Find all $a$ such that for any positive integer $n$, the number $an(n+2)(n+3)(n+4)$ is an integer. (Author: O. Podlipski) [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=427802](similar to Problem 5 of grade 9)[/url] Same problem for grades 10 and 11

Squares $S_1$ and $S_2$ are inscribed in right triangle $ABC$, as shown in the figures below. Find $AC + CB$ if area$(S_1) = 441$ and area$(S_2) = 440$. [asy] size(250); real a=15, b=5; real x=a*b/(a+b), y=a/((a^2+b^2)/(a*b)+1); pair A=(0,b), B=(a,0), C=origin, X=(y,0), Y=(0, y*b/a), Z=foot(Y, A, B), W=foot(X, A, B); draw(A--B--C--cycle); draw(W--X--Y--Z); draw(shift(-(a+b), 0)*(A--B--C--cycle^^(x,0)--(x,x)--(0,x))); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$A$", (A.x-a-b,A.y), dir(point--A)); label("$B$", (B.x-a-b,B.y), dir(point--B)); label("$C$", (C.x-a-b,C.y), dir(point--C)); label("$S_1$", (x/2-a-b, x/2)); label("$S_2$", intersectionpoint(W--Y, X--Z)); dot(A^^B^^C^^(-a-b,0)^^(-b,0)^^(-a-b,b));[/asy]

There is a unique function $f: \mathbb{N} \to \mathbb{R}$ such that $f(1) > 0$ and such that \[\sum_{d \mid n} f(d) f\left(\frac{n}{d}\right) = 1\] for all $n \ge 1$. What is $f(2018^{2019})$?

Find the greatest value of positive integer $ x$ , such that the number $ A\equal{} 2^{182} \plus{} 4^x \plus{} 8^{700}$ is a perfect square .

In acute triangle $ABC$, $AB > AC$. $D$ and $E$ are the midpoints of $AB$, $AC$ respectively. The circumcircle of $ADE$ intersects the circumcircle of $BCE$ again at $P$. The circumcircle of $ADE$ intersects the circumcircle $BCD$ again at $Q$. Prove that $AP = AQ$.

Given a polynomial $f(x)=x^{2020}+\sum_{i=0}^{2019} c_ix^i$, where $c_i \in \{ -1,0,1 \}$. Denote $N$ the number of positive integer roots of $f(x)=0$ (counting multiplicity). If $f(x)=0$ has no negative integer roots, find the maximum of $N$.

There is a wooden $3 \times 3 \times 3$ cube and 18 rectangular $3 \times 1$ paper strips. Each strip has two dotted lines dividing it into three unit squares. The full surface of the cube is covered with the given strips, flat or bent. Each flat strip is on one face of the cube. Each bent strip (bent at one of its dotted lines) is on two adjacent faces of the cube. What is the greatest possible number of bent strips? Justify your answer.

Forty slips are placed into a hat, each bearing a number $ 1$, $ 2$, $ 3$, $ 4$, $ 5$, $ 6$, $ 7$, $ 8$, $ 9$, or $ 10$, with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let $ p$ be the probability that all four slips bear the same number. Let $ q$ be the probability that two of the slips bear a number $ a$ and the other two bear a number $ b\not\equal{} a$. What is the value of $ q/p$? $ \textbf{(A)}\ 162\qquad \textbf{(B)}\ 180\qquad \textbf{(C)}\ 324\qquad \textbf{(D)}\ 360\qquad \textbf{(E)}\ 720$

Let $n$ be a nonzero integer. Prove that $n^4-7n^2+1$ can never be a perfect square.

If $3(4x + 5\pi) = P$, then $6(8x + 10\pi) = $ $ \textbf{(A)}\ 2P\qquad\textbf{(B)}\ 4P\qquad\textbf{(C)}\ 6P\qquad\textbf{(D)}\ 8P\qquad\textbf{(E)}\ 18P $

Suppose that $2^n +1$ is an odd prime for some positive integer $n$. Show that $n$ must be a power of $2$.

Let $a$ be a positive integer and $(a_n)_{n\geqslant 1}$ be a sequence of positive integers satisfying $a_n<a_{n+1}\leqslant a_n+a$ for all $n\geqslant 1$. Prove that there are infinitely many primes which divide at least one term of the sequence. [i]Moldavia Olympiad, 1994[/i]

Let $a,b,c$ be positive real numbers such that $abc\geq 1$. Prove that \[ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq 1. \] [hide="Remark"]This problem derives from the well known inequality given in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=185470#p185470]USAMO 1997, Problem 5[/url]. [/hide]

Define $f : \mathbb{R}^+ \rightarrow \mathbb{R}$ as the strictly increasing function such that $$f(\sqrt{xy})=\frac{f(x)+f(y)}{2}$$ for all positive real numbers $x,y$. Prove that there are some positive real numbers $a$ where $f(a)&lt;0$. [i] (PP-nine) [/i]

Two circumferences, with the distance $d$ between centres, intersect in points $P$ and $Q$ . Two lines are drawn through the point $A$ on the first circumference ($Q\ne A\ne P$) and points $P$ and $Q$ . They intersect the second circumference in the points $B$ and $C$ . a) Prove that the radius of the circle, circumscribed around the triangle$ABC$ , equals $d$. b) Describe the set of the new circle's centres, if thepoint $A$ moves along all the first circumference.

Sabine has a very large collection of shells. She decides to give part of her collection to her sister. On the first day, she lines up all her shells. She takes the shells that are in a position that is a perfect square (the first, fourth, ninth, sixteenth, etc. shell), and gives them to her sister. On the second day, she lines up her remaining shells. Again, she takes the shells that are in a position that is a perfect square, and gives them to her sister. She repeats this process every day. The $27$th day is the first day that she ends up with fewer than $1000$ shells. The $28$th day she ends up with a number of shells that is a perfect square for the tenth time. What are the possible numbers of shells that Sabine could have had in the very beginning?

For a real number $ x$, let $ \|x \|$ denote the distance between $ x$ and the closest integer. Let $ 0 \leq x_n <1 \; (n\equal{}1,2,\ldots)\ ,$ and let $ \varepsilon >0$. Show that there exist infinitely many pairs $ (n,m)$ of indices such that $ n \not\equal{} m$ and \[ \|x_n\minus{}x_m \|< \min \left( \varepsilon , \frac{1}{2|n\minus{}m|} \right).\] [i]V. T. Sos[/i]

Hexagonal pieces numbered by positive integers are placed on the cells of a hexagonal board with side $n$. Two adjacent cells are left empty, and thanks to it some pieces can be moved. Two pieces with common sides exchanged places (see an example in the attachment 2). Prove that if $n \ge 3$ the second arrangement cannot be obtained from the first one by moving piece Note. Moving a piece a requires two adjacent empty cells. For instance, if they are on the right of a (attachment 1, left figure), a can be moved right till it touches an angle (attachment 1, middle figure), and then it can be moved upward right or downward right (attachment 1, right figure)

In $\triangle ABC$, medians $\overline{AD}$ and $\overline{CE}$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC?$ [asy] unitsize(75); pathpen = black; pointpen=black; pair A = MP("A", D((0,0)), dir(200)); pair B = MP("B", D((2,0)), dir(-20)); pair C = MP("C", D((1/2,1)), dir(100)); pair D = MP("D", D(midpoint(B--C)), dir(30)); pair E = MP("E", D(midpoint(A--B)), dir(-90)); pair P = MP("P", D(IP(A--D, C--E)), dir(150)*2.013); draw(A--B--C--cycle); draw(A--D--E--C); [/asy] $\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 13.5 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 14.5 \qquad \textbf{(E)}\ 15 $

Students from three middle schools worked on a summer project. *Seven students from Allen school worked for $3$ days. *Four students from Balboa school worked for $5$ days. *Five students from Carver school worked for $9$ days. The total amount paid for the students' work was $ \$774$. Assuming each student received the same amount for a day's work, how much did the students from Balboa school earn altogether? $\text{(A)}\ 9.00\text{ dollars} \qquad \text{(B)}\ 48.38\text{ dollars} \qquad \text{(C)}\ 180.00\text{ dollars} \qquad \text{(D)}\ 193.50\text{ dollars} \qquad \text{(E)}\ 258.00\text{ dollars}$

Let $k\geq 2$ be a positive integer and $x_1,x_2,\dots ,x_k\in (0,1)$. Also, let $m_1,m_2,\dots ,m_k$ and $n_1,n_2,\dots ,n_k$ be integers. Define $$A=x_1^{m_1}x_2^{m_2}\dots x_k^{m_k},\quad B=x_1^{n_1}x_2^{n_2}\dots x_k^{n_k}.$$ Let $$C=x_1^{\min(m_1,n_1)}x_2^{\min(m_2,n_2)}\dots x_k^{\min(m_k,n_k)}$$ $$D=x_1^{\max(m_1,n_1)}x_2^{\max(m_2,n_2)}\dots x_k^{\max(m_k,n_k)}.$$ Prove that $A+B\leq C+D$. When does equality hold? [i]Dorel Miheț[/i]

Triangle $ABC$ is inscribed in a circle $\omega$ such that $\angle A = 60^o$ and $\angle B = 75^o$. Let the bisector of angle $A$ meet $BC$ and $\omega$ at $E$ and $D$, respectively. Let the reflections of $A$ across $D$ and $C$ be $D'$ and $C'$ , respectively. If the tangent to $\omega$ at $A$ meets line $BC$ at $P$, and the circumcircle of $APD'$ meets line $AC$ at $F \ne A$, prove that the circumcircle of $C'FE$ is tangent to $BC$ at $E$.

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that the following conditions are true for every pair of positive integers $(x, y)$: $(i)$: $x$ and $f(x)$ have the same number of positive divisors. $(ii)$: If $x \nmid y$ and $y \nmid x$, then: $$\gcd(f(x), f(y)) > f(\gcd(x, y))$$

Let $ ABCD $ be a sqare and $ E $ be a point situated on the segment $ BD, $ but not on the mid. Denote by $ H $ and $ K $ the orthocenters of $ ABE, $ respectively, $ ADE. $ Show that $ \overrightarrow{BH}=\overrightarrow{KD} . $

We play the following game in a Cartesian coordinate system in the plane. Given the input $(x,y)$, in one step, we may move to the point $(x,y\pm 2x)$ or to the point $(x\pm 2y,y)$. There is also an additional rule: it is not allowed to make two steps that lead back to the same point (i.e, to step backwards). Prove that starting from the point $\left(1;\sqrt 2\right)$, we cannot return to it in finitely many steps.