Is is true that if the perfect set $F\subseteq [0,1]$ is of zero Lebesgue measure then those functions in $C^1[0,1]$ which are one-to-one on $F$ form a dense subset of $C^1[0,1]$? (We use the metric $$d(f,g)=\sup_{x\in[0,1]} |f(x)-g(x)| + \sup_{x\in[0,1]} |f'(x)-g'(x)|$$ to define the topology in the space $C^1[0,1]$ of continuously differentiable real functions on $[0,1]$.)

[u]Round 5[/u] [b]p13.[/b] Two closed disks of radius $\sqrt2$ are drawn centered at the points $(1,0)$ and $(-1, 0)$. Let P be the region belonging to both disks. Two congruent non-intersecting open disks of radius $r$ have all of their points in $P$ . Find the maximum possible value of $r$ . [b]p14.[/b] A rectangle has positive integer side lengths. The sum of the numerical values of its perimeter and area is $2017$. Find the perimeter of the rectangle. [b]p15.[/b] Find all ordered triples of real numbers $(a,b,c)$ which satisfy $$a +b +c = 6$$ $$a \cdot (b +c) = 6$$ $$(a +b) \cdot c = 6$$ [u]Round 6[/u] [b]p16.[/b] A four digit positive integer is called confused if it is written using the digits $2$, $0$, $1$, and $7$ in some order, each exactly one. For example, the numbers $7210$ and $2017$ are confused. Find the sum of all confused numbers. [b]p17.[/b] Suppose $\vartriangle ABC$ is a right triangle with a right angle at $A$. Let $D$ be a point on segment $BC$ such that $\angle BAD = \angle CAD$. Suppose that $AB = 20$ and $AC = 17$. Compute $AD$. [b]p18.[/b] Let $x$ be a real number. Find the minimum possible positive value of $\frac{|x -20|+|x -17|}{x}$. [u]Round 7[/u] [b]p19.[/b] Find the sum of all real numbers $0 < x < 1$ that satisfy $\{2017x\} = \{x\}$. [b]p20.[/b] Let $a_1,a_2, ,,, ,a_{10}$ be real numbers which sum to $20$ and satisfy $\{a_i\} <0.5$ for $1 \le i\le 10$. Find the sum of all possible values of $\sum_{ 1 \le i <j\le 10} \lfloor a_i +a_j \rfloor .$ Here, $\lfloor x \rfloor$ denotes the greatest integer $x_0$ such that $x_0 \le x$ and $\{x\} =x -\lfloor x \rfloor$. [b]p21.[/b] Compute the remainder when $20^{2017}$ is divided by $17$. [u]Round 8[/u] [b]p22.[/b] Let $\vartriangle ABC$ be a triangle with a right angle at $B$. Additionally, letM be the midpoint of $AC$. Suppose the circumcircle of $\vartriangle BCM$ intersects segment $AB$ at a point $P \ne B$. If $CP = 20$ and $BP = 17$, compute $AC$. [b]p23.[/b] Two vertices on a cube are called neighbors if they are distinct endpoints of the same edge. On a cube, how many ways can a nonempty subset $S$ of the vertices be chosen such that for any vertex $v \in S$, at least two of the three neighbors of $v$ are also in $S$? Reflections and rotations are considered distinct. [b]p24.[/b] Let $x$ be a real number such that $x +\sqrt[4]{5-x^4}=2$. Find all possible values of $x\sqrt[4]{5-x^4}$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158491p28715220]here[/url].and 9-12 [url=https://artofproblemsolving.com/community/c3h3162362p28764144]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There is a point $T$ on a circle with the radius $R$. Points $A{}$ and $B$ are on the tangent to the circle that goes through $T$ such that they are on the same side of $T$ and $TA\cdot TB=4R^2$. The point $S$ is diametrically opposed to $T$. Lines $AS$ and $BS$ intersect the circle again in $P{}$ and $Q{}$. Prove that the lines $PQ$ and $AB{}$ are perpendicular.

[b]a)[/b] Determine $ x\in\mathbb{N} $ and $ y\in\mathbb{Q} $ such that $ \sqrt{x+\sqrt{x}}=y. $ [b]b)[/b] Show that there are infinitely many pairs $ (x,y)\in\mathbb{Q}^2 $ such that $ \sqrt{x+\sqrt{x}} =y . $

You are given $N$ such that $ n \ge 3$. We call a set of $N$ points on a plane acceptable if their abscissae are unique, and each of the points is coloured either red or blue. Let's say that a polynomial $P(x)$ divides a set of acceptable points either if there are no red dots above the graph of $P(x)$, and below, there are no blue dots, or if there are no blue dots above the graph of $P(x)$ and there are no red dots below. Keep in mind, dots of both colors can be present on the graph of $P(x)$ itself. For what least value of k is an arbitrary set of $N$ points divisible by a polynomial of degree $k$?

Assign independent standard normally distributed random variables to the vertices of an n-dimensional cube. Say one vertex is greater than another if the assigned number is greater. Define a random walk on the vertices according to the following rules: a) the starting point is chosen from all the vertices with equal probability, b) during our journey, if we reach a vertex such that there are adjacent vertices which have higher values, we choose the next vertex with equal probability, c) if there is none, we stop. Prove that $\forall\varepsilon>0 \,\exists K\, \forall n>1$ $$P(\lambda> K \log n) <\varepsilon$$ where $\lambda$ is the number of steps of the random walk.

$(HUN 1)$ Let $a$ and $b$ be arbitrary integers. Prove that if $k$ is an integer not divisible by $3$, then $(a + b)^{2k}+ a^{2k} +b^{2k}$ is divisible by $a^2 +ab+ b^2$

We roll a regular 6-sided dice $n$ times. What is the probabilty that the total number of eyes rolled is a multiple of 5?

In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. Denote by $P, Q, R, S$ the orthogonal projections of $O$ onto $AB$ , $BC$ ,$CD$ , $DA$, respectively. Prove that $$PA \cdot AB + RC \cdot CD =\frac12 (AD^2 + BC^2)$$ if and only if $$QB \cdot BC + SD \cdot DA = \frac12(AB ^2 + CD^2)$$

The [i]spikiness[/i] of a sequence $a_1, a_2, \ldots, a_n$ of at least two real numbers is the sum $\textstyle\sum_{i=1}^{n-1} |a_{i+1}-a_i|.$ Suppose $x_1, x_2, \ldots, x_9$ are chosen uniformly at random from the set $[0, 1].$ Let $M$ be the largest possible value of the spikiness of a permutation of $x_1, x_2, \ldots, x_9.$ Compute the expected value of $M.$

Does there exist a bijection $f:\mathbb{N}^{+} \rightarrow \mathbb{N}^{+}$, such that there exist a positive integer $k$, and it's possible to have each positive integer colored by one of $k$ chosen colors, such that for any $x \neq y$ , $f(x)+y$ and $f(y)+x$ are not the same color?

Let $S$ be the set of all positive integers less than $10,000$ whose last four digits in base $2$ are the same as its last four digits in base $5$. What remainder do we get if we divide the sum of all elements of $S$ by $10000$?

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied: i) No line passes through any point of the configuration. ii) No region contains points of both colors. Find the least value of $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines. Proposed by [i]Ivan Guo[/i] from [i]Australia.[/i]

Jonathan applies a normal force that is just enough to keep the rope from slipping. Becky makes a small jump, barely leaving contact with the floor of the box. Upon landing on the box, the force of the impact causes the rope to start slipping from Jonathan’s hand. At what speed does the box smash into the ground? Assume Jonathan’s normal force does not change. (A) $\sqrt{2gH}(\mu_k/\mu_s)$ (B) $\sqrt{2gH}(1-\mu_k/\mu_s)$ (C) $\sqrt{2gH}\sqrt{\mu_k/\mu_s}$ (D) $\sqrt{2gH}\sqrt{1-(\mu_k/\mu_s)}$ (E) $\sqrt{2gH}(\mu_s-\mu_k)$

Let $n$ be a positive integer $\geq 3.$ There are $n$ real numbers $x_1,x_2,\cdots x_n$ that satisfy: \[\left\{\begin{aligned}&\ x_1\ge x_2\ge\cdots \ge x_n;\\& \ x_1+x_2+\cdots+x_n=0;\\& \ x_1^2+x_2^2+\cdots+x_n^2=n(n-1).\end{aligned}\right.\] Find the maximum and minimum value of the sum $S=x_1+x_2.$

Another round, another diagram... [asy] unitsize(1cm); defaultpen(linewidth(0.45)); real[][] arr = { {0,0,0,0}, {0,0,0,0}, {0,0,0,0}, {0,0,0,0}}; for (int i=0; i<4; ++i){ for (int j=0; j<4; ++j){ if(arr[3-j][i] != 0){ label((string) arr[3-j][i], (i+0.5, j+0.5)); } } } label("$+$", (-0.5, 4.5), dir(-45)); label("$-$", (4.5, -0.5), dir(135)); label("\Large 13", (-0.5, 3.5)); label("\Large 28", (-0.5, 2.5)); label("\Large 23", (-0.5, 0.5)); label("\Large 7", (4.5, 3.5)); label("\Large 8", (4.5, 1.5)); label("\Large 8", (4.5, 0.5)); label("\Large 12", (3.5, -0.5)); label("\Large 12", (2.5,-0.5)); label("\Large 7", (0.5, -0.5)); label("\Large 23", (3.5, 4.5 )); label("\Large 25", (2.5,4.5)); label("\Large 28", (1.5,4.5)); label("\Large 13", (0.5,4.5)); for(int i = 1; i <= 3; ++i){ draw((i, 0)--(i, 4)); draw((0, i)--(4, i)); } draw((0,0)--(0,4)--(4,4)--(4,0)--cycle, linewidth(1.5)); draw((-0.8,-0.8)--(0,0), linewidth(1.5)); draw((4,4 )--(4.8,4.8), linewidth(1.5)); [/asy] Use [code]\begin{asy} \end{asy}[/code]environment to render the diagram correctly in a latex document. Remember to write [code]\usepackage{asymptote}[/code] in the preamble. And of course, replace the 0's in the array at the beginning of the code with the numbers you wish to fill it in with.

A bag contains $12$ marbles: $3$ red, $4$ green, and $5$ blue. Repeatedly draw marbles with replacement until you draw two marbles of the same color in a row. What is the expected number of times that you will draw a marble?

We will call the ordered pair $(a,b)$ “parallel”, where $a,b\in \mathbb{N}$, if $\sqrt{ab}\in \mathbb{N}$. Prove that the number of “parallel” pairs $(a,b)$, for which $1\leq a,b\leq 10^6$ is at least $3.10^6(ln\, 10-1)$.

Prove that the group morphisms $f: (\mathbb{C},+)\to(\mathbb{C},+)$ for which there exists a positive $\lambda$ such that $|f(z)| \leq \lambda |z|$ for all $z\in\mathbb{C}$, have the form \[ f(z) = \alpha z + \beta \overline{z} \] for some complex $\alpha$, $\beta$. [i]Cristinel Mortici[/i]

Show that \[ \prod_{1\leq x < y \leq \frac{p\minus{}1}{2}} (x^2\plus{}y^2) \equiv (\minus{}1)^{\lfloor\frac{p\plus{}1}{8}\rfloor} \;(\textbf{mod}\;p\ ) \] for every prime $ p\equiv 3 \;(\textbf{mod}\;4\ )$. [J. Suranyi]

Let the triangle $ABC$ be acute. Let us take in the segment $BC$ two points $F$ and $G$ such that $BG > BF = GC$ and an interior point$ P$ to the triangle on the bisector of $\angle BAC$. Then are drawn through $P$, $PD\parallel AB$ and $PE \parallel AC$, $D \in AC$ and $E \in AB$, $\angle FEP = \angle PDG$. prove that $\vartriangle ABC$ is isosceles.

Find the number of nonnegative integers $k$, $0 \leq k \leq 2188$, and such that $\binom{2188}{k}$ is divisible by 2188.

Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? $\text{(A)}\ \frac{1}{2} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{4}{5} \qquad \text{(E)}\ \frac{5}{6}$ [asy] draw((0,0)--(0,8)--(6,8)--(6,0)--cycle); draw((0,8)--(5,9)--(5,8)); draw((3,-1.5)--(3,10.3),dashed); draw((0,5.5)..(-.75,4.75)..(0,4)); draw((0,4)--(1.5,4),EndArrow); [/asy]

Find all triples $ (x,y,z)$ of real numbers which satisfy the simultaneous equations \[ x \equal{} y^3 \plus{} y \minus{} 8\] \[y \equal{} z^3 \plus{} z \minus{} 8\] \[ z \equal{} x^3 \plus{} x \minus{} 8.\]