Evaluate $\frac{\frac{1}{\frac{1}{10} - \frac{1}{12}}}{\frac{1}{\frac{1}{8} - \frac{1}{6}} + \frac{1}{\frac{1}{5} - \frac{1}{6}}}$.

Find all four digit positive integers such that the sum of the squares of the digits equals twice the sum of the digits.

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [u] Set 1[/u] [b]R1.1 / P1.1[/b] Find $291 + 503 - 91 + 492 - 103 - 392$. [b]R1.2[/b] Let the operation $a$ & $b$ be defined to be $\frac{a-b}{a+b}$. What is $3$ & $-2$? [b]R1.3[/b]. Joe can trade $5$ apples for $3$ oranges, and trade $6$ oranges for $5$ bananas. If he has $20$ apples, what is the largest number of bananas he can trade for? [b]R1.4[/b] A cone has a base with radius $3$ and a height of $5$. What is its volume? Express your answer in terms of $\pi$. [b]R1.5[/b] Guang brought dumplings to school for lunch, but by the time his lunch period comes around, he only has two dumplings left! He tries to remember what happened to the dumplings. He first traded $\frac34$ of his dumplings for Arman’s samosas, then he gave $3$ dumplings to Anish, and lastly he gave David $\frac12$ of the dumplings he had left. How many dumplings did Guang bring to school? [u]Set 2[/u] [b]R2.6 / P1.3[/b] In the recording studio, Kanye has $10$ different beats, $9$ different manuscripts, and 8 different samples. If he must choose $1$ beat, $1$ manuscript, and $1$ sample for his new song, how many selections can he make? [b]R2.7[/b] How many lines of symmetry does a regular dodecagon (a polygon with $12$ sides) have? [b]R2.8[/b] Let there be numbers $a, b, c$ such that $ab = 3$ and $abc = 9$. What is the value of $c$? [b]R2.9[/b] How many odd composite numbers are there between $1$ and $20$? [b]R2.10[/b] Consider the line given by the equation $3x - 5y = 2$. David is looking at another line of the form ax - 15y = 5, where a is a real number. What is the value of a such that the two lines do not intersect at any point? [u]Set 3[/u] [b]R3.11[/b] Let $ABCD$ be a rectangle such that $AB = 4$ and $BC = 3$. What is the length of BD? [b]R3.12[/b] Daniel is walking at a constant rate on a $100$-meter long moving walkway. The walkway moves at $3$ m/s. If it takes Daniel $20$ seconds to traverse the walkway, find his walking speed (excluding the speed of the walkway) in m/s. [b]R3.13 / P1.3[/b] Pratik has a $6$ sided die with the numbers $1, 2, 3, 4, 6$, and $12$ on the faces. He rolls the die twice and records the two numbers that turn up on top. What is the probability that the product of the two numbers is less than or equal to $12$? [b]R3.14 / P1.5[/b] Find the two-digit number such that the sum of its digits is twice the product of its digits. [b]R3.15[/b] If $a^2 + 2a = 120$, what is the value of $2a^2 + 4a + 1$? PS. You should use hide for answers. R16-30 /P6-10/ P26-30 have been posted [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here[/url], and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ n$ be an even positive integer. Prove that there exists a positive inter $ k$ such that \[ k \equal{} f(x) \cdot (x\plus{}1)^n \plus{} g(x) \cdot (x^n \plus{} 1)\] for some polynomials $ f(x), g(x)$ having integer coefficients. If $ k_0$ denotes the least such $ k,$ determine $ k_0$ as a function of $ n,$ i.e. show that $ k_0 \equal{} 2^q$ where $ q$ is the odd integer determined by $ n \equal{} q \cdot 2^r, r \in \mathbb{N}.$ Note: This is variant A6' of the three variants given for this problem.

The $A$-excircle of a triangle $ABC$ touches the side $BC$ at the point $K$ and the extended side $AB$ at the point $L$. The $B$-excircle touches the lines $BA$ and $BC$ at the points $M$ and $N$, respectively. The lines $KL$ and $MN$ meet at the point $X$. Show that the line $CX$ bisects the angle $ACN$.

The vertex $F$ of the parallelogram $ACEF$ lies on the side $BC$ of parallelogram $ABCD$. It is known that $AC = AD$ and $AE = 2CD$. Prove that $\angle CDE = \angle BEF$.

On a board, Ana and Bob start writing $0$s and $1$s alternatively until each of them has written $2021$ digits. Ana starts this procedure and each of them always adds a digit to the right of the already existing ones. Ana wins the game if, after they stop writing, the resulting number (in binary) can be written as the sum of two squares. Otherwise, Bob wins. Determine who has a winning strategy.

We say positive integer $n$ is $\emph{metallic}$ if there is no prime of the form $m^2-n$. What is the sum of the three smallest metallic integers? [i] Proposed by Lewis Chen [/i]

Let $ \left( a_n \right)_{n\ge 1} $ be a non-constant arithmetic progression of positive numbers and $ \left( g_n \right)_{n\ge 1} $ be a non-constant geometric progression of positive numbers satisfying $ a_1=g_1 $ and $ a_{2019} =g_{2019} . $ Specify the set $ \left\{ k\in\mathbb{N} \big| a_k\le g_k \right\} $ and prove that it bijects the natural numbers. [i]Gheorghe Rotariu[/i]

Ana draws a checkered board that has at least 20 rows and at least 24 columns. Then, Beto must completely cover that board, without holes or overlaps, using only pieces of the following two types: Each piece must cover exactly 4 or 3 squares of the board, as shown in the figure, without leaving the board. It is permitted to rotate the pieces and it is not necessary to use all types of pieces. Explain why, regardless of how many rows and how many columns Ana's board has, Beto can always complete his task.

Let $ABC$ be a triangle with $AB<BC<CA$ and let $D$ be a variable point on $BC$. The point $E$ on the circumcircle of $ABC$ is such that $\angle BAD=\angle BED$. The line through $D$ perpendicular to $AB$ meets $AC$ at $F$. Show that the measure of $\angle BEF$ is constant as $D$ varies.

Given is triangle $ABC$ with incenter $I$ and $A$-excenter $J$. Circle $\omega_b$ centered at point $O_b$ passes through point $B$ and is tangent to line $CI$ at point $I$. Circle $\omega_c$ with center $O_c$ passes through point $C$ and touches line $BI$ at point $I$. Let $O_bO_c$ and $IJ$ intersect at point $K$. Find the ratio $IK/KJ$.

Two turtles, the leader and the slave, are crawling along the plane from point $A$ to point $B$. They crawl in turn: first the leader crawls some distance, then the slave crawls some distance in a straight line towards the leading one. Then the leader crawls somewhere again, after which the slave crawls towards the leader, etc. Finally, they both crawl to $B$. Prove that the slave turtle crawled no more than the leading one.

Let $a>2$, $n>1$ integers such that $a^n-2^n$ is a perfect square. Prove that $a$ is a even number.

Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions: (1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$; (2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$; (2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$. Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.

Find all triples of real numbers $(a,b,c)$ for which the set of solutions $x$ of $\sqrt{2x^2 +ax+b} > x-c$ is the set $(-\infty,0]\cup(1,\infty)$.

Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \] are pairwise disjoint.

Given $ n$ points in a line so that any distance occurs at most twice, show that the number of distance occurring exactly once is at least $ \lfloor n/2 \rfloor$. [i]V. T. Sos, L. Szekely[/i]

In acute triangle $\triangle ABC$, $H$ is the orthocenter, $BD$,$CE$ are altitudes. $M$ is the midpoint of $BC$. $P$,$Q$ are on segment $BM$,$DE$, respectively. $R$ is on segment $PQ$ such that $\frac{BP}{EQ}=\frac{CP}{DQ}=\frac{PR}{QR}$. Suppose $L$ is the orthocenter of $\triangle AHR$, then prove: $QM$ passes through the midpoint of $RL$.

Suppose that $(f_n)_{n\in N}$ be the sequence from all functions $f_n:[0,1]\rightarrow \mathbb{R^+}$ s.t. $f_0$ be the continuous function and $\forall x\in [0,1] , \forall n\in \mathbb {N} , f_{n+1}(x)=\int_0^x \frac {1}{1+f_n (t)}dt$. Prove that for every $x\in [0,1]$ the sequence of $(f_n(x))_{n\in N}$ be the convergent sequence and calculate the limitation.

If $\xi_1,\ldots,\xi_n$ denote the $n$-th roots of unity, evaluate $$\prod_{1\leq i<j \leq n} (\xi_{i}-\xi_j )^2 .$$

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]D.16 / L.6[/b] Alex has $100$ Bluffy Funnies in some order, which he wants to sort in order of height. They’re already almost in order: each Bluffy Funny is at most $1$ spot off from where it should be. Alex can only swap pairs of adjacent Bluffy Funnies. What is the maximum possible number of swaps necessary for Alex to sort them? [b]D.17[/b] I start with the number $1$ in my pocket. On each round, I flip a coin. If the coin lands heads heads, I double the number in my pocket. If it lands tails, I divide it by two. After five rounds, what is the expected value of the number in my pocket? [b]D.18 / L.12[/b] Point $P$ inside square $ABCD$ is connected to each corner of the square, splitting the square into four triangles. If three of these triangles have area $25$, $25$, and $15$, what are all the possible values for the area of the fourth triangle? [b]D.19[/b] Mr. Stein and Mr. Schwartz are playing a yelling game. The teachers alternate yelling. Each yell is louder than the previous and is also relatively prime to the previous. If any teacher yells at $100$ or more decibels, then they lose the game. Mr. Stein yells first, at $88$ decibels. What volume, in decibels, should Mr. Schwartz yell at to guarantee that he will win? [b]D.20 / L.15[/b] A semicircle of radius $1$ has line $\ell$ along its base and is tangent to line $m$. Let $r$ be the radius of the largest circle tangent to $\ell$, $m$, and the semicircle. As the point of tangency on the semicircle varies, the range of possible values of $r$ is the interval $[a, b]$. Find $b - a$. [u]Set 5[/u] [b]D.21 / L.14[/b] Hungryman starts at the tile labeled “$S$”. On each move, he moves $1$ unit horizontally or vertically and eats the tile he arrives at. He cannot move to a tile he already ate, and he stops when the sum of the numbers on all eaten tiles is a multiple of nine. Find the minimum number of tiles that Hungryman eats. [img]https://cdn.artofproblemsolving.com/attachments/e/7/c2ecc2a872af6c4a07907613c412d3b86cd7bc.png [/img] [b]D.22 / L.11[/b] How many triples of nonnegative integers $(x, y, z)$ satisfy the equation $6x + 10y +15z = 300$? [b]D.23 / L.16[/b] Anson, Billiam, and Connor are looking at a $3D$ figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a $5 \times 5$ square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure. [b]D.24 / L.13[/b] Tse and Cho are playing a game. Cho chooses a number $x \in [0, 1]$ uniformly at random, and Tse guesses the value of $x(1 - x)$. Tse wins if his guess is at most $\frac{1}{50}$ away from the correct value. Given that Tse plays optimally, what is the probability that Tse wins? [b]D.25 / L.20[/b] Find the largest solution to the equation $$2019(x^{2019x^{2019}-2019^2+2019})^{2019}) = 2019^{x^{2019}+1}.$$ [u]Set 6[/u] [i]This round is an estimation round. No one is expected to get an exact answer to any of these questions, but unlike other rounds, you will get points for being close. In the interest of transparency, the formulas for determining the number of points you will receive are located on the answer sheet, but they aren’t very important when solving these problems.[/i] [b]D.26 / L.26[/b] What is the sum over all MBMT volunteers of the number of times that volunteer has attended MBMT (as a contestant or as a volunteer, including this year)? Last year there were $47$ volunteers; this is the fifth MBMT. [b]D.27 / L.27[/b] William is sharing a chocolate bar with Naveen and Kevin. He first randomly picks a point along the bar and splits the bar at that point. He then takes the smaller piece, randomly picks a point along it, splits the piece at that point, and gives the smaller resulting piece to Kevin. Estimate the probability that Kevin gets less than $10\%$ of the entire chocolate bar. [b]D.28 / L.28[/b] Let $x$ be the positive solution to the equation $x^{x^{x^x}}= 1.1$. Estimate $\frac{1}{x-1}$. [b]D.29 / L.29[/b] Estimate the number of dots in the following box: [img]https://cdn.artofproblemsolving.com/attachments/8/6/416ba6379d7dfe0b6302b42eff7de61b3ec0f1.png[/img] It may be useful to know that this image was produced by plotting $(4\sqrt{x}, y)$ some number of times, where x, y are random numbers chosen uniformly randomly and independently from the interval $[0, 1]$. [b]D.30 / L.30[/b] For a positive integer $n$, let $f(n)$ be the smallest prime greater than or equal to $n$. Estimate $$(f(1) - 1) + (f(2) - 2) + (f(3) - 3) + ...+ (f(10000) - 10000).$$ For $26 \le i \le 30$, let $E_i$ be your team’s answer to problem $i$ and let $A_i$ be the actual answer to problem $i$. Your score $S_i$ for problem $i$ is given by $S_{26} = \max(0, 12 - |E_{26} - A_{26}|/5)$ $S_{27} = \max(0, 12 - 100|E_{27} - A_{27}|)$ $S_{28} = \max(0, 12 - 5|E_{28} - A_{28}|))$ $S_{29} = 12 \max \left(0, 1 - 3 \frac{|E_{29} - A_{29}|}{A_{29}} \right)$ $S_{30} = \max (0, 12 - |E_{30} - A_{30}|/2000)$ PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve the following equation for real numbers: $ \sqrt{4\minus{}x\sqrt{4\minus{}(x\minus{}2)\sqrt{1\plus{}(x\minus{}5)(x\minus{}7)}}}\equal{}\frac{5x\minus{}6\minus{}x^2}{2}$ (all square roots are non negative)

Find all positive integer number $n$ such that $[\sqrt{n}]-2$ divides $n-4$ and $[\sqrt{n}]+2$ divides $n+4$. Note: $[r]$ denotes the integer part of $r$.

Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.