In the land of Cockaigne, people play the following solitaire. It starts from a finite string of zeros and ones, and are granted the following moves: (i) cancel each two consecutive ones; (ii) delete three consecutive zeros; (iii) if the substring within the string is $01$, one may replace this by substring $100$. The moves (i), (ii) and (iii) must be made one at a time. You win if you can reduce the string to a string formed by two digits or less. (For example, starting from $0101$, one can win using move (iii) first in the last two digits, resulting in $01100$, then playing the move (i) on two 'ones', and finally the move (ii) on the three zeros, one will get the empty string.) Among all the $1024$ possible strings of ten-digit binary numbers, how many are there from which it is not possible to win the solitary?

Prove that for positive reals $a,b,c$ satisfying $a+b+c=3$ the following inequality holds: $$ \frac{a}{1+2b^3}+\frac{b}{1+2c^3}+\frac{c}{1+2a^3} \ge 1 $$

Let $ABCD$ be a convex quadrilateral inscribed in a circle of radius $1$. Prove that \[ 0< (AB+BC+CD+AD)-(AC+BD) < 4. \]

Show that $n$ is prime iff $\lim_{r \rightarrow\infty}\,\lim_{s \rightarrow\infty}\,\lim_{t \rightarrow \infty}\,\sum_{u=0}^{s}\left(1-\left(\cos\,\frac{(u!)^{r} \pi}{n} \right)^{2t} \right)=n$ PS : I posted it because it's in the PDF file but not here ...

Let be given an arbitrary tetrahedron $ABCD$ with volume $V$. Consider all lines which pass through the barycenter $S$ of the tetrahedron and intersect the edges $AD,BD,CD$ at points $A',B',C$ respectively. It is known that among the obtained tetrahedra there exists one with the minimal volume. Express this minimal volume in terms of $V$

Given $k \in \mathbb{Z}$ prove that there exist infinite pairs of distinct natural numbers such that \begin{align*} n+s(2n)=m+s(2m) \\ kn+s(n^2)=km+s(m^2). \end{align*} ($s(n)$ denotes the sum of digits of $n$.) [i]Proposed by Mohammadamin Sharifi[/i]

(a) Show that the equation \[\lfloor x \rfloor (x^2 + 1) = x^3,\] where $\lfloor x \rfloor$ denotes the largest integer not larger than $x$, has exactly one real solution in each interval between consecutive positive integers. (b) Show that none of the positive real solutions of this equation is rational.

In trapezoid $ABCD$, $AD$ is perpendicular to $DC$, $AD=AB=3$, and $DC=6$. In addition, E is on $DC$, and $BE$ is parallel to $AD$. Find the area of $\Delta BEC$. [asy] defaultpen(linewidth(0.7)); pair A=(0,3), B=(3,3), C=(6,0), D=origin, E=(3,0); draw(E--B--C--D--A--B); draw(rightanglemark(A, D, C)); label("$A$", A, NW); label("$B$", B, NW); label("$C$", C, SE); label("$D$", D, SW); label("$E$", E, NW); label("$3$", A--D, W); label("$3$", A--B, N); label("$6$", E, S);[/asy] $\textbf{(A)} \: 3\qquad \textbf{(B)} \: 4.5\qquad \textbf{(C)} \: 6\qquad \textbf{(D)} \: 9\qquad \textbf{(E)} \: 18\qquad $

Find all pair of constants $(a,b)$ such that there exists real-coefficient polynomial $p(x)$ and $q(x)$ that satisfies the condition below. [b]Condition[/b]: $\forall x\in \mathbb R,$ $ $ $p(x^2)q(x+1)-p(x+1)q(x^2)=x^2+ax+b$

Mark has a cursed six-sided die that never rolls the same number twice in a row, and all other outcomes are equally likely. Compute the expected number of rolls it takes for Mark to roll every number at least once.

Let $A_1A_2A_3$ be a triangle and $M$ an interior point. The straight lines $MA_1, MA_2, MA_3$ intersect the opposite sides at the points $B_1, B_2, B_3$ respectively (see Fig.). Show that if the areas of triangles $A_2B_1M, A_3B_2M$ and $A_1B_3M$ are equal, then $M$ coincides with the centroid of triangle $A_1A_2A_3$. [img]https://cdn.artofproblemsolving.com/attachments/1/7/b29bdbb1f2b103be1f3cb2650b3bfff352024a.png[/img]

[u]Round 1[/u] [b]p1.[/b] When Shiqiao sells a bale of kale, he makes $x$ dollars, where $$x =\frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8}{3 + 4 + 5 + 6}.$$ Find $x$. [b]p2.[/b] The fraction of Shiqiao’s kale that has gone rotten is equal to $$\sqrt{ \frac{100^2}{99^2} -\frac{100}{99}}.$$ Find the fraction of Shiqiao’s kale that has gone rotten. [b]p3.[/b] Shiqiao is growing kale. Each day the number of kale plants doubles, but $4$ of his kale plants die afterwards. He starts with $6$ kale plants. Find the number of kale plants Shiqiao has after five days. [u]Round 2[/u] [b]p4.[/b] Today the high is $68$ degrees Fahrenheit. If $C$ is the temperature in Celsius, the temperature in Fahrenheit is equal to $1.8C + 32$. Find the high today in Celsius. [b]p5.[/b] The internal angles in Evan’s triangle are all at most $68$ degrees. Find the minimum number of degrees an angle of Evan’s triangle could measure. [b]p6.[/b] Evan’s room is at $68$ degrees Fahrenheit. His thermostat has two buttons, one to increase the temperature by one degree, and one to decrease the temperature by one degree. Find the number of combinations of $10$ button presses Evan can make so that the temperature of his room never drops below $67$ degrees or rises above $69$ degrees. [u]Round 3[/u] [b]p7.[/b] In a digital version of the SAT, there are four spaces provided for either a digit $(0-9)$, a fraction sign $(\/)$, or a decimal point $(.)$. The answer must be in simplest form and at most one space can be a non-digit character. Determine the largest fraction which, when expressed in its simplest form, fits within this space, but whose exact decimal representation does not. [b]p8.[/b] Rounding Rox picks a real number $x$. When she rounds x to the nearest hundred, its value increases by $2.71828$. If she had instead rounded $x$ to the nearest hundredth, its value would have decreased by $y$. Find $y$. [b]p9.[/b] Let $a$ and $b$ be real numbers satisfying the system of equations $$\begin{cases} a + \lfloor b \rfloor = 2.14 \\ \lfloor a \rfloor + b = 2.72 \end{cases}$$ Determine $a + b$. [u]Round 4[/u] [b]p10.[/b] Carol and Lily are playing a game with two unfair coins, both of which have a $1/4$ chance of landing on heads. They flip both coins. If they both land on heads, Lily loses the game, and if they both land on tails, Carol loses the game. If they land on different sides, Carol and Lily flip the coins again. They repeat this until someone loses the game. Find the probability that Lily loses the game. [b]p11.[/b] Dongchen is carving a circular coin design. He carves a regular pentagon of side length $1$ such that all five vertices of the pentagon are on the rim of the coin. He then carves a circle inside the pentagon so that the circle is tangent to all five sides of the pentagon. Find the area of the region between the smaller circle and the rim of the coin. [b]p12.[/b] Anthony flips a fair coin six times. Find the probability that at some point he flips $2$ heads in a row. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3248731p29808147]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let's call a ticket with a number from $000000$ to $999999$ [i]excellent [/i] if the difference between some two adjacent digits is $5$. Find the number of excellent tickets.

[4] Let $x_1,x_2,\ldots,x_{100}$ be defined so that for each $i$, $x_i$ is a (uniformly) random integer between $1$ and $6$ inclusive. Find the expected number of integers in the set $\{x_1,x_1+x_2,\ldots,x_1+x_2+\cdots+x_{100}\}$ that are multiples of $6$.

There are 20 buns with jam and 20 buns with treacle arranged in a row in random order. Alice and Bob take in turn a bun from any end of the row. Alice starts, and wants to finally obtain 10 buns of each type; Bob tries to prevent this. Is it true for any order of the buns that Alice can win no matter what are the actions of Bob? [i]Alexandr Gribalko[/i]

Let $C$ be a circle in the plane. Let $C_1$ and $C_2$ be two non-intersecting circles touching $C$ internally at points $A$ and $B$ respectively (Fig. ). Suppose that $D$ and $E$ are two points on $C_1$ and $C_2$ respectively such that $DE$ is a common tangent of $C_1$ and $C_2$, and both $C_1$ and C2 are on the same side of $DE$. Let $F$ be the intersection point of $AD$ and $BE$. Prove that $F$ lies on $C$. [img]https://cdn.artofproblemsolving.com/attachments/f/c/5c733db462ef8ec3d3f82bbb762f7f087fbd3d.png[/img]

Baron Münchhausen’s watch works properly, but has no markings on its face. The hour, minute and second hands have distinct lengths, and they move uniformly. The Baron claims that since none of the mutual positions of the hands is repeats twice in the period between 8:00 and 19:59, he can use his watch to tell the time during the day. Is his assertion true? [i](5 points)[/i]

Given a natural number $n$. Prove that you can choose $ \phi (n)+1 $ (not necessarily different) divisors $n$ with the sum $n$. Here $ \phi (n)$ denotes the number of natural numbers less than $n$ that are coprime with $n$. (Fedir Yudin)

$MD$ is a chord of length $2$ in a circle of radius $1,$ and $L$ is chosen on the circle so that the area of $\triangle MLD$ is the maximized. Find $\angle MLD.$

A regular $n$-gon has $135$ diagonals. What is the measure of its exterior angle, in degrees? (An exterior angle is the supplement of an interior angle.)

Which one of the following is not equivalent to $ 0.000000375$? $ \textbf{(A)}\ 3.75 \times 10^{\minus{}7} \qquad \textbf{(B)}\ 3 \frac{3}{4} \times 10^{\minus{}7} \qquad \textbf{(C)}\ 375 \times 10^{\minus{}9} \\ \textbf{(D)}\ \frac{3}{8} \times 10^{\minus{}7} \qquad \textbf{(E)}\ \frac{3}{80000000}$

Given an integer $k\geq 2$, determine all functions $f$ from the positive integers into themselves such that $f(x_1)!+f(x_2)!+\cdots f(x_k)!$ is divisibe by $x_1!+x_2!+\cdots x_k!$ for all positive integers $x_1,x_2,\cdots x_k$. $Albania$

Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers.

Acute triangle $ABC$ has circumcenter $O.$ The bisector of $ABC$ and the altitude from $C$ to side $AB$ intersect at $X.$ Suppose that there is a circle passing through $B, O, X,$ and $C.$ If $\angle BAC = n^\circ,$ where $n$ is a positive integer, compute the largest possible value of $n.$

Let $\mathcal{F}$ be the set of functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(2x)=f(x)$ for all $x\in\mathbb{R}.$ [list=a] [*]Determine all functions $f\in\mathcal{F}$ which admit antiderivatives on $\mathbb{R}.$ [*]Give an example of a non-constant function $f\in\mathcal{F}$ which is integrable on any interval $[a,b]\subset\mathbb{R}$ and satisfies \[\int_a^bf(x) \ dx=0\]for all real numbers $a$ and $b.$ [/list][i]Mihai Piticari and Sorin Rădulescu[/i]