prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying "1", so Blair follows by saying "1, 2". Jo then says "1, 2, 3", and so on. What is the 53rd number said? | 8 |
A unit equilateral triangle is given. Divide each side into three equal parts. Remove the equilateral triangles whose bases are the middle one-third segments. Now we have a new polygon. Remove the equilateral triangles whose bases are the middle one-third segments of the sides of the polygon. After repeating these step... | 13 |
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38$, $38$, and $34$. What is the distance between two adjacent parallel lines? | 6 |
Five times $A$'s money added to $B$'s money is more than $51.00$. Three times $A$'s money minus $B$'s money is $21.00$. If $a$ represents $A$'s money in dollars and $b$ represents $B$'s money in dollars, find the smallest integer value of $a$ and $b$ that satisfy these conditions. The answer should be in the form $a > ... | 15 |
Phillip and Paula both pick a rational number, and they notice that Phillip's number is greater than Paula's number by $12$. They each square their numbers to get a new number, and see that the sum of these new numbers is half of $169$. Finally, they each square their new numbers and note that Phillip's latest number i... | 5 |
If 1 pint of paint is needed to paint a statue 6 ft. high, find the number of pints it will take to paint (to the same thickness) 540 statues similar to the original but only 1 ft. high. | 15 |
What is the value of \[ \log_3 7 \cdot \log_5 9 \cdot \log_7 11 \cdot \log_9 13 \cdots \log_{21} 25 \cdot \log_{23} 27 \] Find the value of the expression. | 6 |
A regular octahedron is formed by joining the centers of adjoining faces of a cube. What is the ratio of the volume of the octahedron to the volume of the cube? The original answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of k + m. | 7 |
What is the maximum possible value of $5 - |6x - 80|$ over all integers $x$? | 3 |
If both \(x\) and \(y\) are integers, find the number of pairs of solutions for the equation \((x-8)(x-10) = 2^y\). | 2 |
Find the largest integer that divides each of the numbers in the sequence \(1^5 - 1, 2^5 - 2, 3^5 - 3, \cdots, n^5 - n, \cdots\). | 30 |
A box contains $3$ shiny pennies and $4$ dull pennies. One by one, pennies are drawn at random from the box and not replaced. If the probability is $\frac{a}{b}$ that it will take more than four draws until the third shiny penny appears and $\frac{a}{b}$ is in lowest terms, find the value of $a+b$. | 66 |
A unit cube has vertices \(P_1, P_2, P_3, P_4, P_1', P_2', P_3'\), and \(P_4'\). Vertices \(P_2, P_3\), and \(P_4\) are adjacent to \(P_1\), and for \(1\leq i\leq 4\), vertices \(P_i\) and \(P_i'\) are opposite to each other. A regular octahedron has one vertex in each of the segments \(P_1P_2, P_1P_3, P_1P_4, P_1'P_2'... | 9 |
In a narrow alley of width $w$, a ladder of length $a$ is placed with its foot at point $P$ between the walls. Resting against one wall at $Q$, the distance $k$ above the ground makes a $45^\circ$ angle with the ground. Resting against the other wall at $R$, a distance $h$ above the ground, the ladder makes a $75^\circ... | 1 |
Each integer $1$ through $9$ is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. What digit is most likely to be the units digit of the **sum** of Jack's integer and Jill's integer? | 0 |
A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks, and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at... | 23 |
The roots of the equation $x^2 + 5x - 7 = 0$ are $x_1$ and $x_2$. Find the value of $x_1^3 + 5x_1^2 - 4x_1 + x_1^2 x_2 - 4x_2$. | 20 |
A father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges $4.95$ for the father and $0.45$ for each year of a child's age, where age is defined as the age at the most recent birthday. If the total bill is $9.45$, find the age of the youngest child. | 2 |
Let $f$ be a function defined on positive integers such that $f(1)=4$, $f(2n)=f(n)$ and $f(2n+1)=f(n)+2$ for every positive integer $n$. Find the number of positive integers $k$ less than $2014$ for which $f(k)=8$. | 165 |
Let $ABCD$ be a trapezoid such that $AB \parallel CD$, $|AB|<|CD|$, and $\text{Area}(\triangle ABC)=30$. Let the line through $B$ parallel to $AD$ meet $[AC]$ at $E$. If $|AE|:|EC|=3:2$, then what is the area of trapezoid $ABCD$? | 80 |
The altitudes of a triangle are $12$, $15$, and $20$. If the largest angle in this triangle is $x^\circ$, determine the value of $x$. | 90 |
How many three-digit positive integers $x$ are there with the property that $x$ and $2x$ have only even digits? (One such number is $x = 220$, since $2x = 440$ and each of $x$ and $2x$ has only even digits.) | 18 |
A ladder is formed by removing some consecutive unit squares of a $10 \times 10$ chessboard such that for each $k$-th row ($k \in \{1,2,\dots, 10\}$), the leftmost $k-1$ unit squares are removed. Find the number of rectangles formed by the composition of unit squares that the ladder has. | 715 |
Before Ashley started a three-hour drive, her car’s odometer reading was $29792$, a palindrome. At her destination, the odometer reading was another palindrome. If Ashley never exceeded the speed limit of $75$ miles per hour, what was her greatest possible average speed? The original answer is in the form of \frac{m}{n... | 214 |
Let $s_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3 - 5x^2 + 8x - 13$. In particular, $s_0 = 3$, $s_1 = 5$, and $s_2 = 9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \cdot s_k + b \cdot s_{k-1} + c \cdot s_{k-2}$ for $k = 2$, $3$, $\ldots$ Find the value of $a + b + c$. | 10 |
Let $a_1, a_2, \dots, a_n$ be positive integers such that none of them is a multiple of $5$. What is the largest integer $n < 2005$, such that $a_1^4 + a_2^4 + \cdots + a_n^4$ is divisible by $5$? Find the value of $n$. | 2000 |
Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$, and there was an additional cost of $2$ for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the rem... | 4 |
For a peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5, and 6 on each die are in the ratio \(1:2:3:4:5:6\). What is the probability of rolling a total of 7 on the two dice? Express your answer as a fraction in simplest form. The answer should be in the form \(\frac{k}{m}\). Please provide the value of ... | 71 |
There are $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find the absolute value of the difference $|P_s - P_d|$. The original answer is in the form \(\f... | 2002 |
How many positive integers $n$ satisfy \[ \frac{n+1000}{70} = \lfloor \sqrt{n} \rfloor? \] (Recall that $\lfloor x \rfloor$ is the greatest integer not exceeding $x$.) | 6 |
A polynomial $P$ with degree exactly $3$ satisfies $P(0)=1$, $P(1)=3$, and $P(3)=10$. Find a value that cannot be the value of $P(2)$. | 6 |
In real numbers, find the number of distinct pairs \((x,y)\) that satisfy the system of equations: \[ x^2 = 4y + 1 \] and \[ y^2 = x^3 + 1 \] | 2 |
The angle bisector of the acute angle formed at the origin by the graphs of the lines \(y = x\) and \(y = 3x\) has equation \(y = kx\). The original answer is in the form \(\frac{k+\sqrt{n}}{m}\). Please find the value of k + m + n. | 8 |
Let $N = 34 \times 34 \times 63 \times 270$. If the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$ is $1:x$, what is the value of $x$? | 14 |
At most, how many positive integers less than $51$ are there such that no one is a triple of another one? Provide your answer as an integer. | 38 |
Two strips of width 1 overlap at an angle of \(\alpha\) as shown. Find the area of the overlap (shown shaded) in terms of \(\alpha\). The answer is in the form \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please provide the value of \(k + m\). | 2 |
Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\measuredangle BAD=92^{\circ}$ and $\measuredangle ADC=68^{\circ}$, find $\measuredangle EBC$. If the answer is $x^\circ$, what is the value of $x$? | 68 |
Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed 37th and 64th, respectively. How many sc... | 23 |
Square \(ABCD\) has an area of \(36\), and \(\overline{AB}\) is parallel to the x-axis. Vertices \(A\), \(B\), and \(C\) are on the graphs of \(y = \log_{a}x\), \(y = 2\log_{a}x\), and \(y = 3\log_{a}x\), respectively. What is the value of \(a\)? The original answer is in the form \(k^{m/n}\), please give the value of ... | 10 |
If an integer $n > 8$ is a solution of the equation $x^2 - ax + b = 0$ and the representation of $a$ in the base-$n$ number system is $18$, what is the base-$n$ representation of $b$? | 80 |
Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is $44$ years, find Ann's age. | 24 |
Two boys start moving from the same point $A$ on a circular track but in opposite directions. Their speeds are $5$ ft. per second and $9$ ft. per second. If they start at the same time and finish when they first meet at the point $A$ again, find the number of times they meet, excluding the start and finish. | 13 |
A sled loaded with children starts from rest and slides down a snowy $25^\circ$ (with respect to the horizontal) incline, traveling $85$ meters in $17$ seconds. Ignore air resistance. If the coefficient of kinetic friction between the sled and the slope is $x$, what is the value of $100x$? | 40 |
Let \(a, b, c, d, e\) be non-negative real numbers such that \(a + b + c + d + e > 0\). Find the least real number \(t\) such that \(a + c = tb\), \(b + d = tc\), \(c + e = td\). The original answer is in the form \(\sqrt{m}\). Please find the value of \(m\). | 2 |
A three-quarter sector of a circle with a radius of \(4\) inches, together with its interior, can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? Provide your answer in the form of \(a\pi \sqrt{b}\), whe... | 10 |
In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers? | 7 |
For each integer $n \geq 4$, let $a_n$ denote the base-$n$ number $0.\overline{133}_n$. The product $a_4 a_5 \cdots a_{99}$ can be expressed as $\frac{m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$? | 962 |
What is the smallest integer larger than $(\sqrt{3}+\sqrt{2})^6$? | 970 |
Find the value(s) of $y$ for which the following pair of equations \[x^2+y^2-16=0 \text{ and } x^2-3y+12=0\] may have a real common solution. Please provide the sum of all such $y$ values. | 4 |
The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length of $\overline{AB}$? | 20 |
If $60^a = 3$ and $60^b = 5$, find the value of $12^{\frac{1-a-b}{2(1-b)}}$. | 2 |
If we write $|x^2 - 4| < N$ for all $x$ such that $|x - 2| < 0.01$, what is the smallest value we can use for $N$ if the answer is expressed as $N \times 10000$? | 401 |
The sequence \(\{a_n\}\) is defined as follows: \[ a_n = \sqrt{1 + \left(1 + \frac{1}{n} \right)^2} + \sqrt{1 + \left(1 - \frac{1}{n} \right)^2}. \] Calculate the value of the expression given below: \[ \frac{4}{a_1} + \frac{4}{a_2} + \dots + \frac{4}{a_{96}}. \] The original answer is in the form \(\sqrt{m} - 1\). Ple... | 137 |
In the $xy$-plane, find the number of lines whose $x$-intercept is a positive prime number and whose $y$-intercept is a positive integer that pass through the point $(4,3)$. | 2 |
Points $A, B, Q, D,$ and $C$ lie on the circle shown. The measures of arcs $\widehat{BQ}$ and $\widehat{QD}$ are $42^\circ$ and $38^\circ$, respectively. Find the sum of the measures of angles $P$ and $Q$ in degrees. | 40 |
Circles $A$, $B$, and $C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D$. Find the radius of circle $B$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 17 |
If $(0.2)^x = 2$ and $\log 2 = 0.3010$, then if the value of $x$ to the nearest tenth is $y$, what is the value of $10y$? | -4 |
Let $n$ be the number of ways that $10$ dollars can be changed into dimes and quarters, with at least one of each coin being used. Find the value of $n$. | 19 |
How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both? | 382 |
The floor of a rectangular room is covered with square tiles. The room is 10 tiles long and 5 tiles wide. How many tiles touch the walls of the room? | 26 |
Rectangle $ABCD$, pictured below, shares $50\%$ of its area with square $EFGH$. Square $EFGH$ shares $20\%$ of its area with rectangle $ABCD$. What is the value of $\frac{AB}{AD}$? \[ \begin{asy} unitsize(5mm); defaultpen(linewidth(0.8pt)+fontsize(10pt)); pair A=(0,3), B=(8,3), C=(8,2), D=(0,2), Ep=(0,4), F=(4,4), G=(4... | 10 |
Determine the number of distinct points in the $xy$-plane that are common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$. | 1 |
Two standard six-sided dice are tossed. What is the probability that the sum of the numbers is greater than \(7\)? The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 17 |
What is the sum of real roots of the equation \(x^4 - 7x^3 + 14x^2 - 14x + 4 = 0\)? | 5 |
Let $AD$ be a median of $\triangle ABC$ such that $m(\widehat{ADB})=45^{\circ}$ and $m(\widehat{ACB})=30^{\circ}$. If the measure of $\widehat{ABC}$ is $x^\circ$, what is the value of $x$? | 105 |
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB' = 3AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC' = 3BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA' = 3CA$. Find the ratio of the area of $\triangl... | 37 |
How many ordered triples of positive integers \((a,b,c)\) are there such that \((2a+b)(2b+a)=2^c\)? Provide your answer as an integer. | 0 |
When the circumference of a toy balloon is increased from \(20\) inches to \(25\) inches, the radius is increased by how many inches? If the answer is in the form of \(\frac{a}{b}\), what is the value of \(a + b\)? | 7 |
In the figure, it is given that: - \(\angle C = 90^\circ\) - \(\overline{AD} = \overline{DB}\) - \(DE \perp AB\) - \(\overline{AB} = 20\) - \(\overline{AC} = 12\) Find the area of quadrilateral \(ADEC\). The answer should be in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the... | 119 |
A child has a set of $96$ distinct blocks. Each block is one of $2$ materials (plastic, wood), $3$ sizes (small, medium, large), $4$ colors (blue, green, red, yellow), and $4$ shapes (circle, hexagon, square, triangle). How many blocks in the set are different from the "plastic medium red circle" in exactly two ways? (... | 29 |
Suppose \(2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.\) The original answer is in \(\frac{k}{m}\) format. What is the value of \(k+m\)? | 7 |
Lines from the vertices of a unit square are drawn to the midpoints of the sides as shown in the figure below. What is the area of quadrilateral \(ABCD\)? Express your answer in the form \(\frac{k}{m}\) where the fraction is in simplest terms, and give the value of \(k + m\). | 6 |
Mary chose an even 4-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1, 2, \ldots, \frac{n}{2}, n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$? | 340 |
A circle of radius \(r\) passes through both foci of, and exactly four points on, the ellipse with equation \(x^2 + 16y^2 = 16\). The set of all possible values of \(r\) is an interval \([a, b)\). The original answer is in the form \(\sqrt{a} + b\). Please find the value of \(a + b\). | 23 |
In a table tennis tournament, every participant played every other participant exactly once. There were twice as many right-handed players as left-handed players. However, the number of games won by left-handed players was 40\% more than the number of games won by right-handed players. (Note: There were no ties and no ... | 36 |
For how many ordered triples $(x, y, z)$ of nonnegative integers less than $20$ are there exactly two distinct elements in the set $\{i^x, (1 + i)^y, z\}$, where $i = \sqrt{-1}$? | 225 |
For real numbers $x$, let \[P(x)=1+\cos(x)+i \sin(x)-\cos(2x)-i \sin(2x)+\cos(3x)+i \sin(3x)\] where $i=\sqrt{-1}$. Find the number of values of $x$ with $0 \leq x<2 \pi$ such that $P(x)=0$. | 0 |
Let $ABC$ be a triangle with $m(\widehat{A}) = 70^\circ$ and the incenter $I$. If $|BC| = |AC| + |AI|$, then what is the value of $m(\widehat{B})$ in degrees? If the answer is $x^\circ$, what is the value of $x$? | 35 |
Let \( A = \frac{2^2 + 3 \cdot 2 + 1}{3! \cdot 4!} + \frac{3^2 + 3 \cdot 3 + 1}{4! \cdot 5!} + \frac{4^2 + 3 \cdot 4 + 1}{5! \cdot 6!} + \dots + \frac{10^2 + 3 \cdot 10 + 1}{11! \cdot 12!} \). Find the remainder when \( 11! \cdot 12! \cdot A \) is divided by 11. | 10 |
Triangles $ABC$ and $ABD$ are isosceles with $AB = AC = BD$, and $BD$ intersects $AC$ at $E$. If $BD$ is perpendicular to $AC$, then what is the value of $\angle C + \angle D$ if the answer is in degrees? | 135 |
The base three representation of $x$ is \[ 12112211122211112222. \] What is the first digit (on the left) of the base nine representation of $x$? | 5 |
The circumference of a circle is \(100\) inches. Find the length of the side of a square inscribed in this circle, expressed in inches. The original answer is in the form \(\frac{k\sqrt{m}}{n\pi}\). Please provide the value of \(k + m + n\). | 53 |
For all positive numbers \(x, y, z\), evaluate the product: \[ (x+y+z)^{-1}(x^{-1}+y^{-1}+z^{-1})(xy+yz+xz)^{-1}[(xy)^{-1}+(yz)^{-1}+(xz)^{-1}] \] What is the simplified expression for this product? The simplified expression is in the form \(x^{-a}y^{-b}z^{-c}\). Please provide the value of \(a+b+c\). | 6 |
How many prime divisors does the number $1\cdot 2003 + 2\cdot 2002 + 3\cdot 2001 + \cdots + 2001 \cdot 3 + 2002 \cdot 2 + 2003 \cdot 1$ have? Fill in the blank with the number of prime divisors. | 5 |
A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers. For each ball, the probability that it is tossed into bin \(k\) is \(2^{-k}\) for \(k=1,2,3,\ldots\). What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? Express y... | 4 |
What is the remainder when \(2011^{(2011^{(2011^{(2011^{2011})})})}\) is divided by \(19\)? | 5 |
Given the function $f$ defined as follows: \[ f(0) = 0, \] \[ f(1) = 1, \] and for every integer $n \geq 1$, \[ f(3n-1) = f(n) - 1, \] \[ f(3n) = f(n), \] \[ f(3n+1) = f(n) + 1. \] Find the value of $f(2011)$. | 3 |
A circle with area \(A_1\) is contained in the interior of a larger circle with area \(A_1 + A_2\). If the radius of the larger circle is \(3\), and if \(A_1, A_2, A_1 + A_2\) is an arithmetic progression, find the radius of the smaller circle. The original answer is in the form \(\sqrt{k}\), please give the value of \... | 3 |
The weight of \( \frac{1}{3} \) of a large pizza together with \( 3 \frac{1}{2} \) cups of orange slices is the same as the weight of \( \frac{3}{4} \) of a large pizza together with \( \frac{1}{2} \) cup of orange slices. A cup of orange slices weighs \( \frac{1}{4} \) of a pound. What is the weight, in pounds, of a l... | 14 |
An integer $x$, with $10 \le x \le 99$, is to be chosen. If all choices are equally likely, find the probability that at least one digit of $x$ is a $7$. The original answer is in the form \frac{k}{m}, where k and m are coprime. Please give the value of k + m. | 6 |
Find the smallest positive number from the following expressions: \[ 10 - 3\sqrt{11}, \ 3\sqrt{11} - 10, \ 18 - 5\sqrt{13}, \ 51 - 10\sqrt{26}, \ 10\sqrt{26} - 51 \] The original answer is in the form of k - m\sqrt{n}, please give the value of k + m. | 61 |
What is the sum of all $k \leq 25$ such that one can completely cover a $k \times k$ square with $T$ tetrominos without any overlap? | 84 |
Let $P(x) = x^{2022} + x^{1011} + 1$. Find the polynomial of the form ax^6 + bx^3 + c that divides $P(x)$. Please provide the sum of the coefficients a + b + c. | 3 |
In the eight-term sequence $A, B, C, D, E, F, G, H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. Find the value of $A + H$. | 25 |
Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\overline{AB}$, and $\overline{CD}$ bisects the right angle. The inscribed circles of $\triangle ADC$ and $\triangle BCD$ have radii $r_a$ and $r_b$, respectively. The value of \frac{r_a}{r_b} is given in the form \frac{k}{m}(n-\sqrt{p}). Please calculate ... | 43 |
Given a rectangle \(R_1\) with one side measuring \(2\) inches and an area of \(12\) square inches. Rectangle \(R_2\) has a diagonal of \(15\) inches and is similar to \(R_1\). What is the area of \(R_2\), expressed in the form \(\frac{k}{m}\) square inches? Please provide the value of \(k + m\). | 137 |
In $\triangle ABC$, $\angle ABC = 45^\circ$. Point $D$ is on $\overline{BC}$ such that $2 \cdot BD = CD$ and $\angle DAB = 15^\circ$. Find the value of $\angle ACB$ in degrees. If the answer is $x^\circ$, what is the value of $x$? | 75 |
If \(i^2 = -1\), then find the sum \[ \cos{45^\circ} + i\cos{135^\circ} + \cdots + i^n\cos{(45 + 90n)^\circ} \] \[ + \cdots + i^{40}\cos{3645^\circ} \] The original answer is in the form \(\frac{k}{m}(\sqrt{2})(a - bi)\), please calculate the value of a + b + k + m. | 44 |
An unfair coin lands on heads with a probability of $\frac{1}{4}$. When tossed $n$ times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the value of $n$? | 11 |
A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. The rotational inertia of the rod about its center is $md^2$. Find the ratio $\frac{L}{d}$. The original answer is in the form k\sqrt{m}, please provide the value of k + m. | 5 |
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