Assume that $(a_n)_{n \ge 1}$ is an increasing sequence of positive real numbers such that $\lim \tfrac{a_n}{n}=0$. Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\cdots,n-1$?

Mario has a deck of seven pairs of matching number cards and two pairs of matching Jokers, for a total of $18$ cards. He shuffles the deck, then draws the cards from the top one by one until he holds a pair of matching Jokers. The expected number of complete pairs that Mario holds at the end (including the Jokers) is $\frac{m}{n}$ , where $m$, $n$ are positive integers and gcd $(m,n) = 1$. Find $100m + n$.

A prime $p$ has decimal digits $p_{n}p_{n-1} \cdots p_0$ with $p_{n}>1$. Show that the polynomial $p_{n}x^{n} + p_{n-1}x^{n-1}+\cdots+ p_{1}x + p_0$ cannot be represented as a product of two nonconstant polynomials with integer coefficients

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?

Find $3$ numbers, each of which is equal to the square of the difference of the other two.

Consider all cubic polynomials $f(x)$ such that $f(2018)=2018$, the graph of $f$ intersects the $y$-axis at height $2018$, the coefficients of $f$ sum to $2018$, and $f(2019)>(2018)$. We define the infinimum of a set $S$ as follows. Let $L$ be the set of lower bounds of $S$. That is, $\ell\in L$ if and only if for all $s\in S$, $\ell\leq s$. Then the infinimum of $S$ is $\max(L)$. Of all such $f(x)$, what is the infinimum of the leading coefficient (the coefficient of the $x^3$ term)?

After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds? [asy] unitsize(1.25cm); dotfactor = 10; pen shortdashed=linetype(new real[] {2.7,2.7}); for (int i = 0; i < 6; ++i) { for (int j = 0; j < 6; ++j) { draw((i,0)--(i,6), grey); draw((0,j)--(6,j), grey); } } for (int i = 1; i <= 6; ++i) { draw((-0.1,i)--(0.1,i),linewidth(1.25)); draw((i,-0.1)--(i,0.1),linewidth(1.25)); label(string(5*i), (i,0), 2*S); label(string(i), (0, i), 2*W); } draw((0,0)--(0,6)--(6,6)--(6,0)--(0,0)--cycle,linewidth(1.25)); label(rotate(90) * "Distance (miles)", (-0.5,3), W); label("Time (minutes)", (3,-0.5), S); dot("Naomi", (2,6), 3*dir(305)); dot((6,6)); label("Maya", (4.45,3.5)); draw((0,0)--(1.15,1.3)--(1.55,1.3)--(3.15,3.2)--(3.65,3.2)--(5.2,5.2)--(5.4,5.2)--(6,6),linewidth(1.35)); draw((0,0)--(0.4,0.1)--(1.15,3.7)--(1.6,3.7)--(2,6),linewidth(1.35)+shortdashed); [/asy] $\textbf{(A) }6 \qquad \textbf{(B) }12 \qquad \textbf{(C) }18 \qquad \textbf{(D) }20 \qquad \textbf{(E) }24$

Equilateral triangle $ABC$ has side length $20$. Let $PQRS$ be a square such that $A$ is the midpoint of $\overline{RS}$ and $Q$ is the midpoint of $\overline{BC}$. Compute the area of $PQRS$.

On the side $AC$ of the triangle $ABC$, choose any point $D$ different from the vertices $A$ and C. Let $O_1$ and $O_2$ be circumcenters the triangles $ABD$ and $CBD$, respectively. Prove that the triangles $O_1DO_2$ and $ABC$ are similar.

Does there exist a nonzero algebraic number $\alpha$ with $|\alpha| \neq 1$ such that there exists infinitely many positive integers $n$ for which there's $\beta_n \in \mathbb{C}$ with $\beta_n \in \mathbb{Q}(\alpha)$ and $\beta_n^n = \alpha$?

Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]

Given a regular $2013$-sided polygon, how many isosceles triangles are there whose vertices are vertices vertex of given polygon and haave an angle greater than $120^o$? Nguyen Tien Lam, High School for Natural Science,Hanoi National University.

Let be a natural number $ n\ge 3 $ with the property that $ 1+3n $ is a perfect square. Show that there are three natural numbers $ a,b,c, $ such that the number $$ 1+\frac{3n+3}{a^2+b^2+c^2} $$ is a perfect square.

Let $r_1=2$ and $r_n = \prod^{n-1}_{k=1} r_i + 1$, $n \geq 2.$ Prove that among all sets of positive integers such that $\sum^{n}_{k=1} \frac{1}{a_i} < 1,$ the partial sequences $r_1,r_2, ... , r_n$ are the one that gets nearer to 1.

Let $a,\,b,$ and $c$ be real numbers such that $a+b+c=\frac1{a}+\frac1{b}+\frac1{c}$ and $abc=5$. The value of $$\left(a-\frac1{b}\right)^3+\left(b-\frac1{c}\right)^3+\left(c-\frac1{a}\right)^3$$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.

If $ a > 1$, then the sum of the real solutions of \[\sqrt{a \minus{} \sqrt{a \plus{} x}} \equal{} x\] is equal to $ \textbf{(A)}\ \sqrt{a} \minus{} 1\qquad \textbf{(B)}\ \frac{\sqrt{a} \minus{} 1}{2}\qquad \textbf{(C)}\ \sqrt{a \minus{} 1}\qquad \textbf{(D)}\ \frac{\sqrt{a \minus{} 1}}{2}\qquad \textbf{(E)}\ \frac{\sqrt{4a \minus{} 3} \minus{} 1}{2}$

Let $(F_n)_{n\geq 1} $ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying \[ P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.\] Prove that $P(1983) = F_{1983} - 1.$

Let $ABC$ be a triangle and $M$ be the midpoint of $BC$, and let $AM$ meet the circumcircle of $ABC$ again at $R$. A line passing through $R$ and parallel to $BC$ meet the circumcircle of $ABC$ again at $S$. Let $U$ be the foot from $R$ to $BC$, and $T$ be the reflection of $U$ in $R$. $D$ lies in $BC$ such that $AD$ is an altitude. $N$ is the midpoint of $AD$. Finally let $AS$ and $MN$ meets at $K$. Prove that $AT$ bisector $MK$.

Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that the numbers \[n, f(n),f(f(n)),\dots,f^{m-1}(n)\] are distinct modulo $m$ for all integers $n,m$ with $m>1$. (Here $f^k$ is defined by $f^0(n)=n$ and $f^{k+1}(n)=f(f^{k}(n))$ for $k \ge 0$.)

Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that \[2^{j-i}x_ix_j>2^{s-3}.\]

A game in a casino uses the diagram shown. At the start a ball appears at $S$. Each time the player presses a button, the ball moves to one of the adjacent letters with equal probability. The game ends when one of the following two things happens: (i) The ball returns to $S$, the player loses. (ii) The ball reaches $G$, the player wins. Find the probability that the player wins and the expected duration of a game.

In a circle with centre at $O$ and diameter $AB$, two chords $BD$ and $AC$ intersect at $E$. $F$ is a point on $AB$ such that $EF \perp AB$. $FC$ intersects $BD$ in $G$. If $DE = 5$ and $EG =3$, determine $BG$.

Let $n \geq 3$ be an integer, and consider a set of $n$ points in three-dimensional space such that: [list=i] [*] every two distinct points are connected by a string which is either red, green, blue, or yellow; [*] for every three distinct points, if the three strings between them are not all of the same colour, then they are of three different colours; [*] not all the strings have the same colour. [/list] Find the maximum possible value of $n$.

[b]p1.[/b] In the picture below, the two parallel cuts divide the square into three pieces of equal area. The distance between the two parallel cuts is $d$. The square has length $s$. Find and prove a formula that expresses $s$ as a function of $d$. [img]https://cdn.artofproblemsolving.com/attachments/c/b/666074d28de50cdbf338a2c667f88feba6b20c.png[/img] [b]p2.[/b] Let $S$ be a subset of $\{1, 2, 3, . . . 10, 11\}$. We say that $S$ is lucky if no two elements of $S$ differ by $4$ or $7$. (a) Give an example of a lucky set with five elements. (b) Is it possible to find a lucky set with six elements? Explain why or why not.[/quote] [b]p3.[/b] Find polynomials $p(x)$ and $q(x)$ with real coefficients such that (a) $p(x) - q(x) = x^3 + x^2 - x - 1$ for all real $x$, (b) $p(x) > 0$ for all real $x$, (c) $q(x) > 0$ for all real $x$. [b]p4.[/b] A permutation on $\{1, 2, 3, …, n\}$ is a rearrangement of the symbols. For example $32154$ is a permutation on $\{1, 2, 3, 4, 5\}$. Given a permutation $a_1a_2a_3…a_n$, an inversion is a pair of $a_i$ and $a_j$ such that $a_i > a_j$ but $i < j$. For example, $32154$ has $4$ inversions. Suppose you are only allowed to exchange adjacent symbols. For any permutation, show that the minimum number of exchanges required to put all the symbols in their natural positions (that is, $123 …n$) is the number of inversions. [b]p5.[/b] We say a number $N$ is a nontrivial sum of consecutive positive integers if it can be written as the sum of $2$ or more consecutive positive integers. What is the set of numbers from $1000$ to $2000$ that are NOT nontrivial sums of consecutive positive integers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].