Found problems: 29
2012 BMT Spring, round 2
[b]p1.[/b] $4$ balls are distributed uniformly at random among $6$ bins. What is the expected number of empty bins?
[b]p2.[/b] Compute ${150 \choose 20 }$ (mod $221$).
[b]p3.[/b] On the right triangle $ABC$, with right angle at$ B$, the altitude $BD$ is drawn. $E$ is drawn on $BC$ such that AE bisects angle $BAC$ and F is drawn on $AC$ such that $BF$ bisects angle $CBD$. Let the intersection of $AE$ and $BF$ be $G$. Given that $AB = 15$,$ BC = 20$, $AC = 25$, find $\frac{BG}{GF}$ .
[b]p4.[/b] What is the largest integer $n$ so that $\frac{n^2-2012}{n+7}$ is also an integer?
[b]p5.[/b] What is the side length of the largest equilateral triangle that can be inscribed in a regular pentagon with side length $1$?
[b]p6.[/b] Inside a LilacBall, you can find one of $7$ different notes, each equally likely. Delcatty must collect all $7$ notes in order to restore harmony and save Kanto from eternal darkness. What is the expected number of LilacBalls she must open in order to do so?
PS. You had better use hide for answers.
2021 BMT, 10
Let $N$ be the number of ways to draw 22 straight edges between 10 labeled points, of which no three are collinear, such that no triangle with vertices among these 10 points is created, and there is at most one edge between any two labeled points. Compute $\dfrac{N}{9!}$.
2020 BMT Fall, 10
For $k\ge 1$, define $a_k=2^k$. Let $$S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right).$$ Compute $\lfloor 100S\rfloor$.
2020 BMT Fall, 6
Given that $\tbinom{n}{k}=\tfrac{n!}{k!(n-k)!}$, the value of $$\sum_{n=3}^{10}\frac{\binom{n}{2}}{\binom{n}{3}\binom{n+1}{3}}$$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
2020 BMT Fall, 5
Let $f:\mathbb{R}^+\to \mathbb{R}^+$ be a function such that for all $x,y \in \mathbb{R}+,\, f(x)f(y)=f(xy)+f\left(\frac{x}{y}\right)$, where $\mathbb{R}^+$ represents the positive real numbers. Given that $f(2)=3$, compute the last two digits of $f\left(2^{2^{2020}}\right)$.
2020 BMT Fall, 8
Compute the smallest value $C$ such that the inequality $$x^2(1+y)+y^2(1+x)\le \sqrt{(x^4+4)(y^4+4)}+C$$ holds for all real $x$ and $y$.
2012 BMT Spring, round 4
[b]p1.[/b] Denote $S_n = 1 + \frac12 + \frac13 + ...+ \frac{1}{n}$. What is $144169\cdot S_{144169} - (S_1 + S_2 + ... + S_{144168})$?
[b]p2.[/b] Let $A,B,C$ be three collinear points, with $AB = 4$, $BC = 8$, and $AC = 12$. Draw circles with diameters $AB$, $BC$, and $AC$. Find the radius of the two identical circles that will lie tangent to all three circles.
[b]p3.[/b] Let $s(i)$ denote the number of $1$’s in the binary representation of $i$. What is $$\sum_{x=1}{314}\left( \sum_{i=0}^{2^{576}-2} x^{s(i)} \right)
\,\, mod \,\,629 ?$$
[b]p4.[/b] Parallelogram $ABCD$ has an area of $S$. Let $k = 42$. $E$ is drawn on AB such that $AE =\frac{AB}{k}$ . $F$ is drawn on $CD$ such that $CF = \frac{CD}{k}$ . $G$ is drawn on $BC$ such that $BG = \frac{BC}{k}$ . $H$ is drawn on $AD$ such that $DH = \frac{AD}{k}$ . Line $CE$ intersects $BH$ at $M$, and $DG$ at $N$. Line $AF$ intersects $DG$ at $P$, and $BH$ at $Q$. If $S_1$ is the area of quadrilateral $MNPQ$, find $\frac{S_1}{S}$.
[b]p5.[/b] Let $\phi$ be the Euler totient function. What is the sum of all $n$ for which $\frac{n}{\phi(n)}$ is maximal for $1 \le n \le 500$?
[b]p6.[/b] Link starts at the top left corner of an $12 \times 12$ grid and wants to reach the bottom right corner. He can only move down or right. A turn is defined a down move immediately followed by a right move, or a right move immediately followed by a down move. Given that he makes exactly $6$ turns, in how many ways can he reach his destination?
PS. You had better use hide for answers.
2023 BMT, 19-21
[center][u]Guts Round[/u] / [u]Set 7[/u][/center]
[b]p19.[/b] Let $N_{21}$ be the answer to question 21.
Suppose a jar has $3N_{21}$ colored balls in it: $N_{21}$ red, $N_{21}$ green, and $N_{21}$ blue balls. Jonathan takes one ball at a time out of the jar uniformly at random without replacement until all the balls left in the jar are the same color. Compute the expected number of balls left in the jar after all balls are the same color.
[b]p20.[/b] Let $N_{19}$ be the answer to question 19.
For every non-negative integer $k$, define $$f_k(x) = x(x - 1) + (x + 1)(x - 2) + ...+ (x + k)(x - k - 1),$$ and let $r_k$ and $s_k$ be the two roots of $f_k(x)$. Compute the smallest positive integer $m$ such that $|r_m - s_m| > 10N_{19}$.
[b]p21.[/b] Let $N_{20}$ be the answer to question 20.
In isosceles trapezoid $ABCD$ (where $\overline{BC}$ and $\overline{AD}$ are parallel to each other), the angle bisectors of $A$ and $D$ intersect at $F$, and the angle bisectors of points $B$ and $C$ intersect at $H$. Let $\overline{BH}$ and $\overline{AF}$ intersect at $E$, and let $\overline{CH}$ and $\overline{DF}$ intersect at $G$. If $CG = 3$, $AE = 15$, and $EG = N_{20}$, compute the area of the quadrilateral formed by the four tangency points of the largest circle that can fit inside quadrilateral $EFGH$.
2021 BMT, 1
Towa has a hand of three different red cards and three different black cards. How many ways can Towa pick a set of three cards from her hand that uses at least one card of each color?
2020 BMT Fall, 9
There is a unique triple $(a,b,c)$ of two-digit positive integers $a,\,b,$ and $c$ that satisfy the equation $$a^3+3b^3+9c^3=9abc+1.$$ Compute $a+b+c$.
2021 BMT, 7
For a given positive integer $n$, you may perform a series of steps. At each step, you may apply an operation: you may increase your number by one, or if your number is divisible by 2, you may divide your number by 2. Let $\ell(n)$ be the minimum number of operations needed to transform the number $n$ to 1 (for example, $\ell(1) = 0$ and $\ell(7) = 4$). How many positive integers $n$ are there such that $\ell(n) \leq 12$?
2021 BMT, 5
How many three-digit numbers $\underline{abc}$ have the property that when it is added to $\underline{cba}$, the number obtained by reversing its digits, the result is a palindrome? (Note that $\underline{cba}$ is not necessarily a three-digit number since before reversing, $c$ may be equal to $0$.)
2020 BMT Fall, 1
Marisela is putting on a juggling show! She starts with $1$ ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute after she starts juggling.
2012 BMT Spring, Championship
[b]p1.[/b] If $n$ is a positive integer such that $2n+1 = 144169^2$, find two consecutive numbers whose squares add up to $n + 1$.
[b]p2.[/b] Katniss has an $n$-sided fair die which she rolls. If $n > 2$, she can either choose to let the value rolled be her score, or she can choose to roll a $n - 1$ sided fair die, continuing the process. What is the expected value of her score assuming Katniss starts with a $6$ sided die and plays to maximize this expected value?
[b]p3.[/b] Suppose that $f(x) = x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f$, and that $f(1) = f(2) = f(3) = f(4) = f(5) = f(6) = 7$. What is $a$?
[b]p4.[/b] $a$ and $b$ are positive integers so that $20a+12b$ and $20b-12a$ are both powers of $2$, but $a+b$ is not. Find the minimum possible value of $a + b$.
[b]p5.[/b] Square $ABCD$ and rhombus $CDEF$ share a side. If $m\angle DCF = 36^o$, find the measure of $\angle AEC$.
[b]p6.[/b] Tom challenges Harry to a game. Tom first blindfolds Harry and begins to set up the game. Tom places $4$ quarters on an index card, one on each corner of the card. It is Harry’s job to flip all the coins either face-up or face-down using the following rules:
(a) Harry is allowed to flip as many coins as he wants during his turn.
(b) A turn consists of Harry flipping as many coins as he wants (while blindfolded). When he is happy with what he has flipped, Harry will ask Tom whether or not he was successful in flipping all the coins face-up or face-down. If yes, then Harry wins. If no, then Tom will take the index card back, rotate the card however he wants, and return it back to Harry, thus starting Harry’s next turn. Note that Tom cannot touch the coins after he initially places them before starting the game.
Assuming that Tom’s initial configuration of the coins weren’t all face-up or face-down, and assuming that Harry uses the most efficient algorithm, how many moves maximum will Harry need in order to win? Or will he never win?
PS. You had better use hide for answers.
2023 BMT, 25
Let triangle $\vartriangle ABC$ have side lengths $AB = 6$, $BC = 8$, and $CA = 10$. Let $S_1$ be the largest square fitting inside of $\vartriangle ABC$ (sharing points on edges is allowed). Then, for $i \ge 2$, let $S_i$ be the largest square that fits inside of $\vartriangle ABC$ while remaining outside of all other squares $S_1$,$...$, $S_{i-1}$ (with ties broken arbitrarily). For all $i \ge 1$, let $m_i$ be the side length of $S_i$ and let $S$ be the set of all $m_i$. Let $x$ be the $2023$rd largest value in $S$. Compute $\log_2 \left( \frac{1}{x}\right).$
Submit your answer as a decimal $E$ to at most $3$ decimal places. If the correct answer is $A$, your score for this question will be $\max(0, 25 -2|A - E|)$, rounded to the nearest integer
2020 BMT Fall, 3
The graph of the degree $2021$ polynomial $P(x)$, which has real coefficients and leading coefficient $1$, meets the $x$-axis at the points $(1,0),\, (2,0),\,(3,0),\dots,\, (2020,0)$ and nowhere else. The mean of all possible values of $P(2021)$ can be written in the form $a!/b$, where $a$ and $b$ are positive integers and $a$ is as small as possible. Compute $a+b$.
2012 BMT Spring, Consolation
[b]p1.[/b] How many ways can we arrange the elements $\{1, 2, ..., n\}$ to a sequence $a_1, a_2, ..., a_n$ such that there is only exactly one $a_i$, $a_{i+1}$ such that $a_i > a_{i+1}$?
[b]p2. [/b]How many distinct (non-congruent) triangles are there with integer side-lengths and perimeter $2012$?
[b]p3.[/b] Let $\phi$ be the Euler totient function, and let $S = \{x| \frac{x}{\phi (x)} = 3\}$. What is $\sum_{x\in S} \frac{1}{x}$?
[b]p4.[/b] Denote $f(N)$ as the largest odd divisor of $N$. Compute $f(1) + f(2) + f(3) +... + f(29) + f(30)$.
[b]p5.[/b] Triangle $ABC$ has base $AC$ equal to $218$ and altitude $100$. Squares $s_1, s_2, s_3, ...$ are drawn such that $s_1$ has a side on $AC$ and has one point each touching $AB$ and $BC$, and square $s_k$ has a side on square $s_{k-}1$ and also touches $AB$ and $BC$ exactly once each. What is the sum of the area of these squares?
[b]p6.[/b] Let $P$ be a parabola $6x^2 - 28x + 10$, and $F$ be the focus. A line $\ell$ passes through $F$ and intersects the parabola twice at points $P_1 = (2,-22)$, $P_2$. Tangents to the parabola with points at $P_1, P_2$ are then drawn, and intersect at a point $Q$. What is $m\angle P_1QP_2$?
PS. You had better use hide for answers.
2021 BMT, 6
Compute the sum of all positive integers $n$ such that $n^n$ has 325 positive integer divisors. (For example, $4^4=256$ has 9 positive integer divisors: 1, 2, 4, 8, 16, 32, 64, 128, 256.)
2021 BMT, 2
Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob?
2022 BMT, 4
Big Chungus has been thinking of a new symbol for BMT, and the drawing below is what he came up with. If each of the $16$ small squares in the grid are unit squares, what is the area of the shaded region?
2012 BMT Spring, round 1
[b]p1.[/b] Find all prime factors of $8051$.
[b]p2.[/b] Simplify $$[\log_{xyz}(x^z)][1 + \log_x y + \log_x z],$$ where $x = 628$, $y = 233$, $z = 340$.
[b]p3.[/b] In prokaryotes, translation of mRNA messages into proteins is most often initiated at start codons on the mRNA having the sequence AUG. Assume that the mRNA is single-stranded and consists of a sequence of bases, each described by a single letter A,C,U, or G.
Consider the set of all pieces of bacterial mRNA six bases in length. How many such mRNA sequences have either no A’s or no U’s?
[b]p4.[/b] What is the smallest positive $n$ so that $17^n + n$ is divisible by $29$?
[b]p5.[/b] The legs of the right triangle shown below have length $a = 255$ and $b = 32$. Find the area of the smaller rectangle (the one labeled $R$).
[img]https://cdn.artofproblemsolving.com/attachments/c/d/566f2ce631187684622dfb43f36c7e759e2f34.png[/img]
[b]p6.[/b] A $3$ dimensional cube contains ”cubes” of smaller dimensions, ie: faces ($2$-cubes),edges ($1$-cubes), and vertices ($0$-cubes). How many 3-cubes are in a $5$-cube?
PS. You had better use hide for answers.
2021 BMT, 9
Let $p=101.$ The sum
\[\sum_{k=1}^{10}\frac1{\binom pk}\]
can be written as a fraction of the form $\dfrac a{p!},$ where $a$ is a positive integer. Compute $a\pmod p.$
2012 BMT Spring, round 3
[b]p1.[/b] Let $A(S)$ denote the average value of a set $S$. Let $T$ be the set of all subsets of the set $\{1, 2, 3, 4, ... , 2012\}$, and let $R$ be $\{A(K)|K \in T \}$. Compute $A(R)$.
[b]p2.[/b] Consider the minute and hour hands of the Campanile, our clock tower. During one single day ($12:00$ AM - $12:00$ AM), how many times will the minute and hour hands form a right-angle at the center of the clock face?
[b]p3.[/b] In a regular deck of $52$ face-down cards, Billy flips $18$ face-up and shuffles them back into the deck. Before giving the deck to Penny, Billy tells her how many cards he has flipped over, and blindfolds her so she can’t see the deck and determine where the face-up cards are. Once Penny is given the deck, it is her job to split the deck into two piles so that both piles contain the same number of face-up cards. Assuming that she knows how to do this, how many cards should be in each pile when he is done?
[b]p4.[/b] The roots of the equation $x^3 + ax^2 + bx + c = 0$ are three consecutive integers. Find the maximum value of $\frac{a^2}{b+1}$.
[b]p5.[/b] Oski has a bag initially filled with one blue ball and one gold ball. He plays the following game: first, he removes a ball from the bag. If the ball is blue, he will put another blue ball in the bag with probability $\frac{1}{437}$ and a gold ball in the bag the rest of the time. If the ball is gold, he will put another gold ball in the bag with probability $\frac{1}{437}$ and a blue ball in the bag the rest of the time. In both cases, he will put the ball he drew back into the bag. Calculate the expected number of blue balls after $525600$ iterations of this game.
[b]p6.[/b] Circles $A$ and $B$ intersect at points $C$ and $D$. Line $AC$ and circle $B$ meet at $E$, line $BD$ and circle $A$ meet at $F$, and lines $EF$ and $AB$ meet at $G$. If $AB = 10$, $EF = 4$, $FG = 8$, find $BG$.
PS. You had better use hide for answers.
2020 BMT Fall, 4
Let $\varphi$ be the positive solution to the equation $$x^2=x+1.$$ For $n\ge 0$, let $a_n$ be the unique integer such that $\varphi^n-a_n\varphi$ is also an integer. Compute $$\sum_{n=0}^{10}a_n.$$
2020 BMT Fall, 7
Let $a,\,b,$ and $c$ be real numbers such that $a+b+c=\frac1{a}+\frac1{b}+\frac1{c}$ and $abc=5$. The value of $$\left(a-\frac1{b}\right)^3+\left(b-\frac1{c}\right)^3+\left(c-\frac1{a}\right)^3$$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.