problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
The projection of $\begin{pmatrix} 0 \\ 1 \\ 4 \end{pmatrix}$ onto a certain vector $\mathbf{w}$ is $\begin{pmatrix} 1 \\ -1/2 \\ 1/2 \end{pmatrix}.$ Find the projection of $\begin{pmatrix} 3 \\ 3 \\ -2 \end{pmatrix}$ onto $\mathbf{w}.$ | \begin{pmatrix} 1/3 \\ -1/6 \\ 1/6 \end{pmatrix} |
The surface area of the circumscribed sphere of cube \( K_1 \) is twice the surface area of the inscribed sphere of cube \( K_2 \). Let \( V_1 \) denote the volume of the inscribed sphere of cube \( K_1 \), and \( V_2 \) denote the volume of the circumscribed sphere of cube \( K_2 \). What is the ratio \( \frac{V_1}{V_... | \frac{2\sqrt{2}}{27} |
In the diagram below, $AB = AC = 115,$ $AD = 38,$ and $CF = 77.$ Compute $\frac{[CEF]}{[DBE]}.$
[asy]
unitsize(0.025 cm);
pair A, B, C, D, E, F;
B = (0,0);
C = (80,0);
A = intersectionpoint(arc(B,115,0,180),arc(C,115,0,180));
D = interp(A,B,38/115);
F = interp(A,C,(115 + 77)/115);
E = extension(B,C,D,F);
draw(C--B... | \frac{19}{96} |
A parking area near Peter's house is 4 stories tall. There are 4 open parking spots on the first level. There are 7 more open parking spots on the second level than on the first level, and there are 6 more open parking spots on the third level than on the second level. There are 14 open parking spots on the fourth leve... | The number of open parking spots on the second level is 4 spots + 7 spots = <<4+7=11>>11 spots.
The number of open parking spots on the third level is 11 spots + 6 spots = <<11+6=17>>17 spots.
There are 4 spots + 11 spots + 17 spots + 14 spots = <<4+11+17+14=46>>46 open parking spots in all.
#### 46 |
Given a positive integer $k$, let \|k\| denote the absolute difference between $k$ and the nearest perfect square. For example, \|13\|=3 since the nearest perfect square to 13 is 16. Compute the smallest positive integer $n$ such that $\frac{\|1\|+\|2\|+\cdots+\|n\|}{n}=100$. | 89800 |
An equilateral triangle and a circle intersect so that each side of the triangle contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the triangle to the area of the circle? Express your answer as a common fraction in terms of $\pi$. | \frac{9\sqrt{3}}{4\pi} |
The numbers \(a\) and \(b\) are such that \(|a| \neq |b|\) and \(\frac{a+b}{a-b} + \frac{a-b}{a+b} = 6\). Find the value of the expression \(\frac{a^{3} + b^{3}}{a^{3} - b^{3}} + \frac{a^{3} - b^{3}}{a^{3} + b^{3}}\). | \frac{18}{7} |
Given that point \( F \) is the right focus of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a > b > 0\)), and the eccentricity of the ellipse is \(\frac{\sqrt{3}}{2}\), a line \( l \) passing through point \( F \) intersects the ellipse at points \( A \) and \( B \) (point \( A \) is above the \( x \)-a... | -\sqrt{2} |
Vasya wrote natural numbers on the pages of an 18-page notebook. On each page, he wrote at least 10 different numbers, and on any consecutive three pages, there are no more than 20 different numbers in total. What is the maximum number of different numbers Vasya could have written on the pages of the notebook? | 190 |
Two people agreed to meet at a specific location between 12 PM and 1 PM. The condition is that the first person to arrive will wait for the second person for 15 minutes and then leave. What is the probability that these two people will meet if each of them chooses their moment of arrival at the agreed location randomly... | 7/16 |
Let $a, b, c$ be integers. Define $f(x)=a x^{2}+b x+c$. Suppose there exist pairwise distinct integers $u, v, w$ such that $f(u)=0, f(v)=0$, and $f(w)=2$. Find the maximum possible value of the discriminant $b^{2}-4 a c$ of $f$. | 16 |
The five solutions to the equation\[(z-1)(z^2+2z+4)(z^2+4z+6)=0\] may be written in the form $x_k+y_ki$ for $1\le k\le 5,$ where $x_k$ and $y_k$ are real. Let $\mathcal E$ be the unique ellipse that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ and $(x_5,y_5)$. The eccentricity of $\mathcal E$ ca... | 7 |
Let $a,$ $b,$ and $c$ be distinct real numbers such that
\[\frac{a^3 + 6}{a} = \frac{b^3 + 6}{b} = \frac{c^3 + 6}{c}.\]Find $a^3 + b^3 + c^3.$ | -18 |
Find the largest prime divisor of 11! + 12! | 13 |
Let $p(x) = x^2 + bx + c,$ where $b$ and $c$ are integers. If $p(x)$ is factor of both $x^4 + 6x^2 + 25$ and $3x^4 + 4x^ 2+ 28x + 5,$ what is $p(1)$? | 4 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\cos A= \frac {c}{a}\cos C$, $b+c=2+ \sqrt {2}$, and $\cos B= \frac {3}{4}$, find the area of $\triangle ABC$. | \frac { \sqrt {7}}{2} |
There is an equilateral triangle $ABC$ on the plane. Three straight lines pass through $A$ , $B$ and $C$ , respectively, such that the intersections of these lines form an equilateral triangle inside $ABC$ . On each turn, Ming chooses a two-line intersection inside $ABC$ , and draws the straight line determined... | 45853 |
A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F,$ in such a way that t... | 512 |
Polygon $ABCDEF$ is a regular hexagon. What is the measure in degrees of angle $ABF$? | 30 |
Let \( s(n) \) denote the sum of the digits of the natural number \( n \). Solve the equation \( n + s(n) = 2018 \). | 2008 |
Zane purchases 2 polo shirts from the 40% off rack at the men’s store. The polo shirts are $50 each at the regular price. How much did he pay for the shirts? | At regular price, the polo shirts are 2 x $50 = $<<2*50=100>>100.
At 40% off, he gets a discount of , $100 x 40% = $<<100*40*.01=40>>40.
With this discount, the polo shirts cost $100 - $40 = $<<100-40=60>>60.
#### 60 |
In the given figure, hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$? | $12$ |
In 3 years, Jayden will be half of Ernesto's age. If Ernesto is 11 years old, how many years old is Jayden now? | Ernesto = 11 + 3 = <<11+3=14>>14
Jayden = 14/2 = <<14/2=7>>7 in 3 years
Now = 7 - 3 = <<7-3=4>>4
Jayden is 4 years old.
#### 4 |
The common ratio of the geometric sequence \( a + \log_{2} 3, a + \log_{4} 3, a + \log_{8} 3 \) is? | \frac{1}{3} |
Kevin has an elm tree in his yard that is $11\frac{2}{3}$ feet tall and an oak tree that is $17\frac{5}{6}$ feet tall. How much taller is the oak tree than the elm tree? Express your answer as a simplified mixed number. | 6\frac{1}{6}\text{ feet} |
Solve the quadratic equation $(x-h)^2 + 4h = 5 + x$ and find the sum of the squares of its roots. If the sum is equal to $20$, what is the absolute value of $h$?
**A)** $\frac{\sqrt{22}}{2}$
**B)** $\sqrt{22}$
**C)** $\frac{\sqrt{44}}{2}$
**D)** $2$
**E)** None of these | \frac{\sqrt{22}}{2} |
Given Mr. Thompson can choose between two routes to commute to his office: Route X, which is 8 miles long with an average speed of 35 miles per hour, and Route Y, which is 7 miles long with an average speed of 45 miles per hour excluding a 1-mile stretch with a reduced speed of 15 miles per hour. Calculate the time dif... | 1.71 |
Quadrilateral $ALEX,$ pictured below (but not necessarily to scale!)
can be inscribed in a circle; with $\angle LAX = 20^{\circ}$ and $\angle AXE = 100^{\circ}:$ | 80 |
Selina takes a sheet of paper and cuts it into 10 pieces. She then takes one of these pieces and cuts it into 10 smaller pieces. She then takes another piece and cuts it into 10 smaller pieces and finally cuts one of the smaller pieces into 10 tiny pieces. How many pieces of paper has the original sheet been cut into?
... | 37 |
Tyler rolls two $ 4025 $ sided fair dice with sides numbered $ 1, \dots , 4025 $ . Given that the number on the first die is greater than or equal to the number on the second die, what is the probability that the number on the first die is less than or equal to $ 2012 $ ? | 1006/4025 |
The U.S. produces about 8 million tons of apples each year. Initially, $30\%$ of the apples are mixed with other products. If the production increases by 1 million tons, the percentage mixed with other products increases by $5\%$ for each additional million tons. Of the remaining apples, $60\%$ is used to make apple ju... | 2.24 |
For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible IDs that the tourname... | 9100 |
In a town where 60% of the citizens own a pet, half own a dog and 30 own a cat. How many citizens are in the town? | Half the pet-owners have a cat because 1 - 1/2 = 1/2
There are 60 pet owners because 30 / (1/2) = <<30/(1/2)=60>>60
The town has 100 citizens because 60 / .6 = <<60/.6=100>>100
#### 100 |
Simplify $(2x - 5)(x + 7) - (x + 5)(2x - 1)$. | -30 |
Compute $63 \times 57$ in your head. | 3591 |
Let $(F_n)$ be the sequence defined recursively by $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ for $n\geq 2$. Find all pairs of positive integers $(x,y)$ such that
$$5F_x-3F_y=1.$$ | (2,3);(5,8);(8,13) |
A positive integer divisor of $10!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 10 |
What is the first nonzero digit to the right of the decimal point of the fraction $\frac{1}{129}$? | 7 |
Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the fourth row is 6. The sum of the interior numbers of the fifth row is 14. What is the sum of the interior numbers of the seventh row? | 62 |
What is the greatest divisor of 372 that is smaller than 50 and also a factor of 72? | 12 |
In triangle $ABC,$ $\angle C = \frac{\pi}{2}.$ Find
\[\arctan \left( \frac{a}{b + c} \right) + \arctan \left( \frac{b}{a + c} \right).\] | \frac{\pi}{4} |
Triangle $ABC$ has $\angle BAC=90^\circ$ . A semicircle with diameter $XY$ is inscribed inside $\triangle ABC$ such that it is tangent to a point $D$ on side $BC$ , with $X$ on $AB$ and $Y$ on $AC$ . Let $O$ be the midpoint of $XY$ . Given that $AB=3$ , $AC=4$ , and $AX=\tfrac{9}{4}$ , compute th... | 39/32 |
Find the area of the triangle with vertices $(3,-5),$ $(-2,0),$ and $(1,-6).$ | \frac{15}{2} |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} |
How many distinct prime factors does 56 have? | 2 |
Given that in $\triangle ABC$, $AB=4$, $AC=6$, $BC= \sqrt{7}$, and the center of its circumcircle is $O$, find $\overset{⇀}{AO}· \overset{⇀}{BC} =$ ___. | 10 |
How many unordered pairs of coprime numbers are there among the integers 2, 3, ..., 30? Recall that two integers are called coprime if they do not have any common natural divisors other than one. | 248 |
Let $ABCD$ be a unit square. Let $Q_1$ be the midpoint of $\overline{CD}$. For $i=1,2,\dots,$ let $P_i$ be the intersection of $\overline{AQ_i}$ and $\overline{BD}$, and let $Q_{i+1}$ be the foot of the perpendicular from $P_i$ to $\overline{CD}$. What is
\[\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i \, ?\] | \frac{1}{4} |
The Bank of Springfield's Super High Yield savings account compounds annually at a rate of one percent. If Lisa invests 1000 dollars in one of these accounts, then how much interest will she earn after five years? (Give your answer to the nearest dollar.) | 51 |
In $\triangle ABC$, if $BC=4$, $\cos B= \frac{1}{4}$, then $\sin B=$ _______, the minimum value of $\overrightarrow{AB} \cdot \overrightarrow{AC}$ is: _______. | -\frac{1}{4} |
Let
\[\mathbf{M} = \begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix}\]be a matrix with complex entries such that $\mathbf{M}^2 = \mathbf{I}.$ If $abc = 1,$ then find the possible values of $a^3 + b^3 + c^3.$ | 2,4 |
A circle inscribed in triangle \( ABC \) touches side \( AB \) at point \( M \), and \( AM = 1 \), \( BM = 4 \). Find \( CM \) given that \( \angle BAC = 120^\circ \). | \sqrt{273} |
There are 5 balls of the same shape and size in a bag, including 3 red balls and 2 yellow balls. Now, balls are randomly drawn from the bag one at a time until two different colors of balls are drawn. Let the random variable $\xi$ be the number of balls drawn at this time. Find $E(\xi)=$____. | \frac{5}{2} |
Given the complex numbers \( z_1 \) and \( z_2 \) such that \( \left| z_1 + z_2 \right| = 20 \) and \( \left| z_1^2 + z_2^2 \right| = 16 \), find the minimum value of \( \left| z_1^3 + z_2^3 \right| \). | 3520 |
Let $x$ be a multiple of $7200$. Determine the greatest common divisor of $g(x) = (5x+3)(11x+2)(17x+5)(4x+7)$ and $x$. | 30 |
The arithmetic mean of 12 scores is 82. When the highest and lowest scores are removed, the new mean becomes 84. If the highest of the 12 scores is 98, what is the lowest score? | 46 |
Gauss is a famous German mathematician, known as the "Prince of Mathematics". There are 110 achievements named after his name "Gauss". For $x\in R$, let $[x]$ represent the largest integer not greater than $x$, and let $\{x\}=x-[x]$ represent the non-negative fractional part of $x$. Then, $y=[x]$ is called the Gauss fu... | 3027+ \sqrt{3} |
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is \(5:4\). Find the minimum possible value of their common perimeter. | 524 |
Evaluate the expression $\frac{16^{24}}{64^{8}}$.
A) $16^2$
B) $16^4$
C) $16^8$
D) $16^{16}$
E) $16^{24}$ | 16^8 |
Let \( p, q, r, \) and \( s \) be the roots of the polynomial
\[ x^4 + 10x^3 + 20x^2 + 15x + 6 = 0. \]
Find the value of
\[ \frac{1}{pq} + \frac{1}{pr} + \frac{1}{ps} + \frac{1}{qr} + \frac{1}{qs} + \frac{1}{rs}. \] | \frac{10}{3} |
Find the sum of the first eight prime numbers that have a units digit of 3. | 394 |
Let $a,$ $b,$ and $c$ be positive real numbers. Find the minimum value of
\[\frac{a}{b} + \frac{b}{c} + \frac{c}{a}.\] | 3 |
Aunt Angela has 70 jellybeans in a jar. She wants to divide them equally and give them to her 3 nephews and 2 nieces. How many jellybeans did each nephew or niece receive? | Aunt Angela has a total of 3 + 2 = <<3+2=5>>5 nephews and nieces.
She gave each nephew or niece 70 / 5 = <<70/5=14>>14 jellybeans.
#### 14 |
8 coins are simultaneously flipped. What is the probability that heads are showing on at most 2 of them? | \dfrac{37}{256} |
How many four-digit whole numbers are there such that the leftmost digit is an odd prime, the second digit is a multiple of 3, and all four digits are different? | 616 |
Gretchen has ten socks, two of each color: red, blue, green, yellow, and purple. She randomly draws five socks. What is the probability that she has exactly two pairs of socks with the same color? | \frac{5}{42} |
Given an integer \( n \geq 2 \), let \( f(n) \) be the second largest positive divisor of \( n \). For example, \( f(12)=6 \) and \( f(13)=1 \). Determine the largest positive integer \( n \) such that \( f(n)=35 \). | 175 |
How many integers fall between $\sqrt7$ and $\sqrt{77}$ on a number line? | 6 |
$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD.$ The crease is $EF,$ where $E$ is on $AB$ and $F$ is on $CD.$ The dimensions $AE=8, BE=17,$ and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n,$ where $m$ and $n$ are relatively prime posi... | 293 |
Q. A light source at the point $(0, 16)$ in the co-ordinate plane casts light in all directions. A disc(circle along ith it's interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\sqrt{n}$ where $m, n$ are positive integers and $... | 21 |
Lilith originally had five dozen water bottles that she needed to sell at $2 each to get exactly enough money to buy her friend a birthday gift. However, at the store, Lilith realized she could not sell at $2 because the regular price was $1.85 per water bottle in her town, and she had to reduce her price to $1.85 as w... | Lilith had 5 dozen water bottles, and since a dozen has 12 water bottles, the total number of water bottles she had was 12 bottles/dozen * 5 dozen = <<12*5=60>>60 bottles
To buy her friend the birthday gift, Lilith originally had to sell her water bottles for a total of 60 bottles * $2/bottle = $<<60*2=120>>120
When sh... |
Given the hyperbola $\frac{x^{2}}{m} + \frac{y^{2}}{n} = 1 (m < 0 < n)$ with asymptote equations $y = \pm \sqrt{2}x$, calculate the hyperbola's eccentricity. | \sqrt{3} |
Grace just started her own business. Each week, she charges 300 dollars. Grace's client will pay her every 2 weeks. How many weeks will it take for Grace to get 1800 dollars? | Every two weeks, Grace will get 300*2=<<300*2=600>>600 dollars.
It will take Grace 1800/600=3 2-week intervals to get 1800 dollars.
It will take Grace 3*2=<<3*2=6>>6 weeks in total.
#### 6 |
A certain company implements an annual salary system, where an employee's annual salary consists of three components: basic salary, housing allowance, and medical expenses, as specified below:
| Item | Salary in the First Year (in ten thousand yuan) | Calculation Method After One Year |
|---------------|-----... | 20\% |
What value of $x$ will give the maximum value for $-x^2- 6x + 12$? | -3 |
Let $a$ and $b$ be positive real numbers such that $3a^2 + 2b^2 = 3a + 2b$. Find the minimum value of $A =\sqrt{\frac{a}{b(3a+2)}} + \sqrt{\frac{b}{a(2b+3)}} $ | \frac{2}{\sqrt{5}} |
The Greenhill Soccer Club has 25 players, including 4 goalies. During an upcoming practice, the team plans to have a competition in which each goalie will try to stop penalty kicks from every other player, including the other goalies. How many penalty kicks are required for every player to have a chance to kick against... | 96 |
Given real numbers \( x \) and \( y \) satisfy
\[
\left\{
\begin{array}{l}
x - y \leq 0, \\
x + y - 5 \geq 0, \\
y - 3 \leq 0
\end{array}
\right.
\]
If the inequality \( a(x^2 + y^2) \leq (x + y)^2 \) always holds, then the maximum value of the real number \( a \) is $\qquad$. | 25/13 |
In a board game, I move on a linear track. For move 1, I stay still. For subsequent moves $n$ where $2 \le n \le 30$, I move forward two steps if $n$ is prime and three steps backward if $n$ is composite. How many steps in total will I need to make to return to my original starting position after all 30 moves? | 37 |
If the seven digits 1, 1, 3, 5, 5, 5, and 9 are arranged to form a seven-digit positive integer, what is the probability that the integer is divisible by 25? | \frac{1}{14} |
How many positive integers less than $1000$ are either a perfect cube or a perfect square? | 37 |
There are two boxes, A and B, each containing four cards labeled with the numbers 1, 2, 3, and 4. One card is drawn from each box, and each card is equally likely to be chosen;
(I) Find the probability that the product of the numbers on the two cards drawn is divisible by 3;
(II) Suppose that Xiao Wang and Xiao Li draw... | \frac{8}{9} |
Given $a\in \mathbb{R}$, $b\in \mathbb{R}$, if the set $\{a, \frac{b}{a}, 1\} = \{a^{2}, a-b, 0\}$, calculate the value of $a^{2019}+b^{2019}$. | -1 |
A pair of dogs are barking back and forth at each other from across the street. The poodle barks twice for every one time the terrier barks. The terrier’s owner hushes it every second time it barks. She has to say “hush” six times before the dogs stopped barking. How many times did the poodle bark? | The terrier barked twice as many times as it was told to hush, so it barked 6 * 2 = <<6*2=12>>12 times.
The poodle barked two times for every terrier bark, so it barked 12 * 2 = <<12*2=24>>24 times.
#### 24 |
Find $x$ such that $\log_x 81=\log_2 16$. | 3 |
On one side of the acute angle \(A\), points \(P\) and \(Q\) are marked such that \(AP = 4\), \(AQ = 12\). On the other side, points \(M\) and \(N\) are marked at distances of 6 and 10 from the vertex. Find the ratio of the areas of triangles \(MNO\) and \(PQO\), where \(O\) is the intersection point of the lines \(MQ\... | 1:5 |
The circle, which has its center on the hypotenuse $AB$ of the right triangle $ABC$, touches the two legs $AC$ and $BC$ at points $E$ and $D$ respectively.
Find the angle $ABC$, given that $AE = 1$ and $BD = 3$. | 30 |
Given that the random variable $\xi$ follows the normal distribution $N(1, 4)$, if $P(\xi > 4) = 0.1$, then $P(-2 \leq \xi \leq 4)$ equals _______. | 0.8 |
Given the function $f(x)=2\sin x\cos x+2\sqrt{3}\cos^{2}x-\sqrt{3}$, $x\in R$.
(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) In acute triangle $ABC$, if $f(A)=1$, $\overrightarrow{AB}\cdot\overrightarrow{AC}=\sqrt{2}$, find the area of $\triangle ABC$. | \frac{\sqrt{2}}{2} |
Suppose $a$, $b,$ and $c$ are positive numbers satisfying: \begin{align*}
a^2/b &= 1, \\
b^2/c &= 2, \text{ and}\\
c^2/a &= 3.
\end{align*} Find $a$. | 12^{1/7} |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% |
If in an arithmetic sequence, the sum of the first three terms is 34, the sum of the last three terms is 146, and the sum of all terms is 390, then the number of terms in the sequence is __________. | 13 |
(1) Given $\cos(15°+\alpha) = \frac{15}{17}$, with $\alpha \in (0°, 90°)$, find the value of $\sin(15°-\alpha)$.
(2) Given $\cos\alpha = \frac{1}{7}$, $\cos(\alpha-\beta) = \frac{13}{14}$, and $0 < \beta < \alpha < \frac{\pi}{2}$, find the value of $\beta$. | \frac{\pi}{3} |
The minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30}\cdot 5^3}$ as a decimal. | 30 |
Three cards, each with a positive integer written on it, are lying face-down on a table. Casey, Stacy, and Tracy are told that
(a) the numbers are all different,
(b) they sum to $13$, and
(c) they are in increasing order, left to right.
First, Casey looks at the number on the leftmost card and says, "I don't have enou... | 4 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $b=3$, $c=2\sqrt{3}$, and $A=30^{\circ}$, find the values of angles $B$, $C$, and side $a$. | \sqrt{3} |
The bases of an isosceles trapezoid are in the ratio 3:2. A circle is constructed on the larger base as its diameter, and this circle intersects the smaller base such that the segment cut off on the smaller base is equal to half of the smaller base. In what ratio does the circle divide the non-parallel sides of the tra... | 1:2 |
The denominator of the fraction $15 \cdot 18$ in simplest form is 30. Find the sum of all such positive rational numbers less than 10. | 400 |
Let $\mathbf{M}$ be a matrix such that
\[\mathbf{M} \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 3 \\ 0 \end{pmatrix} \quad \text{and} \quad \mathbf{M} \begin{pmatrix} -3 \\ 5 \end{pmatrix} = \begin{pmatrix} -1 \\ -1 \end{pmatrix}.\]Compute $\mathbf{M} \begin{pmatrix} 5 \\ 1 \end{pmatrix}.$ | \begin{pmatrix} 11 \\ -1 \end{pmatrix} |
Determine the value of $a^3 + b^3$ given that $a+b=12$ and $ab=20$, and also return the result for $(a+b-c)(a^3+b^3)$, where $c=a-b$. | 4032 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.