Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
3,500 |
Determine the number of subsets $S$ of $\{1,2, \ldots, 1000\}$ that satisfy the following conditions: - $S$ has 19 elements, and - the sum of the elements in any non-empty subset of $S$ is not divisible by 20 .
|
8 \cdot\binom{50}{19}
| 0 |
3,501 |
The mathematician John is having trouble remembering his girlfriend Alicia's 7-digit phone number. He remembers that the first four digits consist of one 1, one 2, and two 3s. He also remembers that the fifth digit is either a 4 or 5. While he has no memory of the sixth digit, he remembers that the seventh digit is 9 minus the sixth digit. If this is all the information he has, how many phone numbers does he have to try if he is to make sure he dials the correct number?
|
240
| 89.0625 |
3,502 |
Find all the integers $n>1$ with the following property: the numbers $1,2, \ldots, n$ can be arranged in a line so that, of any two adjacent numbers, one is divisible by the other.
|
2, 3, 4, 6
| 13.28125 |
3,503 |
Let $S$ be a set of size 3. How many collections $T$ of subsets of $S$ have the property that for any two subsets $U \in T$ and $V \in T$, both $U \cap V$ and $U \cup V$ are in $T$ ?
|
74
| 0 |
3,504 |
Let $S$ be the set of all positive factors of 6000. What is the probability of a random quadruple $(a, b, c, d) \in S^{4}$ satisfies $$\operatorname{lcm}(\operatorname{gcd}(a, b), \operatorname{gcd}(c, d))=\operatorname{gcd}(\operatorname{lcm}(a, b), \operatorname{lcm}(c, d)) ?$$
|
\frac{41}{512}
| 0 |
3,505 |
In the Year 0 of Cambridge there is one squirrel and one rabbit. Both animals multiply in numbers quickly. In particular, if there are $m$ squirrels and $n$ rabbits in Year $k$, then there will be $2 m+2019$ squirrels and $4 n-2$ rabbits in Year $k+1$. What is the first year in which there will be strictly more rabbits than squirrels?
|
13
| 73.4375 |
3,506 |
Milan has a bag of 2020 red balls and 2021 green balls. He repeatedly draws 2 balls out of the bag uniformly at random. If they are the same color, he changes them both to the opposite color and returns them to the bag. If they are different colors, he discards them. Eventually the bag has 1 ball left. Let $p$ be the probability that it is green. Compute $\lfloor 2021 p \rfloor$.
|
2021
| 60.9375 |
3,507 |
Complex numbers $a, b, c$ form an equilateral triangle with side length 18 in the complex plane. If $|a+b+c|=36$, find $|b c+c a+a b|$.
|
432
| 37.5 |
3,508 |
A regular hexagon $A B C D E F$ has side length 1 and center $O$. Parabolas $P_{1}, P_{2}, \ldots, P_{6}$ are constructed with common focus $O$ and directrices $A B, B C, C D, D E, E F, F A$ respectively. Let $\chi$ be the set of all distinct points on the plane that lie on at least two of the six parabolas. Compute $$\sum_{X \in \chi}|O X|$$ (Recall that the focus is the point and the directrix is the line such that the parabola is the locus of points that are equidistant from the focus and the directrix.)
|
35 \sqrt{3}
| 0 |
3,509 |
Let $x<y$ be positive real numbers such that $\sqrt{x}+\sqrt{y}=4$ and $\sqrt{x+2}+\sqrt{y+2}=5$. Compute $x$.
|
\frac{49}{36}
| 6.25 |
3,510 |
Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function such that for any integers $x, y$, we have $f\left(x^{2}-3 y^{2}\right)+f\left(x^{2}+y^{2}\right)=2(x+y) f(x-y)$. Suppose that $f(n)>0$ for all $n>0$ and that $f(2015) \cdot f(2016)$ is a perfect square. Find the minimum possible value of $f(1)+f(2)$.
|
246
| 0 |
3,511 |
Determine the number of unordered triples of distinct points in the $4 \times 4 \times 4$ lattice grid $\{0,1,2,3\}^{3}$ that are collinear in $\mathbb{R}^{3}$ (i.e. there exists a line passing through the three points).
|
376
| 21.09375 |
3,512 |
Find the number of pentominoes (5-square polyominoes) that span a 3-by-3 rectangle, where polyominoes that are flips or rotations of each other are considered the same polyomino.
|
6
| 12.5 |
3,513 |
Given two distinct points $A, B$ and line $\ell$ that is not perpendicular to $A B$, what is the maximum possible number of points $P$ on $\ell$ such that $A B P$ is an isosceles triangle?
|
5
| 41.40625 |
3,514 |
Let $f: \mathbb{Z}^{2} \rightarrow \mathbb{Z}$ be a function such that, for all positive integers $a$ and $b$, $$f(a, b)= \begin{cases}b & \text { if } a>b \\ f(2 a, b) & \text { if } a \leq b \text { and } f(2 a, b)<a \\ f(2 a, b)-a & \text { otherwise }\end{cases}$$ Compute $f\left(1000,3^{2021}\right)$.
|
203
| 10.9375 |
3,515 |
I have written a strictly increasing sequence of six positive integers, such that each number (besides the first) is a multiple of the one before it, and the sum of all six numbers is 79 . What is the largest number in my sequence?
|
48
| 59.375 |
3,516 |
Compute the number of real solutions $(x, y, z, w)$ to the system of equations: $$\begin{array}{rlrl} x & =z+w+z w x & z & =x+y+x y z \\ y & =w+x+w x y & w & =y+z+y z w \end{array}$$
|
5
| 14.0625 |
3,517 |
Determine the value of $$1 \cdot 2-2 \cdot 3+3 \cdot 4-4 \cdot 5+\cdots+2001 \cdot 2002$$
|
2004002
| 54.6875 |
3,518 |
A convex quadrilateral is drawn in the coordinate plane such that each of its vertices $(x, y)$ satisfies the equations $x^{2}+y^{2}=73$ and $x y=24$. What is the area of this quadrilateral?
|
110
| 13.28125 |
3,519 |
Yannick picks a number $N$ randomly from the set of positive integers such that the probability that $n$ is selected is $2^{-n}$ for each positive integer $n$. He then puts $N$ identical slips of paper numbered 1 through $N$ into a hat and gives the hat to Annie. Annie does not know the value of $N$, but she draws one of the slips uniformly at random and discovers that it is the number 2. What is the expected value of $N$ given Annie's information?
|
\frac{1}{2 \ln 2-1}
| 0 |
3,520 |
For integers $a, b, c, d$, let $f(a, b, c, d)$ denote the number of ordered pairs of integers $(x, y) \in \{1,2,3,4,5\}^{2}$ such that $a x+b y$ and $c x+d y$ are both divisible by 5. Find the sum of all possible values of $f(a, b, c, d)$.
|
31
| 0 |
3,521 |
Let $AD, BE$, and $CF$ be segments sharing a common midpoint, with $AB < AE$ and $BC < BF$. Suppose that each pair of segments forms a $60^{\circ}$ angle, and that $AD=7, BE=10$, and $CF=18$. Let $K$ denote the sum of the areas of the six triangles $\triangle ABC, \triangle BCD, \triangle CDE, \triangle DEF, \triangle EFA$, and $\triangle FAB$. Compute $K \sqrt{3}$.
|
141
| 2.34375 |
3,522 |
Two integers are relatively prime if they don't share any common factors, i.e. if their greatest common divisor is 1. Define $\varphi(n)$ as the number of positive integers that are less than $n$ and relatively prime to $n$. Define $\varphi_{d}(n)$ as the number of positive integers that are less than $d n$ and relatively prime to $n$. What is the least $n$ such that $\varphi_{x}(n)=64000$, where $x=\varphi_{y}(n)$, where $y=\varphi(n)$?
|
41
| 6.25 |
3,523 |
For each positive integer $1 \leq m \leq 10$, Krit chooses an integer $0 \leq a_{m}<m$ uniformly at random. Let $p$ be the probability that there exists an integer $n$ for which $n \equiv a_{m}(\bmod m)$ for all $m$. If $p$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.
|
1540
| 1.5625 |
3,524 |
A regular hexagon PROFIT has area 1. Every minute, greedy George places the largest possible equilateral triangle that does not overlap with other already-placed triangles in the hexagon, with ties broken arbitrarily. How many triangles would George need to cover at least $90 \%$ of the hexagon's area?
|
46
| 0 |
3,525 |
The pairwise products $a b, b c, c d$, and $d a$ of positive integers $a, b, c$, and $d$ are $64,88,120$, and 165 in some order. Find $a+b+c+d$.
|
42
| 33.59375 |
3,526 |
Bob knows that Alice has 2021 secret positive integers $x_{1}, \ldots, x_{2021}$ that are pairwise relatively prime. Bob would like to figure out Alice's integers. He is allowed to choose a set $S \subseteq\{1,2, \ldots, 2021\}$ and ask her for the product of $x_{i}$ over $i \in S$. Alice must answer each of Bob's queries truthfully, and Bob may use Alice's previous answers to decide his next query. Compute the minimum number of queries Bob needs to guarantee that he can figure out each of Alice's integers.
|
11
| 10.15625 |
3,527 |
A positive integer $n$ is picante if $n$ ! ends in the same number of zeroes whether written in base 7 or in base 8 . How many of the numbers $1,2, \ldots, 2004$ are picante?
|
4
| 42.96875 |
3,528 |
Let $A B C$ be a triangle with $A B=3, B C=4$, and $C A=5$. Let $A_{1}, A_{2}$ be points on side $B C$, $B_{1}, B_{2}$ be points on side $C A$, and $C_{1}, C_{2}$ be points on side $A B$. Suppose that there exists a point $P$ such that $P A_{1} A_{2}, P B_{1} B_{2}$, and $P C_{1} C_{2}$ are congruent equilateral triangles. Find the area of convex hexagon $A_{1} A_{2} B_{1} B_{2} C_{1} C_{2}$.
|
\frac{12+22 \sqrt{3}}{15}
| 0 |
3,529 |
Ann and Anne are in bumper cars starting 50 meters apart. Each one approaches the other at a constant ground speed of $10 \mathrm{~km} / \mathrm{hr}$. A fly starts at Ann, flies to Anne, then back to Ann, and so on, back and forth until it gets crushed when the two bumper cars collide. When going from Ann to Anne, the fly flies at $20 \mathrm{~km} / \mathrm{hr}$; when going in the opposite direction the fly flies at $30 \mathrm{~km} / \mathrm{hr}$ (thanks to a breeze). How many meters does the fly fly?
|
55
| 0 |
3,530 |
Bob is coloring lattice points in the coordinate plane. Find the number of ways Bob can color five points in $\{(x, y) \mid 1 \leq x, y \leq 5\}$ blue such that the distance between any two blue points is not an integer.
|
80
| 56.25 |
3,531 |
A permutation of a finite set is a one-to-one function from the set to itself; for instance, one permutation of $\{1,2,3,4\}$ is the function $\pi$ defined such that $\pi(1)=1, \pi(2)=3$, $\pi(3)=4$, and $\pi(4)=2$. How many permutations $\pi$ of the set $\{1,2, \ldots, 10\}$ have the property that $\pi(i) \neq i$ for each $i=1,2, \ldots, 10$, but $\pi(\pi(i))=i$ for each $i$?
|
945
| 46.875 |
3,532 |
Compute the number of complex numbers $z$ with $|z|=1$ that satisfy $$1+z^{5}+z^{10}+z^{15}+z^{18}+z^{21}+z^{24}+z^{27}=0$$
|
11
| 0 |
3,533 |
Find the smallest positive integer $n$ such that $$\underbrace{2^{2^{2^{2}}}}_{n 2^{\prime} s}>\underbrace{((\cdots((100!)!)!\cdots)!)!}_{100 \text { factorials }}$$
|
104
| 0 |
3,534 |
Let $A B C$ be a triangle where $A B=9, B C=10, C A=17$. Let $\Omega$ be its circumcircle, and let $A_{1}, B_{1}, C_{1}$ be the diametrically opposite points from $A, B, C$, respectively, on $\Omega$. Find the area of the convex hexagon with the vertices $A, B, C, A_{1}, B_{1}, C_{1}$.
|
\frac{1155}{4}
| 0 |
3,535 |
Two circles $\Gamma_{1}$ and $\Gamma_{2}$ of radius 1 and 2, respectively, are centered at the origin. A particle is placed at $(2,0)$ and is shot towards $\Gamma_{1}$. When it reaches $\Gamma_{1}$, it bounces off the circumference and heads back towards $\Gamma_{2}$. The particle continues bouncing off the two circles in this fashion. If the particle is shot at an acute angle $\theta$ above the $x$-axis, it will bounce 11 times before returning to $(2,0)$ for the first time. If $\cot \theta=a-\sqrt{b}$ for positive integers $a$ and $b$, compute $100 a+b$.
|
403
| 0.78125 |
3,536 |
Let $A B C$ be an acute isosceles triangle with orthocenter $H$. Let $M$ and $N$ be the midpoints of sides $\overline{A B}$ and $\overline{A C}$, respectively. The circumcircle of triangle $M H N$ intersects line $B C$ at two points $X$ and $Y$. Given $X Y=A B=A C=2$, compute $B C^{2}$.
|
2(\sqrt{17}-1)
| 0 |
3,537 |
A 5 by 5 grid of unit squares is partitioned into 5 pairwise incongruent rectangles with sides lying on the gridlines. Find the maximum possible value of the product of their areas.
|
2304
| 0 |
3,538 |
Let $a_{0}, a_{1}, a_{2}, \ldots$ denote the sequence of real numbers such that $a_{0}=2$ and $a_{n+1}=\frac{a_{n}}{1+a_{n}}$ for $n \geq 0$. Compute $a_{2012}$.
|
\frac{2}{4025}
| 100 |
3,539 |
2019 points are chosen independently and uniformly at random on the interval $[0,1]$. Tairitsu picks 1000 of them randomly and colors them black, leaving the remaining ones white. Hikari then computes the sum of the positions of the leftmost white point and the rightmost black point. What is the probability that this sum is at most 1 ?
|
\frac{1019}{2019}
| 5.46875 |
3,540 |
After the Guts round ends, HMMT organizers will collect all answers submitted to all 66 questions (including this one) during the individual rounds and the guts round. Estimate $N$, the smallest positive integer that no one will have submitted at any point during the tournament. An estimate of $E$ will receive $\max (0,24-4|E-N|)$ points.
|
139
| 0 |
3,541 |
Let \(A B C\) be an acute triangle with circumcenter \(O\) such that \(A B=4, A C=5\), and \(B C=6\). Let \(D\) be the foot of the altitude from \(A\) to \(B C\), and \(E\) be the intersection of \(A O\) with \(B C\). Suppose that \(X\) is on \(B C\) between \(D\) and \(E\) such that there is a point \(Y\) on \(A D\) satisfying \(X Y \parallel A O\) and \(Y O \perp A X\). Determine the length of \(B X\).
|
\frac{96}{41}
| 0 |
3,542 |
Roger initially has 20 socks in a drawer, each of which is either white or black. He chooses a sock uniformly at random from the drawer and throws it away. He repeats this action until there are equal numbers of white and black socks remaining. Suppose that the probability he stops before all socks are gone is $p$. If the sum of all distinct possible values of $p$ over all initial combinations of socks is $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$
|
20738
| 0 |
3,543 |
If four fair six-sided dice are rolled, what is the probability that the lowest number appearing on any die is exactly 3?
|
\frac{175}{1296}
| 75 |
3,544 |
Evaluate $\sin (\arcsin (0.4)+\arcsin (0.5)) \cdot \sin (\arcsin (0.5)-\arcsin (0.4))$ where for $x \in[-1,1]$, $\arcsin (x)$ denotes the unique real number $y \in[-\pi, \pi]$ such that $\sin (y)=x$.
|
0.09 \text{ OR } \frac{9}{100}
| 20.3125 |
3,545 |
Let $A B C D$ be a quadrilateral, and let $E, F, G, H$ be the respective midpoints of $A B, B C, C D, D A$. If $E G=12$ and $F H=15$, what is the maximum possible area of $A B C D$?
|
180
| 79.6875 |
3,546 |
Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $\angle D=90^{\circ}$. Suppose that there is a point $E$ on $CD$ such that $AE=BE$ and that triangles $AED$ and $CEB$ are similar, but not congruent. Given that $\frac{CD}{AB}=2014$, find $\frac{BC}{AD}$.
|
\sqrt{4027}
| 0 |
3,547 |
The numbers $2^{0}, 2^{1}, \cdots, 2^{15}, 2^{16}=65536$ are written on a blackboard. You repeatedly take two numbers on the blackboard, subtract one from the other, erase them both, and write the result of the subtraction on the blackboard. What is the largest possible number that can remain on the blackboard when there is only one number left?
|
131069
| 0 |
3,548 |
Let acute triangle $ABC$ have circumcenter $O$, and let $M$ be the midpoint of $BC$. Let $P$ be the unique point such that $\angle BAP=\angle CAM, \angle CAP=\angle BAM$, and $\angle APO=90^{\circ}$. If $AO=53, OM=28$, and $AM=75$, compute the perimeter of $\triangle BPC$.
|
192
| 0 |
3,549 |
Define a power cycle to be a set $S$ consisting of the nonnegative integer powers of an integer $a$, i.e. $S=\left\{1, a, a^{2}, \ldots\right\}$ for some integer $a$. What is the minimum number of power cycles required such that given any odd integer $n$, there exists some integer $k$ in one of the power cycles such that $n \equiv k$ $(\bmod 1024) ?$
|
10
| 20.3125 |
3,550 |
Let $Z$ be as in problem 15. Let $X$ be the greatest integer such that $|X Z| \leq 5$. Find $X$.
|
2
| 36.71875 |
3,551 |
Geoff walks on the number line for 40 minutes, starting at the point 0. On the $n$th minute, he flips a fair coin. If it comes up heads he walks $\frac{1}{n}$ in the positive direction and if it comes up tails he walks $\frac{1}{n}$ in the negative direction. Let $p$ be the probability that he never leaves the interval $[-2,2]$. Estimate $N=\left\lfloor 10^{4} p\right\rfloor$. An estimate of $E$ will receive $\max \left(0,\left\lfloor 20-20\left(\frac{|E-N|}{160}\right)^{1 / 3}\right\rfloor\right)$ points.
|
8101
| 0 |
3,552 |
Let $P$ be a point selected uniformly at random in the cube $[0,1]^{3}$. The plane parallel to $x+y+z=0$ passing through $P$ intersects the cube in a two-dimensional region $\mathcal{R}$. Let $t$ be the expected value of the perimeter of $\mathcal{R}$. If $t^{2}$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, compute $100 a+b$.
|
12108
| 0 |
3,553 |
Let $a_{1}, a_{2}, \ldots, a_{n}$ be a sequence of distinct positive integers such that $a_{1}+a_{2}+\cdots+a_{n}=2021$ and $a_{1} a_{2} \cdots a_{n}$ is maximized. If $M=a_{1} a_{2} \cdots a_{n}$, compute the largest positive integer $k$ such that $2^{k} \mid M$.
|
62
| 0 |
3,554 |
Let $x_{1}, \ldots, x_{100}$ 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 $\{x_{1}, x_{1}+x_{2}, \ldots, x_{1}+x_{2}+\ldots+x_{100}\}$ that are multiples of 6.
|
\frac{50}{3}
| 98.4375 |
3,555 |
Let $ABCDEF$ be a regular hexagon. Let $P$ be the circle inscribed in $\triangle BDF$. Find the ratio of the area of circle $P$ to the area of rectangle $ABDE$.
|
\frac{\pi \sqrt{3}}{12}
| 2.34375 |
3,556 |
Let $ABC$ be an equilateral triangle of side length 6 inscribed in a circle $\omega$. Let $A_{1}, A_{2}$ be the points (distinct from $A$) where the lines through $A$ passing through the two trisection points of $BC$ meet $\omega$. Define $B_{1}, B_{2}, C_{1}, C_{2}$ similarly. Given that $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ appear on $\omega$ in that order, find the area of hexagon $A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}$.
|
\frac{846\sqrt{3}}{49}
| 0 |
3,557 |
Let $A=\{a_{1}, a_{2}, \ldots, a_{7}\}$ be a set of distinct positive integers such that the mean of the elements of any nonempty subset of $A$ is an integer. Find the smallest possible value of the sum of the elements in $A$.
|
1267
| 0.78125 |
3,558 |
Let $\omega$ be a circle, and let $ABCD$ be a quadrilateral inscribed in $\omega$. Suppose that $BD$ and $AC$ intersect at a point $E$. The tangent to $\omega$ at $B$ meets line $AC$ at a point $F$, so that $C$ lies between $E$ and $F$. Given that $AE=6, EC=4, BE=2$, and $BF=12$, find $DA$.
|
2 \sqrt{42}
| 2.34375 |
3,559 |
Estimate the number of positive integers $n \leq 10^{6}$ such that $n^{2}+1$ has a prime factor greater than $n$. Submit a positive integer $E$. If the correct answer is $A$, you will receive $\max \left(0,\left\lfloor 20 \cdot \min \left(\frac{E}{A}, \frac{10^{6}-E}{10^{6}-A}\right)^{5}+0.5\right\rfloor\right)$ points.
|
757575
| 0 |
3,560 |
Let $f(x)=x^{3}-3x$. Compute the number of positive divisors of $$\left\lfloor f\left(f\left(f\left(f\left(f\left(f\left(f\left(f\left(\frac{5}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right)\rfloor$$ where $f$ is applied 8 times.
|
6562
| 0 |
3,561 |
What is the largest real number $\theta$ less than $\pi$ (i.e. $\theta<\pi$ ) such that $\prod_{k=0}^{10} \cos \left(2^{k} \theta\right) \neq 0$ and $\prod_{k=0}^{10}\left(1+\frac{1}{\cos \left(2^{k} \theta\right)}\right)=1 ?
|
\frac{2046 \pi}{2047}
| 0 |
3,562 |
Given that $a, b, c$ are positive integers satisfying $$a+b+c=\operatorname{gcd}(a, b)+\operatorname{gcd}(b, c)+\operatorname{gcd}(c, a)+120$$ determine the maximum possible value of $a$.
|
240
| 22.65625 |
3,563 |
Find the number of sequences $a_{1}, a_{2}, \ldots, a_{10}$ of positive integers with the property that $a_{n+2}=a_{n+1}+a_{n}$ for $n=1,2, \ldots, 8$, and $a_{10}=2002$.
|
3
| 81.25 |
3,564 |
A path of length $n$ is a sequence of points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)$ with integer coordinates such that for all $i$ between 1 and $n-1$ inclusive, either (1) $x_{i+1}=x_{i}+1$ and $y_{i+1}=y_{i}$ (in which case we say the $i$th step is rightward) or (2) $x_{i+1}=x_{i}$ and $y_{i+1}=y_{i}+1$ (in which case we say that the $i$th step is upward). This path is said to start at $\left(x_{1}, y_{1}\right)$ and end at $\left(x_{n}, y_{n}\right)$. Let $P(a, b)$, for $a$ and $b$ nonnegative integers, be the number of paths that start at $(0,0)$ and end at $(a, b)$. Find $\sum_{i=0}^{10} P(i, 10-i)$.
|
1024
| 63.28125 |
3,565 |
$r$ and $s$ are integers such that $3 r \geq 2 s-3 \text { and } 4 s \geq r+12$. What is the smallest possible value of $r / s$ ?
|
1/2
| 5.46875 |
3,566 |
In a 3 by 3 grid of unit squares, an up-right path is a path from the bottom left corner to the top right corner that travels only up and right in steps of 1 unit. For such a path $p$, let $A_{p}$ denote the number of unit squares under the path $p$. Compute the sum of $A_{p}$ over all up-right paths $p$.
|
90
| 15.625 |
3,567 |
Suppose that $x$ and $y$ are positive real numbers such that $x^{2}-xy+2y^{2}=8$. Find the maximum possible value of $x^{2}+xy+2y^{2}$.
|
\frac{72+32 \sqrt{2}}{7}
| 4.6875 |
3,568 |
Let $D$ be the set of divisors of 100. Let $Z$ be the set of integers between 1 and 100, inclusive. Mark chooses an element $d$ of $D$ and an element $z$ of $Z$ uniformly at random. What is the probability that $d$ divides $z$?
|
\frac{217}{900}
| 85.9375 |
3,569 |
In triangle $ABC, \angle A=2 \angle C$. Suppose that $AC=6, BC=8$, and $AB=\sqrt{a}-b$, where $a$ and $b$ are positive integers. Compute $100 a+b$.
|
7303
| 7.03125 |
3,570 |
Suppose that $(a_{1}, \ldots, a_{20})$ and $(b_{1}, \ldots, b_{20})$ are two sequences of integers such that the sequence $(a_{1}, \ldots, a_{20}, b_{1}, \ldots, b_{20})$ contains each of the numbers $1, \ldots, 40$ exactly once. What is the maximum possible value of the sum $\sum_{i=1}^{20} \sum_{j=1}^{20} \min (a_{i}, b_{j})$?
|
5530
| 0 |
3,571 |
The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at $(5,1)$ and put poison in two pipes, one along the line $y=x$ and one along the line $x=7$. However, they need to get the job done quickly before Captain Hammer catches them. What's the shortest distance they can travel to visit both pipes and then return to their headquarters?
|
4 \sqrt{5}
| 15.625 |
3,572 |
$x, y$ are positive real numbers such that $x+y^{2}=x y$. What is the smallest possible value of $x$?
|
4
| 75.78125 |
3,573 |
How many four-digit numbers are there in which at least one digit occurs more than once?
|
4464
| 99.21875 |
3,574 |
Let $x$ and $y$ be positive real numbers such that $x^{2}+y^{2}=1$ and \left(3 x-4 x^{3}\right)\left(3 y-4 y^{3}\right)=-\frac{1}{2}$. Compute $x+y$.
|
\frac{\sqrt{6}}{2}
| 10.15625 |
3,575 |
Find the number of ordered quadruples of positive integers $(a, b, c, d)$ such that $a, b, c$, and $d$ are all (not necessarily distinct) factors of 30 and $abcd>900$.
|
1940
| 92.96875 |
3,576 |
On floor 0 of a weird-looking building, you enter an elevator that only has one button. You press the button twice and end up on floor 1. Thereafter, every time you press the button, you go up by one floor with probability $\frac{X}{Y}$, where $X$ is your current floor, and $Y$ is the total number of times you have pressed the button thus far (not including the current one); otherwise, the elevator does nothing. Between the third and the $100^{\text {th }}$ press inclusive, what is the expected number of pairs of consecutive presses that both take you up a floor?
|
97
| 0 |
3,577 |
Compute the number of ordered quintuples of nonnegative integers $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ such that $0 \leq a_{1}, a_{2}, a_{3}, a_{4}, a_{5} \leq 7$ and 5 divides $2^{a_{1}}+2^{a_{2}}+2^{a_{3}}+2^{a_{4}}+2^{a_{5}}$.
|
6528
| 84.375 |
3,578 |
A restricted path of length $n$ is a path of length $n$ such that for all $i$ between 1 and $n-2$ inclusive, if the $i$th step is upward, the $i+1$st step must be rightward. Find the number of restricted paths that start at $(0,0)$ and end at $(7,3)$.
|
56
| 6.25 |
3,579 |
Two fair coins are simultaneously flipped. This is done repeatedly until at least one of the coins comes up heads, at which point the process stops. What is the probability that the other coin also came up heads on this last flip?
|
1/3
| 96.09375 |
3,580 |
Find the number of pairs of union/intersection operations $\left(\square_{1}, \square_{2}\right) \in\{\cup, \cap\}^{2}$ satisfying the condition: for any sets $S, T$, function $f: S \rightarrow T$, and subsets $X, Y, Z$ of $S$, we have equality of sets $f(X) \square_{1}\left(f(Y) \square_{2} f(Z)\right)=f\left(X \square_{1}\left(Y \square_{2} Z\right)\right)$.
|
11
| 0 |
3,581 |
Define $P=\{\mathrm{S}, \mathrm{T}\}$ and let $\mathcal{P}$ be the set of all proper subsets of $P$. (A proper subset is a subset that is not the set itself.) How many ordered pairs $(\mathcal{S}, \mathcal{T})$ of proper subsets of $\mathcal{P}$ are there such that (a) $\mathcal{S}$ is not a proper subset of $\mathcal{T}$ and $\mathcal{T}$ is not a proper subset of $\mathcal{S}$; and (b) for any sets $S \in \mathcal{S}$ and $T \in \mathcal{T}, S$ is not a proper subset of $T$ and $T$ is not a proper subset of $S$ ?
|
7
| 13.28125 |
3,582 |
For any integer $n$, define $\lfloor n\rfloor$ as the greatest integer less than or equal to $n$. For any positive integer $n$, let $$f(n)=\lfloor n\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+\left\lfloor\frac{n}{3}\right\rfloor+\cdots+\left\lfloor\frac{n}{n}\right\rfloor.$$ For how many values of $n, 1 \leq n \leq 100$, is $f(n)$ odd?
|
55
| 93.75 |
3,583 |
How many positive integers $2 \leq a \leq 101$ have the property that there exists a positive integer $N$ for which the last two digits in the decimal representation of $a^{2^{n}}$ is the same for all $n \geq N$ ?
|
36
| 0 |
3,584 |
Determine the number of subsets $S$ of $\{1,2,3, \ldots, 10\}$ with the following property: there exist integers $a<b<c$ with $a \in S, b \notin S, c \in S$.
|
968
| 63.28125 |
3,585 |
A set of 6 distinct lattice points is chosen uniformly at random from the set $\{1,2,3,4,5,6\}^{2}$. Let $A$ be the expected area of the convex hull of these 6 points. Estimate $N=\left\lfloor 10^{4} A\right\rfloor$. An estimate of $E$ will receive $\max \left(0,\left\lfloor 20-20\left(\frac{|E-N|}{10^{4}}\right)^{1 / 3}\right\rfloor\right)$ points.
|
104552
| 0 |
3,586 |
In how many ways can the numbers $1,2, \ldots, 2002$ be placed at the vertices of a regular 2002-gon so that no two adjacent numbers differ by more than 2? (Rotations and reflections are considered distinct.)
|
4004
| 39.0625 |
3,587 |
Let $S=\left\{p_{1} p_{2} \cdots p_{n} \mid p_{1}, p_{2}, \ldots, p_{n}\right.$ are distinct primes and $\left.p_{1}, \ldots, p_{n}<30\right\}$. Assume 1 is in $S$. Let $a_{1}$ be an element of $S$. We define, for all positive integers $n$ : $$ \begin{gathered} a_{n+1}=a_{n} /(n+1) \quad \text { if } a_{n} \text { is divisible by } n+1 \\ a_{n+1}=(n+2) a_{n} \quad \text { if } a_{n} \text { is not divisible by } n+1 \end{gathered} $$ How many distinct possible values of $a_{1}$ are there such that $a_{j}=a_{1}$ for infinitely many $j$ 's?
|
512
| 3.90625 |
3,588 |
Given that $a, b, c$ are positive real numbers and $\log _{a} b+\log _{b} c+\log _{c} a=0$, find the value of $\left(\log _{a} b\right)^{3}+\left(\log _{b} c\right)^{3}+\left(\log _{c} a\right)^{3}$.
|
3
| 98.4375 |
3,589 |
Find the numbers $\mathbf{1 5 3 , 3 7 0 , 3 7 1 , 4 0 7}$.
|
153, 370, 371, 407
| 32.8125 |
3,590 |
Caroline starts with the number 1, and every second she flips a fair coin; if it lands heads, she adds 1 to her number, and if it lands tails she multiplies her number by 2. Compute the expected number of seconds it takes for her number to become a multiple of 2021.
|
4040
| 2.34375 |
3,591 |
Triangle $ABC$ has side lengths $AB=19, BC=20$, and $CA=21$. Points $X$ and $Y$ are selected on sides $AB$ and $AC$, respectively, such that $AY=XY$ and $XY$ is tangent to the incircle of $\triangle ABC$. If the length of segment $AX$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, compute $100 a+b$.
|
6710
| 0 |
3,592 |
Let $r=H_{1}$ be the answer to this problem. Given that $r$ is a nonzero real number, what is the value of $r^{4}+4 r^{3}+6 r^{2}+4 r ?$
|
-1
| 4.6875 |
3,593 |
Consider a number line, with a lily pad placed at each integer point. A frog is standing at the lily pad at the point 0 on the number line, and wants to reach the lily pad at the point 2014 on the number line. If the frog stands at the point $n$ on the number line, it can jump directly to either point $n+2$ or point $n+3$ on the number line. Each of the lily pads at the points $1, \cdots, 2013$ on the number line has, independently and with probability $1 / 2$, a snake. Let $p$ be the probability that the frog can make some sequence of jumps to reach the lily pad at the point 2014 on the number line, without ever landing on a lily pad containing a snake. What is $p^{1 / 2014}$? Express your answer as a decimal number.
|
0.9102805441016536
| 0 |
3,594 |
Let $ABC$ be a triangle with circumcenter $O$, incenter $I, \angle B=45^{\circ}$, and $OI \parallel BC$. Find $\cos \angle C$.
|
1-\frac{\sqrt{2}}{2}
| 0 |
3,595 |
Let $S(x)$ denote the sum of the digits of a positive integer $x$. Find the maximum possible value of $S(x+2019)-S(x)$.
|
12
| 8.59375 |
3,596 |
The Fibonacci numbers are defined by $F_{1}=F_{2}=1$ and $F_{n+2}=F_{n+1}+F_{n}$ for $n \geq 1$. The Lucas numbers are defined by $L_{1}=1, L_{2}=2$, and $L_{n+2}=L_{n+1}+L_{n}$ for $n \geq 1$. Calculate $\frac{\prod_{n=1}^{15} \frac{F_{2 n}}{F_{n}}}{\prod_{n=1}^{13} L_{n}}$.
|
1149852
| 0 |
3,597 |
Given $\frac{e}{f}=\frac{3}{4}$ and $\sqrt{e^{2}+f^{2}}=15$, find $ef$.
|
108
| 89.0625 |
3,598 |
Calculate the sum of the coefficients of $P(x)$ if $\left(20 x^{27}+2 x^{2}+1\right) P(x)=2001 x^{2001}$.
|
87
| 18.75 |
3,599 |
Define a monic irreducible polynomial with integral coefficients to be a polynomial with leading coefficient 1 that cannot be factored, and the prime factorization of a polynomial with leading coefficient 1 as the factorization into monic irreducible polynomials. How many not necessarily distinct monic irreducible polynomials are there in the prime factorization of $\left(x^{8}+x^{4}+1\right)\left(x^{8}+x+1\right)$ (for instance, $(x+1)^{2}$ has two prime factors)?
|
5
| 9.375 |
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