For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$

In the grid below, fill each gray cell with one of the numbers from the provided bank, with each number used once, and fill each white cell with a positive one-digit number. The number in a gray cell must equal the sum of the numbers in all touching white cells, where two cells sharing a vertex are considered touching. All of the terms in each of these sums must be distinct, meaning that two white cells with the same digit may not touch the same gray cell. Bank: 15, 23, 28, 35, 36, 38, 40, 42, 44 [asy] unitsize(1.2cm); defaultpen(fontsize(30pt)); int[][] x = { {0,5,0,0,0,8,0}, {4,0,0,0,0,0,0}, {0,3,0,0,0,0,1}, {0,0,0,0,0,7,0}, {0,8,0,0,1,0,4}, {0,0,0,0,0,2,0}}; void square(int a, int b) { filldraw((a,b)--(a+1,b)--(a+1,b+1)--(a,b+1)--cycle,mediumgray); } square(1,0); square(3,1); square(5,1); square(1,2); square(3,3); square(5,3); square(1,4); square(5,4); square(3,5); for(int i = 0; i < 8; ++i) { draw((i,0)--(i,6)); } for(int i = 0; i < 7; ++i) { draw((0,i)--(7,i)); } for(int k = 0; k<6; ++k){ for(int l = 0; l<7; ++l){ if(x[k][l]!=0){ label(string(x[k][l]),(l+0.5,-k+5.5)); } } } [/asy] There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the constraints above. (Note: in any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)

Suppose the curve $ C: y \equal{} ax^3 \plus{} 4x\ (a\neq 0)$ has a common tangent line at the point $ P$ with the hyperbola $ xy \equal{} 1$ in the first quadrant. (1) Find the value of $ a$ and the coordinate of the point $ P$. (2) Find the volume formed by the revolution of the solid of the figure bounded by the line segment $ OP$ and the curve $ C$ about the line $ OP$. [color=green][Edited.][/color]

Six straight lines are drawn on the plane. It is known that for any three of them there is a fourth of the same set of lines, such that all four will touch some circle. Do all six lines necessarily touch the same circle? (I. Bogdanov)

A sign at the fish market says, '50\% off, today only: half-pound packages for just \$3 per package.' What is the regular price for a full pound of fish, in dollars? $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15$

A circle with center O inscribed in a convex quadrilateral ABCD is tangent to the lines AB, BC, CD, DA at points K, L, M, N respectively. Assume that the lines KL and MN are not parallel and intersect at the point S. Prove that BD is perpendicular OS. I think it is very good and beautiful problem. I solved it without help. I'm wondering is it a well known theorem? Also I'm interested who is the creator of this problem? I'll be glad to see simple solution of this problem.

The digits of a $3$-digit number are interchanged so that none of the digits retain their original position. The difference of the two numbers is a $2$-digit number and is a perfect square. Find the difference.

Prove that for any set containing $2047$ positive integers, there exists $1024$ positive integers in the set such that the sum of these positive integers is divisible by $1024$.

Find the number of ways in which three numbers can be selected from the set $\{1,2,\cdots ,4n\}$, such that the sum of the three selected numbers is divisible by $4$.

A $100 \times100$ chessboard has a non-negative real number in each of its cells. A chessboard is [b]balanced[/b] if and only if the numbers sum up to one for each column of cells as well as each row of cells. Find the largest positive real number $x$ so that, for any balanced chessboard, we can find $100$ cells of it so that these cells all have number greater or equal to $x$, and no two of these cells are on the same column or row. [i]Proposed by CSJL.[/i]

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

Let $S$ be the set of sequences of length 2018 whose terms are in the set $\{1, 2, 3, 4, 5, 6, 10\}$ and sum to 3860. Prove that the cardinality of $S$ is at most \[2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.\]

Felix picks four points uniformly at random inside a unit circle $\mathcal{C}$. He then draws the four possible triangles which can be formed using these points as vertices. Finally, he randomly chooses of the six possible pairs of the triangles he just drew. What is the probability that the center of the circle $\mathcal{C}$ is contained in the union of the interiors of the two triangles that Felix chose?

Three circles with radii $1$, $2$, and $3$ are pairwise tangent to each other. Find the radius of the circle that is externally tangent to all three of these circles. [i]Proposed by Tamar Turashvili, Georgia [/i]

Let $a_1, a_2, \cdots, a_n$ be a sequence of real numbers with $a_1+a_2+\cdots+a_n=0$. Define the score $S(\sigma)$ of a permutation $\sigma=(b_1, \cdots b_n)$ of $(a_1, \cdots a_n)$ to be the minima of the sum $$(x_1-b_1)^2+\cdots+(x_n-b_n)^2$$ over all real numbers $x_1\le \cdots \le x_n$. Prove that $S(\sigma)$ attains the maxima over all permutations $\sigma$, if and only if for all $1\le k\le n$, $$b_1+b_2+\cdots+b_k\ge 0.$$ [i]Proposed by Anzo Teh Zhao Yang[/i]

Given the integer $n > 1$ and the real number $a > 0$ determine the maximum of $\sum_{i=1}^{n-1} x_i x_{i+1}$ taken over all nonnegative numbers $x_i$ with sum $a.$

Let $a < b < c < d < e$ be positive integers. Prove that $$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$ where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?

Suppose that $P(x)$ is a polynomial with the property that there exists another polynomial $Q(x)$ to satisfy $P(x)Q(x)=P(x^2)$. $P(x)$ and $Q(x)$ may have complex coefficients. If $P(x)$ is a quintic with distinct complex roots $r_1,\dots,r_5$, find all possible values of $|r_1|+\dots+|r_5|$.

Facundo and Luca have been given a cake that is shaped like the quadrilateral in the figure. [img]https://cdn.artofproblemsolving.com/attachments/3/2/630286edc1935e1a8dd9e704ed4c813c900381.png[/img] They are going to make two straight cuts on the cake, thus obtaining $4$ portions in the shape of a quadrilateral. Then Facundo will be left with two portions that do not share any side, the other two will be for Luca. Show how they can cut the cuts so that both children get the same amount of cake. Justify why cutting in this way achieves the objective.

Is there a positive integer $m$ such that the equation \[ {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c} \] has infinitely many solutions in positive integers $a,b,c$?

Let $ABC$ be a triangle with $AB<AC<BC$ with circumcircle $\Gamma_1$. The circle $\Gamma_2$ has center $D$ lying on $\Gamma_1$ and touches $BC$ at $E$ and the extension of $AB$ at $F$. Let $\Gamma_1$ and $\Gamma_2$ meet at $K, G$ and the line $KG$ meets $EF$ and $CD$ at $M, N$. Show that $BCNM$ is cyclic.

A tortoise is given an $80$-second head start in a race. When Achilles catches up to where the tortoise was when he (Achilles) began running, he finds that while he is now $40$ meters ahead of the starting line, the tortoise is now $5$ meters ahead of him. At this point, how long will it be, in seconds, before Achilles passes the tortoise? [i]2015 CCA Math Bonanza Team Round #3[/i]

A square is divided into $16$ equal squares, obtaining the set of $25$ different vertices. What is the least number of vertices one must remove from this set, so that no $4$ points of the remaining set are the vertices of any square with sides parallel to the sides of the initial square?

Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.

Find all primes $p$ such that $\dfrac{2^{p+1}-4}{p}$ is a perfect square.