prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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If three standard, six-faced dice are rolled, what is the probability that the sum of the three numbers rolled is 9? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 241 |
Let $f(r) = \sum_{j=2}^{2008} \frac{1}{j^r} = \frac{1}{2^r}+ \frac{1}{3^r}+ \dots + \frac{1}{2008^r}$. Find $\sum_{k=2}^{\infty} f(k)$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 4015 |
Let $x$ and $y$ be positive real numbers such that $x + \frac{1}{y} = 5.$ Find the minimum value of $\frac{1}{x} + y.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 9 |
A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. Find the probability that the roots of the polynomial \[x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2\]are all real.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 37 |
Find the minimum value of \[\frac{\sin^6 x + \cos^6 x + 1}{\sin^4 x + \cos^4 x + 1}\]over all real values $x.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 11 |
Let $x,$ $y,$ and $z$ be three positive real numbers whose sum is 1. If no one of these numbers is more than twice any other, then find the minimum value of the product $xyz.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 33 |
Markov plays a game for three turns. On each turn, he either rolls a fair, six sided die or flips a fair coin. If he rolls a 1 or 2 on the die, he will switch to the coin on the next turn, and if he flips a tails on the coin, he will switch to the die on the next turn. If Markov starts by rolling the die, what is the p... | 25 |
Compute \[\sum_{n = 2}^{10000} \frac{1}{n \sqrt{n - 1} + (n - 1) \sqrt{n}}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 199 |
An infinite geometric series has first term $328$ and a sum of $2009$. What is its common ratio?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 90 |
Circles with centers $A$ and $B$ have radii 3 and 8, respectively. A common internal tangent touches the circles at $C$ and $D$, as shown. Lines $AB$ and $CD$ intersect at $E$, and $AE = 5$. What is $CD$? [asy] unitsize(0.2cm); pair A,B,C,D,E; A=(0,0); B=(18.3,0); E=(5,0); C=(1.8,-2.4); D=(13.5,6.4); draw(Circle(A,3),l... | 47 |
The parabola $y = x^2$ is tangent to the graph of $y = x^4 + ax^3 + x^2 + bx + 1$ at two points. Find the positive difference between the $x$-coordinates of the points of tangency.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 7 |
The equation $x^3 - 4x^2 + 5x - \frac{19}{10} = 0$ has real roots $r,$ $s,$ and $t.$ Find the length of the long diagonal of a box with sides of length $r,$ $s,$ and $t.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 7 |
Let $\tau = \frac{1 + \sqrt{5}}{2}.$ Find \[\sum_{n = 0}^\infty \frac{\lfloor \tau^n \rceil}{2^n}.\]Note: For a real number $x,$ $\lfloor x \rceil$ denotes the integer closest to $x.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 13 |
The admission fee for an exhibition is $ \$25$ per adult and $ \$12$ per child. Last Tuesday, the exhibition collected $ \$1950$ in admission fees from at least one adult and at least one child. Of all the possible ratios of adults to children at the exhibition last Tuesday, which one is closest to $ 1$?The answer is i... | 52 |
In the figure shown, $AC=13$ and $DC=2$ units. What is the length of the segment $BD$? Express your answer in simplest radical form. [asy] draw((0,5)--(12,5)); draw((12,5)--(12,0)); draw((12,0)--(0,5)); draw((12,5)--(10.1,0.8)); draw((10.48,1.64)--(9.47,2.06)); draw((9.47,2.06)--(9.09,1.22)); draw((11,5)--(11,4)); draw... | 23 |
The digits 2, 3, 4, 7 and 8 will be put in random order to make a positive five-digit integer. What is the probability that the resulting integer will be divisible by 11? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 11 |
Compute \[\sum_{n = 1}^\infty \frac{2n - 1}{n(n + 1)(n + 2)}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 7 |
Let $a,$ $b,$ $c$ be nonnegative real numbers such that $a + b + c = 1.$ Find the maximum value of \[\frac{ab}{a + b} + \frac{ac}{a + c} + \frac{bc}{b + c}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 3 |
A cube 4 units on each side is composed of 64 unit cubes. Two faces of the larger cube that share an edge are painted blue, and the cube is disassembled into 64 unit cubes. Two of the unit cubes are selected uniformly at random. What is the probability that one of two selected unit cubes will have exactly two painted f... | 15 |
In square $ABCD$, $AD$ is 4 centimeters, and $M$ is the midpoint of $\overline{CD}$. Let $O$ be the intersection of $\overline{AC}$ and $\overline{BM}$. What is the ratio of $OC$ to $OA$? Express your answer as a common fraction. [asy] size (3cm,3cm); pair A,B,C,D,M; D=(0,0); C=(1,0); B=(1,1); A=(0,1); draw(A--B--C--D-... | 3 |
Let $\theta$ be an acute angle such that \[\sin 5 \theta = \sin^5 \theta.\]Compute $\tan 2 \theta.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 4 |
Determine the exact value of the series \[\frac{1}{5 + 1} + \frac{2}{5^2 + 1} + \frac{4}{5^4 + 1} + \frac{8}{5^8 + 1} + \frac{16}{5^{16} + 1} + \dotsb.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 5 |
In convex quadrilateral $ABCD$, $\angle A = \angle C$, $AB=CD=180$, and $AD \ne BC$. The perimeter of $ABCD$ is 640. Find $\cos A$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 16 |
The parametric curve $(x,y) = (\cos 2t, \cos^2 t),$ for real numbers $t,$ describes a line segment. Find the length of this line segment.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 6 |
A sequence of numbers $a_1,$ $a_2,$ $a_3,$ $a_4,$ $\dots,$ has the property that each term after the first is one less than the product of its two neighbors. If $a_1 = 1492$ and $a_2 = 1776,$ then determine $a_{2003}.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 3269 |
Find the minimum value of the function \[f(x) = \sqrt{-x^2 + 4x + 21} - \sqrt{-x^2 + 3x + 10}.\]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 3 |
For the three identical squares shown, points $A$, $B$ and $C$ are vertices, and $AB$ = $2\sqrt{5}$ cm. What is the length of $AC$, in centimeters? Express your answer in simplest radical form. [asy] draw((0,0)--(0,10)--(30,10)--(30,0)--cycle); draw((10,0)--(10,10)); draw((20,0)--(20,10)); draw((0,10)--(20,0),linewidth... | 12 |
The first four terms in an arithmetic sequence are $x + y, x - y, xy,$ and $x/y,$ in that order. What is the fifth term?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 163 |
Find the maximum value of \[\begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 + \sin \theta & 1 \\ 1 + \cos \theta & 1 & 1 \end{vmatrix},\]as $\theta$ ranges over all angles.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 3 |
On the complex plane, the parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z}$ has area $\frac{35}{37}.$ If the real part of $z$ is positive, let $d$ be the smallest possible value of $\left| z + \frac{1}{z} \right|.$ Compute $d^2.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Ple... | 87 |
Find the sum of the squares of the solutions to \[\left| x^2 - x + \frac{1}{2008} \right| = \frac{1}{2008}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 1505 |
In the diagram, $R$ is on $QS$ and $QR=8$. Also, $PR=12$, $\angle PRQ=120^\circ$, and $\angle RPS = 90^\circ$. What is the area of $\triangle QPS$? [asy] //C19 size(10cm); // ADJUST import olympiad; // Point coordinates pair r = (0, 0); pair q = (-8, 0); pair p = 12 * dir(60); pair s = (24, 0); // draw lines draw(p--s-... | 99 |
Compute the infinite sum $S,$ where \[S = \frac{1}{2} - \frac{2}{4} + \frac{3}{8} - \frac{4}{16} + \frac{5}{32} - \dots + \frac{n}{2^n} (-1)^{n + 1} + \dotsb.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 11 |
The ratio of losses to wins for Kyle's team is 3 to 2. If the team had played the same number of games, but had won twice as many of its games, what would the ratio of losses to wins have been? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of... | 5 |
Compute the sum and enter your answer as a common fraction: $$ \begin{array}{r r@{}c@{}l} & 1 &.& 11111111\ldots \\ & 0 &.& 11111111\ldots \\ & 0 &.& 01111111\ldots \\ & 0 &.& 00111111\ldots \\ & 0 &.& 00011111\ldots \\ & 0 &.& 00001111\ldots \\ & 0 &.& 00000111\ldots \\ + &&\vdots \\ \hline &&& ~~~? \end{array} $$The ... | 181 |
An equilateral triangle $PQR$ is inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$ so that $Q$ is at $(0,b),$ and $\overline{PR}$ is parallel to the $x$-axis, as shown below. Also, foci $F_1$ and $F_2$ lie on sides $\overline{QR}$ and $\overline{PQ},$ respectively. Find $\frac{PQ}{F_1 F_2}.$ [asy] units... | 13 |
Compute \[\sum_{n = 1}^\infty \frac{2n + 1}{n(n + 1)(n + 2)}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 9 |
Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 21 |
Two different 2-digit numbers are randomly chosen and multiplied together. What is the probability that the resulting product is even?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 156 |
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{BC}$ and $\overline{AC}$, respectively. If $\overline{AD}$ and $\overline{BE}$ intersect at $T$ so that $AT/DT=3$ and $BT/ET=4$, what is $CD/BD$? [asy] pair A,B,C,D,I,T; A=(0,0); B=(6,8); C=(11,0); D=(9.33,2.66); I=(7.5,0); T=(6.5,2); label("$T$",T,NW); label("$D$... | 15 |
A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater than the number of the green point, but les... | 5 |
Two eight-sided dice each have faces numbered 1 through 8. When the dice are rolled, each face has an equal probability of appearing on the top. What is the probability that the product of the two top numbers is greater than their sum? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where ... | 7 |
Given that $\sec x + \tan x = \frac{4}{3},$ enter all possible values of $\sin x.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 32 |
Two jokers are added to a $52$ card deck and the entire stack of $54$ cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 55 |
Find the minimum possible value of \[\sqrt{58 - 42x} + \sqrt{149 - 140 \sqrt{1 - x^2}}\]where $-1 \le x \le 1.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 110 |
Ben rolls two fair six-sided dice. What is the expected value of the larger of the two numbers rolled? Express your answer as a fraction. (If the two numbers are the same, we take that number to be the "larger" number.)The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 197 |
Let $z$ be a complex number that satisfies \[|z - 3i| + |z - 4| = 5.\]Find the minimum value of $|z|.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 17 |
To take quizzes, each of 30 students in a class is paired with another student. If the pairing is done randomly, what is the probability that Margo is paired with her best friend, Irma? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 30 |
Two numbers are independently selected from the set of positive integers less than or equal to 5. What is the probability that the sum of the two numbers is greater than their product? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 34 |
Billy is hiking in Colorado. He walks eastward four miles, then turns $60$ degrees northward and walks six miles. How far is he from his starting point? Express your answer in simplest radical form.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 21 |
An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the least possible value of the $ 50$th term and let $ G$ be the greatest possible value of the $ 50$th term. What is the value of $ G - L$?The answer is in the form rac{... | 8279 |
If Greg rolls four fair six-sided dice, what is the probability that he rolls more 1's than 6's?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 1717 |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 1,$ $\|\mathbf{b}\| = 5,$ $\|\mathbf{c}\| = 3,$ and \[\mathbf{a} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{c}.\]If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b},$ then find $\sin \theta.$The answer is in the form rac{... | 8 |
What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le1$ and $|x|+|y|+|z-1|\le1$?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 7 |
Find $XY$ in the triangle below. [asy] unitsize(1inch); pair P,Q,R; P = (0,0); Q= (1,0); R = (0,1); draw (P--Q--R--P,linewidth(0.9)); draw(rightanglemark(Q,P,R,3)); label("$X$",P,S); label("$Y$",Q,S); label("$Z$",R,N); label("$12\sqrt{2}$",R/2,W); label("$45^\circ$",(0.7,0),N); [/asy]The answer is in the form k\sqrt{m}... | 14 |
In triangle $ABC,$ $M$ is the midpoint of $\overline{BC},$ $AB = 12,$ and $AC = 16.$ Let $E$ be on $\overline{AC},$ and $F$ be on $\overline{AB},$ and let $G$ be the intersection of $\overline{EF}$ and $\overline{AM}.$ If $AE = 2AF,$ then find $\frac{EG}{GF}.$ [asy] unitsize(0.3 cm); pair A, B, C, E, F, G, M; real x = ... | 5 |
Juan rolls a fair regular octahedral die marked with the numbers 1 through 8. Then Amal rolls a fair six-sided die. What is the probability that the product of the two rolls is a multiple of 3?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 3 |
Triangle $ABC$ has $AB=BC=5$ and $AC=6$. Let $E$ be the foot of the altitude from $B$ to $\overline{AC}$ and let $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Compute the area of triangle $DEC$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 133 |
If $a,$ $b,$ $c,$ $x,$ $y,$ and $z$ are positive and $a^2 + b^2 + c^2 = 25,$ $x^2 + y^2 + z^2 = 36,$ and $ax + by + cz = 30,$ compute \[\frac{a + b + c}{x + y + z}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 11 |
A $30^\circ$-$60^\circ$-$90^\circ$ triangle is drawn on the exterior of an equilateral triangle so the hypotenuse of the right triangle is one side of the equilateral triangle. If the shorter leg of the right triangle is 6 units, what is the distance between the two vertices that the triangles do not have in common? Ex... | 13 |
What is the ratio of the volume of cone $A$ to the volume of cone $B$? Express your answer as a common fraction. [asy] size(260); draw(ellipse((0,0),14.8,6),S); label("Cone $A$", (0,-6), S); draw((0,28.3)--(0,0),dashed); label("$28.3$",(0,14),SW); draw((-14.8,0)--(0,28.3)--(14.8,0)); draw("$14.8$",(-14.8,0)--(0,0),N,da... | 431 |
A circle having center $(0,k)$, with $k>6$, is tangent to the lines $y=x$, $y=-x$ and $y=6$. What is the radius of this circle?The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 14 |
For each value of $x,$ $f(x)$ is defined to be the minimum value of the three numbers $2x + 2,$ $\frac{1}{2} x + 1,$ and $-\frac{3}{4} x + 7.$ Find the maximum value of $f(x).$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 22 |
There is a machine with 8 toys in it that each cost between 25 cents and 2 dollars, with each toy being 25 cents more expensive than the next most expensive one. Each time Sam presses the big red button on the machine, the machine randomly selects one of the remaining toys and gives Sam the option to buy it. If Sam has... | 13 |
A pirate is searching for buried treasure on 6 islands. On each island, there is a $\frac{1}{4}$ chance that the island has buried treasure and no traps, a $\frac{1}{12}$ chance that the island has traps but no treasure, and a $\frac{2}{3}$ chance that the island has neither traps nor treasure. What is the probability ... | 59 |
Each day, two out of the three teams in a class are randomly selected to participate in a MATHCOUNTS trial competition. What is the probability that Team A is selected on at least two of the next three days? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provid... | 47 |
Let $a,b,c$ be positive real numbers such that $a+b+c=10$ and $ab+bc+ca=25$. Let $m=\min\{ab,bc,ca\}$. Find the largest possible value of $m$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 34 |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 1$ and \[f(xy + f(x)) = xf(y) + f(x)\]for all real numbers $x$ and $y.$ Let $n$ be the number of possible values of $f \left( \frac{1}{2} \right),$ and let $s$ be the sum of all possible values of $f \left( \frac{1}{2} \right).$ Find $n \times s.$The a... | 3 |
Let $a,$ $b,$ and $c$ be nonnegative real numbers such that $a^2 + b^2 + c^2 = 1.$ Find the maximum value of \[2ab \sqrt{2} + 2bc.\]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 4 |
Let $z$ be a complex number such that $z^{23} = 1$ and $z \neq 1.$ Find \[\sum_{n = 0}^{22} \frac{1}{1 + z^n + z^{2n}}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 49 |
A bag contains 4 balls. Each ball is a single color, and 3 distinct colors are represented in the bag. What is the probability that the first 2 balls selected without replacement are the same color? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the val... | 7 |
Allison, Brian and Noah each have a 6-sided cube. All of the faces on Allison's cube have a 5. The faces on Brian's cube are numbered 1, 2, 3, 4, 5 and 6. Three of the faces on Noah's cube have a 2 and three of the faces have a 6. All three cubes are rolled. What is the probability that Allison's roll is greater than e... | 4 |
Compute the sum \[\sum_{i = 0}^\infty \sum_{j = 0}^\infty \frac{1}{(i + j + 1)(i + j + 2)(i + j + 3)(i + j + 4)(i + j + 5)(i + j + 6)(i + j + 7)}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 3601 |
A circle rests in the interior of the parabola with equation $y = x^2,$ so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 3 |
A right circular cone is sliced into four pieces by planes parallel to its base, as shown in the figure. All of these pieces have the same height. What is the ratio of the volume of the second-largest piece to the volume of the largest piece? Express your answer as a common fraction. [asy] size(150); pair A, B, C, D, E... | 56 |
$ABC$ is a triangle: $A=(0,0), B=(36,15)$ and both the coordinates of $C$ are integers. What is the minimum area $\triangle ABC$ can have? $\textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ \frac{3}{2} \qquad \textbf{(D)}\ \frac{13}{2}\qquad \textbf{(E)}\ \text{there is no minimum}$The answer is in ... | 5 |
Triangles $ABC$ and $AEF$ are such that $B$ is the midpoint of $\overline{EF}.$ Also, $AB = EF = 1,$ $BC = 6,$ $CA = \sqrt{33},$ and \[\overrightarrow{AB} \cdot \overrightarrow{AE} + \overrightarrow{AC} \cdot \overrightarrow{AF} = 2.\]Find the cosine of the angle between vectors $\overrightarrow{EF}$ and $\overrightarr... | 5 |
Allen and Bethany each arrive at a party at a random time between 1:00 and 2:00. Each stays for 15 minutes, then leaves. What is the probability that Allen and Bethany see each other at the party?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 23 |
A fair coin is flipped $7$ times. What is the probability that at least $5$ consecutive flips come up heads?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 17 |
Integers $a$, $b$, $c$, and $d$, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that $ad-bc$ is even?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 13 |
Bob and Meena play a two-person game which is won by the first person to accumulate 10 points. At each turn Bob gains a point with probability of $\frac{1}{3}$ . If he doesn't get a point, then Meena gets a point. Meena is now ahead 9 to 8. What is the probability that Meena will win? Express your answer as a common fr... | 17 |
In the triangle shown, for $\angle A$ to be the largest angle of the triangle, it must be that $m<x<n$. What is the least possible value of $n-m$, expressed as a common fraction? [asy] draw((0,0)--(1,0)--(.4,.5)--cycle); label("$A$",(.4,.5),N); label("$B$",(1,0),SE); label("$C$",(0,0),SW); label("$x+9$",(.5,0),S); labe... | 23 |
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to hi... | 7 |
Let $A$ be a point on the circle $x^2 + y^2 - 12x + 31 = 0,$ and let $B$ be a point on the parabola $y^2 = 4x.$ Find the smallest possible distance $AB.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n. | 6 |
Billy and Bobbi each selected a positive integer less than 200. Billy's number is a multiple of 18, and Bobbi's number is a multiple of 24. What is the probability that they selected the same number? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the va... | 45 |
If $y - x,$ $y - 2x,$ and $y - kx$ are all factors of \[x^3 - 3x^2 y + p_1 xy^2 + p_2 y^3,\]then find $k$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 5 |
Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that \[f(x) f(y) = f(xy) + 2005 \left( \frac{1}{x} + \frac{1}{y} + 2004 \right)\]for all $x,$ $y > 0.$ Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.... | 4013 |
Let $0 \le x,$ $y,$ $z \le 1.$ Find the maximum value of \[f(x,y,z) = x^2 y + y^2 z + z^2 x - xy^2 - yz^2 - zx^2.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 5 |
The expression \[\sin^3 2x \cos 6x + \cos^3 2x \sin 6x\]can be written in the equivalent form $a \sin bx$ for some positive constants $a$ and $b.$ Find $a + b.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 39 |
Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral? $\textbf{(A)}\ \sqrt{15} \qquad \textbf{(B)}\ \sqrt{21} \qquad \textbf{(C)}\ 2\sqrt{6} \qquad \textbf{(D)}\ 5 \qquad \textbf{(... | 8 |
Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\frac{n}{d}$ where $n$ and $d$ are integers with $1 \le d \le 5$. What is the probability that \[(\text{cos}(a\pi)+i\text{sin}(b\pi))^4\]is a real number?The answer is in the form rac{m... | 31 |
Calculate $\frac{1}{4} \cdot \frac{2}{5} \cdot \frac{3}{6} \cdot \frac{4}{7} \cdots \frac{49}{52} \cdot \frac{50}{53}$. Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 23427 |
Let $a,$ $b,$ $c,$ $d$ be positive real numbers such that $a + b + c + d = 1.$ Find the minimum value of \[\frac{a}{b + c + d} + \frac{b}{a + c + d} + \frac{c}{a + b + d} + \frac{d}{a + b + c}.\]The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 7 |
Two numbers between $0$ and $1$ on a number line are to be chosen at random. What is the probability that the second number chosen will exceed the first number chosen by a distance greater than $\frac 14$ unit on the number line? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m,... | 41 |
The equation \[2000x^6+100x^5+10x^3+x-2=0\]has two real roots. Compute the square of the difference between them.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 561 |
Each of the symbols $\star$ and $*$ represents an operation in the set $\{+,-,\times,\div\}$, and $\frac{12\star 2}{9*3}=4$. What is the value of $\frac{10\star 7}{24*9}$? Express your answer as a common fraction.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 17 |
An ellipse in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(3,7)$ and the other focus is at $(d,7).$ Compute $d.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 52 |
There are 2011 positive numbers with both their sum and the sum of their reciprocals equal to 2012. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 10057 |
All triangles have the same value, and all circles have the same value. What is the sum of two circles? \begin{align*} \Delta + \bigcirc + \Delta + \bigcirc + \Delta + \bigcirc + \Delta + \Delta&= 21\\ \bigcirc + \Delta+\bigcirc+\Delta+\bigcirc + \Delta + \bigcirc + \bigcirc &= 16\\ \bigcirc + \bigcirc &= \ \, ? \end{a... | 25 |
We flip a fair coin 10 times. What is the probability that we get heads in at least 6 of the 10 flips?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 705 |
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