problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
In right triangle $DEF$, we have $\sin D = \frac{5}{13}$ and $\sin E = 1$. Find $\sin F$. | \frac{12}{13} |
What is the maximum number of queens that can be placed on an $8 \times 8$ chessboard so that each queen can attack at least one other queen? | 16 |
Given two boxes, each containing the chips numbered $1$, $2$, $4$, $5$, a chip is drawn randomly from each box. Calculate the probability that the product of the numbers on the two chips is a multiple of $4$. | \frac{1}{2} |
Peter carried $500 to the market. He bought 6 kilos of potatoes for $2 per kilo, 9 kilos of tomato for $3 per kilo, 5 kilos of cucumbers for $4 per kilo, and 3 kilos of bananas for $5 per kilo. How much is Peter’s remaining money? | The price of potatoes is 6 * 2 = $<<6*2=12>>12.
The price of tomatoes is 9 * 3 = $<<9*3=27>>27.
The price of cucumbers is 5 * 4 = $<<5*4=20>>20.
The price of bananas is 3 * 5 = $<<3*5=15>>15.
The total price Peter spent is 12 + 27 + 20 + 15 = $<<12+27+20+15=74>>74.
The amount left with Peter is $500 - $74 = $<<500-74=4... |
Emberly takes her mornings walks every day. If each walk takes her 1 hour covering 4 miles, and she didn't walk for 4 days in March, calculate the total number of miles she's walked. | In March, she walked for 31-4 = <<31-4=27>>27 days.
If she walks for an hour a day, the total number of hours she'll walk in March is 27 days*1 hour a day= <<27*1=27>>27 hours.
If each walk is 4 miles, the number of miles she walked in March is 27*4 = <<27*4=108>>108 miles.
#### 108 |
Given that cos(15°+α) = $\frac{3}{5}$, where α is an acute angle, find: $$\frac{tan(435° -α)+sin(α-165° )}{cos(195 ° +α )\times sin(105 ° +α )}$$. | \frac{5}{36} |
An artist wants to completely cover a rectangle with identically sized squares which do not overlap and do not extend beyond the edges of the rectangle. If the rectangle is \(60 \frac{1}{2} \mathrm{~cm}\) long and \(47 \frac{2}{3} \mathrm{~cm}\) wide, what is the minimum number of squares required? | 858 |
In a certain cross country meet between 2 teams of 5 runners each, a runner who finishes in the $n$th position contributes $n$ to his teams score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible?
(A) 10 (B) 13 (C) 27 (D) 120 (E) 126
| 13 |
If the functions \( f(x) \) and \( g(x) \) are defined for all real numbers, and they satisfy the equation \( f(x-y) = f(x) g(y) - g(x) f(y) \), with \( f(-2) = f(1) \neq 0 \), then find \( g(1) + g(-1) \). | -1 |
The product of two positive integers plus their sum is 119. The integers are relatively prime and each is less than 30. What is the sum of the two integers? | 20 |
In $\triangle ABC$, angle bisectors $BD$ and $CE$ intersect at $I$, with $D$ and $E$ located on $AC$ and $AB$ respectively. A perpendicular from $I$ to $DE$ intersects $DE$ at $P$, and the extension of $PI$ intersects $BC$ at $Q$. If $IQ = 2 IP$, find $\angle A$. | 60 |
In a school cafeteria line, there are 16 students alternating between boys and girls (starting with a boy, followed by a girl, then a boy, and so on). Any boy, followed immediately by a girl, can swap places with her. After some time, all the girls end up at the beginning of the line and all the boys are at the end. Ho... | 36 |
A garden is filled with 105 flowers of various colors. There are twice as many red flowers as orange. There are five fewer yellow flowers than red. If there are 10 orange flowers, how many pink and purple flowers are there if they have the same amount and there are no other colors? | The number of red flowers is twice as many as orange so 2 * 10 orange = <<2*10=20>>20 red flowers
The number of yellow flowers is 5 less than red so 20 red - 5 = <<20-5=15>>15 yellow flowers
The number of red, orange and yellow flowers is 20 red + 10 orange + 15 yellow = <<20+10+15=45>>45 flowers
The number of pink and... |
Let $s (n)$ denote the sum of digits of a positive integer $n$. Using six different digits, we formed three 2-digits $p, q, r$ such that $$p \cdot q \cdot s(r) = p\cdot s(q) \cdot r = s (p) \cdot q \cdot r.$$ Find all such numbers $p, q, r$. | (12, 36, 48), (21, 63, 84) |
The midsegment of a trapezoid divides it into two quadrilaterals. The difference in the perimeters of these two quadrilaterals is 24, and the ratio of their areas is $\frac{20}{17}$. Given that the height of the trapezoid is 2, what is the area of this trapezoid? | 148 |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and $(\sin A + \sin B)(a-b) = (\sin C - \sin B)c$.
1. Find the measure of angle $A$.
2. If $a=4$, find the maximum area of $\triangle ABC$. | 4\sqrt{3} |
Let \[f(x) = \left\{
\begin{array}{cl} x^2-4 & \text{ if }x < 7, \\
x-13 & \text{ if } x \geq 7.
\end{array}
\right.\] What is $f(f(f(17)))$? | -1 |
A sphere is inscribed in a cube with edge length 9 inches. Then a smaller cube is inscribed in the sphere. How many cubic inches are in the volume of the inscribed cube? Express your answer in simplest radical form. | 81\sqrt{3} |
The number \( C \) is defined as the sum of all the positive integers \( n \) such that \( n-6 \) is the second largest factor of \( n \). What is the value of \( 11C \)? | 308 |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\diamondsuit$? | \dfrac{1}{52} |
It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. | 40 |
Consider a unit square $WXYZ$ with midpoints $M_1$, $M_2$, $M_3$, and $M_4$ on sides $WZ$, $XY$, $YZ$, and $XW$ respectively. Let $R_1$ be a point on side $WZ$ such that $WR_1 = \frac{1}{4}$. A light ray starts from $R_1$ and reflects off at point $S_1$ (which is the intersection of the ray $R_1M_2$ and diagonal $WY$).... | \frac{1}{24} |
$\zeta_1, \zeta_2,$ and $\zeta_3$ are complex numbers such that
\[\zeta_1+\zeta_2+\zeta_3=1\]\[\zeta_1^2+\zeta_2^2+\zeta_3^2=3\]\[\zeta_1^3+\zeta_2^3+\zeta_3^3=7\]
Compute $\zeta_1^{7} + \zeta_2^{7} + \zeta_3^{7}$.
| 71 |
Let $\mathbf{a} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -5 \\ 2 \end{pmatrix}.$ Find the area of the triangle with vertices $\mathbf{0},$ $\mathbf{a},$ and $\mathbf{b}.$ | \frac{11}{2} |
Rationalize the denominator of $\frac{1+\sqrt{3}}{1-\sqrt{3}}$. When you write your answer in the form $A+B\sqrt{C}$, where $A$, $B$, and $C$ are integers, what is $ABC$? | 6 |
Let $w_1$ and $w_2$ denote the circles $x^2+y^2+10x-24y-87=0$ and $x^2 +y^2-10x-24y+153=0,$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_2$ and internally tangent to $w_1.$ Given that $m^2=\frac pq,$ where $p$ and ... | 169 |
The interior angles of a convex polygon form an arithmetic sequence, with the smallest angle being $120^\circ$ and the common difference being $5^\circ$. Determine the number of sides $n$ for the polygon. | n = 9 |
Find the center of the hyperbola $4x^2 - 24x - 25y^2 + 250y - 489 = 0.$ | (3,5) |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $(2c-a)\cos B=b\cos A$.
(1) Find angle $B$;
(2) If $b=6$ and $c=2a$, find the area of $\triangle ABC$. | 6 \sqrt{3} |
If $g(x) = 2x^2+2x-1$, what is the value of $g(g(2))$? | 263 |
At 12 o'clock, the angle between the hour hand and the minute hand is 0 degrees. After that, at what time do the hour hand and the minute hand form a 90-degree angle for the 6th time? (12-hour format) | 3:00 |
Five years ago, there were 500 old books in the library. Two years ago, the librarian bought 300 books. Last year, the librarian bought 100 more books than she had bought the previous year. This year, the librarian donated 200 of the library's old books. How many books are in the library now? | Last year the librarian bought 300 + 100 = <<300+100=400>>400 books.
Thus, there were 500 + 300 + 400 = <<500+300+400=1200>>1200 books as of last year.
Therefore, there are 1200 - 200 = <<1200-200=1000>>1000 books now in the library.
#### 1000 |
George and Henry started a race from opposite ends of the pool. After a minute and a half, they passed each other in the center of the pool. If they lost no time in turning and maintained their respective speeds, how many minutes after starting did they pass each other the second time? | 4\frac{1}{2} |
For $-1<r<1$, let $S(r)$ denote the sum of the geometric series \[12+12r+12r^2+12r^3+\cdots .\]Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2016$. Find $S(a)+S(-a)$. | 336 |
Jolene and Tia are playing a two-player game at a carnival. In one bin, there are five red balls numbered 5, 10, 15, 20, and 25. In another bin, there are 25 green balls numbered 1 through 25. In the first stage of the game, Jolene chooses one of the red balls at random. Next, the carnival worker removes the green ball... | 13/40 |
The minimum value of the function \( y = \sin^4{x} + \cos^4{x} + \sec^4{x} + \csc^4{x} \). | \frac{17}{2} |
On the set of solutions to the system of constraints
$$
\left\{\begin{array}{l}
2-2 x_{1}-x_{2} \geqslant 0 \\
2-x_{1}+x_{2} \geqslant 0 \\
5-x_{1}-x_{2} \geqslant 0 \\
x_{1} \geqslant 0, \quad x_{2} \geqslant 0
\end{array}\right.
$$
find the minimum value of the function $F = x_{2} - x_{1}$. | -2 |
The measures of angles \( X \) and \( Y \) are both positive, integer numbers of degrees. The measure of angle \( X \) is a multiple of the measure of angle \( Y \), and angles \( X \) and \( Y \) are supplementary angles. How many measures are possible for angle \( X \)? | 17 |
What three-digit integer is equal to the sum of the factorials of its digits, where one of the digits is `3`, contributing `3! = 6` to the sum? | 145 |
The reality game show Survivor is played with 16 people divided into two tribes of 8. In the first episode, two people get homesick and quit. If every person has an equal chance of being one of the two quitters, and the probability that one person quits is independent of the probability that any other person quits, wha... | \frac{7}{15} |
Seven distinct points are identified on the circumference of a circle. How many different triangles can be formed if each vertex must be one of these 7 points? | 35 |
Let \( S = \{1, 2, \ldots, 2005\} \). If any set of \( n \) pairwise co-prime numbers in \( S \) always contains at least one prime number, what is the minimum value of \( n \)? | 16 |
Alina and Masha wanted to create an interesting version of the school tour of the Olympiad. Masha proposed several problems and rejected every second problem of Alina (exactly half). Alina also proposed several problems and only accepted every third problem of Masha (exactly one-third). In the end, there were 10 proble... | 15 |
Given the function $f(x)=e^{ax}-x-1$, where $a\neq 0$. If $f(x)\geqslant 0$ holds true for all $x\in R$, then the set of possible values for $a$ is \_\_\_\_\_\_. | \{1\} |
Let $(2-x)^{6}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{6}x^{6}$, then the value of $|a_{1}|+|a_{2}|+\ldots+|a_{6}|$ is \_\_\_\_\_\_. | 665 |
How many positive two-digit integers leave a remainder of 2 when divided by 8? | 12 |
Regular pentagon $ABCDE$ and regular hexagon $AEFGHI$ are drawn on opposite sides of line segment $AE$ such that they are coplanar. What is the degree measure of exterior angle $DEF$? [asy]
draw((0,2.5)--(0,7.5)--(4,10)--(8,7.5)--(8,2.5)--(4,0)--cycle,linewidth(1));
draw((8,2.5)--(11.5,-1)--(9,-5)--(5,-4.5)--(4,0),line... | 132 |
Compute the smallest positive integer $n$ such that $n + i,$ $(n + i)^2,$ and $(n + i)^3$ are the vertices of a triangle in the complex plane whose area is greater than 2015. | 9 |
Find the remainder when $2^{2^{2^2}}$ is divided by 500. | 36 |
Jacob takes four tests in his physics class and earns 85, 79, 92 and 84. What must he earn on his fifth and final test to have an overall average of 85? | Since wants needs an 85 as an average and he takes five tests, he must have 85 * 5 = <<85*5=425>>425 total points.
Currently he has 85 + 79 + 92 + 84 = <<85+79+92+84=340>>340 points.
So for his fifth test he needs 425 - 340 = <<425-340=85>>85 points.
#### 85 |
A chihuahua, pitbull, and great dane weigh a combined weight of 439 pounds. The pitbull weighs 3 times as much as the chihuahua. The great dane weighs 10 more pounds than triple the pitbull. How much does the great dane weigh? | Let x represent the weight of the chihuahua
Pitbull:3x
Great dane:10+3(3x)=10+9x
Total:x+3x+10+9x=439
13x+10=439
13x=429
x=<<33=33>>33 pounds
Great dane:10+9(33)=307 pounds
#### 307 |
Let
\[f(x) = \left\{
\begin{array}{cl}
x^2 + 3 & \text{if $x < 15$}, \\
3x - 2 & \text{if $x \ge 15$}.
\end{array}
\right.\]
Find $f^{-1}(10) + f^{-1}(49).$ | \sqrt{7} + 17 |
Given that Steve's empty swimming pool holds 30,000 gallons of water when full and will be filled by 5 hoses, each supplying 2.5 gallons of water per minute, calculate the time required to fill the pool. | 40 |
In a convex quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(BD\), \(\angle A=65^\circ\), \(\angle B=80^\circ\), and \(\angle C=75^\circ\). What is \(\angle CAD\) (in degrees)? | 15 |
Seven students stand in a row for a photo, among them, students A and B must stand next to each other, and students C and D must not stand next to each other. The total number of different arrangements is. | 960 |
Given the sequence $\{a\_n\}$ satisfies $a\_1=2$, $a_{n+1}-2a_{n}=2$, and the sequence $b_{n}=\log _{2}(a_{n}+2)$. If $S_{n}$ is the sum of the first $n$ terms of the sequence $\{b_{n}\}$, then the minimum value of $\{\frac{S_{n}+4}{n}\}$ is ___. | \frac{9}{2} |
John is trying to save money by buying cheap calorie-dense food. He can buy 10 burritos for $6 that have 120 calories each. He could also buy 5 burgers that are 400 calories each for $8. How many more calories per dollar does he get from the burgers? | The burritos have 10*120=<<10*120=1200>>1200 calories
That means he gets 1200/6=<<1200/6=200>>200 calories per dollar
The burgers get 5*400=<<5*400=2000>>2000 calories
So he gets 2000/8=<<2000/8=250>>250 calories per dollar
So he gets 250-200=<<250-200=50>>50 more calories per dollar from the burgers
#### 50 |
In triangle $ ABC$ , $ 3\sin A \plus{} 4\cos B \equal{} 6$ and $ 4\sin B \plus{} 3\cos A \equal{} 1$ . Then $ \angle C$ in degrees is | 30 |
Let $f(x) = x^3 - 9x^2 + 27x - 25$ and let $g(f(x)) = 3x + 4$. What is the sum of all possible values of $g(7)$? | 39 |
Suppose $x$ is an integer that satisfies the following congruences:
\begin{align*}
2+x &\equiv 3^2 \pmod{2^4}, \\
3+x &\equiv 2^3 \pmod{3^4}, \\
4+x &\equiv 3^3 \pmod{2^3}.
\end{align*}
What is the remainder when $x$ is divided by $24$? | 23 |
[asy]size(8cm);
real w = 2.718; // width of block
real W = 13.37; // width of the floor
real h = 1.414; // height of block
real H = 7; // height of block + string
real t = 60; // measure of theta
pair apex = (w/2, H); // point where the strings meet
path block = (0,0)--(w,0)--(w,h)--(0,h)--cycle; // construct the bl... | 13 |
A regular octagon is inscribed in a circle of radius 2. Alice and Bob play a game in which they take turns claiming vertices of the octagon, with Alice going first. A player wins as soon as they have selected three points that form a right angle. If all points are selected without either player winning, the game ends i... | 2 \sqrt{2}, 4+2 \sqrt{2} |
The real function $g$ has the property that, whenever $x,$ $y,$ $m$ are positive integers such that $x + y = 3^m,$ the equation
\[g(x) + g(y) = 2m^2\]holds. What is $g(2187)$? | 98 |
Point P moves on the parabola y^2 = 4x with focus F, and point Q moves on the line x-y+5=0. Find the minimum value of ||PF+|PQ||. | 3\sqrt{2} |
The arithmetic mean of a set of $60$ numbers is $42$. If three numbers of the set, namely $40$, $50$, and $60$, are discarded, the arithmetic mean of the remaining set of numbers is:
**A)** 41.3
**B)** 41.4
**C)** 41.5
**D)** 41.6
**E)** 41.7 | 41.6 |
In March of this year, the Municipal Bureau of Industry and Commerce conducted a quality supervision and random inspection of beverages in the circulation field within the city. The results showed that the qualification rate of a newly introduced X beverage in the market was 80%. Now, three people, A, B, and C, gather ... | 0.44 |
Given that $f(x)$ is an odd function on $\mathbb{R}$ and satisfies $f(x+4)=f(x)$, when $x \in (0,2)$, $f(x)=2x^2$. Evaluate $f(2015)$. | -2 |
A tour group has three age categories of people, represented in a pie chart. The central angle of the sector corresponding to older people is $9^{\circ}$ larger than the central angle for children. The percentage of total people who are young adults is $5\%$ higher than the percentage of older people. Additionally, the... | 120 |
Find constants $A$, $B$, and $C$, such that
$$\frac{-x^2+3x-4}{x^3+x}= \frac{A}{x} +\frac{Bx+C}{x^2+1} $$Enter your answer as the ordered triplet $(A,B,C)$. | (-4,3,3) |
Given the sequence \(\left\{a_{n}\right\}\), which satisfies
\[
a_{1}=0,\left|a_{n+1}\right|=\left|a_{n}-2\right|
\]
Let \(S\) be the sum of the first 2016 terms of the sequence \(\left\{a_{n}\right\}\). Determine the maximum value of \(S\). | 2016 |
For a real number $a$ and an integer $n(\geq 2)$ , define $$ S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}} $$ Find every value of $a$ s.t. sequence $\{S_n(a)\}_{n\geq 2}$ converges to a positive real. | 2019 |
Find the smallest real number \( a \) such that for any non-negative real numbers \( x \), \( y \), and \( z \) that sum to 1, the following inequality holds:
$$
a(x^{2} + y^{2} + z^{2}) + xyz \geq \frac{a}{3} + \frac{1}{27}.
$$ | \frac{2}{9} |
On a chessboard, $n$ white rooks and $n$ black rooks are arranged such that rooks of different colors do not attack each other. Find the maximum possible value of $n$. | 16 |
In the geometric sequence $\{a_n\}$, $(a_1 \cdot a_2 \cdot a_3 = 27)$, $(a_2 + a_4 = 30)$, $(q > 0)$
Find:
$(1)$ $a_1$ and the common ratio $q$;
$(2)$ The sum of the first $6$ terms $(S_6)$. | 364 |
A right triangle is inscribed in the ellipse given by the equation $x^2 + 9y^2 = 9$. One vertex of the triangle is at the point $(0,1)$, and one leg of the triangle is fully contained within the x-axis. Find the squared length of the hypotenuse of the inscribed right triangle, expressed as the ratio $\frac{m}{n}$ with ... | 11 |
Let $n$ be an odd integer with exactly 11 positive divisors. Find the number of positive divisors of $8n^3$. | 124 |
Two balls are randomly chosen from a box containing 20 balls numbered from 1 to 20. Calculate the probability that the sum of the numbers on the two balls is divisible by 3. | \frac{32}{95} |
A bag contains 50 fewer baseball cards than football cards. There are 4 times as many football cards as hockey cards. If there are 200 hockey cards in the bag, how many cards are there altogether? | If there are 200 hockey cards in the bag and 4 times as many football cards as hockey cards, there are 4*200 = <<200*4=800>>800 football cards.
The total number of football and hockey cards in the bag is 800+200 = <<800+200=1000>>1000 cards.
The bag contains 50 fewer baseball cards than football cards, meaning there ar... |
The distance from the intersection point of the diameter of a circle with a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius of the circle.
$$
AB = 18, EO = 7, AE = 2 BE, R = ?
$$ | 11 |
Find the sum of all positive integers $n$ such that $1.5n - 6.3 < 7.5$. | 45 |
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible? | 132 |
The graph of the parabola defined by the equation $y=(x-2)^2+3$ is rotated 180 degrees about its vertex, then shifted 3 units to the left, then shifted 2 units down. The resulting parabola has zeros at $x=a$ and $x=b$. What is $a+b$? | -2 |
Given \( x_{1}, x_{2}, \cdots, x_{1993} \) satisfy:
\[
\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{1992}-x_{1993}\right|=1993,
\]
and
\[
y_{k}=\frac{x_{1}+x_{2}+\cdots+x_{k}}{k} \quad (k=1,2,\cdots,1993),
\]
what is the maximum possible value of \( \left|y_{1}-y_{2}\right|+\left|y_{2}-y_{3}\right|... | 1992 |
George and Harry want to fill a pool with buckets. George can carry two buckets each round, and Harry can carry three buckets each round. If it takes 110 buckets to fill the pool, how many rounds will it take to fill the pool? | Each round 2+3=5 buckets will be filled.
It will take 110/5=<<110/5=22>>22 rounds.
#### 22 |
Twelve congruent pentagonal faces, each of a different color, are used to construct a regular dodecahedron. How many distinguishable ways are there to construct this dodecahedron? (Two colored dodecahedrons are distinguishable if neither can be rotated to look just like the other.) | 7983360 |
Find the sum of all distinct possible values of $x^2-4x+100$ , where $x$ is an integer between 1 and 100, inclusive.
*Proposed by Robin Park* | 328053 |
Alice and Bob are playing a game with dice. They each roll a die six times and take the sums of the outcomes of their own rolls. The player with the higher sum wins. If both players have the same sum, then nobody wins. Alice's first three rolls are 6, 5, and 6, while Bob's first three rolls are 2, 1, and 3. The probabi... | 3895 |
Consider a rectangle with dimensions 6 units by 8 units. Points $A$, $B$, and $C$ are located on the sides of this rectangle such that the coordinates of $A$, $B$, and $C$ are $(0,2)$, $(6,0)$, and $(3,8)$ respectively. What is the area of triangle $ABC$ in square units? | 21 |
Let $G_{1} G_{2} G_{3}$ be a triangle with $G_{1} G_{2}=7, G_{2} G_{3}=13$, and $G_{3} G_{1}=15$. Let $G_{4}$ be a point outside triangle $G_{1} G_{2} G_{3}$ so that ray $\overrightarrow{G_{1} G_{4}}$ cuts through the interior of the triangle, $G_{3} G_{4}=G_{4} G_{2}$, and $\angle G_{3} G_{1} G_{4}=30^{\circ}$. Let $G... | \frac{169}{23} |
Given that $$(x+y+z)(xy+xz+yz)=18$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=6$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | 4 |
Expand $-(3-c)(c+2(3-c))$. What is the sum of the coefficients of the expanded form? | -10 |
A rotating disc is divided into five equal sectors labeled $A$, $B$, $C$, $D$, and $E$. The probability of the marker stopping on sector $A$ is $\frac{1}{5}$, the probability of it stopping in $B$ is $\frac{1}{5}$, and the probability of it stopping in sector $C$ is equal to the probability of it stopping in sectors $D... | \frac{1}{5} |
A card is chosen at random from a standard deck of 52 cards, and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a diamond or an ace? | \frac{88}{169} |
Radii of five concentric circles $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$ form a geometric progression with common ratio $q$ in this order. What is the maximal value of $q$ for which it's possible to draw a broken line $A_0A_1A_2A_3A_4$ consisting of four equal segments such that $A_i$ lies on $\omega_... | \frac{1 + \sqrt{5}}{2} |
If the line that passes through the points $(2,7)$ and $(a, 3a)$ has a slope of 2, what is the value of $a$? | 3 |
The bakery makes 3 batches of baguettes a day. Each batch has 48 baguettes. After the first batch came out, he sold 37. After the second batch, he sold 52. After the third batch he sold 49. How many baguettes are left? | He makes 3 batches of baguettes a day, 48 per batch so he makes 3*48 = <<3*48=144>>144
He sells 37, then 52 then 49 baguettes for a total of 37+52+49 = <<37+52+49=138>>138
He had 144 to sell and he sold 138 so he has 144-138 = <<144-138=6>>6 left over
#### 6 |
Let triangle $ABC$ be a right triangle with right angle at $C.$ Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C.$ If $\frac{DE}{BE} = \frac{8}{15},$ then find $\tan B.$ | \frac{4 \sqrt{3}}{11} |
Find all odd natural numbers greater than 500 but less than 1000, for which the sum of the last digits of all divisors (including 1 and the number itself) is equal to 33. | 729 |
The exchange rate of the cryptocurrency Chukhoyn was one dollar on March 1, and then increased by one dollar each day. The exchange rate of the cryptocurrency Antonium was also one dollar on March 1, and then each day thereafter, it was equal to the sum of the previous day's rates of Chukhoyn and Antonium divided by th... | 92/91 |
Max bought a new dirt bike and paid $10\%$ of the cost upfront, which was $\$150$. What was the price of the bike? | \$ 1500 |
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