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 $f(P).$

Let $a_0,a_1,a_2,\dots$ be a periodic sequence of real numbers(that is, there is a fixed positive integer $k$ such that $a_n=a_{n+k}$ for every integer $n\geq 0$). The following equality is true, for all $n\geq 0$: $a_{n+2}=\frac{1}{n+2} (a_n - \frac{n+1}{a_{n+1}})$ if $a_0=2020$, determine the value of $a_1$.

There is given a convex quadrilateral $ ABCD$. Prove that there exists a point $ P$ inside the quadrilateral such that \[ \angle PAB \plus{} \angle PDC \equal{} \angle PBC \plus{} \angle PAD \equal{} \angle PCD \plus{} \angle PBA \equal{} \angle PDA \plus{} \angle PCB = 90^{\circ} \] if and only if the diagonals $ AC$ and $ BD$ are perpendicular. [i]Proposed by Dusan Djukic, Serbia[/i]

Given is an acute triangle $ABC$ and the median $AM.$ Draw $BH\perp AC.$ The line which goes through $A$ and is perpendicular to $AM$ intersects $BH$ at $E.$ On the opposite ray of the ray $AE$ choose $F$ such that $AE=AF.$ Prove that $CF\perp AB.$

A father, a mother and son hold a family tournament, playing a two person board game with no ties. The tournament rules are: (i) The weakest player chooses the first two contestants. (ii) The winner of any game plays the next game against the person left out. (iii) The first person to win two games wins the tournament. The father is the weakest player, the son the strongest, and it is assumed that any player's probability of winning an individual game from another player does not change during the tournament. Prove that the father's optimal strategy for winning the tournament is to play the first game with his wife.

In $ \triangle ABC$ , $ AB \equal{} 13$, $ AC \equal{} 5$, and $ BC \equal{} 12$. Points $ M$ and $ N$ lie on $ \overline{AC}$ and $ \overline{BC}$, respectively, with $ CM \equal{} CN \equal{} 4$. Points $ J$ and $ K$ are on $ \overline{AB}$ so that $ \overline{MJ}$ and $ \overline{NK}$ are perpendicular to $ \overline{AB}$. What is the area of pentagon $ CMJKN$? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair C=(0,0), B=(12,0), A=(0,5), M=(0,4), Np=(4,0); pair K=foot(Np,A,B), J=foot(M,A,B); draw(A--B--C--cycle); draw(M--J); draw(Np--K); label("$C$",C,SW); label("$A$",A,NW); label("$B$",B,SE); label("$N$",Np,S); label("$M$",M,W); label("$J$",J,NE); label("$K$",K,NE);[/asy]$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ \frac{81}{5} \qquad \textbf{(C)}\ \frac{205}{12} \qquad \textbf{(D)}\ \frac{240}{13} \qquad \textbf{(E)}\ 20$

Find the smallest positive integer $n,$ such that $3^k+n^k+ (3n)^k+ 2014^k$ is a perfect square for all natural numbers $k,$ but not a perfect cube, for all natural numbers $k.$

Let $p_1, p_2, p_3, \ldots$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between 0 and 1. For positive integer $k$, define \[ x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \\[.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases} \] where $\{x\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 < x_0 < 1$ for which the sequence $x_0, x_1, x_2, \ldots$ eventually becomes 0.

The figure shows a large circle with radius $2$ m and four small circles with radii $1$ m. It is to be painted using the three shown colours. What is the cost of painting the figure? [img]https://1.bp.blogspot.com/-oWnh8uhyTIo/XzP30gZueKI/AAAAAAAAMUY/GlC3puNU_6g6YRf6hPpbQW8IE8IqMP3ugCLcBGAsYHQ/s0/2018%2BMohr%2Bp2.png[/img]

Inside the parallelogram $ABCD$ are the circles $\gamma_1$ and $\gamma_2$, which are externally tangent at the point $K$. The circle $\gamma_1$ touches the sides $AD$ and $AB$ of the parallelogram, and the circle $\gamma_2$ touches the sides $CD$ and $CB$. Prove that the point $K$ lies on the diagonal $AC$ of the paralelogram.

Let $k$ and $n$ be positive integers, with $k \le n$. A certain class has n students, and among any $k$ of them there is always one that is friends with the other $k- 1$. Find all values of $k$ and $n$ for which there must necessarily be a student who is friends with everyone else in the class.

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? $\textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 40$

The natural number $n>1$ is called “heavy”, if it is coprime with the sum of its divisors. What’s the maximal number of consecutive “heavy” numbers?

Five people are sitting at a round table. Let $f \ge 0$ be the number of people sitting next to at least one female and $m \ge 0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11 $

The representation of real number $a$ as a decimal infinite fraction contain all $10$ digits. For a positive integer $n$ let $v_n$ be the number of all segments of length $n$ that occur. Prove that, if $v_n \leq n + 8$ for some positive integer $n$, then the number $a$ is rational.

Let $x, y, z$ be positive real numbers. Prove that $$4(x^2+y^2+z^2)\geq3(xy+yz+zx).$$

Let $T$ be a finite set of positive integers greater than 1. A subset $S$ of $T$ is called [i]good[/i] if for every $t \in T$ there exists some $s \in S$ with $\gcd(s, t) > 1$. Prove that the number of good subsets of $T$ is odd.

Let $a_1,a_2,\cdots ,a_n$ be $n$ positive numbers such that $\sum \limits_{i=1}^n\sqrt{a_i}=\sqrt{n}$. Then$$\prod \limits_{i=1}^{n-1}\left(1+\frac{1}{a_i}\right)^{a_{i+1}}\left(1+\frac{1}{a_n}\right)^{a_1}\geq 1+\frac{n}{\sum \limits_{i=1}^na_i}$$

Given is a set $A$ of $n$ elements and positive integers $k, m$ such that $4 \leq k <n$ and $m \leq \min \{k-3, \frac {n} {2}\}$. Let $A_1, A_2, \ldots, A_l$ be subsets of $A$, all with size $k$, such that $|A_i \cap A_j| \leq m$ for all $i \neq j$. Prove that there exists a subset $B$ of $A$ with at least $\sqrt[m+1]{n}+m$ elements which doesn't contain entirely any of the subsets $A_1, A_2, \ldots, A_l$.

Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}...p_k^{e_k}$ is the prime factorization of $n>1$, then \[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.\] For every $m \ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$ in the range $1 \le N \le 400$ is the sequence $(f_1(N), f_2(N), f_3(N),...)$ unbounded? [b]Note:[/b] a sequence of positive numbers is unbounded if for every integer $B$, there is a member of the sequence greater than $B$. $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 19 $