task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | The number of real roots of the equation $\frac{x}{100} = \sin x$ is:
(32nd United States of America Mathematical Olympiad, 1981) | 63 | 42 | 2 |
math | Given the function $f(x)=\sin ^{2}x+\cos x+ \frac {5}{8}a- \frac {3}{2}$ defined on the closed interval $[0, \frac {π}{2}]$, its minimum value is $2$. Find the corresponding value of $a$. | a=4 | 66 | 3 |
math | Select two digits from 1, 3, and 5 to complete a three-digit number. One of the selected digits must be odd. Given that 0 or 2 is chosen as one of the digits in the number, find the number of distinct odd three-digit numbers that can be formed. | 18 | 61 | 2 |
math | What digit can the product of two distinct prime numbers end with? | 1,3,7,9 | 13 | 7 |
math | Given the curve $C\_1$ with the polar coordinate equation: $ρ = 6 \sin θ - 8 \cos θ$, and the curve $C\_2$ with the parametric equations: $\begin{cases} x = 8 \cos φ \ y = 3 \sin φ \end{cases} (φ \text{ is the parameter})$,
1. Convert $C\_1$ and $C\_2$ into rectangular coordinate equations and describe the curves they... | \frac{8\sqrt{5}}{5} | 202 | 12 |
math | A certain cafeteria serves ham and cheese sandwiches, ham and tomato sandwiches, and tomato and cheese sandwiches. It is common for one meal to include multiple types of sandwiches. On a certain day, it was found that 80 customers had meals which contained both ham and cheese; 90 had meals containing both ham and tomat... | 230 | 95 | 3 |
math | Regular octagon $ABCDEFGH$ is divided into eight smaller isosceles triangles, with vertex angles at the center of the octagon, such as $\triangle ABJ$, by constructing lines from each vertex to the center $J$. By connecting every second vertex (skipping one vertex in between), we obtain a larger equilateral triangle $\... | \frac{1}{4} | 97 | 7 |
math | (Selected Topics on Inequalities) If $ab>0$, and points $A(a,0)$, $B(0,b)$, and $C(-2,-2)$ are collinear, then the minimum value of $ab$ is \_\_\_\_\_. | 16 | 57 | 2 |
math | Point \((x,y)\) is randomly picked from the rectangular region with vertices at \((0,0), (3000,0), (3000,4500),\) and \((0,4500)\). What is the probability that \(x < 3y\)? Express your answer as a common fraction. | \frac{11}{18} | 75 | 9 |
math | Given that Carl has 24 fence posts and places one on each of the four corners, with 3 yards between neighboring posts, where the number of posts on the longer side is three times the number of posts on the shorter side, determine the area, in square yards, of Carl's lawn. | 243 | 61 | 3 |
math | Let $(a_1, a_2, a_3,\ldots,a_{13})$ be a permutation of $(1,2,3,\ldots,13)$ for which
$$a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7 \mathrm{\ and \ } a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13}.$$
Find the number of such permutations. | 924 | 120 | 3 |
math | The lines with equations $ax-2y=c$ and $2x+by=c$ are parallel and intersect at $(2, -4)$. Find the value of $c$. | 0 | 38 | 1 |
math | Suppose that $a_0=1$ and that $a_{n+1}=a_n+e^{-a_n}$ for $n=0,1,2,\dots.$ Does $a_n-\log n$ have a finite limit as $n\to\infty?$ (Here $\log n=\log_en=\ln n.$ ) | \lim_{n \to \infty} (a_n - \log n) = 0 | 84 | 21 |
math | Delete all perfect squares from the sequence of positive integers $n$, resulting in a new sequence $m(n)$. What is the $2005^{th}$ term of this new sequence $m(n)$? | 2050 | 44 | 4 |
math | Determine the greatest common divisor (GCD) of the following numbers:
188094, 244122, 395646. | 6 | 38 | 1 |
math | Given the equation
\[ x^{2} + p x + q = 0 \]
with roots \( x_{1} \) and \( x_{2} \), determine the coefficients \( r \) and \( s \) as functions of \( p \) and \( q \) such that the equation
\[ y^{2} + r y + s = 0 \]
has roots \( y_{1} \) and \( y_{2} \) which are related to the roots \( x_{1} \) and \( x_{2} \) by... | p + q = 0 | 196 | 6 |
math | Let \( a \) and \( b \) be positive real numbers. Find the maximum value of
\[ 2(a - x)(x - \sqrt{x^2 + b^2}) \] in terms of \( a \) and \( b \). | b^2 | 54 | 3 |
math | Determine the number of triples $(x, y, z)$ of real numbers that satisfy the following system:
\begin{align*}
x &= 2020 - 2021 \operatorname{sign}(y + z), \\
y &= 2020 - 2021 \operatorname{sign}(x + z), \\
z &= 2020 - 2021 \operatorname{sign}(x + y).
\end{align*}
Note: For a real number $a$, the $\operatorname{sign}(a)... | 3 | 202 | 1 |
math | Eight women of different heights are at a conference. Their heights in centimeters are 150, 155, 160, 165, 170, 175, 180, and 185. Each woman decides to only shake hands with women whose height difference is 10 centimeters or less. How many handshakes take place? | 7 | 86 | 1 |
math | If \\((3-2x)^{5}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{5}x^{5}\\), then \\(a_{0}+a_{1}+2a_{2}+3a_{3}+4a_{4}+5a_{5}=\\) \_\_\_\_\_\_. | 233 | 86 | 3 |
math | A wooden cube, whose edges are two centimeters long, rests on a horizontal surface. It's illuminated by a point source of light that is $y$ centimeters directly above an upper vertex of the cube. The cube casts a shadow on the horizontal surface, and the area of the shadow, excluding the area beneath the cube, is 112 s... | 8770 | 91 | 4 |
math | Which of the following statements are correct? (Fill in all the correct serial numbers)
① Acute angles are first quadrant angles;
② Second quadrant angles are obtuse angles;
③ The set of first quadrant angles is $\{ \beta | k \cdot 360° < \beta < 90° + k \cdot 360°, k \in \mathbb{Z} \}$;
④ The set of first and ... | ①③ | 147 | 4 |
math | When the digits in the number $2017$ are reversed, we obtain the number $7102$. Factorize $7102$ into the product of three distinct primes $p$, $q$, and $r$. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$, and $p_3$ such that $p_1 + p_2 + p_3 = p + q + r$? | 0 \text{ other positive integers} | 101 | 8 |
math | For let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For instance, $\clubsuit(9)=9$ and $\clubsuit(234)=2+3+4=9$. Determine the number of two-digit values of $x$ for which $\clubsuit(\clubsuit(x))=2$. | 9 | 72 | 1 |
math | What is the residue of $7^{2050}$, modulo 19? | 11 | 19 | 2 |
math | Let $S = \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $S$. Let $N$ be the sum of all of these differences. Find $N.$ | 16398 | 58 | 5 |
math | How many different $5 \times 5$ arrays whose entries are 1's, -1's, and 0's have the property that the sum of the entries in each row is 0, the sum of the entries in each column is 0, and each row and column should contain exactly one 0? | 933120 | 66 | 6 |
math | A square and an isosceles triangle are inscribed in a circle and share a common vertex $D$. Designate the vertices of the square as $A$, $B$, $C$, and $D$ in clockwise order. The isosceles triangle has its unequal side along $DA$. The other vertex of the triangle is at point $E$, lying on the circle. If the angle $EDA$... | \angle DEA = 45^\circ | 103 | 9 |
math | In triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively. Given that $a\cos B = 4$ and $b\sin A = 3$.
(I) Find $\tan B$ and the value of side $a$;
(II) If the area of triangle $ABC$ is $S = 9$, find the perimeter of triangle $ABC$. | 11 + \sqrt{13} | 101 | 9 |
math | Given the equation of the line $3x+2y-1=0$, find one of its directional vectors. | (2,-3) | 24 | 5 |
math | Given $$\cos\alpha= \frac {4}{5}$$ and $$\cos(\alpha+\beta)= \frac {5}{13}$$, where $\alpha$ and $\beta$ are acute angles.
(1) Find the value of $\sin2\alpha$;
(2) Find the value of $\sin\beta$. | \frac {33}{65} | 74 | 9 |
math | Let $\square B_1B_2B_3B_4$ be a square, and define $B_{n+4}$ to be the midpoint of the line segment $B_nB_{n+1}$ for all positive integers $n$. Find the measure of $\measuredangle B_{101}B_{102}B_{100}$.
A) $45^\circ$
B) $60^\circ$
C) $90^\circ$
D) $120^\circ$
E) $135^\circ$ | 90^\circ | 122 | 4 |
math | Simplify $$\sqrt {a^{ \frac {1}{2}} \sqrt {a^{ \frac {1}{2}} \sqrt {a}}}.$$ | a^{ \frac {1}{2}} | 33 | 9 |
math | In the era of "Internet$+$", there is a method of producing passwords using the "factorization method": factorizing a polynomial. For example, factorizing the polynomial $x^{3}-x$ results in $x\left(x+1\right)\left(x-1\right)$. When $x=20$, $x-1=19$, $x+1=21$, we can obtain the numerical password $201921$, or $192021$.... | m=11; \ n=6 | 177 | 9 |
math | Find the equation of the line that passes through the intersection point of the lines $2x-y=0$ and $x+y-6=0$, and is perpendicular to the line $2x+y-1=0$. | x-2y+6=0 | 46 | 8 |
math | The probability it will rain on Monday is $40\%$, and the probability it will rain on Tuesday is $30\%$. Additionally, if it does not rain on Monday, the chance it will not rain on Tuesday increases to $50\%$. What is the probability that it will rain on both Monday and Tuesday? | 12\% | 69 | 4 |
math | On a circle, $n$ points are marked. It turned out that exactly half of the triangles formed by these points are acute-angled.
Find all possible values of $n$ for which this is possible. | 4 \text{ or } 5 | 43 | 8 |
math | A cube with a side length of 4 inches is made of the same material as a cube with a side length of 3 inches, which weighs 5 pounds and is valued at $300. What is the value of the 4-inch cube? | \$711 | 53 | 4 |
math | The hypotenuse of a right triangle measures 13 inches and one angle is $30^{\circ}$. What is the number of square inches in the area of the triangle? | 21.125\sqrt{3} | 39 | 11 |
math | Consider the lines:
\[
y = 4x + 3, \quad 2y = 8x + 6, \quad 3y = -12x + 2, \quad y = -\frac{1}{4}x + 7, \quad 4y = -x + 4.
\]
Determine the number of pairs of lines that are either parallel or perpendicular. | 6 | 88 | 1 |
math | Sides $\overline{AM}$ and $\overline{CD}$ of regular dodecagon $ABCDEFGHIJKL$ are extended to meet at point $P$. What is the degree measure of angle $P$? | 90^\circ | 46 | 4 |
math | A university adopts a major preference policy in its college entrance examination enrollment. A candidate chooses 3 majors from the 10 majors provided by the university as his/her first, second, and third major preferences. Among these majors, majors A and B cannot be chosen simultaneously. The number of different ways... | 672 | 79 | 3 |
math | There are lily pads in a row numbered $0$ to $11$, in that order. There are predators on lily pads $3$ and $6$, and a morsel of food on lily pad $10$. Fiona the frog starts on pad $0$, and from any given lily pad, has a $\frac{1}{2}$ chance to hop to the next pad, and an equal chance to jump $2$ pads. What is the proba... | \frac{15}{256} | 121 | 10 |
math | Given that $b$ is an even multiple of $97$, find the greatest common divisor of $3b^2 + 41b + 74$ and $b + 19$. | 1 | 43 | 1 |
math | Given the proposition $p: \exists x \in \mathbb{R}, \sin x > a$, if $\lnot p$ is a true proposition, determine the range of values for the real number $a$. | a \geqslant 1 | 46 | 8 |
math | If the line $x+y-1=0$ bisects the circumference of the circle $x^2+y^2-2ax-2(a^2+1)y+4=0$, then $a=$ ___. | 0 \text{ or } -1 | 48 | 8 |
math | Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $(\overrightarrow{a}+\overrightarrow{b})⊥(\overrightarrow{a}-\overrightarrow{b})$, $|\overrightarrow{a}+\overrightarrow{b}|=2$, and the range of $\overrightarrow{a}•\overrightarrow{b}$ is $[-2,\frac{2}{3}]$, find the range of the angle $\theta... | [\frac{\pi}{3},\frac{2\pi}{3}] | 115 | 16 |
math | 1. Compute:
(1) $-27 + (-32) + (-8) + 27$
(2) $(-5) + |-3|$
(3) If the opposite of $x$ is 3, and $|y| = 5$, find the value of $x + y$
(4) $(-1 \frac{1}{2}) + (1 \frac{1}{4}) + (-2 \frac{1}{2}) - (-3 \frac{1}{4}) - (1 \frac{1}{4})$
(5) If $|a - 4| + |b + 5| = 0$, find $a - b$. | 9 | 160 | 1 |
math | If the function $f(x)=\ln x+ax+\frac{1}{x}$ is a monotonically decreasing function on $\left[1,+\infty \right)$, then the range of $a$ is ______. | (-\infty, -\frac{1}{4}] | 49 | 13 |
math | Number $125$ is written as the sum of several pairwise distinct and relatively prime numbers, greater than $1$ . What is the maximal possible number of terms in this sum? | 8 | 41 | 1 |
math | Given that Henry walks $\frac{2}{3}$ of the remaining distance towards his gym or home from the point where he last changed his mind, and this process continues indefinitely, determine the absolute difference between the distances of points $A$ and $B$ from Henry's home. | 1 | 57 | 1 |
math | Calculate the limit of the function:
$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\operatorname{tg} 3 x}{\operatorname{tg} x}$$ | \frac{1}{3} | 43 | 7 |
math | Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x - y)) = g(x) g(y) - g(x) + g(y) - x^2y^2\] for all $x, y \in \mathbb{R}$. Find the sum of all possible values of $g(1)$. | -1 | 84 | 2 |
math | If for any $x \in D$, the inequality $f_{1}(x) \leqslant f(x) \leqslant f_{2}(x)$ holds, then the function $f(x)$ is called a "compromise function" of $f_{1}(x)$ to $f_{2}(x)$ over the interval $D$. Given that the function $f(x)=(k-1)x-1$, $g(x)=0$, $h(x)=(x+1)\ln x$, and $f(x)$ is a "compromise function" of $g(x)$ to ... | \{2\} | 161 | 5 |
math | If \(a + 3b = 27\) and \(5a + 2b = 40\), what is the value of \(a + b\)? | \frac{161}{13} | 37 | 10 |
math | A piece of graph paper is folded once so that $(3,3)$ is matched with $(7,1)$, and $(9,4)$ is matched with $(m,n)$. Find $m+n$.
A) 8.3
B) 9.3
C) 9.\overline{3}
D) 10.3 | 9.\overline{3} | 75 | 7 |
math | Exactly at noon, a truck left the village and headed towards the city, and at the same time, a car left the city and headed towards the village. If the truck had left 45 minutes earlier, they would have met 18 kilometers closer to the city. If the car had left 20 minutes earlier, they would have met $k$ kilometers clos... | 8 | 84 | 1 |
math | A sequence $a_1, a_2, \ldots$ satisfies $a_1 = \dfrac 52$ and $a_{n + 1} = {a_n}^2 - 2$ for all $n \ge 1.$ Let $M$ be the integer which is closest to $a_{2023}.$ The last digit of $M$ equals $$ \mathrm a. ~ 0\qquad \mathrm b.~2\qquad \mathrm c. ~4 \qquad \mathrm d. ~6 \qquad \mathrm e. ~8 $$ | 4 | 146 | 1 |
math | There are 20 cards with numbers $1, 2, \cdots, 19, 20$ on them. These cards are placed in a box, and 4 people each draw one card. The two people who draw the smaller numbers will be in one group, and the two who draw the larger numbers will be in another group. If two of the people draw the numbers 5 and 14 respectivel... | $\frac{7}{51}$ | 105 | 8 |
math | Compute: $4\;^{\frac{3}{2}}$ | 8 | 15 | 1 |
math | Consider the geometric sequence \( \left(a+\log _{2} 3\right),\left(a+\log _{4} 3\right),\left(a+\log _{8} 3\right) \). What is the common ratio of this sequence? | \frac{1}{3} | 58 | 7 |
math | Given a randomly selected number $x$ in the interval $[0,\pi]$, determine the probability of the event "$-1 \leqslant \tan x \leqslant \sqrt {3}$". | \dfrac{7}{12} | 46 | 8 |
math | The distance between two squares of an infinite chessboard is defined as the minimum number of moves a king needs to travel between these squares. Three squares are marked on the board, where each pair of distances between them equals 100. How many squares are there from which the distance to all three marked squares i... | 1 | 67 | 1 |
math | Given the set $A=\{m+2, 2m^2+m\}$, if $3 \in A$, then the value of $m$ is \_\_\_\_\_\_. | -\frac{3}{2} | 42 | 7 |
math | In $\triangle ABC$, $\angle C= \frac{\pi}{2}$, $\angle B= \frac{\pi}{6}$, and $AC=2$. $M$ is the midpoint of $AB$. $\triangle ACM$ is folded along $CM$ such that the distance between $A$ and $B$ is $2\sqrt{2}$. The surface area of the circumscribed sphere of the tetrahedron $M-ABC$ is \_\_\_\_\_\_. | 16\pi | 104 | 4 |
math | Consider a parabola with vertex V and a focus F. There exists a point B on the parabola such that BF = 25 and BV = 24. Determine the sum of all possible values of the length FV. | \frac{50}{3} | 50 | 8 |
math | Sasha chose a natural number \( N > 1 \) and wrote down all of its natural divisors in ascending order: \( d_{1}<\ldots<d_{s} \) (such that \( d_{1}=1 \) and \( d_{s}=N \)). Then for each pair of adjacent numbers, he calculated their greatest common divisor; the sum of the \( s-1 \) resulting numbers turned out to be e... | N = 3 | 108 | 4 |
math | For two lines $l_{1}: (3+a)x + 4y = 5 - 3a$ and $l_{2}: 2x + (5+a)y = 8$ to be parallel, find the value of $a$. | a = -7 | 53 | 4 |
math | A hexagon inscribed in a circle has three consecutive sides, each of length 4, and three consecutive sides, each of length 6. Compute the length of the chord that divides the hexagon into two trapezoids, one with three sides each of length 4, and the other with three sides each of length 6. | 10 | 70 | 2 |
math | In the Cartesian coordinate system, points whose x and y coordinates are both integers are called lattice points. How many lattice points \((x, y)\) satisfy the inequality \((|x|-1)^{2}+(|y|-1)^{2}<2\)? | 16 | 56 | 2 |
math | Given that the temperature \( T \) of an object throughout the day is a function of time \( t \): \( T(t) = a t^3 + b t^2 + c t + d \) (\(a \neq 0\)). The temperature is measured in \(^\circ \text{C}\), and time is measured in hours. Here, \( t = 0 \) represents 12:00 PM, and positive values of \( t \) represent times ... | 14:00 \, \text{(2 hours after noon)}, \, 62^\circ \text{C} | 306 | 27 |
math | Given that \(a\) and \(b\) are natural numbers not exceeding 10, find the number of pairs \((a, b)\) that satisfy the equation \(ax = b\) such that the solution \(x\) is less than \(\frac{1}{2}\) but greater than \(\frac{1}{3}\). | 5 | 70 | 1 |
math | There are four students, A, B, C, and D, who are divided into two volunteer groups to participate in two activities outside the school. Calculate the probability that students B and C will participate in the same activity. | \dfrac{1}{3} | 45 | 7 |
math | In $\triangle ABC$, $\angle A = 90^\circ$ and $\tan C = 3$. If $BC = 90$, what is the length of $AB$, and what is the perimeter of triangle ABC? | 36\sqrt{10} + 90 | 48 | 12 |
math | If the one-variable quadratic equation $kx^{2}-2x-1=0$ has two real roots with respect to $x$, then the range of values for $k$ is ______. | k \geq -1 \text{ and } k \neq 0 | 41 | 17 |
math | Given a permutation \( \left(a_{0}, a_{1}, \ldots, a_{n}\right) \) of the sequence \(0, 1, \ldots, n\). A transposition of \(a_{i}\) with \(a_{j}\) is called legal if \(a_{i}=0\) for \(i>0\), and \(a_{i-1}+1=a_{j}\). The permutation \( \left(a_{0}, a_{1}, \ldots, a_{n}\right) \) is called regular if after a number of l... | n = 2, \ n = 2^k - 1 | 204 | 15 |
math | Given vectors $\overrightarrow{a}=(\cos \alpha, \sin \alpha)$, $\overrightarrow{b}=(\cos \beta, \sin \beta)$, $\overrightarrow{c}=(1,2)$, and $\overrightarrow{a} \cdot \overrightarrow{b} = \frac{\sqrt{2}}{2}$,
(1) Find $\cos(\alpha - \beta)$;
(2) If $\overrightarrow{a}$ is parallel to $\overrightarrow{c}$ and $0 < \bet... | \frac{3\sqrt{10}}{10} | 133 | 14 |
math | In quadrilateral $ABCD$ , diagonals $AC$ and $BD$ intersect at $O$ . If the area of triangle $DOC$ is $4$ and the area of triangle $AOB$ is $36$ , compute the minimum possible value of the area of $ABCD$ . | 80 | 78 | 2 |
math | Some people know each other in a group of people, where "knowing" is a symmetric relation. For a person, we say that it is $social$ if it knows at least $20$ other persons and at least $2$ of those $20$ know each other. For a person, we say that it is $shy$ if it doesn't know at least $20$ other persons and... | 40 | 153 | 2 |
math | Given points $A(1,2)$, $B(3,4)$, and lines $l_1: x=0$, $l_2: y=0$, and $l_3: x+3y-1=0$. Let $P_i$ be the point on line $l_i$ ($i=1,2,3$) such that the sum of the squared distances from $P_i$ to points $A$ and $B$ is minimized. Find the area of triangle $P_1P_2P_3$. | \frac{3}{2} | 118 | 7 |
math | **a)** Show that the expression $ x^3-5x^2+8x-4 $ is nonegative, for every $ x\in [1,\infty ) . $ **b)** Determine $ \min_{a,b\in [1,\infty )} \left( ab(a+b-10) +8(a+b) \right) . $ | 8 | 85 | 1 |
math | A solid cube has dimensions \(a \times b \times c\), where \(a\), \(b\), and \(c\) are integers and \(a \geq b \geq c \geq 1\). We know that \(4(ab + bc + ca) = 3abc\). Determine the number of ordered triples \((a, b, c)\). | 3 | 80 | 1 |
math | Given the function $f(x) = ax^{2} - |x-a|$, where $a \in \mathbb{R}$.
$(1)$ Discuss the parity of the function $f(x)$.
$(2)$ When $-1 \leq a \leq 1$, if for any $x \in [1,3]$, $f(x) + bx \leq 0$ always holds, find the maximum value of $a^{2} + 3b$. | 10 | 106 | 2 |
math | \(A, B, C, D\) are consecutive vertices of a parallelogram. Points \(E, F, P, H\) lie on sides \(AB\), \(BC\), \(CD\), and \(AD\) respectively. Segment \(AE\) is \(\frac{1}{3}\) of side \(AB\), segment \(BF\) is \(\frac{1}{3}\) of side \(BC\), and points \(P\) and \(H\) bisect the sides they lie on. Find the ratio of t... | 37/72 | 131 | 5 |
math | Given an equilateral triangle $ABC$ with side length $a$ and area $s$, the radius of the inscribed circle is $r=\frac{2s}{3a}$. By analogy, for a regular tetrahedron $S-ABC$ with the base area $S$, the radius of the inscribed sphere is $R$, and the volume is $V$, then $R=$_______. | \frac{3V}{4S} | 86 | 9 |
math | A regular tetrahedron $SABC$ of volume $V$ is given. The midpoints $D$ and $E$ are taken on $SA$ and $SB$ respectively and the point $F$ is taken on the edge $SC$ such that $SF: FC = 1: 3$ . Find the volume of the pentahedron $FDEABC$ . | \frac{15}{16}V | 100 | 10 |
math | Given the function $f(x) = (m^2 - m - 1)x^{-5m-3}$ is a power function, and it is increasing on the interval $(0, +\infty)$, determine the value of $m$. | -1 | 52 | 2 |
math | Given $$\{\beta |\beta = \frac {\pi }{6}+2k\pi ,k\in Z\}$$, find the values of $\beta$ that satisfy $(-2\pi, 2\pi)$. | \frac {\pi }{6} \text{ or } - \frac {11}{6}\pi | 51 | 23 |
math | Find the volume of a regular tetrahedron with the side of its base equal to $\sqrt{3}$ and the angle between its lateral face and the base equal to $60^{\circ}$. | 0.5 | 43 | 3 |
math | The values of $x$ and $y$ are always positive, and $3x^2$ and $y$ vary inversely. If $y$ is 30 when $x$ is 3, find the value of $y$ when $x$ is 6. | 7.5 | 60 | 3 |
math | Camy made a list of every possible distinct five-digit positive integer that can be formed using each of the digits 1, 3, 4, 5 and 9 exactly once in each integer. What is the sum of the integers on Camy's list? | 5,\!866,\!608 | 55 | 11 |
math | A rectangular playground that is 12 feet wide and 25 feet long is tiled with 300 two-foot square tiles. A bug starts walking from one corner of the rectangle to the diagonally opposite corner in a straight line. Including both the starting and ending tiles, calculate the number of tiles the bug visits. | 18 | 66 | 2 |
math | Let $P$ be the sum of all integers $c$ for which the polynomial $x^2+cx+4032c$ can be factored over the integers. Compute $|P|$. | 0 | 44 | 1 |
math | The cards in a stack of $2n$ cards are numbered consecutively from 1 through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A$. The remaining cards form pile $B$. The cards are then restacked by taking cards alternately from the tops of pile $B$ and $A$, respectively. Card number ... | 402 | 158 | 3 |
math | A sphere intersects the $xy$-plane in a circle centered at $(3,5,0)$ with a radius of 2. The sphere also intersects the $yz$-plane in a circle centered at $(0,5,-8),$ with radius $r.$ Find $r.$ | \sqrt{59} | 59 | 6 |
math | In $\triangle ABC$, $C-A=\frac{\pi }{2}$, $\sin B=\frac{1}{3}$.
(1) Find the value of $\sin A$;
(2) Given $AC=\sqrt{6}$, find the area of $\triangle ABC$. | 3\sqrt{2} | 60 | 6 |
math | The problem involves finding the value of the expressions $\lg 2 + \lg 5$ and $4(-100)^4$. | 400000000 | 29 | 9 |
math | In \( \triangle ABC \), if \( |\overrightarrow{AB}| = 2 \), \( |\overrightarrow{BC}| = 3 \), and \( |\overrightarrow{CA}| = 4 \), find the value of \( \overrightarrow{AB} \cdot \overrightarrow{BC} + \overrightarrow{BC} \cdot \overrightarrow{CA} + \overrightarrow{CA} \cdot \overrightarrow{AB} \). | -\frac{29}{2} | 96 | 8 |
math | Define the munificence of a polynomial \(p(x)\) as the maximum value of \(|p(x)|\) on the interval \([-2, 2]\). Find the smallest possible munificence of a monic cubic polynomial of the form \(p(x) = x^3 + ax^2 + bx + c\). | 2 | 69 | 1 |
math | Given the progression \(8^{\frac{2}{11}}, 8^{\frac{3}{11}}, 8^{\frac{4}{11}}, \dots, 8^{\frac{(n+1)}{11}}\), calculate the least positive integer \(n\) such that the product of the first \(n\) terms of the progression exceeds 1,000,000. | 11 | 89 | 2 |
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