A calculator can square a number or add $1$ to it. It cannot add $1$ two times in a row. By several operations it transformed a number $x$ into a number $S > x^n + 1$ ($x, n,S$ are positive integers). Prove that $S > x^n + x - 1$.

Prove that if $(a,b,c,d)$ are positive integers such that $(a+2^{\frac13}b+2^{\frac23}c)^2=d$ then $d$ is a perfect square (i.e is the square of a positive integer).

A diagonal is drawn in an isosceles trapezoid. By the contour of each of the resulting two triangles creeps its own beetle. The velocities of the beetles are constant and identical. Beetles don't change directions around their contours, and along the diagonal of the trapezoid they crawl in different directions. Prove that for any starting positions of the beetles they will ever meet.

Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms? ${{ \textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 72}\qquad\textbf{(E)}\ 80} $

Given three letters $X, Y, Z$, we can construct letter sequences arbitrarily, such as $XZ, ZZYXYY, XXYZX$, etc. For any given sequence, we can perform following operations: $T_1$: If the right-most letter is $Y$, then we can add $YZ$ after it, for example, $T_1(XYZXXY) = (XYZXXYYZ).$ $T_2$: If The sequence contains $YYY$, we can replace them by $Z$, for example, $T_2(XXYYZYYYX) = (XXYYZZX).$ $T_3$: We can replace $Xp$ ($p$ is any sub-sequence) by $XpX$, for example, $T_3(XXYZ) = (XXYZX).$ $T_4$: In a sequence containing one or more $Z$, we can replace the first $Z$ by $XY$, for example, $T_4(XXYYZZX) = (XXYYXYZX).$ $T_5$: We can replace any of $XX, YY, ZZ$ by $X$, for example, $T_5(ZZYXYY) = (XYXX)$ or $(XYXYY)$ or $(ZZYXX).$ Using above operations, can we get $XYZZ$ from $XYZ \ ?$

Given an acute triangle $ABC$ with altitudes $AA_1$, $BB_1$, and $CC_1$ ($A_1 \in BC$, $B_1 \in AC$, $C_1 \in AB$) and circumcircle $k$, the rays $B_1A_1$, $C_1B_1$, and $A_1C_1$ meet $k$ at points $A_2$, $B_2$, and $C_2$, respectively. Find the maximum possible value of \[ \sin \angle ABB_2 \cdot \sin \angle BCC_2 \cdot \sin \angle CAA_2 \] and all acute triangles $ABC$ for which it is achieved.

Two walls and the ceiling of a room meet at right angles at point $P$. A fly is in the air one meter from one wall, eight meters from the other wall, and $9$ meters from point $P$. How many meters is the fly from the ceiling? $\textbf{(A) }\sqrt{13}\qquad\textbf{(B) }\sqrt{14}\qquad\textbf{(C) }\sqrt{15}\qquad\textbf{(D) }4\qquad\textbf{(E) }\sqrt{17}$

The quadrilateral $ABCD$ is inscribed in circle $\Omega$ with center $O$, not lying on either of the diagonals. Suppose that the circumcircle of triangle $AOC$ passes through the midpoint of the diagonal $BD$. Prove that the circumcircle of triangle $BOD$ passes through the midpoint of diagonal $AC$. [i](A. Zaslavsky)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

Find all quadruplets $(a, b, c, d)$ of real numbers satisfying the system $(a + b)(a^2 + b^2) = (c + d)(c^2 + d^2)$ $(a + c)(a^2 + c^2) = (b + d)(b^2 + d^2)$ $(a + d)(a^2 + d^2) = (b + c)(b^2 + c^2)$ (Slovakia)

An isosceles right-angled triangle $ABC$ is given. On the extensions of sides $AB$ and $AC$, behind vertices $B$ and $C$ equal segments $BK$ and $CL$ were laid. $E$ and F are the points of intersection of the segment $KL$ and the lines perpendicular to the $KC$ , passing through the points $B$ and $A$, respectively. Prove that $EF = FL$.

Let $k>1$ be an integer. A function $f:\mathbb{N^*}\to\mathbb{N^*}$ is called $k$-[i]tastrophic[/i] when for every integer $n>0$, we have $f_k(n)=n^k$ where $f_k$ is the $k$-th iteration of $f$: \[f_k(n)=\underbrace{f\circ f\circ\cdots \circ f}_{k\text{ times}}(n)\] For which $k$ does there exist a $k$-tastrophic function?

Let $A(x,y)$ be the number of points $(m,n)$ in the plane with integer coordinates $m$ and $n$ satisfying $m^2+n^2\le x^2+y^2$. Let $g=\sum_{k=1}^\infty e^{-k^2}$. Express $$\int^\infty_{-\infty}\int^\infty_{-\infty}A(x,y)e^{-x^2-y^2}dxdy$$ as a polynomial in $g$.

[u]Round 9[/u] [b]p25.[/b] What is the largest integer that cannot be expressed as the sum of nonnegative multiples of $7$, $11$, and $13$? [b]p26.[/b] Evaluate $12{3 \choose3}+ 11{4\choose 3}+ 10{5\choose 3}+ ...+ 2{13\choose 3}+{14 \choose 3}$. [b]p27.[/b] Worker Bob drives to work at $30$ mph half the time and $60$ mph half the time. He returns home along the same route at $30$ mph half the distance and $60$ mph half the distance. What is his average speed along the entire trip, in mph? [u]Round 10[/u] [b]p28.[/b] In quadrilateral $ABCD$, diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$ with $BP = 4$, $P D = 6$, $AP = 8$, $P C = 3$, and $AB = 6$. What is the length of $AD$? [b]p29.[/b] Find all positive integers $x$ such that$ x^2 + 17x + 17$ is a square number. [b]p30.[/b] Zach has ten weighted coins that turn up heads with probabilities $\frac{2}{11^2}$ ,$\frac{2}{10^2}$ ,$\frac{2}{9^2}$ $, . . $.,$\frac{2}{2^2}$ . If he flips all ten coins simultaneously, then what is the probability that he will get an even number of heads? [u]Round 11[/u] [b]p31.[/b] Given a sequence $a_1, a_2, . . .$ such that $a_1 = 3$ and $a_{n+1} = a^2_n - 2a_n + 2$ for $n \ge 1$, find the remainder when the product a1a2 · · · a2012 is divided by 100. [b]p32.[/b] Let $ABC$ be an equilateral triangle and let $O$ be its circumcircle. Let $D$ be a point on $\overline{BC}$, and extend $\overline{AD}$ to intersect $O$ at $P$. If $BP = 5$ and $CP = 4$, then what is the value of $DP$? [b]p33.[/b] Surya and Hao take turns playing a game on a calendar. They start with the date January $1$ and they can either increase the month to a later month or increase the day to a later day in that month but not both. The first person to adjust the date to December $31$ is the winner. If Hao goes first, then what is the first date that he must choose to ensure that he does not lose? [u]Round 12[/u] [b]p34.[/b] On May $5$, $1868$, exactly $144$ years before today, Memorial Day in the United States was officially proclaimed. The first Memorial Day took place that year on May $30$ at Waterloo, New York. On May $5$, $2012$, at $12:00$ PM, how many results did the search “memorial day” on Google return? The search phrase is in quotes, so Google will only return sites that have the words memorial and day next to each other in that order. Let $N = max-\{0, \rfloor 15.5 \times \frac{ Your\,\,\, Answer}{Actual \,\,\,Answer} \rfloor \}$. You will earn the number of points equal to $min\{N, max\{0, 30 - N\}\}$. [b]p35.[/b] Estimate the side length of a regular pentagon whose area is $2012$. You will earn the number of points equal to $max\{0, 15 - \lfloor 5 \times |Your \,\,\,Answer - Actual \,\,\,Answer| \rfloor \}$. [b]p36.[/b] Write down one integer between $1$ and $15$, inclusive. (If you do not, then you will receive $0$ points.) Let the number that you submit be $x$. Let $\overline{x}$ be the arithmetic mean of all of the valid numbers submitted by all of the teams. If $x > \overline{x}$, then you will receive $0$ points; otherwise, you will receive $x$ points. PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134177p28401527]here [/url] and 6-8 [url=https://artofproblemsolving.com/community/c3h3134466p28406321]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all pairs of twice differentiable functions $f,g \colon \mathbb{R} \to \mathbb{R}$, with their second derivative being continuous, such that the following holds for all $x,y \in \mathbb{R}$: \[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]

Given $a+b+c=5$ and $ 1 \le a, b, c \le 2 $, what is the minimum possible value of $\frac{1}{a+b}+\frac{1}{b+c}$?

[b]8.1[/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. Draw another straight line so that after that the rectangle was divided into the largest possible number of parts. [b]8.2[/b] Engineers always tell the truth, but businessmen always lie. F and G are engineers. A declares that, B asserts that, C asserts that, D says that, E insists that, F denies that G is an businessman. C also announces that D is a businessman. If A is a businessman, then how much total businessmen in this company? [b]8.3 [/b]There is a straight road through the field. A tourist stands on the road at a point ?. It can walk along the road at a speed of 6 km/h and across the field at a speed of 3 km/h. Find the locus of the points where the tourist can get there within an hour's walk. [b]8.4 / 7.5 [/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ . [b]8.5.[/b] In some state, every two cities are connected by a road. Each road is only allowed to move in one direction. Prove that there is a city from which you can travel around everything. state, having visited each city exactly once. [b]8.6[/b] Find all eights of prime numbers such that the sum of the squares of the numbers in the eight is 992 less than their quadruple product. [hide=original wording]Найдите все восьмерки простых чисел такие, что сумма квадратов чисел в восьмерке на 992 меньше, чем их учетверенное произведение.[/hide] PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].