Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
16,600 | Given a point P $(x, y)$ on the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$, find the minimum distance from point P to the line $2x + y - 10 = 0$. | \sqrt{5} | 68.75 |
16,601 | A circle has a radius of 6. What is the area of the smallest square that can entirely contain this circle, and what is the circumference of the circle? | 12\pi | 84.375 |
16,602 | If $4 \in \{a^2-3a, a\}$, then the value of $a$ equals ____. | -1 | 15.625 |
16,603 | Given that $\tan \alpha = 2$, where $\alpha$ is an angle in the first quadrant, find the value of $\sin 2\alpha + \cos \alpha$. | \dfrac{4 + \sqrt{5}}{5} | 96.875 |
16,604 | Cara is sitting at a circular table with her seven friends. How many different possible pairs of people could Cara be sitting between? | 21 | 84.375 |
16,605 | If \(a, c, d\) are integers, \(b\) is a positive integer, and they satisfy \(a + b = c\), \(b + c = d\), and \(c + d = a\), then the maximum value of \(a + b + c + d\) is: | -5 | 89.84375 |
16,606 | Given that three balls are randomly and independently tossed into bins numbered with the positive integers such that for each ball, the probability that it is tossed into bin i is $3^{-i}$ for i = 1,2,3,..., find the probability that all balls end up in consecutive bins. | 1/702 | 51.5625 |
16,607 | In 2005, the ages of a brother and sister were 16 and 10 years old, respectively. In which year was the brother's age twice that of the sister's? | 2001 | 96.875 |
16,608 | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c\sin\frac{A+C}{2}=b\sin C$.
$(1)$ Find angle $B$;
$(2)$ Let $BD$ be the altitude from $B$ to side $AC$, and $BD=1$, $b=\sqrt{3}$. Find the perimeter of $\triangle ABC$. | 3 + \sqrt{3} | 48.4375 |
16,609 | The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? | \frac{1}{2} | 46.09375 |
16,610 | Given the function $f(x)=-\frac{1}{3}x^{3}+bx^{2}+cx+bc$ has an extreme value of $-\frac{4}{3}$ at $x=1$, find the value of $b$. | -1 | 25 |
16,611 | Convert the binary number $101101110_{(2)}$ to a decimal number and then to an octal number $({\ }\_{(8)})$. | 556_{(8)} | 13.28125 |
16,612 | Quantities $a$ and $b$ vary inversely. When $a$ is $800$, $b$ is $0.5$. If the product of $a$ and $b$ increases by $200$ when $a$ is doubled, what is $b$ when $a$ is $1600$? | 0.375 | 4.6875 |
16,613 | Given $( \sqrt {x}+ \dfrac {2}{x^{2}})^{n}$, the ratio of the coefficient of the fifth term to the coefficient of the third term in its expansion is $56:3$.
(Ⅰ) Find the constant term in the expansion;
(Ⅱ) When $x=4$, find the term with the maximum binomial coefficient in the expansion. | \dfrac {63}{256} | 27.34375 |
16,614 | Estimate the number of primes among the first thousand primes divide some term of the sequence
\[2^0+1,2^1+1,2^2+1,2^3+1,\ldots.\]
An estimate of $E$ earns $2^{1-0.02|A-E|}$ points, where $A$ is the actual answer.
*2021 CCA Math Bonanza Lightning Round #5.4* | 10 | 1.5625 |
16,615 | Given the equation of the parabola $y^{2}=4x$, and a line $l$ passing through its focus $F$ intersecting the parabola at points $A$ and $B$. If $S_{\triangle AOF}=3S_{\triangle BOF}$ (where $O$ is the origin), calculate the length of $|AB|$. | \dfrac {16}{3} | 25 |
16,616 | If $a$ and $b$ are two positive numbers, and the three numbers $a$, $b$, and $-4$ can be rearranged to form both an arithmetic sequence and a geometric sequence, then the value of $a+b$ is __________. | 10 | 52.34375 |
16,617 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $2c-2a\cos B=b$.
$(1)$ Find the size of angle $A$;
$(2)$ If the area of $\triangle ABC$ is $\frac{\sqrt{3}}{4}$, and $c^{2}+ab\cos C+a^{2}=4$, find $a$. | \frac{\sqrt{7}}{2} | 25.78125 |
16,618 | Calculate the value of the polynomial $f(x) = 3x^6 + 4x^5 + 5x^4 + 6x^3 + 7x^2 + 8x + 1$ at $x=0.4$ using Horner's method, and then determine the value of $v_1$. | 5.2 | 57.03125 |
16,619 | Given plane vectors $\overrightarrow{a}=(-1,2)$ and $\overrightarrow{b}=(1,-4)$.
$(1)$ If $4\overrightarrow{a}+\overrightarrow{b}$ is perpendicular to $k\overrightarrow{a}-\overrightarrow{b}$, find the value of the real number $k$.
$(2)$ If $\theta$ is the angle between $4\overrightarrow{a}+\overrightarrow{b}$ and ... | -\frac{3}{4} | 87.5 |
16,620 | Mark rolls 5 fair 8-sided dice. What is the probability that at least three of the dice show the same number? | \frac{1052}{8192} | 0 |
16,621 | A school has between 130 and 210 students enrolled. Every afternoon, all the students gather to participate in a singing session. The students are divided into eight distinct groups. If two students are absent from school, the groups can all have the same number of students. What is the sum of all possible numbers of s... | 1870 | 1.5625 |
16,622 | If $\tan(\alpha-\beta) = \frac{1}{2}$ and $\tan(\alpha+\beta) = \frac{1}{3}$, calculate the value of $\tan 2\beta$. | - \frac{1}{7} | 75.78125 |
16,623 | Given an ellipse with foci at $(1,0)$ and $(-1,0)$, it intersects with the line $y=x-2$. Determine the maximum value of the eccentricity of this ellipse. | \frac{\sqrt{10}}{5} | 48.4375 |
16,624 | Let \( A \) and \( B \) be points on the curve \( xy = 1 \) (where \( x > 0 \) and \( y > 0 \)) in the Cartesian coordinate system \( xOy \). Given the vector \( \vec{m} = (1, |OA|) \), find the minimum value of the dot product \( \vec{m} \cdot \overrightarrow{OB} \). | 2 \sqrt[4]{2} | 41.40625 |
16,625 | For the power function $y = (m^2 - m - 1)x^{2m+1}$, if it is a decreasing function for $x \in (0, +\infty)$, then the value of the real number $m$ is ______. | -1 | 39.0625 |
16,626 | The positive integers $m$ and $n$ satisfy $8m + 9n = mn + 6$. Find the maximum value of $m$. | 75 | 80.46875 |
16,627 | What is the value of \(6102 - 2016\)? | 4086 | 75 |
16,628 | During the "Cool Summer Happy Shopping" promotion held in a certain shopping mall, Xiao Yang bought $m$ items of type A goods priced at $5$ yuan each, and $n$ items of type B goods priced at $17 yuan each, spending a total of $203$ yuan. Then the maximum value of $m+n$ is ______. | 31 | 48.4375 |
16,629 | Given a decreasing arithmetic sequence $\{a_n\}$, if $a_1 + a_{100} = 0$, find the value of $n$ when the sum of the first $n$ terms, $S_n$, is maximized. | 50 | 96.875 |
16,630 | A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. But during the first week the quarry only sells $60\%$ of the gravel at full price. The following week the quarry drops the price by $10\%$ , and, again, it sells $60\%$ of the remaining gravel. Each week, the... | 3000 | 77.34375 |
16,631 | How many different four-digit numbers, divisible by 4, can be made from the digits 1, 2, 3, and 4,
a) if each digit can be used only once?
b) if each digit can be used multiple times? | 64 | 79.6875 |
16,632 | In $\triangle ABC$, $a=5$, $b=8$, $C=60^{\circ}$, the value of $\overrightarrow{BC}\cdot \overrightarrow{CA}$ is $\_\_\_\_\_\_$. | -20 | 77.34375 |
16,633 | The inclination angle of the line $x+ \sqrt {3}y+c=0$ is \_\_\_\_\_\_. | \frac{5\pi}{6} | 52.34375 |
16,634 | A store offers an initial discount of 25% on all its merchandise. Subsequently, an additional discount of 10% is applied to the already reduced prices. The store advertises that the final price of the goods is 35% less than the original price. What is the actual percentage difference between the store's claimed discoun... | 2.5\% | 97.65625 |
16,635 | The circumference of a circle is 100. The diameter of this circle is equal to: | $\frac{100}{\pi}$ | 0 |
16,636 | Given the function $f(x)= \begin{vmatrix} x & \frac {1}{x} \\ -2 & 1\end{vmatrix}$ ($x>0$), its range is set $A$,
(1) If the universal set $U=\mathbb{R}$, find $C_UA$;
(2) For any $x \in (0, \frac {1}{2}]$, the inequality $f(x)+a \geq 0$ always holds, find the range of the real number $a$;
(3) Let $P$ be any poin... | -1 | 73.4375 |
16,637 | A line $l$ passing through the focus of the parabola $y=4x^2$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $y_1+y_2=2$, then the length of segment $AB$ equals \_\_\_\_\_\_. | \frac{17}{8} | 39.0625 |
16,638 | Molly and Jake play a game where Molly picks an integer between 1 and 720 inclusive, and Jake divides 720 by that integer and states whether the quotient is an integer. How many integers could Molly pick such that Jake’s quotient is an integer? | 30 | 60.15625 |
16,639 | Add 74.6893 to 23.152 and round to the nearest hundredth. | 97.84 | 96.875 |
16,640 | In how many ways can the digits of $45,\!502,\!2$ be arranged to form a 6-digit number? (Remember, numbers cannot begin with 0.) | 150 | 56.25 |
16,641 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. Given vectors $\overrightarrow{m}=(\sin B+\sin C, \sin A-\sin B)$ and $\overrightarrow{n}=(\sin B-\sin C, \sin(B+C))$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(1) Find the magnitude of angle $C$... | \frac{4\sqrt{3}-3}{10} | 26.5625 |
16,642 | If \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\) are integers for which \(729x^3+64 = (ax^2 + bx + c)(dx^2 + ex + f)\) for all \(x\), then what is \(a^2+b^2+c^2+d^2+e^2+f^2\)? | 8210 | 3.90625 |
16,643 | Given the polynomial function $f(x) = 2x^5 - 5x^4 - 4x^3 + 3x^2 - 6x + 7$, using the Horner's method when $x = 5$, we can obtain $v_2 = \_$. | 21 | 92.96875 |
16,644 | Find the greatest possible value of $pq + r$ , where p, q, and r are (not necessarily distinct) prime numbers satisfying $pq + qr + rp = 2016$ .
| 1008 | 53.125 |
16,645 | It is known that Xiao Li has a probability of hitting the bullseye of $40\%$ each time he shoots. Now, the method of random simulation is used to estimate the probability that Xiao Li hits the bullseye exactly twice in three shots. First, a calculator is used to generate random integers between $0$ and $9$, with $0$, $... | 0.30 | 0.78125 |
16,646 | Given that point $A(-2,α)$ lies on the terminal side of angle $α$ and $\sin α=- \dfrac{ \sqrt{5}}{5}$.
(1) Find the values of $α$, $\cos α$, and $\tan α$.
(2) Find the value of $\dfrac{\cos ( \dfrac{π}{2}+α)\sin (-π-α)}{\cos ( \dfrac{11π}{2}-α)\sin ( \dfrac{9π}{2}+α)}$. | \dfrac {1}{2} | 68.75 |
16,647 | Determine the sum of all real numbers $x$ that are not in the domain of the function $$g(x) = \frac{1}{2+\frac{1}{3+\frac{1}{x}}}.$$ | -\frac{13}{21} | 67.96875 |
16,648 | Let the function $f(x)=\begin{cases} 2^{x}+\int_{0}^{1}{3(\sqrt{x}-x^{2})dx} & (x\geqslant 4) \\ f(x+2) & (x < 4) \end{cases}$. Evaluate the expression $f(\log_{2}3)$. | 49 | 60.9375 |
16,649 | Calculate the value of $n$ such that
\[(1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 30^\circ) = 2^n.\] | 15 | 17.96875 |
16,650 | Determine the smallest positive integer $n$ such that $n^2$ is divisible by 50 and $n^3$ is divisible by 294. | 210 | 81.25 |
16,651 | The Fibonacci numbers are defined by $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n>2$ . It is well-known that the sum of any $10$ consecutive Fibonacci numbers is divisible by $11$ . Determine the smallest integer $N$ so that the sum of any $N$ consecutive Fibonacci numbers is divisible by $12$ . | 24 | 96.09375 |
16,652 | Define a function $f(x)$ on $\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \in [-3,3)$, $f(x)=\begin{cases} -(x+2)^2, & -3 \leqslant x < -1 \\ x, & -1 \leqslant x < 3 \end{cases}$. Find $f(4)$ and the value of $f(1)+f(2)+f(3)+\ldots+f(2016)+f(2017)$. | 337 | 32.03125 |
16,653 | A farmer buys 600 cows. He sells 500 of them for the price he paid for all 600 cows. The remaining 100 cows are sold for 10% more per cow than the price of the 500 cows. Calculate the percentage gain on the entire transaction. | 22\% | 69.53125 |
16,654 | A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | \sqrt{55} | 22.65625 |
16,655 | Given the curve $y=\sin x$ ($0 \leq x \leq \pi$) and the line $y= \frac {1}{2}$, calculate the area of the enclosed shape formed by these two functions. | \sqrt {3}- \frac {\pi}{3} | 0 |
16,656 | A pawn moves on a $6 \times 6$ chessboard. It starts from the bottom-left square. At each step, it can either jump to the square directly to its right or the square directly above it. It must reach the top-right square, such that it never lies strictly above the diagonal connecting the starting square and the destinati... | 132 | 64.84375 |
16,657 | Arnaldo claimed that one billion is the same as one million millions. Professor Piraldo corrected him and said, correctly, that one billion is the same as one thousand millions. What is the difference between the correct value of one billion and Arnaldo's assertion?
(a) 1000
(b) 999000
(c) 1000000
(d) 999000000
(e) 999... | 999000000000 | 88.28125 |
16,658 | Suppose (4-4): Coordinate Systems and Parametric Equations
In the rectangular coordinate system $xOy$, the parametric equations of the curve $C$ are $\begin{cases} x=3\cos \alpha \\ y=\sin \alpha \end{cases}$, where $\alpha$ is the parameter. Establish a polar coordinate system with the coordinate origin $O$ as the po... | \frac{18\sqrt{2}}{5} | 25.78125 |
16,659 | The minimum value of the polynomial $x^2 + y^2 - 6x + 8y + 7$ is ______. | -18 | 89.84375 |
16,660 | Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 7\}$. What is the probability that the greatest common divisor of these two numbers is one? Express your answer as a common fraction. | \frac{17}{21} | 0.78125 |
16,661 | What is the value of $\sqrt{64 \times \sqrt{49}}$? | 8\sqrt{7} | 32.8125 |
16,662 | An isosceles triangle with a base of $\sqrt{2}$ has medians intersecting at a right angle. Calculate the area of this triangle. | \frac{3}{2} | 3.90625 |
16,663 | In the 2016 art exam of a certain high school, there were 6 contestants, including 3 females and 3 males. Now, these six contestants are to perform their talents in sequence. If any two of the three males cannot perform consecutively, and the female contestant A cannot be the first to perform, then calculate the number... | 132 | 28.90625 |
16,664 | Griffin and Hailey run for $45$ minutes on a circular track. Griffin runs counterclockwise at $260 m/min$ and uses the outer lane with a radius of $50$ meters. Hailey runs clockwise at $310 m/min$ and uses the inner lane with a radius of $45$ meters, starting on the same radial line as Griffin. Determine how many times... | 86 | 1.5625 |
16,665 | Given that point $P$ is a moving point on the parabola $y^{2}=4x$, the minimum value of the sum of the distance from point $P$ to line $l$: $2x-y+3=0$ and the $y$-axis is ___. | \sqrt{5}-1 | 12.5 |
16,666 | Given that point $P$ is a moving point on circle $C$: $x^{2}+y^{2}-2x-4y+1=0$, the maximum distance from point $P$ to a certain line $l$ is $6$. If a point $A$ is taken arbitrarily on line $l$ to form a tangent line $AB$ to circle $C$, with $B$ being the point of tangency, then the minimum value of $AB$ is _______. | 2\sqrt{3} | 92.1875 |
16,667 | Given that the function $f\left(x\right)$ is an even function on $R$, and $f\left(x+2\right)$ is an odd function. If $f\left(0\right)=1$, then $f\left(1\right)+f\left(2\right)+\ldots +f\left(2023\right)=\_\_\_\_\_\_$. | -1 | 14.84375 |
16,668 | Samantha has 10 green marbles and 5 purple marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 7 marbles. What is the probability that exactly four of the marbles that she removes are green? Express your answer as a decimal. | 0.256 | 8.59375 |
16,669 | Find the sum of the first eight prime numbers that have a units digit of 3. | 404 | 0 |
16,670 | Let $T$ be the sum of all the real coefficients of the expansion of $(1 + ix)^{2018}$. What is $\log_2(T)$? | 1009 | 53.90625 |
16,671 | Evaluate the expression $-20 + 15 \times (4^{\div -1} \times 2)$. | -12.5 | 45.3125 |
16,672 | The length of the longer side of rectangle \(R'\) is \(15\%\) more than the length of a side of square \(S'\). The length of the shorter side of rectangle \(R'\) is \(5\%\) less than the length of a side of square \(S'\). What is the ratio of the area of rectangle \(R'\) to the area of square \(S'\)? Express your answe... | \frac{10925}{10000} | 0 |
16,673 | The graph of $y = ax^2 + bx + c$ has a maximum value of 75, and passes through the points $(-3,0)$ and $(3,0)$. Find the value of $a + b + c$ at $x = 2$. | \frac{125}{3} | 42.96875 |
16,674 | If \( x_{i}=\frac{i}{101} \), then the value of \( S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \) is | 51 | 75.78125 |
16,675 | A dodecahedron consists of two pentagonal-based pyramids glued together along their pentagonal bases, forming a polyhedron with 12 faces. Consider an ant at the top vertex of one of the pyramids, selecting randomly one of the five adjacent vertices, designated as vertex A. From vertex A, the ant then randomly selects o... | \frac{1}{5} | 38.28125 |
16,676 | If \(\sum_{n = 0}^{\infty} \cos^{2n} \theta = \frac{9}{2}\), what is the value of \(\cos{4\theta}\)? | -\frac{31}{81} | 92.96875 |
16,677 | Given non-zero plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}-\overrightarrow{c}|=1$. If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{π}{3}$, calculate the minimum value of $|\overrightarrow{a}-\overright... | \sqrt{3} - 1 | 12.5 |
16,678 | In the rectangular coordinate system $xOy$, the parametric equations of line $l$ are given by $\begin{cases} x=2t \\ y=-2-t \end{cases}$ (where $t$ is the parameter). In the polar coordinate system (using the same length unit as the rectangular coordinate system and with the origin $O$ as the pole and the polar axis co... | \frac{12\sqrt{5}}{5} | 82.03125 |
16,679 | Given the function $f(x)=\sqrt{3}\sin x\cos x+\cos^{2}x$.
(1) Find the value of $f(\frac{\pi }{24})$;
(2) If the function $f(x)$ is monotonically increasing in the interval $[-m,m]$, find the maximum value of the real number $m$. | \frac{\pi}{6} | 84.375 |
16,680 | If $A = 3009 \div 3$, $B = A \div 3$, and $Y = A - 2B$, then what is the value of $Y$? | 335 | 32.03125 |
16,681 | What is the value of $\frac{3}{5} + \frac{2}{3} + 1\frac{1}{15}$? | 2\frac{1}{3} | 18.75 |
16,682 | Find the largest prime $p$ such that $p$ divides $2^{p+1} + 3^{p+1} + 5^{p+1} + 7^{p+1}$ . | 29 | 75.78125 |
16,683 | What is the smallest positive multiple of $17$ that is $3$ more than a multiple of $76$? | 459 | 12.5 |
16,684 | On Arbor Day, 20 students in a class planted trees along one side of a straight road, with each person planting one tree, and the distance between two adjacent trees being 10 meters. Initially, it is required to place all the saplings next to one of the tree pits so that the total distance traveled back and forth by ea... | 2000 | 34.375 |
16,685 | Let $ABC$ be an isosceles right triangle with $\angle A=90^o$ . Point $D$ is the midpoint of the side $[AC]$ , and point $E \in [AC]$ is so that $EC = 2AE$ . Calculate $\angle AEB + \angle ADB$ . | 135 | 62.5 |
16,686 | Mathematical operation refers to the ability to solve mathematical problems based on clear operation objects and operation rules. Because of operations, the power of numbers is infinite; without operations, numbers are just symbols. Logarithmic operation and exponential operation are two important types of operations. ... | \frac{1}{2} | 83.59375 |
16,687 | Given that $0 < α < \frac {π}{2}$, and $\cos (2π-α)-\sin (π-α)=- \frac { \sqrt {5}}{5}$.
(1) Find the value of $\sin α+\cos α$
(2) Find the value of $\frac {2\sin α\cos α-\sin ( \frac {π}{2}+α)+1}{1-\cot ( \frac {3π}{2}-α)}$. | \frac {\sqrt {5}-9}{5} | 0 |
16,688 | What is the smallest positive multiple of $225$ that can be written using
digits $0$ and $1$ only? | 11111111100 | 99.21875 |
16,689 | A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point? | $\sqrt{13}$ | 0 |
16,690 | Given that Chloe's telephone numbers have the form $555-ab-cdef$, where $a$, $b$, $c$, $d$, $e$, and $f$ are distinct digits, in descending order, and are chosen between $1$ and $8$, calculate the total number of possible telephone numbers that Chloe can have. | 28 | 100 |
16,691 | The numbers $\sqrt{3v-2}$, $\sqrt{3v+1}$, and $2\sqrt{v}$ are the side lengths of a triangle. What is the measure of the largest angle? | 90 | 28.125 |
16,692 | Given that $\binom{24}{5}=42504$, and $\binom{24}{6}=134596$, find $\binom{26}{6}$. | 230230 | 67.96875 |
16,693 | Given a sequence $\{a_n\}$ that satisfies $3a_{n+1}+a_n=0$ and $a_2=-\frac{4}{3}$, find the sum of the first $10$ terms of $\{a_n\}$. | 3(1-3^{-10}) | 0 |
16,694 | The set \( M = \left\{(x, y) \mid \log_{4} x + \log_{4} y \leq 1, x, y \in \mathbf{N}^{*}\right\} \) has how many subsets? | 256 | 93.75 |
16,695 | Compute $\cos \left( \arctan \frac{3}{4} \right)$ and $\tan \left( \arctan \frac{3}{4} \right)$. | \frac{3}{4} | 92.1875 |
16,696 | Find the smallest positive integer \( k \) such that \( (k-10)^{5026} \geq 2013^{2013} \). | 55 | 15.625 |
16,697 | In the expansion of $(1+x){(x-\frac{2}{x})}^{3}$, calculate the coefficient of $x$. | -6 | 85.15625 |
16,698 | Points $F$, $G$, $H$, $I$, and $J$ are positioned in 3-dimensional space with $FG = GH = HI = IJ = JF = 3$ and $\angle FGH = \angle HIJ = \angle JIF = 90^\circ$. The plane of triangle $FGH$ is parallel to $\overline{IJ}$. Determine the area of triangle $GIJ$. | 4.5 | 0.78125 |
16,699 | What is the result of the correct calculation for the product $0.08 \times 3.25$? | 0.26 | 98.4375 |
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