problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+2$, where $b<150$, determine the number of possible triangles. | 12 |
A certain university needs $40L$ of helium gas to make balloon decorations for its centennial celebration. The chemistry club voluntarily took on this task. The club's equipment can produce a maximum of $8L$ of helium gas per day. According to the plan, the club must complete the production within 30 days. Upon receivi... | 4640 |
Mark writes the expression $\sqrt{d}$ for each positive divisor $d$ of 8 ! on the board. Seeing that these expressions might not be worth points on HMMT, Rishabh simplifies each expression to the form $a \sqrt{b}$, where $a$ and $b$ are integers such that $b$ is not divisible by the square of a prime number. Compute th... | 3480 |
Jonathan's full name contains 8 letters for the first name and 10 letters for the surname. His sister's name has 5 letters for the first name and 10 letters for the second name. What's the total number of letters in their names? | Jonathan has 8 +10 = <<8+10=18>>18 letters in his name.
His sister has 5+10 = <<5+10=15>>15 letters in her name.
The total number of letters in their names is 18+15 = <<18+15=33>>33
#### 33 |
In triangle $PQR$, we have $\angle P = 90^\circ$, $QR = 15$, and $\tan R = 5\cos Q$. What is $PQ$? | 6\sqrt{6} |
A man named Juan has three rectangular solids, each having volume 128. Two of the faces of one solid have areas 4 and 32. Two faces of another solid have areas 64 and 16. Finally, two faces of the last solid have areas 8 and 32. What is the minimum possible exposed surface area of the tallest tower Juan can construct b... | 688 |
On the side $BC$ of the triangle $ABC$, a point $D$ is chosen such that $\angle BAD = 50^\circ$, $\angle CAD = 20^\circ$, and $AD = BD$. Find $\cos \angle C$. | \frac{\sqrt{3}}{2} |
Given a circle with radius $8$, two intersecting chords $PQ$ and $RS$ intersect at point $T$, where $PQ$ is bisected by $RS$. Assume $RS=10$ and the point $P$ is on the minor arc $RS$. Further, suppose that $PQ$ is the only chord starting at $P$ which is bisected by $RS$. Determine the cosine of the central angle subte... | 8\sqrt{39} |
John rents a car to visit his family. It cost $150 to rent the car. He also had to buy 8 gallons of gas to fill it up and gas is $3.50 per gallon. The final expense is $.50 per mile. If he drove 320 miles how much did it cost? | The gas cost 8*3.5=$<<8*3.5=28>>28
The mileage expenses cost 320*.5=$<<320*.5=160>>160
So in total he paid 150+28+160=$<<150+28+160=338>>338
#### 338 |
At a gym, the blue weights are 2 pounds each, and the green weights are 3 pounds each. Harry put 4 blue weights and 5 green weights onto a metal bar. The bar itself weighs 2 pounds. What is the total amount of weight, in pounds, of Harry's custom creation? | The blue weights weigh 4*2=<<4*2=8>>8 pounds
The green weights weigh 5*3=<<5*3=15>>15 pounds
The weights weigh 8+15=<<8+15=23>>23 pounds.
The total is then 23+2=<<23+2=25>>25.
#### 25 |
Define a function $A(m, n)$ in line with the Ackermann function and compute $A(3, 2)$. | 11 |
Adam has 18 magnets. He gave away a third of the magnets, and he still had half as many magnets as Peter. How many magnets does Peter have? | Adam gave away 18/3 = <<18/3=6>>6 magnets.
Adam had 18-6 = <<18-6=12>>12 magnets left.
Peter has 12*2 = <<12*2=24>>24 magnets.
#### 24 |
In an election, there are two candidates, A and B, who each have 5 supporters. Each supporter, independent of other supporters, has a \(\frac{1}{2}\) probability of voting for his or her candidate and a \(\frac{1}{2}\) probability of being lazy and not voting. What is the probability of a tie (which includes the case i... | 63/256 |
Jeff decides to play with a Magic 8 Ball. Each time he asks it a question, it has a 1/3 chance of giving him a positive answer. If he asks it 7 questions, what is the probability that it gives him exactly 3 positive answers? | \frac{560}{2187} |
Given the lines $l_{1}$: $\left(3+a\right)x+4y=5-3a$ and $l_{2}$: $2x+\left(5+a\right)y=8$, if $l_{1}$ is parallel to $l_{2}$, determine the value of $a$. | -7 |
Distribute 5 volunteers from the Shanghai World Expo to work in the pavilions of China, the United States, and the United Kingdom. Each pavilion must have at least one volunteer, with the requirement that two specific volunteers, A and B, do not work in the same pavilion. How many different distribution schemes are pos... | 114 |
Let \( x = 19.\overline{87} \). If \( 19.\overline{87} = \frac{a}{99} \), find \( a \).
If \( \frac{\sqrt{3}}{b \sqrt{7} - \sqrt{3}} = \frac{2 \sqrt{21} + 3}{c} \), find \( c \).
If \( f(y) = 4 \sin y^{\circ} \) and \( f(a - 18) = b \), find \( b \). | 25 |
There are 1000 candies in a row. Firstly, Vasya ate the ninth candy from the left, and then ate every seventh candy moving to the right. After that, Petya ate the seventh candy from the left of the remaining candies, and then ate every ninth one of them, also moving to the right. How many candies are left after this? | 761 |
When rolling a fair 6-sided die, what is the probability of a 2 or 4 being rolled? | \frac{1}{3} |
A marathon of 42 km started at 11:30 AM and the winner finished at 1:45 PM on the same day. What was the average speed of the winner, in km/h? | 18.6 |
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term. | M > 1 |
A car travels 192 miles on 6 gallons of gas. How far can it travel on 8 gallons of gas? | 256 |
Compute \[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{25} \rfloor.\] | 75 |
In a certain academic knowledge competition, where the total score is 100 points, if the scores (ξ) of the competitors follow a normal distribution (N(80,σ^2) where σ > 0), and the probability that ξ falls within the interval (70,90) is 0.8, then calculate the probability that it falls within the interval [90,100]. | 0.1 |
Given that $\alpha$ and $\beta$ are the roots of $x^2 - 3x + 1 = 0,$ find $7 \alpha^5 + 8 \beta^4.$ | 1448 |
A parabola with equation $y = x^2 + bx + c$ passes through the points $(2,3)$ and $(4,3)$. What is $c$? | 11 |
The sequence of integers $ a_1 $ , $ a_2 $ , $ \dots $ is defined as follows: $ a_1 = 1 $ and $ n> 1 $ , $ a_ {n + 1} $ is the smallest integer greater than $ a_n $ and such, that $ a_i + a_j \neq 3a_k $ for any $ i, j $ and $ k $ from $ \{1, 2, \dots, n + 1 \} $ are not necessarily different.
Define ... | 3006 |
The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four cents to label locker number $10$. If it costs $137.94$ to label all the lockers, how ... | 2001 |
The distance between locations A and B is 135 kilometers. Two cars, a large one and a small one, travel from A to B. The large car departs 4 hours earlier than the small car, but the small car arrives 30 minutes earlier than the large car. The speed ratio of the small car to the large car is 5:2. Find the speeds of bot... | 18 |
Points $P$ and $R$ are located at (2, 1) and (12, 15) respectively. Point $M$ is the midpoint of segment $\overline{PR}$. Segment $\overline{PR}$ is reflected over the $x$-axis. What is the sum of the coordinates of the image of point $M$ (the midpoint of the reflected segment)? | -1 |
Triangles $ABC$ and $AFG$ have areas $3012$ and $10004$, respectively, with $B=(0,0),$ $C=(335,0),$ $F=(1020, 570),$ and $G=(1030, 580).$ Find the sum of all possible $x$-coordinates of $A$. | 1800 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $b\sin C + c\sin B = 4a\sin B\sin C$ and $b^2 + c^2 - a^2 = 8$. Find the area of $\triangle ABC$. | \frac{2\sqrt{3}}{3} |
Eden, Mary and Iris gather sand to fill their sandbox. Eden carried 4 buckets of sand. Mary carried 3 more buckets of sand than Eden. Iris carried 1 less bucket of sand than Mary. If each bucket contains 2 pounds of sand, how many pounds of sand did they collect in total? | Mary carried 4 + 3 = <<4+3=7>>7 buckets of sand.
Iris carried 7 – 1 = <<7-1=6>>6 buckets of sand.
All together, they carried 4 + 7 + 6 = <<4+7+6=17>>17 buckets of sand.
The sand weighed a total of 17 * 2 = <<17*2=34>>34 pounds.
#### 34 |
A judge oversaw seventeen court cases. Two were immediately dismissed from court. Two-thirds of the remaining cases were ruled innocent, one ruling was delayed until a later date, and the rest were judged guilty. On how many cases did the judge rule guilty? | The judge dismissed 2 cases, leaving 17 - 2 = <<17-2=15>>15 cases.
Of the remaining 15 cases, 2 / 3 * 15 = <<2/3*15=10>>10 were ruled innocent.
One ruling was delayed, so the judge ruled guilty on 15 - 10 - 1 = <<15-10-1=4>>4 cases.
#### 4 |
If $a+\frac {a} {3}=\frac {8} {3}$, what is the value of $a$? | 2 |
Find the value of the function \( f(x) \) at the point \( x_{0} = 4500 \), if \( f(0) = 1 \) and for any \( x \) the equality \( f(x + 3) = f(x) + 2x + 3 \) holds. | 6750001 |
A traveler visited a village where each person either always tells the truth or always lies. The villagers stood in a circle, and each person told the traveler whether the neighbor to their right was truthful or deceitful. Based on these statements, the traveler was able to determine what fraction of the villagers are... | 1/2 |
Al, Betty, and Clare split $\$1000$ among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of $\$1500$. Betty and Clare have both doubled their money, whereas Al has managed to lose $\$100$. What was Al's original portion? | 400 |
Among the integers from 1 to 100, how many integers can be divided by exactly two of the following four numbers: 2, 3, 5, 7? | 27 |
Given the digits 1, 2, 3, 4, and 5, create a five-digit number without repetition, with 5 not in the hundred's place, and neither 2 nor 4 in the unit's or ten-thousand's place, and calculate the total number of such five-digit numbers. | 32 |
Find $q(x)$ if the graph of $\frac{x^3-2x^2-5x+3}{q(x)}$ has vertical asymptotes at $2$ and $-2$, no horizontal asymptote, and $q(3) = 15$. | 3x^2 - 12 |
Completely factor the following expression: \[(9x^5+25x^3-4)-(x^5-3x^3-4).\] | 4x^3(2x^2+7) |
The diagonals of trapezoid \(ABCD\) intersect at point \(M\). The areas of triangles \(ABM\) and \(CDM\) are 18 and 50 units, respectively. What is the area of the trapezoid? | 128 |
What is the sum of the digits of the greatest prime number that is a divisor of 8,191? | 10 |
Given that $x$ is a positive integer less than 100, how many solutions does the congruence $x + 13 \equiv 55 \pmod{34}$ have? | 3 |
Given that $\sin(\alpha + \frac{\pi}{5}) = \frac{1}{3}$ and $\alpha$ is an obtuse angle, find the value of $\cos(\alpha + \frac{9\pi}{20})$. | -\frac{\sqrt{2} + 4}{6} |
What is the total number of digits used when the first 3003 positive even integers are written? | 11460 |
If $x$, $y$, and $z$ are positive with $xy=20\sqrt[3]{2}$, $xz = 35\sqrt[3]{2}$, and $yz=14\sqrt[3]{2}$, then what is $xyz$? | 140 |
The maximum value of $k$ such that the inequality $\sqrt{x-3}+\sqrt{6-x}\geq k$ has a real solution. | \sqrt{6} |
A regular octahedron has a side length of 1. What is the distance between two opposite faces? | \sqrt{6} / 3 |
For a positive integer $n$, the factorial notation $n!$ represents the product of the integers from $n$ to $1$. What value of $N$ satisfies the following equation? $5!\cdot 9!=12\cdot N!$ | 10 |
Suppose I have 6 shirts and 5 ties. How many shirt-and-tie outfits can I make? | 30 |
Ivan has a piggy bank that can hold 100 pennies and 50 dimes. How much, in dollars, does Ivan have if he has filled his two piggy banks with those coins? | Ivan has 50 x 10 = <<50*10=500>>500 cents from the 50 dimes in his one piggy bank.
So, he has a total of 100 + 500 = <<100+500=600>>600 cents in one of his piggy banks.
Since 100 cents is equal to 1 dollar, then Ivan has 600/100 = 6 dollars from one of his piggy banks.
Therefore, Ivan has a total of $6 x 2 piggy banks ... |
There are 5 different books to be distributed among three people, with each person receiving at least 1 book and at most 2 books. Calculate the total number of different distribution methods. | 90 |
Let \( x_1, x_2, \ldots, x_{100} \) be natural numbers greater than 1 (not necessarily distinct). In an \(80 \times 80\) table, numbers are arranged as follows: at the intersection of the \(i\)-th row and the \(k\)-th column, the number \(\log _{x_{k}} \frac{x_{i}}{16}\) is written. Find the minimum possible value of t... | -19200 |
On the section of the river from $A$ to $B$, the current is so small that it can be ignored; on the section from $B$ to $C$, the current affects the movement of the boat. The boat covers the distance downstream from $A$ to $C$ in 6 hours, and upstream from $C$ to $A$ in 7 hours. If the current on the section from $A$ t... | 7.7 |
In $\triangle ABC$, medians $\overline{AM}$ and $\overline{BN}$ are perpendicular. If $AM = 15$ and $BN = 20$, find the length of side $AB$. | \frac{50}{3} |
Alex wrote all natural divisors of a natural number \( n \) on the board in ascending order. Dima erased several of the first and several of the last numbers of the resulting sequence so that 151 numbers remained. What is the maximum number of these 151 divisors that could be fifth powers of natural numbers? | 31 |
Given complex numbers \( z, z_{1}, z_{2} \left( z_{1} \neq z_{2} \right) \) such that \( z_{1}^{2}=z_{2}^{2}=-2-2 \sqrt{3} \mathrm{i} \), and \(\left|z-z_{1}\right|=\left|z-z_{2}\right|=4\), find \(|z|=\ \ \ \ \ .\) | 2\sqrt{3} |
Given $α \in \left(0, \frac{\pi}{2}\right)$, $\cos \left(α+ \frac{\pi}{3}\right) = -\frac{2}{3}$, then $\cos α =$ \_\_\_\_\_\_. | \frac{\sqrt{15}-2}{6} |
Let $[x]$ denote the greatest integer not exceeding $x$. Find the last two digits of $\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\cdots+\left[\frac{2^{2014}}{3}\right]$. | 15 |
Given a cube \( ABCD-A_1B_1C_1D_1 \) with edge length 1, a point \( M \) is taken on the diagonal \( A_1D \) and a point \( N \) is taken on \( CD_1 \) such that the line segment \( MN \) is parallel to the diagonal plane \( A_1ACC_1 \), find the minimum value of \( |MN| \). | \frac{\sqrt{3}}{3} |
Among all the simple fractions where both the numerator and the denominator are two-digit numbers, find the smallest fraction that is greater than $\frac{3}{5}$. Provide the numerator of this fraction in your answer. | 59 |
Circle $C$ with radius 2 has diameter $\overline{AB}$. Circle D is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$, and can be written in the form $... | 254 |
What is the value of the expression $2 \times 3 + 2 \times 3$? | 12 |
A large rectangle is tiled by some $1\times1$ tiles. In the center there is a small rectangle tiled by some white tiles. The small rectangle is surrounded by a red border which is fi ve tiles wide. That red border is surrounded by a white border which is fi ve tiles wide. Finally, the white border is surrounded by a ... | 350 |
If $|x-2|=p$, where $x<2$, then what is $x-p$ in terms of $p$? | 2-2p |
Greg's PPO algorithm obtained 90% of the possible reward on the CoinRun environment. CoinRun's maximum reward is half as much as the maximum ProcGen reward of 240. How much reward did Greg's PPO algorithm get? | Half of much as ProcGen's maximum reward is 240/2=<<240/2=120>>120 reward
90% of CoinRun's maximum reward is 120*.9=<<120*.9=108>>108 reward
#### 108 |
What is \(\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) \)? | 1{,}010{,}000 |
The function $f(x)$ satisfies
\[f(x) + 2f(1 - x) = 3x^2\]for all real numbers $x.$ Find $f(3).$ | -1 |
Find all natural numbers whose own divisors can be paired such that the numbers in each pair differ by 545. An own divisor of a natural number is a natural divisor different from one and the number itself. | 1094 |
Erin is sorting through the library books to decide which ones to replace. She finds 8 less than 6 times as many obsolete books as damaged books. If she removes 69 books total, how many books were damaged? | Let o be the number of obsolete books and d be the number of damaged books. We know that o + d = 69 and o = 6d - 8.
Substituting the first equation into the second equation, we get 6d - 8 + d = 69
Combining like terms, we get 7d - 8 = 69
Adding 8 to both sides, we get 7d = 77
Dividing both sides by 7, we get d = 11
###... |
A straight one-way city street has 8 consecutive traffic lights. Every light remains green for 1.5 minutes, yellow for 3 seconds, and red for 1.5 minutes. The lights are synchronized so that each light turns red 10 seconds after the preceding one turns red. What is the longest interval of time, in seconds, during which... | 20 |
In the diagram, each circle is divided into two equal areas and $O$ is the center of the larger circle. The area of the larger circle is $64\pi.$ What is the total area of the shaded regions? [asy]
size(100);
import graph;
fill(Arc((0,0),2,180,360)--cycle,mediumgray);fill(Arc((0,1),1,0,180)--cycle,mediumgray);
draw(Cir... | 40\pi |
Aunt Wang needs 3 minutes to cut a paper-cut for window decoration. After cutting each one, she rests for 1 minute. She starts cutting at 9:40. After cutting 10 paper-cuts, it is \_\_\_\_ hour \_\_\_\_ minute. | 10:19 |
In a similar game setup, there are 30 boxes, each containing one of the following values: \begin{tabular}{|c|c|}\hline\$.01&\$1,000\\\hline\$1&\$5,000\\\hline\$5&\$10,000\\\hline\$10&\$25,000\\\hline\$25&\$50,000\\\hline\$50&\$75,000\\\hline\$75&\$100,000\\\hline\$100&\$200,000\\\hline\$200&\$300,000\\\hline\$300&\$400... | 18 |
Let $\mathcal{P}$ be a parabola, and let $V_{1}$ and $F_{1}$ be its vertex and focus, respectively. Let $A$ and $B$ be points on $\mathcal{P}$ so that $\angle AV_{1}B=90^{\circ}$. Let $\mathcal{Q}$ be the locus of the midpoint of $AB$. It turns out that $\mathcal{Q}$ is also a parabola, and let $V_{2}$ and $F_{2}$ deno... | \frac{7}{8} |
John earned $18 on Saturday but he only managed to earn half that amount on Sunday. He earned $20 the previous weekend. How much more money does he need to earn to give him the $60 he needs to buy a new pogo stick? | John earnt $18 / 2 = $<<18/2=9>>9 on Sunday.
In total, John made $18 + $9 + $20 = $<<18+9+20=47>>47 over the two weekends.
John needs $60 - $47 = $<<60-47=13>>13 extra to buy the pogo stick.
#### 13 |
(1) Given $\cos \alpha =\frac{\sqrt{5}}{3}, \alpha \in \left(-\frac{\pi }{2},0\right)$, find $\sin (\pi -\alpha)$;
(2) Given $\cos \left(\theta+ \frac{\pi}{4}\right)= \frac{4}{5}, \theta \in \left(0, \frac{\pi}{2}\right)$, find $\cos \left(\frac{\pi }{4}-\theta \right)$. | \frac{3}{5} |
The base of a triangle is of length $b$, and the altitude is of length $h$. A rectangle of height $x$ is inscribed in the triangle with the base of the rectangle in the base of the triangle. The area of the rectangle is:
$\textbf{(A)}\ \frac{bx}{h}(h-x)\qquad \textbf{(B)}\ \frac{hx}{b}(b-x)\qquad \textbf{(C)}\ \frac{bx... | \frac{bx}{h}(h-x) |
In an arithmetic sequence \(\left\{a_{n}\right\}\), if \(\frac{a_{11}}{a_{10}} < -1\), and the sum of its first \(n\) terms \(S_{n}\) has a maximum value. Then, when \(S_{n}\) attains its smallest positive value, \(n =\) ______ . | 19 |
Find the minimum value of
\[\frac{\sin^6 x + \cos^6 x + 1}{\sin^4 x + \cos^4 x + 1}\]over all real values $x.$ | \frac{5}{6} |
Given a fixed circle $\odot P$ with a radius of 1, the distance from the center $P$ to a fixed line $l$ is 2. Point $Q$ is a moving point on $l$, and circle $\odot Q$ is externally tangent to circle $\odot P$. Circle $\odot Q$ intersects $l$ at points $M$ and $N$. For any diameter $MN$, there is always a fixed point $A... | 60 |
For how many integer values of $x$ is $5x^{2}+19x+16 > 20$ not satisfied? | 5 |
Let $S$ be a set. We say $S$ is $D^\ast$ *-finite* if there exists a function $f : S \to S$ such that for every nonempty proper subset $Y \subsetneq S$ , there exists a $y \in Y$ such that $f(y) \notin Y$ . The function $f$ is called a *witness* of $S$ . How many witnesses does $\{0,1,\cdots,5\}$ have... | 120 |
Determine the value of $$1 \cdot 2-2 \cdot 3+3 \cdot 4-4 \cdot 5+\cdots+2001 \cdot 2002$$ | 2004002 |
The average score on last week's Spanish test was 90. Marco scored 10% less than the average test score and Margaret received 5 more points than Marco. What score did Margaret receive on her test? | The average test score was 90 and Marco scored 10% less so 90*.10 = <<90*.10=9>>9 points lower
The average test score was 90 and Marco scored 9 points less so his test score was 90-9 = <<90-9=81>>81
Margret received 5 more points than Marco whose test score was 81 so she made 5+81 = <<5+81=86>>86 on her test
#### 86 |
Multiply $555.55$ by $\frac{1}{3}$ and then subtract $333.33$. Express the result as a decimal to the nearest hundredth. | -148.15 |
Find the time, in seconds, after 12 o'clock, when the area of $\triangle OAB$ will reach its maximum for the first time. | \frac{15}{59} |
A sector with acute central angle $\theta$ is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is
$\textbf{(A)}\ 3\cos\theta \qquad \textbf{(B)}\ 3\sec\theta \qquad \textbf{(C)}\ 3 \cos \frac12 \theta \qquad \textbf{(D)}\ 3 \sec \frac12 \theta \qquad \textbf{(E)}\ 3$
| 3 \sec \frac{1}{2} \theta |
What is the sum of all of the solutions of the equation $\frac{4x}{20}=\frac{5}{x}$? | 0 |
Given the sequence ${a_n}$ that satisfies the equation $a_{n+1}+(-1)^{n}a_{n}=3n-1,(n∈N^{*})$, determine the sum of the first 40 terms of the sequence ${a_n}$. | 1240 |
The lateral edges of a triangular pyramid are mutually perpendicular, and the sides of the base are $\sqrt{85}$, $\sqrt{58}$, and $\sqrt{45}$. The center of the sphere, which touches all the lateral faces, lies on the base of the pyramid. Find the radius of this sphere. | 14/9 |
In a circle centered at $O$, point $A$ is on the circle, and $\overline{BA}$ is tangent to the circle at $A$. Triangle $ABC$ is right-angled at $A$ with $\angle ABC = 45^\circ$. The circle intersects $\overline{BO}$ at $D$. Chord $\overline{BC}$ also extends to meet the circle at another point, $E$. What is the value o... | \frac{2 - \sqrt{2}}{2} |
The function $f(x)$ satisfies
\[b^2 f(a) = a^2 f(b)\]for all real numbers $a$ and $b.$ If $f(2) \neq 0,$ find
\[\frac{f(5) - f(1)}{f(2)}.\] | 6 |
Given that for any positive integer \( n \), \( 9^{2n} - 8^{2n} - 17 \) is always divisible by \( m \), find the largest positive integer \( m \). | 2448 |
In a pentagon ABCDE, there is a vertical line of symmetry. Vertex E is moved to \(E(5,0)\), while \(A(0,0)\), \(B(0,5)\), and \(D(5,5)\). What is the \(y\)-coordinate of vertex C such that the area of pentagon ABCDE becomes 65 square units? | 21 |
At the first site, higher-class equipment was used, while at the second site, first-class equipment was used, with higher-class being less than first-class. Initially, 30% of the equipment from the first site was transferred to the second site. Then, 10% of the equipment at the second site was transferred to the first ... | 17 |
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible? | 729 |
Given a group with the numbers $-3, 0, 5, 8, 11, 13$, and the following rules: the largest isn't first, and it must be within the first four places, the smallest isn't last, and it must be within the last four places, and the median isn't in the first or last position, determine the average of the first and last number... | 5.5 |
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