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 | Find the integer $n$, $-90 \le n \le 90$, such that $\sin n^\circ = \sin 782^\circ$. | -62 | 36 | 3 |
math | Given the function f(x) = $\frac{\ln{x} + 1}{x - 1}$, where x ∈ (1, +∞).
(1) Find the monotonic intervals of f(x);
(2) If f(x) > $\frac{k}{x}$ holds true in (1, +∞), find the maximum value of the integer k. | 3 | 76 | 1 |
math | Given that the right focus of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{5} = 1$ is $(3, 0)$, determine the eccentricity of the hyperbola. | \frac{3}{2} | 54 | 7 |
math | Let the set $M=\{x|-1\leq x<2\}$, and $N=\{x|x\leq k+3\}$, if $M\cap N \neq \emptyset$. Then, the range of the real number $k$ is. | k\geq-4 | 61 | 6 |
math | Given the vectors $\boldsymbol{a}=(0,1)$, $\boldsymbol{b}=\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$, and $\boldsymbol{c}=\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$. Real numbers $x, y, z$ satisfy $x \boldsymbol{a}+y \boldsymbol{b}+z \boldsymbol{c}=(1,2)$. Find the minimum value of $x^{2}+y^{2}+z^{2}$. | \frac{10}{3} | 134 | 8 |
math | Given the function $f(x) = \ln x$, the curve $y = g(x)$ is symmetric to the curve $y = f(x)$ with respect to the line $y = x$. If there exists a line passing through the origin that is tangent to both $y = f(x)$ and $y = g(ax)$, then the value of the real number $a$ is ______. | \frac{1}{e^2} | 82 | 9 |
math | For all m and n satisfying \( 1 \leq n \leq m \leq 5 \), the polar equation \( \rho = \frac{1}{1 - C_{m}^{n} \cos \theta} \) represents how many different hyperbolas? | 10 | 60 | 2 |
math | What integer $n$ satisfies $0 \leq n < 31$ and $$-527 \equiv n \pmod{31}~?$$ | 0 | 36 | 1 |
math | Determine the number of real solutions to the equation
\[(x^{10} + 1)(x^8 + x^6 + x^4 + x^2 + 1) = 12x^9.\] | 0 | 50 | 1 |
math | Given the line $y=k(x-1)$ intersects the line segment with endpoints $A(3,2)$ and $B(0,1)$, determine the range of $k$. | (-\infty, -1] \cup [1, +\infty) | 39 | 18 |
math | Magda cut out two identical isosceles triangles, each with a perimeter of $100 \mathrm{~cm}$. First, she created a quadrilateral by placing the triangles together by their legs. Then, she created another quadrilateral by placing them together by their bases. In the first case, the perimeter of the quadrilateral was $4 ... | r = 34 \, \text{cm}, \, z = 32 \, \text{cm} | 99 | 26 |
math | The center of the upper base of a cube with edge length $a$ is connected to the midpoints of the sides of the lower base, which are also connected in a sequential order. Calculate the total surface area of the resulting pyramid. | 2a^2 | 48 | 4 |
math | Given that the inequality (a - a^2)•(x^2 + 1) + x ≤ 0 holds true for all x ∈ (0, 2], determine the range of values for a. | (-\infty, \frac{1 - \sqrt{3}}{2}] \cup [\frac{1 + \sqrt{3}}{2}, +\infty) | 45 | 38 |
math | Susie's parents open a savings account for her with a principal of $\$2500$, invested at an annual interest rate of $6\%$, compounded yearly. Calculate the total amount in the account when Susie turns 21 years old. | \$8017.84 | 53 | 8 |
math | Given two perpendicular chords AB and CD passing through the focus of the parabola $y^2=4x$, calculate the value of $\frac {1}{|AB|} + \frac {1}{|CD|}$. | \frac{1}{2} | 48 | 7 |
math | Given the equation $x^2+px+q=0$ has two real roots that are reciprocals of each other, find the conditions that $p$ and $q$ must satisfy. | p \geq 2 \text{ or } p \leq -2 \text{ and } q = 1 | 41 | 26 |
math | Find the number of positive integers $n \le 1500$ that can be expressed in the form
\[\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor = n\]
for some real number $x.$ | 856 | 63 | 3 |
math | In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a^{2}=bc\cos A+ac\cos B$.
$(1)$ Find the value of $\frac{a}{c}$;
$(2)$ If $a=1$, $\cos B=\frac{3}{4}$, and $D$ is the midpoint of $AC$, find $BD$. | BD = \sqrt{2} | 105 | 7 |
math | The ratio of the interior angles of two regular polygons is $4:3$. Furthermore, one of the polygons has exactly $5$ sides. Determine the number of such pairs of polygons where both polygons have more than three sides. | 0 | 46 | 1 |
math | When two identical dice are thrown at the same time, calculate the total number of possible outcomes. | 36 | 19 | 2 |
math | The base of a quadrilateral pyramid is a rectangle with a diagonal of length $b$ and an angle of $60^{\circ}$ between the diagonals. Each of the lateral edges makes an angle of $45^{\circ}$ with the plane of the base. Find the volume of the pyramid. | \frac{b^3 \sqrt{3}}{24} | 64 | 15 |
math | Determine the equation of the line that passes through the center of the circle given by $x^2+y^2+2x-4y=0$ and is perpendicular to the line $2x+3y=0$. | 3x - 2y + 7 = 0 | 48 | 12 |
math | A new rule is proposed where a year ending in double zeros will be considered a leap year only if it leaves a remainder of either 300 or 700 when divided by 1000. How many leap years will there be that end in double zeros between 2000 and 5000? | 6 | 69 | 1 |
math | What is the smallest two-digit positive integer \( k \) for which the product \( 45k \) is a perfect square? | 20 | 28 | 2 |
math | Calculate: 1g2 + ($\sqrt {2}$ - 1)<sup>0</sup> + lg5 = ___. | 2 | 29 | 1 |
math | In a bag, there are a total of 10 balls of the same size, including black, white, and red balls. It is known that the probability of drawing a black ball from the bag is $\frac{2}{5}$, and the probability of drawing at least 1 white ball when drawing 2 balls from the bag is $\frac{7}{9}$. Find the number of red balls i... | 1 | 88 | 1 |
math | In the Cartesian coordinate system $xOy$, a polar coordinate system is established with $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. Given the curve $C$: $\rho\cos^2\theta=4a\sin\theta$ ($a>0$), and the parametric equations of line $l$ are $$\begin{cases} x=-2+ \frac { \sqrt {2}}{2}t \\... | a=\frac{1}{4} | 217 | 8 |
math | How many ordered pairs of real numbers \((x, y)\) satisfy the following system of equations?
\[\left\{ \begin{aligned} x + 4y &= 4 \\ \left| |x| - 2|y| \right| &= 2 \end{aligned}\right.\] | 2 | 66 | 1 |
math | Given the circle $(x-a)^2+(y-a)^2=8$, determine the range of the real number $a$ for which there always exists two points on the circle that are at a distance of $\sqrt{2}$ from the origin. | (-3, -1) \cup (1, 3) | 51 | 14 |
math | A digital clock shows the time 4:56. How many minutes will pass until the clock next shows a time in which all of the digits are consecutive and are in increasing order? | 458 | 38 | 3 |
math | Let line $l_1: x + my + 6 = 0$ and line $l_2: (m - 2)x + 3y + 2m = 0$. When $m = \_\_\_\_\_\_$, $l_1 \parallel l_2$. | -1 | 64 | 2 |
math | In a certain area, there are 100,000 households, among which there are 99,000 ordinary households and 1,000 high-income households. A simple random sample of 990 households is drawn from the ordinary households, and 100 households are drawn from the high-income households for a survey. It was found that a total of 120 ... | 5.7\% | 148 | 5 |
math | Calculate the definite integral:
$$
\int_{0}^{2 \pi} \sin ^{2}\left(\frac{x}{4}\right) \cos ^{6}\left(\frac{x}{4}\right) d x
$$ | \frac{5\pi}{64} | 51 | 10 |
math | Find the equation of the line that is tangent to both of the parabolas $y^2 = 4x$ and $x^2 = -32y.$ Enter your answer in the form "$y = mx + b$". | y = \frac{1}{2} x + 2 | 51 | 13 |
math | What is the sum of the different prime factors of $210630$? | 93 | 19 | 2 |
math | There are 35 squares and triangles in total, with 120 edges. There are \_\_\_\_\_\_ squares and \_\_\_\_\_\_ triangles. | 20 | 38 | 2 |
math | There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points? | 2016 | 53 | 4 |
math | X is the set \(\{1, 2, 3, ... , n\}\). How many ordered 6-tuples \((A_1, A_2, ... , A_6)\) of subsets of X are there such that every element of X belongs to 0, 3, or 6 subsets in the 6-tuple? | 22^n | 76 | 3 |
math | Given the vectors $\overrightarrow{m}=(x,y)$ and $\overrightarrow{n}=(x-y)$, let $P$ be a moving point on the curve $\overrightarrow{m}\cdot \overrightarrow{n}=1 (x > 0)$. If the distance from point $P$ to the line $x-y+1=0$ is always greater than $\lambda$, find the maximum value of the real number $\lambda$. | \frac{\sqrt{2}}{2} | 92 | 10 |
math | In $\triangle ABC$, if $a + c = 2b$, then find the value of $\tan \frac{A}{2} \cdot \tan \frac{C}{2}$. | 1/3 | 41 | 3 |
math | Given that Eva is using 24 fence posts to fence her rectangular vegetable patch, with each post spaced 6 yards apart, including the corners, and the longer side of the patch has three times as many posts as the shorter side, calculate the area of Eva’s vegetable patch. | 576 | 57 | 3 |
math | Simplify the expression $$(x-1)^{4}+4(x-1)^{3}+6(x-1)^{2}+4(x-1)+1$$ | x^{4} | 39 | 4 |
math | The Minions need to make jam within the specified time. Kevin can finish the job 4 days earlier if he works alone, while Dave would finish 6 days late if he works alone. If Kevin and Dave work together for 4 days and then Dave completes the remaining work alone, the job is completed exactly on time. How many days would... | 12 | 85 | 2 |
math | Given the function $f(x)= \frac {1}{2}x^{2}+mx+m\ln x$
(I) Discuss the monotonicity of the function $f(x)$;
(II) When $m=1$, if the equation $f(x)= \frac {1}{2}x^{2}+ac$ has a unique real solution in the interval $[\frac {1}{e},+\infty)$, find the range of the real number $a$;
(III) When $m > 0$, if for any two r... | \frac {1}{2} | 197 | 7 |
math | What is the minimum number of colors needed to paint the cells of a $5 \times 5$ square so that among any three consecutive cells in a row, column, or diagonal, no three cells are the same color? | 5 | 46 | 1 |
math | Given $(2x-1)^5 = a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0$ is an identity in terms of $x$. Find:
(1) The value of $a_0 + a_1 + a_2 + a_3 + a_4 + a_5$;
(2) The value of $a_0 - a_1 + a_2 - a_3 + a_4 - a_5$;
(3) The value of $a_0 + a_2 + a_4$. | -121 | 143 | 4 |
math | A deck consists of six red and six green cards, with each card labeled from $A$ to $F$. Calculate the probability of drawing a winning pair from this deck. | \frac{6}{11} | 35 | 8 |
math | A line passing through M(2,4) intersects the parabola $y^2=8x$ at only one point. How many such lines are there? | 2 | 35 | 1 |
math | In triangle ABC, A, B, and C are the internal angles, and sinC = 2sin(A - B), B = π/6.
1. Find the size of angle A.
2. If cos(A + x) = -1/3, find the value of cos(A - 2x). | \frac{7}{9} | 66 | 7 |
math | Given that the sequence $\{a_n\}$ is an arithmetic sequence, and its sum of the first $n$ terms is $S_n$, $\{b_n\}$ is a geometric sequence, and $a_1=b_1=2$, $a_4+b_4=27$, $S_4-b_4=10$.
$(1)$ Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$;
$(2)$ Does there exist $m(m\in\mathbb{N}^*)$ such that $\frac{... | m=5 | 179 | 3 |
math | Given proposition $p: \forall x\in \{x\left|\right.-3 \lt x \lt 1\}$, the inequality $x^{2}+4x+9-m \gt 0$ always holds true; proposition $q: \exists x\in \{x\left|\right.x \gt 0\}$, $x^{2}-2mx+1 \lt 0$ holds true.<br/>$(1)$ If proposition $p$ is a true proposition, find the range of real number $m$;<br/>$(2)$ If exactl... | (-\infty, 1] \cup [5, +\infty) | 147 | 18 |
math | In the diagram, line $AB$ is parallel to line $EF$ and $AFC$ is a straight line. Points $D$, $E$ are on line $AB$, and $G$ is on $EF$. Given that $\angle AFC = 180^\circ$, $\angle AFE = 100^\circ$, and $\angle EFA = 60^\circ$, determine the value of $x,$ which is $\angle DGC$.
[asy]
draw((0,0)--(-.5,5)--(8,5)--(6.5,0)-... | 80^\circ | 352 | 4 |
math | Given the function $f(x) = xe^x - k$ ($x \in \mathbb{R}$) has exactly two zeros, determine the range of the real number $k$. | (-\frac{1}{e}, 0) | 40 | 11 |
math | Peyton puts 30 L of oil and 15 L of vinegar into a large empty can. He then adds 15 L of oil to create a new mixture. What percentage of the new mixture is oil? | 75\% | 47 | 4 |
math | Let $X \sim B(4, p)$, and $P(X=2)=\frac{8}{27}$, find the probability of success in one trial. | \frac{2}{3} | 37 | 7 |
math | Given that the domain of the function $f(x+1)$ is $[1,9)$, determine the domain of the function $y=f(x-1)+ \sqrt {7-x}$ is _______. | [3,7] | 43 | 5 |
math | A circular sector with a central angle of $120^{\circ}$ has an inscribed circle. Find the radius of the inscribed circle, given that the radius of the original circle is $R$. | \sqrt{3} R (2 - \sqrt{3}) | 43 | 14 |
math | Using each of the following 4 numbers exactly once, arranged in any order, and using the operations "+", "-", "*", "/", and parentheses, make the number 24:
(1) 7, 12, 9, 12;
(2) 3, 9, 5, 9. | (9 - 3) \times (9 - 5) = 24 | 67 | 18 |
math | It is known that a certain mobile phone brand in the United States has an annual fixed cost of $400,000 for producing a certain model of mobile phone, and an additional investment of $160,000 is required for every 10,000 units produced. Suppose the company produces and sells $x$ ten thousand units of this model within ... | 6104 | 244 | 4 |
math | A bus arrives randomly sometime between 1:00 and 2:30, waits for 20 minutes, and then leaves. If Laura also arrives randomly between 1:00 and 2:30, what is the probability that the bus will be there when Laura arrives? | \frac{16}{81} | 61 | 9 |
math | Calculate the volumes of the solids formed by the rotation of the regions bounded by the graphs of the functions about the x-axis.
$$
y = 2x - x^2, \quad y = -x + 2, \quad x = 0
$$ | V = \frac{9}{5} \pi | 55 | 11 |
math | It is known that the cost price of a certain product is 40 yuan per item, and the selling price is 60 yuan per item, with a weekly sales volume of 300 items. Market research shows that for every 1 yuan increase in price, 10 fewer items will be sold each week. At what price should the product be set to maximize the stor... | 6250 | 82 | 4 |
math | Segments $a$, $b$, and $c$ are connected end to end to form a triangle, with $a=2$ and $b=3$. Determine the length of $c$. | 4 | 40 | 1 |
math | Christina decides to write all possible rearrangements of the letters in her name. If she can write twelve rearrangements every minute for the first half of the rearrangements and eighteen rearrangements every minute for the second half, how many hours will it take her to write all possible rearrangements of her name? | 420 \text{ hours} | 66 | 8 |
math | Given the parabola $C: x^{2}=2py (p > 0)$, with a focus at point $F$, and a point $M$ on the line $x-y-2=0$, let $MA$ and $MB$ be the two tangents of $C$ drawn through $M$ ($A$ and $B$ are the points of tangency). If the minimum value of $|AF| \cdot |BF|$ is $8$, find the value of $p$. | 4 | 107 | 1 |
math | The domain of the function $y= \frac {x+7}{kx^{2}+4kx+3}$ is $\mathbb{R}$, find the range of values for the real number $k$. | [0, \frac {3}{4}) | 47 | 10 |
math | Given regular pentagon $ABCDE,$ a circle can be drawn that is tangent to $\overline{DC}$ at $D$ and to $\overline{AB}$ at $A.$ In degrees, what is the measure of minor arc $AD$? [asy]import olympiad; import geometry; size(100); defaultpen(linewidth(0.8));
pair[] pentagon = new pair[5];
pentagon[0] = dir(36);
pentagon.c... | 144^\circ | 273 | 5 |
math | Wei has updated the design of his company's logo to include more circles within the same square frame. Each circle is still tangent to two sides of the square and to its adjacent circles. If the square's side length remains 20 inches, how many square inches will be shaded now?
[asy]
size(100);
draw((0,0)--(4,0)--(4,4)... | 400 - 100\pi | 334 | 10 |
math | A rhombus has side lengths of integer lengths and its area is numerically equal to the number of units in its perimeter. What is the number of units in the perimeter of this rhombus? | 32 | 41 | 2 |
math | Given the function $y=-x^2+4x-3$, determine its interval of monotonic decrease. | [2, +\infty) | 23 | 8 |
math | The function $f(x) = x(x - m)^2$ reaches its maximum value at $x = -2$. Determine the value of $m$. | -6 | 32 | 2 |
math | The function $f(x) = (m^2 - 1)x^m$ is a power function, and it is increasing on the interval $(0, +\infty)$. The value of the real number $m$ is ______. | m = \sqrt{2} | 51 | 7 |
math | Calculate $7\cdot7! + 6\cdot6! + 5\cdot5! + 5!$. | 40320 | 26 | 5 |
math | The greatest common divisor of two positive integers is $(x+5)$ and their least common multiple is $x(x+5)$. If one of the integers is 36, what is the smallest possible value of the other one? | 1 | 48 | 1 |
math | Calculate: $53\cdot\left(3\frac{1}{4} - 3\frac{3}{4}\right) \div \left(1\frac{2}{3} + 2\frac{2}{5}\right)$, and express the result as a mixed number. | -6\frac{57}{122} | 65 | 12 |
math | Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $30.$ One marble is taken out of each box randomly. The probability that both marbles are black is $3/5,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positive... | 29 | 88 | 2 |
math | Given that the generatrix of a cone makes an angle of $\frac{\pi}{3}$ with the axis, and the length of the generatrix is 3, the maximum value of the sectional area passing through the vertex is \_\_\_\_\_\_. | \frac{9}{2} | 53 | 7 |
math | How many of the following numbers are factors of 34 or multiples of 7?
1, 2, 3, 4, 8, 14, 17, 29, 56, 91 | 6 | 51 | 1 |
math | A sequence $b_1$, $b_2$, $\ldots$ of non-negative integers is defined by the rule $b_{n+2} = |b_{n+1} - b_n|$ for $n \geq 1$. If $b_1 = 1019$, $b_2 < 1019$, and $b_{3006} = 2$, how many different values of $b_2$ are possible? | 509 | 102 | 3 |
math | The perimeter of a square with side length $y$ inches is equal to the circumference of a circle with radius 5 centimeters. If 1 inch equals 2.54 centimeters, what is the value of $y$ in inches? Express your answer as a decimal to the nearest hundredth. | 3.09 | 63 | 4 |
math | Given that acute angles $\alpha$ and $\beta$ satisfy $\frac{sin2\alpha}{cos2\alpha-1}=\frac{1-tan\beta}{1+tan\beta}$, then $\cos (\alpha +\beta)=$____. | -\frac{\sqrt{2}}{2} | 55 | 10 |
math | Two delegations traveled from place A to place B by bus, with each bus having a capacity of 35 people. After filling several buses, the first delegation had 15 people left over, and the second delegation had a number of people left over that, combined with the 15 from the first delegation, exactly filled one more bus. ... | 15 | 130 | 2 |
math | A car dealer sold an average of 7 cars per day over a certain period. Ignoring the day with the lowest sales, the average number of cars sold on the remaining days was 8. Excluding the day with the highest sales, this number was 5. Finally, if neither the day with the lowest nor the day with the highest sales is consid... | 42 | 95 | 2 |
math | Determine the monotonic decreasing interval of the function $y=\sin \left( \frac {\pi}{3}-2x\right)$. | $\left[k\pi- \frac {\pi}{12},k\pi+ \frac {5\pi}{12}\right]$ | 30 | 30 |
math | Two equal parallel chords are drawn 10 inches apart in a circle of radius 10 inches. Calculate the area of the part of the circle that lies between the chords. | \frac{100\pi}{3} - 50\sqrt{3} | 36 | 20 |
math | In a right triangle with integer length sides, the hypotenuse has length 65 units. Determine the length of the shorter leg. | 25 | 28 | 2 |
math | Given an arithmetic sequence $\{a\_n\}$ with first term $a\_1=1$, common difference $d > 0$, and the second, fifth, and fourteenth terms are the second, third, and fourth terms of a geometric sequence $\{b\_n\}$, respectively.
1. Find the general term formulas for the sequences $\{a\_n\}$ and $\{b\_n\}$.
2. Suppose a ... | 3^{2016} | 182 | 7 |
math | If the sequences $\{a_{n}\}$ and $\{b_{n}\}$ are both arithmetic sequences, and $\frac{{{a_1}+{a_2}+⋯+{a_n}}}{{{b_1}+{b_2}+⋯+{b_n}}}=\frac{{5n+3}}{{n+2}}$, then $\frac{{{a_7}}}{{{b_7}}}=\_\_\_\_\_\_$. | \frac{{68}}{{15}} | 96 | 9 |
math | Piravena needs to make a trip from city $D$ to $E$, then from $E$ to $F$, then from $F$ to $D$. The cities form a right-angled triangle with $F$ as the right angle. The distances are as follows: from $D$ to $F$ is 2000 km, and from $D$ to $E$ is 4500 km. The cost per kilometer for a bus is $\$0.18$, and for an airplane... | 10531 \text{ km} | 145 | 10 |
math | Points \( A_{1}, B_{1}, C_{1} \) are the intersections of the extensions of the altitudes of the acute-angled triangle \( A B C \) with the circumcircle of \( A B C \). The incircle of the triangle \( A_{1} B_{1} C_{1} \) touches one of the sides of \( A B C \), and one of the angles of the triangle \( A B C \) is \( 5... | 60^\circ \text{ and } 70^\circ | 121 | 14 |
math | In the Cartesian coordinate system $(xOy)$, the parametric equations of circle $C$ are given by $\begin{cases} x = -5 + \sqrt{2}\cos t \\ y = 3 + \sqrt{2}\sin t \end{cases}, (t \text{ is the parameter})$, and in the polar coordinate system established with the origin $O$ as the pole and the non-negative semi-axis of $x... | 4 | 213 | 1 |
math | The *Olimpia* country is formed by $n$ islands. The most populated one is called *Panacenter*, and every island has a different number of inhabitants. We want to build bridges between these islands, which we'll be able to travel in both directions, under the following conditions:
a) No pair of islands is joined by... | (n-1)! | 147 | 4 |
math | Find the integrals:
1) $\int \frac{d x}{\sqrt{x}+\sqrt[4]{x}}$
2) $\int \frac{x+\sqrt{1+x}}{\sqrt[3]{1+x}} d x$
3) $\int \frac{\sqrt{x}}{x^{2} \sqrt{x-1}} d x$ | 2 \sqrt{\frac{x - 1}{x}} + C | 75 | 14 |
math | Calculate the arithmetic mean of the set $\{8, 88, 888, 8888, \ldots, 888888888\}$ and determine the digit that does not appear in the 9-digit number $N$, which is the calculated mean. | 2 | 65 | 1 |
math | Given α is in the interval ($$\frac{π}{2}$$, π), and 3cos2α = cos($$\frac{π}{4}$$+α), calculate the value of sin2α. | -\frac{17}{18} | 46 | 9 |
math | What is the perimeter, in cm, of quadrilateral \( EFGH \) if \( \overline{EF} \perp \overline{FG} \), \( \overline{HG} \perp \overline{FG} \), \( EF = 15 \) cm, \( HG = 6 \) cm, and \( FG = 20 \) cm? | 41 + \sqrt{481} \text{ cm} | 84 | 15 |
math | A necklace consists of 100 beads of red, blue, and green colors. It is known that among any five consecutive beads, there is at least one blue bead, and among any seven consecutive beads, there is at least one red bead. What is the maximum number of green beads that can be in this necklace? (The beads in the necklace a... | 65 | 88 | 2 |
math | Determine the interval that contains the root of the function $f(x) = 4 - 4x - e^x$ (where $e$ is the base of the natural logarithm). | (0, 1) | 41 | 6 |
math | Given $|\vec{a}|=5$, $|\vec{b}|=3$, and $\vec{a} \cdot \vec{b}=-9$, find the projection of $\vec{a}$ onto $\vec{b}$. | -3 | 51 | 2 |
math | What is the sum of the number of faces, edges, and vertices of a square pyramid that has a square base? Additionally, find the number of diagonals in the square base. | 18 | 37 | 2 |
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