$a_n,b_n,c_n$ are three sequences of positive integers satisfying $$\prod_{d|n}a_d=2^n-1,\prod_{d|n}b_d=\frac{3^n-1}{2},\prod_{d|n}c_d=\gcd(2^n-1,\frac{3^n-1}{2})$$ for all $n\in \mathbb{N}$. Prove that $\gcd(a_n,b_n)|c_n$ for all $n\in \mathbb{N}$

The perpendicular bisectors of the sides $AB$ and $CD$ of the rhombus $ABCD$ are drawn. It turned out that they divided the diagonal $AC$ into three equal parts. Find the altitude of the rhombus if $AB = 1$.

Let $ABC$ be a triangle with $AB=24$, $BC=30$, and $AC=36$. Point $M$ lies on $BC$ such that $BM=12$, and point $N$ lies on $AC$ such that $CN=20$. Let $X$ be the intersection of $AM$ and $BN$ and let line $CX$ intersect $AB$ at point $L$. Compute $$\frac{AX}{XM}+\frac{BX}{XN}+\frac{CX}{XL}.$$

Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A\cap B \neq \emptyset$ and $\left |A \right | \neq \left |B \right |$.

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$, \[xf(y+x)+(y+x)f(y)=f(x^2+y^2)+2f(xy)\] [i]Proposed by Aritra Mondal[/i]

A river flows between two houses $ A$ and $ B$, the houses standing some distances away from the banks. Where should a bridge be built on the river so that a person going from $ A$ to $ B$, using the bridge to cross the river may do so by the shortest path? Assume that the banks of the river are straight and parallel, and the bridge must be perpendicular to the banks.

Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$. [i]Proposed by DVDthe1st[/i]

The positive integer $8833$ has the property that $8833 = 88^2 + 33^2.$ Find the (unique) other four-digit positive integer $\overline{abcd}$ where $\overline{abcd} = (\overline{ab})^2 + (\overline{cd})^2.$ [i]Proposed by Allen Yang[/i]

A fair coin is tossed repeatedly until there is a run of an odd number of heads followed by a tail. Determine the expected number of tosses.

Let $S$ be the graph of $y=x^3$, and $T$ be the graph of $y=\sqrt[3]{y}$. Let $S^*$ be $S$ rotated around the origin $15$ degrees clockwise, and $T^*$ be T rotated around the origin 45 degrees counterclockwise. $S^*$ and $T^*$ will intersect at a point in the first quadrant a distance $M+\sqrt{N}$ from the origin where $M$ and $N$ are positive integers. Find $M+N$.

Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.

Find all pairs $(m,n)$ of positive integers numbers with $m>1$ such that: For any positive integer $b \le m$ that is not coprime with $m$, its posible choose positive integers $a_1, a_2, \cdots, a_n$ all coprimes with $m$ such that: $$m+a_1b+a_2b^2+\cdots+a_nb^n$$ Is a perfect power. Note: A perfect power is a positive integer represented by $a^k$, where $a$ and $k$ are positive integers with $k>1$

Let $ABC$ be a triangle, and its incenter touches the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $AD$ intersects the incircle of $ABC$ at $M$ distinct from $D$. Let $DF$ intersects the circumcircle of $CDM$ at $N$ distinct from $D$. Let $CN$ intersects $AB$ at $G$. Prove that $EC = 3GF$.

An $n$-sided regular polygon with side length $1$ is rotated by $\frac{180^o}{n}$ about its center. The intersection points of the original polygon and the rotated polygon are the vertices of a $2n$-sided regular polygon with side length $\frac{1-tan^2 10^o}{2}$. What is the value of $n$?

A subset $Q \subset H^s(\mathbb{R})$ is said to be equicontinuous if for any $\varepsilon>0$, $\exists \delta>0$ such that \[\|f(x+h)-f(x)\|_{H^s}<\varepsilon, \quad \forall \vert h\vert<\delta, \quad f \in Q.\] Fix $r<s$, given a bounded sequence of functions $f_n \in H^s(\mathbb{R}$. If $f_n$ converges in $H^r(\mathbb{R})$ and equicontinuous in $H^s(\mathbb{R})$, show that it also converges in $H^s(\mathbb{R})$.

Given the positive integer $n$. Let $X = \{1, 2,..., n\}$. For each nonempty subset $A$ of $X$, set $r(A) = max_A - min_A$, where $max_A, min_A$ are the greatest and smallest elements of $A$, respectively. Find the mean value of $r(A)$ when $A$ runs on subsets of $X$.

Let $ q $ be an even positive number. Prove that for every natural number $ n $ number $q^{(q+1)^n}+1$ is divisible by $ (q + 1)^{n+1} $ but not divisible by $ (q + 1)^{n+2} $.

Let $ \alpha\equal{}2\plus{}\sqrt{3}$. Prove that $ \alpha^n\minus{}[\alpha^n]\equal{}1\minus{}\alpha^{\minus{}n}$ for all $ n \in \mathbb{N}_0$.

In the $xyz$-space take four points $P(0,\ 0,\ 2),\ A(0,\ 2,\ 0),\ B(\sqrt{3},-1,\ 0),\ C(-\sqrt{3},-1,\ 0)$. Find the volume of the part satifying $x^2+y^2\geq 1$ in the tetrahedron $PABC$. 50 points

There is a set of $ n$ coins with distinct integer weights $ w_1, w_2, \ldots , w_n$. It is known that if any coin with weight $ w_k$, where $ 1 \leq k \leq n$, is removed from the set, the remaining coins can be split into two groups of the same weight. (The number of coins in the two groups can be different.) Find all $ n$ for which such a set of coins exists.

Let $x,y$ be two positive real numbers such that $x^4-y^4=x-y$. Prove that $$\frac{x-y}{x^6-y^6}\leq \frac{4}{3}(x+y).$$

What is the largest integer $n < 2018$ such that for all integers $b > 1$, $n$ has at least as many $1$'s in its base-$4$ representation as it has in its base-$b$ representation?

Let $T=\text{TNFTPP}$. Consider the sequence $(1, 2007)$. Inserting the difference between $1$ and $2007$ between them, we get the sequence $(1, 2006, 2007)$. Repeating the process of inserting differences between numbers, we get the sequence $(1, 2005, 2006, 1, 2007)$. A third iteration of this process results in $(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)$. A total of $2007$ iterations produces a sequence with $2^{2007}+1$ terms. If the integer $4T$ (that is, $4$ times the integer $T$) appears a total of $N$ times among these $2^{2007}+1$ terms, find the remainder when $N$ gets divided by $2007$.

Three equal mass satellites $A$, $B$, and $C$ are in coplanar orbits around a planet as shown in the figure. The magnitudes of the angular momenta of the satellites as measured about the planet are $L_A$, $L_B$, and $L_C$. Which of the following statements is correct? [asy] // Code created by riben size(250); dotfactor=12; draw(circle((0,0),1.5),linewidth(2)); draw(circle((0,0),6),dashdotted); draw(circle((0,0),14),dashed); draw(ellipse((4,0),10,8),linewidth(1)); pair A,B,C; A=(-7,12.12); B=(5,7.9); C=(5.7,-1.87); dot(A); dot(B); dot(C); label("A",A,NW*1.5); label("B",B,NW*1.5); label("C",C,E*1.5); filldraw((-1.500, 0.078)-- (-1.428, 0.080)-- (-1.337, 0.094)-- (-1.295, 0.157)-- (-1.246, 0.209)-- (-1.186, 0.227)-- (-1.143, 0.290)-- (-1.148, 0.357)-- (-1.135, 0.469)-- (-1.057, 0.505)-- (-0.996, 0.563)-- (-0.936, 0.526)-- (-0.852, 0.557)-- (-0.773, 0.587)-- (-0.772, 0.716)-- (-0.765, 0.828)-- (-0.781, 0.955)-- (-0.732, 1.035)-- (-0.648, 1.083)-- (-0.605, 1.162)-- (-0.604, 1.246)-- (-0.645, 1.295)-- (-0.736, 1.270)-- (-0.796, 1.229)-- (-0.851, 1.193)-- (-0.941, 1.135)-- (-1.014, 1.076)-- (-1.105, 0.995)-- (-1.154, 0.921)-- (-1.227, 0.841)-- (-1.288, 0.760)-- (-1.349, 0.669)-- (-1.398, 0.556)-- (-1.453, 0.465)-- (-1.485, 0.357)-- (-1.510, 0.239)--cycle,gray); filldraw((-0.119, 1.245)-- (-0.130, 1.193)-- (-0.146, 1.095)-- (-0.202, 1.056)-- (-0.327, 1.033)-- (-0.262, 1.031)-- (-0.278, 0.979)-- (-0.193, 0.949)-- (-0.108, 0.943)-- (-0.013, 0.941)-- (0.032, 0.915)-- (0.026, 0.840)-- (0.015, 0.779)-- (0.019, 0.705)-- (0.074, 0.646)-- (0.113, 0.582)-- (0.162, 0.533)-- (0.167, 0.463)-- (0.241, 0.400)-- (0.311, 0.412)-- (0.416, 0.410)-- (0.465, 0.342)-- (0.541, 0.410)-- (0.611, 0.347)-- (0.679, 0.242)-- (0.728, 0.132)-- (0.732, 0.048)-- (0.671, -0.037)-- (0.615, -0.104)-- (0.540, -0.172)-- (0.409, -0.209)-- (0.324, -0.244)-- (0.253, -0.293)-- (0.188, -0.314)-- (0.162, -0.389)-- (0.181, -0.486)-- (0.270, -0.534)-- (0.340, -0.537)-- (0.380, -0.596)-- (0.424, -0.688)-- (0.418, -0.772)-- (0.352, -0.825)-- (0.281, -0.883)-- (0.241, -0.926)-- (0.145, -0.981)-- (0.044, -1.044)-- (-0.006, -1.107)-- (-0.007, -1.190)-- (0.077, -1.216)-- (0.162, -1.213)-- (0.253, -1.163)-- (0.323, -1.128)-- (0.404, -1.075)-- (0.510, -1.015)-- (0.605, -0.980)-- (0.671, -0.931)-- (0.731, -0.920)-- (0.817, -0.852)-- (0.898, -0.798)-- (0.963, -0.777)-- (0.964, -0.708)-- (1.024, -0.645)-- (1.025, -0.571)-- (0.976, -0.488)-- (0.912, -0.425)-- (0.878, -0.347)-- (0.823, -0.289)-- (0.779, -0.225)-- (0.744, -0.193)-- (0.756, -0.100)-- (0.816, -0.033)-- (0.837, 0.047)-- (0.838, 0.122)-- (0.824, 0.200)-- (0.800, 0.307)-- (0.796, 0.381)-- (0.872, 0.416)-- (0.967, 0.414)-- (1.016, 0.360)-- (1.096, 0.381)-- (1.117, 0.428)-- (1.058, 0.506)-- (0.998, 0.564)-- (0.954, 0.591)-- (0.914, 0.617)-- (0.860, 0.676)-- (0.800, 0.716)-- (0.751, 0.775)-- (0.757, 0.859)-- (0.797, 0.921)-- (0.823, 0.987)-- (0.889, 1.096)-- (0.850, 1.160)-- (0.780, 1.176)-- (0.700, 1.183)-- (0.645, 1.125)-- (0.579, 1.039)-- (0.518, 0.986)-- (0.438, 0.956)-- (0.343, 0.967)-- (0.289, 1.049)-- (0.249, 1.117)-- (0.195, 1.176)-- (0.125, 1.192)-- (0.030, 1.208)-- (-0.040, 1.220)--cycle,gray); [/asy] (A) $L_\text{A} > L_\text{B} > L_\text{C}$ (B) $L_\text{C} > L_\text{B} > L_\text{A}$ (C) $L_\text{B} > L_\text{C} > L_\text{A}$ (D) $L_\text{B} > L_\text{A} > L_\text{C}$ (E) The relationship between the magnitudes is different at various instants in time.

A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$