12,4)--(11,4)--(11,3)--(10,3)--cycle); draw((12,3)--(11,3)--(11,0)); draw((12,2)--(10,2)); draw((12,1)--(10,1)); draw((14,0)--(17,0)--(17,4)--(16,4)--(16,2)--(14,2)--cycle); draw((17,3)--(16,3)); draw((17,2)--(16,2)--(16,0)); draw((17,1)--(14,1)); draw((15,0)--(15,2)); draw((19,0)--(22,0)--(22,4)--(20,4)--(20,1)--(19,1)--cycle); draw((22,3)--(20,3)); draw((22,2)--(20,2)); draw((22,1)--(20,1)--(20,0)); draw((21,0)--(21,4)); label("(A)",(1.5,0),S); label("(B)",(6.5,0),S); label("(C)",(11,0),S); label("(D)",(15.5,0),S); label("(E)",(20.5,0),S);[/asy]

Let $n \ge 3$ be an integer. A [i]circle dance[/i] is a dance that is performed according to the following rule: On the floor, $n$ points are marked at equal distances along a large circle. At each of these points is a sheet of paper with an arrow pointing either clockwise or counterclockwise. One of the points is labeled "Start". The dancer starts at this point. In each step, he first changes the direction of the arrow at his current position and then moves to the next point in the new direction of the arrow. a) Show that each circle dance visits each point infinitely often. b) How many different circle dances are there? Two circle dances are considered to be the same if they differ only by a finite number of steps at the beginning and then always visit the same points in the same order. (The common sequence of steps may begin at different times in the two dances.) [i](Birgit Vera Schmidt)[/i]

Let $a < b < c < d < e$ be positive integers. Prove that $$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$ where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?

Let $p$ be an odd prime and let $$F(n)=1+2n+3n^2+\ldots+(p-1)n^{p-2}.$$Prove that if $a$ and $b$ are distinct integers in $\{0,1,2,\ldots,p-1\}$ then $F(a)$ and $F(b)$ are not congruent modulo $p$.

The lines $ x \equal{} \frac 14y \plus{} a$ and $ y \equal{} \frac 14x \plus{} b$ intersect at the point $ (1,2)$. What is $ a \plus{} b$? $ \textbf{(A) } 0 \qquad \textbf{(B) } \frac 34 \qquad \textbf{(C) } 1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } \frac 94$

Define a sequence $(a_n)$ by $a_0 =0$ and $a_n = 1 +\sin(a_{n-1}-1)$ for $n\geq 1$. Evaluate $$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k.$$