For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of several distinct positive integers not exceeding $2n$?

Define \[ N=\sum\limits_{k=1}^{60}e_k k^{k^k} \] where $e_k \in \{-1, 1\}$ for each $k$. Prove that $N$ cannot be the fifth power of an integer.

Inés chose four different digits from the set $\{1,2,3,4,5,6,7,8,9\}$. He formed with them all possible four-digit numbers and added all those four-digit numbers. The result is $193314$. Find the four digits Inés chose.

Suppose $f(x) = x^3 + \log_2(x + \sqrt{x^2+1})$. For any $a,b \in \mathbb{R}$, to satisfy $f(a) + f(b) \ge 0$, the condition $a + b \ge 0$ is $ \textbf{(A)}\ \text{necessary and sufficient}\qquad\textbf{(B)}\ \text{not necessary but sufficient}\qquad\textbf{(C)}\ \text{necessary but not sufficient}\qquad$ $\textbf{(D)}\ \text{neither necessary nor sufficient}\qquad$

Let $n\geqslant 2$ be a fixed integer. Consider $n$ real numbers $a_1,a_2,\ldots,a_n$ not all equal and let\[d:=\max_{1\leqslant i<j\leqslant n}|a_i-a_j|;\qquad s=\sum_{1\leqslant i<j\leqslant n}|a_i-a_j|.\]Determine in terms of $n{}$ the smalest and largest values the quotient $s/d$ may achieve. [i]Selected from the Kvant Magazine[/i]

Let $\mathbb{Z}_{\geq 0}$ be the set of nonnegative integers. Let $f: \mathbb{Z}_{\geq0} \to \mathbb{Z}_{\geq0}$ be a function such that, for all $a,b \in \mathbb{Z}_{\geq0}$: \[f(a)^2+f(b)^2+f(a+b)^2=1+2f(a)f(b)f(a+b).\] Furthermore, suppose there exists $n \in \mathbb{Z}_{\geq0}$ such that $f(n)=577$. Let $S$ be the sum of all possible values of $f(2017)$. Find the remainder when $S$ is divided by $2017$. [i]Proposed by Zack Chroman[/i]

Ada and Emily are playing a game that ends when either player wins, after some number of rounds. Each round, either nobody wins, Ada wins, or Emily wins. The probability that neither player wins each round is $\frac{1}{5}$ and the probability that Emily wins the game as a whole is $\frac{3}{4}.$ If the probability that in a given round Emily wins is $\frac{m}{n}$ such that $m$ and $n$ are relatively prime integers, then find $m+n.$ [i]Proposed by Ada Tsui[/i]

Prove that for every $n \in N$ there exists exactly one sequence of $2n + 1$ consecutive numbers, such that the sum of the squares of the first $n+1$ numbers is equal to the sum of the squares of the last $n$ numbers. Also express the smallest number of that sequence in terms of $n$.

Find the number of ordered palindromic partitions of an integer $n$.

Let $S=\{1,2,3,\ldots,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime.

Let $ \overline{AB}$ be a diameter of a circle and $ C$ be a point on $ \overline{AB}$ with $ 2 \cdot AC \equal{} BC$. Let $ D$ and $ E$ be points on the circle such that $ \overline{DC} \perp \overline{AB}$ and $ \overline{DE}$ is a second diameter. What is the ratio of the area of $ \triangle DCE$ to the area of $ \triangle ABD$? [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,0), C=(-1/3.0), B=(1,0), A=(-1,0); pair D=dir(aCos(C.x)), E=(-D.x,-D.y); draw(A--B--D--cycle); draw(D--E--C); draw(unitcircle,white); drawline(D,C); dot(O); clip(unitcircle); draw(unitcircle); label("$E$",E,SSE); label("$B$",B,E); label("$A$",A,W); label("$D$",D,NNW); label("$C$",C,SW); draw(rightanglemark(D,C,B,2));[/asy]$ \textbf{(A)} \ \frac {1}{6} \qquad \textbf{(B)} \ \frac {1}{4} \qquad \textbf{(C)}\ \frac {1}{3} \qquad \textbf{(D)}\ \frac {1}{2} \qquad \textbf{(E)}\ \frac {2}{3}$

Given: (i) $a$, $b > 0$; (ii) $a$, $A_1$, $A_2$, $b$ is an arithmetic progression; (iii) $a$, $G_1$, $G_2$, $b$ is a geometric progression. Show that \[A_1 A_2 \ge G_1 G_2.\]

Following the previous set up, find the speed $v$ of the small block after it leaves the slope. (A) $v=v_\text{0}$ (B) $v=\frac{m}{m+M}v_\text{0}$ (C) $v=\frac{M}{m+M}v_\text{0}$ (D) $v=\frac{M-m}{m}v_\text{0}$ (E) $v=\frac{M-m}{m+M}v_\text{0}$

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

Given $10$ points in the space such that each $4$ points are not lie on a plane. Connect some points with some segments such that there are no triangles or quadrangles. Find the maximum number of the segments.

A very large number of small particles forms a spherical cloud. Initially they are at rest, have uniform mass density per unit volume $\rho_0$, and occupy a region of radius $r_0$. The cloud collapses due to gravitation; the particles do not interact with each other in any other way. How much time passes until the cloud collapses fully? (The constant $0.5427$ is actually $\sqrt{\frac{3 \pi}{32}}$.) $\textbf{(A) } \frac{0.5427}{r_0^2 \sqrt{G \rho_0}}\\ \\ \textbf{(B) } \frac{0.5427}{r_0 \sqrt{G \rho_0}}\\ \\ \textbf{(C) } \frac{0.5427}{\sqrt{r_0} \sqrt{G \rho_0}}\\ \\ \textbf{(D) } \frac{0.5427}{\sqrt{G \rho_0}}\\ \\ \textbf{(E) } \frac{0.5427}{\sqrt{G \rho_0}}r_0$

Let $n$ be the number of polynomial functions from the integers modulo $2010$ to the integers modulo $2010$. $n$ can be written as $n = p_1 p_2 \cdots p_k$, where the $p_i$s are (not necessarily distinct) primes. Find $p_1 + p_2 + \cdots + p_n$.

Prove that if positive integers $x,y,z$ satisfy the equation $$x^n + y^n = z^n,$$ then $\min\, (x,y) \ge n$.

Toss the coin $n$ times, assume that each time, only appear only head or tail Let $a(n)$ denote number of way that head appear in multiple of $3$ times among $n$ times Let $b(n)$ denote numbe of way that head appear in multiple of $6$ times among $n$ times $(1)$ Find $a(2016)$ and $b(2016)$ $(2)$ Find the number of positive integer $n\leq 2016$ that $2b(n)-a(n)\geq 0$

2. Let P be a point in the interior of triangle ABC . Extend AP, BP, and CP to meet BC, AC, and AB at D, E, and F, respectively. If triangle APF, triangle BPD and triangle CPE have equal areas, prove that P is the centroid of triangle ABC .

Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i>0$ for all $i$) and set $b_n=\frac1{na_n^2}$ for $n\ge1$. Prove that $\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}$ is convergent.

Let $ABC$ be a triangle and let $I$ be the incenter. Let $N$, $M$ be the midpoints of the sides $AB$ and $CA$ respectively. The lines $BI$ and $CI$ meet $MN$ at $K$ and $L$ respectively. Prove that $AI+BI+CI>BC+KL$. [i]Greece[/i]

Suppose $ABCD$ is a convex quadrilateral.Points $P,Q,R$ and $S$ are four points on the line segments $AB,BC,CD$ and $DA$ respectively.The line segments $PR$ and $QS$ meet at $T$.Suppose that each of the quadrilaterals $APTS,BQTP,CRTQ$ and $DSTR$ have an incircle.Prove that the quadrilateral $ABCD$ also has an incircle.

In triangle $ABC$, $AD$ is angle bisector ($D$ is on $BC$) if $AB+AD=CD$ and $AC+AD=BC$, what are the angles of $ABC$?

In trapezoid $ABCD,$ leg $\overline{BC}$ is perpendicular to bases $\overline{AB}$ and $\overline{CD},$ and diagonals $\overline{AC}$ and $\overline{BD}$ are perpendicular. Given that $AB=\sqrt{11}$ and $AD=\sqrt{1001},$ find $BC^2.$