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24,700 | When any set of $k$ consecutive positive integers necessarily includes at least one positive integer whose digit sum is a multiple of 11, we call each of these sets of $k$ consecutive positive integers a "dragon" of length $k." Find the shortest dragon length. | 39 | 0 |
24,701 | Calculate the volumes of the bodies bounded by the surfaces.
$$
z = 2x^2 + 18y^2, \quad z = 6
$$ | 6\pi | 1.5625 |
24,702 | Given points $A(-2,0)$ and $B(2,0)$, the slope of line $PA$ is $k_1$, and the slope of line $PB$ is $k_2$, with the product $k_1k_2=-\frac{3}{4}$.
$(1)$ Find the equation of the locus $C$ for point $P$.
$(2)$ Let $F_1(-1,0)$ and $F_2(1,0)$. Extend line segment $PF_1$ to meet the locus $C$ at another point $Q$. Let point $R$ be the midpoint of segment $PF_2$, and let $O$ be the origin. Let $S$ represent the sum of the areas of triangles $QF_1O$ and $PF_1R$. Find the maximum value of $S$. | \frac{3}{2} | 1.5625 |
24,703 | Given that $0 < \alpha < \frac{\pi}{2}$ and $0 < \beta < \frac{\pi}{2}$, if $\sin\left(\frac{\pi}{3}-\alpha\right) = \frac{3}{5}$ and $\cos\left(\frac{\beta}{2} - \frac{\pi}{3}\right) = \frac{2\sqrt{5}}{5}$,
(I) find the value of $\sin \alpha$;
(II) find the value of $\cos\left(\frac{\beta}{2} - \alpha\right)$. | \frac{11\sqrt{5}}{25} | 23.4375 |
24,704 | In right triangle $\triangle ABC$ with hypotenuse $\overline{AB}$, $AC = 15$, $BC = 20$, and $\overline{CD}$ is the altitude to $\overline{AB}$. Let $\omega$ be the circle having $\overline{CD}$ as a diameter. Let $I$ be a point outside $\triangle ABC$ such that $\overline{AI}$ and $\overline{BI}$ are both tangent to circle $\omega$. Find the ratio of the perimeter of $\triangle ABI$ to the length $AB$ and express it in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. | 97 | 0 |
24,705 | Matt now needs to place five identical, dotless dominoes (shaded 1 by 2 rectangles) on a 6 by 5 grid. He wishes to create a path from the upper left-hand corner $C$ to the lower right-hand corner $D$. In the path, consecutive dominoes must touch at their sides and not just at their corners. No domino may be placed diagonally; each domino must cover exactly two of the grid squares shown. Determine how many such distinct domino arrangements are possible. | 126 | 3.90625 |
24,706 | Let \( D \) be a point inside the acute triangle \( \triangle ABC \). Given that \( \angle ADB = \angle ACB + 90^\circ \) and \( AC \cdot BD = AD \cdot BC \), find the value of \( \frac{AB \cdot CD}{AC \cdot BD} \). | \sqrt{2} | 4.6875 |
24,707 | Express the quotient $2033_4 \div 22_4$ in base 4. | 11_4 | 0 |
24,708 | On every card of a deck of cards a regular 17-gon is displayed with all sides and diagonals, and the vertices are numbered from 1 through 17. On every card all edges (sides and diagonals) are colored with a color 1,2,...,105 such that the following property holds: for every 15 vertices of the 17-gon the 105 edges connecting these vertices are colored with different colors on at least one of the cards. What is the minimum number of cards in the deck? | 34 | 0 |
24,709 | Given that $\cos (\frac{\pi}{4}-a) = \frac{12}{13}$ and $(\frac{\pi}{4}-a)$ is a first-quadrant angle, find the value of $\frac{\sin (\frac{\pi}{2}-2a)}{\sin (\frac{\pi}{4}+a)}$: \_\_\_\_\_\_. | \frac{119}{144} | 0 |
24,710 | Compute without using a calculator: $12!/11!$ | 12 | 100 |
24,711 | Find the number of integers $ c$ such that $ \minus{}2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 \plus{} c$ is a multiple of $ 2^{2007}$ . | 670 | 0 |
24,712 | Find the sum of the roots of the equation $\tan^2x - 8\tan x + 2 = 0$ that are between $x = 0$ and $x = 2\pi$ radians. | 3\pi | 48.4375 |
24,713 | The triangle $ABC$ is isosceles with $AB=BC$ . The point F on the side $[BC]$ and the point $D$ on the side $AC$ are the feets of the the internals bisectors drawn from $A$ and altitude drawn from $B$ respectively so that $AF=2BD$ . Fine the measure of the angle $ABC$ . | 36 | 2.34375 |
24,714 | Let the roots of the cubic equation \(27x^3 - 81x^2 + 63x - 14 = 0\) be in geometric progression. Find the difference between the square of the largest root and the square of the smallest root. | \frac{5}{3} | 0.78125 |
24,715 | In triangle $XYZ,$ points $G,$ $H,$ and $I$ are on sides $\overline{YZ},$ $\overline{XZ},$ and $\overline{XY},$ respectively, such that $YG:GZ = XH:HZ = XI:IY = 2:3.$ Line segments $\overline{XG},$ $\overline{YH},$ and $\overline{ZI}$ intersect at points $S,$ $T,$ and $U,$ respectively. Compute $\frac{[STU]}{[XYZ]}.$ | \frac{9}{55} | 0 |
24,716 | One angle of a parallelogram is 150 degrees, and two consecutive sides have lengths of 10 inches and 20 inches. What is the area of the parallelogram? Express your answer in simplest radical form. | 100\sqrt{3} | 0 |
24,717 | Let \( x, y, z \) be positive real numbers such that \( xyz = 3 \). Compute the minimum value of
\[ x^2 + 4xy + 12y^2 + 8yz + 3z^2. \] | 162 | 0 |
24,718 | Given that the area of a cross-section of sphere O is $\pi$, and the distance from the center O to this cross-section is 1, then the radius of this sphere is __________, and the volume of this sphere is __________. | \frac{8\sqrt{2}}{3}\pi | 92.1875 |
24,719 | If $x^{2y}=16$ and $x = 16$, what is the value of $y$? Express your answer as a common fraction. | \frac{1}{4} | 0 |
24,720 | Consider two lines $p$ and $q$ in a coordinate plane with equations $y = -3x + 9$ and $y = -6x + 9$, respectively. Determine the probability that a point randomly selected in the first quadrant and below line $p$ will fall between $p$ and $q$. | 0.5 | 76.5625 |
24,721 | The average weight of 8 boys is 160 pounds, and the average weight of 6 girls is 130 pounds. Calculate the average weight of these 14 children. | 147 | 0 |
24,722 | What integer $n$ satisfies $0 \leq n < 151$ and $$100n \equiv 93 \pmod {151}~?$$ | 29 | 0 |
24,723 | Compute the sum of the geometric series $-3 + 6 - 12 + 24 - \cdots - 768$. | 514 | 0 |
24,724 | What is the smallest value of $k$ for which it is possible to mark $k$ cells on a $9 \times 9$ board such that any placement of a three-cell corner touches at least two marked cells? | 56 | 0 |
24,725 | Find the product of the roots and the sum of the roots of the equation $24x^2 + 60x - 600 = 0$. | -2.5 | 1.5625 |
24,726 | In recent years, live streaming e-commerce has gradually become an emerging marketing model, bringing new growth points to the e-commerce industry. At the beginning of the first year, a certain live streaming platform had an initial capital of 5 million yuan. Due to the participation of some well-known hosts, the platform's annual average capital growth rate can reach 40%. At the end of each year, operating costs of $a$ million yuan are deducted, and the remaining funds are reinvested in the live streaming platform.
$(1)$ If $a=100$, how many million yuan will the live streaming platform have at the end of the third year after deducting operating costs?
$(2)$ How much should the maximum annual operating cost be controlled at most, in order for the live streaming platform to reach 3000 million yuan at the end of the sixth year after deducting operating costs? (Give the result accurate to 0.1 million yuan) | 46.8 | 0 |
24,727 | Given that Jo and Blair take turns counting from 1, with Jo adding 2 to the last number said and Blair subtracting 1 from the last number said, determine the 53rd number said. | 79 | 0 |
24,728 | Given a rectangular yard containing two congruent isosceles right triangles in the form of flower beds and a trapezoidal remainder, with the parallel sides of the trapezoid having lengths $15$ and $25$ meters. | \frac{1}{5} | 0 |
24,729 | Suppose $a$, $b$, $c$, and $d$ are positive integers satisfying $a + b + c + d = 3000$. Calculate $a!b!c!d! = m \cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by 10. What is the smallest possible value of $n$?
A) 745
B) 748
C) 751
D) 754
E) 757 | 748 | 78.90625 |
24,730 | $21$ Savage has a $12$ car garage, with a row of spaces numbered $1,2,3,\ldots,12$ . How many ways can he choose $6$ of them to park his $6$ identical cars in, if no $3$ spaces with consecutive numbers may be all occupied?
*2018 CCA Math Bonanza Team Round #9* | 357 | 57.03125 |
24,731 | \(5, 6, 7\) | 21 | 0.78125 |
24,732 | Let \( a_0 = -3 \), \( b_0 = 2 \), and for \( n \geq 0 \), let:
\[
\begin{align*}
a_{n+1} &= 2a_n + 2b_n + 2\sqrt{a_n^2 + b_n^2}, \\
b_{n+1} &= 2a_n + 2b_n - 2\sqrt{a_n^2 + b_n^2}.
\end{align*}
\]
Find \( \frac{1}{a_{2023}} + \frac{1}{b_{2023}} \). | \frac{1}{3} | 0.78125 |
24,733 | From a deck of 32 cards which includes three colors (red, yellow, and blue) with each color having 10 cards numbered from $1$ to $10$, plus an additional two cards (a small joker and a big joker) both numbered $0$, a subset of cards is selected. The score for each card is calculated as $2^{k}$, where $k$ is the number on the card. If the sum of these scores equals $2004$, the subset is called a "good" hand. How many "good" hands are there?
(2004 National Girls' Olympiad problem) | 1006009 | 49.21875 |
24,734 | A four-digit number \((xyzt)_B\) is called a stable number in base \(B\) if \((xyzt)_B = (dcba)_B - (abcd)_B\), where \(a \leq b \leq c \leq d\) are the digits \(x, y, z, t\) arranged in ascending order. Determine all the stable numbers in base \(B\).
(Problem from the 26th International Mathematical Olympiad, 1985) | (1001)_2, (3021)_4, (3032)_5, (3B/5, B/5-1, 4B/5-1, 2B/5)_B, 5 | B | 0 |
24,735 | Let \(a\), \(b\), and \(c\) be positive real numbers such that \(a + b + c = 5.\) Find the minimum value of
\[
\frac{9}{a} + \frac{16}{b} + \frac{25}{c}.
\] | 30 | 0 |
24,736 | Given that 600 athletes are numbered from 001 to 600 and divided into three color groups (red: 001 to 311, white: 312 to 496, and yellow: 497 to 600), calculate the probability of randomly drawing an athlete wearing white clothing. | \frac{8}{25} | 0 |
24,737 | Henry walks $\tfrac{3}{4}$ of the way from his home to his gym, which is $2$ kilometers away from Henry's home, and then walks $\tfrac{3}{4}$ of the way from where he is back toward home. Determine the difference in distance between the points toward which Henry oscillates from home and the gym. | \frac{6}{5} | 0 |
24,738 | What is the greatest integer less than 150 for which the greatest common factor of that integer and 24 is 3? | 147 | 40.625 |
24,739 | Calculate the sum $2^{-2} + 2^{-3} + 2^{-4} + 2^{-5} + 2^{-6} + 2^{-7} \pmod{17}$.
Express your answer as an integer from $0$ to $16$, inclusive. | 10 | 4.6875 |
24,740 | Given a 12-hour digital clock with a glitch where every '2' is displayed as a '7', determine the fraction of the day that the clock shows the correct time. | \frac{55}{72} | 0.78125 |
24,741 | How many ten digit positive integers with distinct digits are multiples of $11111$ ? | 3456 | 72.65625 |
24,742 | Let $PQRST$ be a convex pentagon with $PQ \parallel RT, QR \parallel PS, QS \parallel PT, \angle PQR=100^\circ, PQ=4, QR=7,$ and $PT = 21.$ Given that the ratio between the area of triangle $PQR$ and the area of triangle $RST$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ | 232 | 0 |
24,743 | Find the area of the region enclosed by the graph of \( |x-75| + |y| = \left|\frac{x}{3}\right| \). | 703.125 | 7.03125 |
24,744 | A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines so that the folded flaps just touch at the corners as indicated by the dotted lines. Find the area of the resulting trapezoid. | 55/2 | 0 |
24,745 | Solve for $x$:
\[\arcsin 3x - \arccos (2x) = \frac{\pi}{6}.\] | -\frac{1}{\sqrt{7}} | 0 |
24,746 | For any $x \in (0, +\infty)$, the inequality $(x-a+\ln \frac{x}{a})(-2x^2+ax+10) \leq 0$ always holds. Then, the range of the real number $a$ is ______. | \sqrt{10} | 7.03125 |
24,747 | Consider a square flag with a red cross of uniform width and a blue triangular central region on a white background. The cross is symmetric with respect to each of the diagonals of the square. Let's say the entire cross, including the blue triangle, occupies 45% of the area of the flag. Calculate the percentage of the flag's area that is blue if the triangle is an equilateral triangle centered in the flag and the side length of the triangle is half the width of the red cross arms. | 1.08\% | 0 |
24,748 | A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting segments can be used as connections to gas stoves, given that regulations require the stove to be no closer than 75 cm to the main gas pipe. | 1/4 | 22.65625 |
24,749 | A rectangular garden measuring 60 feet by 20 feet is enclosed by a fence. In a redesign to maximize the area using the same amount of fencing, its shape is changed to a circle. How many square feet larger or smaller is the new garden compared to the old one? | 837.62 | 0 |
24,750 | If two numbers are randomly chosen without replacement from the set $\{6, 8, 9, 12\}$, what is the probability that their product will be a multiple of 36? Express your answer as a common fraction. | \frac{1}{3} | 1.5625 |
24,751 | The first operation divides the bottom-left square of diagram $\mathrm{a}$ into four smaller squares, as shown in diagram b. The second operation further divides the bottom-left smaller square of diagram b into four even smaller squares, as shown in diagram c; continuing this process, after the sixth operation, the resulting diagram will contain how many squares in total? | 29 | 0 |
24,752 | (from The Good Soldier Svejk) Senior military doctor Bautze exposed $abccc$ malingerers among $aabbb$ draftees who claimed not to be fit for the military service. He managed to expose all but one draftees. (He would for sure expose this one too, if the lucky guy was not taken by a stroke at the very moment when the doctor yelled at him "Turn around !. . . ") How many malingerers were exposed by the vigilant doctor?
Each digit substitutes a letter. The same digits substitute the same letters, while distinct digits substitute distinct letters.
*(1 point)* | 10999 | 6.25 |
24,753 | Find the largest \( n \) such that the sum of the fourth powers of any \( n \) prime numbers greater than 10 is divisible by \( n \). | 240 | 0 |
24,754 | In triangle $PQR,$ $S$ is on $\overline{PQ}$ such that $PS:SQ = 4:1,$ and $T$ is on $\overline{QR}$ such that $QT:TR = 4:1.$ If lines $ST$ and $PR$ intersect at $U,$ then find $\frac{ST}{TU}.$ | \frac{1}{3} | 0 |
24,755 | How many numbers should there be in a lottery for the probability of getting an ambo to be $\frac{5}{473}$, when drawing five numbers? | 44 | 6.25 |
24,756 | Find the positive integer $n$ such that \[ \underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1 \] where $f(n)$ denotes the $n$ th positive integer which is not a perfect square.
*Proposed by David Stoner* | 6077248 | 0 |
24,757 | For $k\ge 1$ , define $a_k=2^k$ . Let $$ S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right). $$ Compute $\lfloor 100S\rfloor$ . | 157 | 42.1875 |
24,758 | The area of each of Art's cookies is given by the formula for the area of a trapezoid: A = 1/2h(b1 + b2). The total area of dough used by Art to make a batch of cookies is 240 square inches, and this is equivalent to the number of cookies multiplied by the area of each cookie. Let n represent the number of cookies in a batch. The equation is thus expressed as: 240 = n * (1/2 * 4 * (4 + 6)). Solve for n. | 24 | 0 |
24,759 | From the consecutive natural numbers 1, 2, 3, …, 2014, select $n$ numbers such that for any two numbers chosen, one is not five times the other. Find the maximum value of $n$ and explain the reason. | 1665 | 0 |
24,760 | Is it possible to divide an equilateral triangle into 1,000,000 convex polygons such that any straight line intersects no more than 40 of them? | 1707 | 0 |
24,761 | The sequences where each term is a real number are denoted as $\left\{a_{n}\right\}$, with the sum of the first $n$ terms recorded as $S_{n}$. Given that $S_{10} = 10$ and $S_{30} = 70$, what is the value of $S_{40}$? | 150 | 0 |
24,762 | Twenty tiles are numbered 1 through 20 and are placed into box $C$. Twenty other tiles numbered 15 through 34 are placed into box $D$. One tile is randomly drawn from each box. What is the probability that the tile from box $C$ is less than 18 and the tile from box $D$ is either odd or greater than 30? Express your answer as a common fraction. | \frac{17}{40} | 0 |
24,763 | (1) In an arithmetic sequence $\{ a_n \}$, given that $a_1 + a_7 + a_{13} = 6$, find the value of $S_{13}$.
(2) Determine the interval of decrease for the function $y = \log_{\frac{1}{2}}(x^2 + 2x - 3)$.
(3) In triangle $ABC$, given $a = \sqrt{3}$, $b = 1$, and $\angle A = \frac{\pi}{3}$, find the value of $\cos B$.
(4) Point $A$ lies on circle $O$ with diameter $BC$ and is distinct from points $B$ and $C$. Point $P$ is outside the plane $ABC$ and plane $PBC$ is perpendicular to plane $ABC$. Given $BC = 3$, $PB = 2\sqrt{2}$, and $PC = \sqrt{5}$, find the surface area of the circumscribed sphere of the tetrahedron $P-ABC$. | 10\pi | 0.78125 |
24,764 | Let $M$ be the intersection of diagonals of the convex quadrilateral $ABCD$, where $m(\widehat{AMB})=60^\circ$. Let the points $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of the triangles $ABM$, $BCM$, $CDM$, $DAM$, respectively. Calculate the ratio of the area of quadrilateral $ABCD$ to the area of quadrilateral $O_1O_2O_3O_4$. | 3/2 | 0 |
24,765 | Given the equation $2x + 3k = 1$ with $x$ as the variable, if the solution for $x$ is negative, then the range of values for $k$ is ____. | \frac{1}{3} | 0 |
24,766 | Find the last two digits (in order) of \( 7^{30105} \). | 43 | 10.15625 |
24,767 | Alice, Bob, and Charlie each flip a fair coin repeatedly until they each flip heads. In a separate event, three more people, Dave, Eve, and Frank, each flip a biased coin (with a probability of $\frac{1}{3}$ of getting heads) until they first flip heads. Determine the probability that both groups will stop flipping their coins on the same round. | \frac{1}{702} | 0.78125 |
24,768 | A regular decagon $B_1B_2B_3B_4B_5B_6B_7B_8B_9B_{10}$ is inscribed in a circle of area $1$ square units. Point $Q$ lies inside the circle such that the region bounded by $\overline{QB_1},\overline{QB_2},$ and the minor arc $\widehat{B_1B_2}$ of the circle has area $\tfrac{1}{10},$ while the region bounded by $\overline{QB_3},\overline{QB_4},$ and the minor arc $\widehat{B_3B_4}$ of the circle has area $\tfrac{1}{12}.$ There is a positive integer $m$ such that the area of the region bounded by $\overline{QB_7},\overline{QB_8},$ and the minor arc $\widehat{B_7B_8}$ of the circle is equal to $\tfrac{1}{11}-\tfrac{\sqrt{3}}{m}.$ Find $m.$ | 110\sqrt{3} | 0 |
24,769 | Find the greatest root of the equation $g(x) = 21x^4 - 20x^2 + 3$. | \frac{\sqrt{21}}{7} | 8.59375 |
24,770 | There are three cards with numbers on both the front and back sides: one with 0 and 1, another with 2 and 3, and a third with 4 and 5. A student uses these cards to form a three-digit even number. How many different three-digit even numbers can the student make? | 16 | 3.90625 |
24,771 | A class has $25$ students. The teacher wants to stock $N$ candies, hold the Olympics and give away all $N$ candies for success in it (those who solve equally tasks should get equally, those who solve less get less, including, possibly, zero candies). At what smallest $N$ this will be possible, regardless of the number of tasks on Olympiad and the student successes? | 600 | 0 |
24,772 | Consider the function
\[ f(x) = \max \{-8x - 29, 3x + 2, 7x - 4\} \] defined for all real $x$. Let $q(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with $x$-coordinates $a_1$, $a_2$, $a_3$. Find $a_1 + a_2 + a_3$. | -\frac{163}{22} | 0 |
24,773 | The coordinates of vertex \( C(x, y) \) of triangle \( \triangle ABC \) satisfy the inequalities \( x^{2}+y^{2} \leq 8+2y \) and \( y \geq 3 \). The side \( AB \) is on the x-axis. Given that the distances from point \( Q(0,1) \) to the lines \( AC \) and \( BC \) are both 1, find the minimum area of \( \triangle ABC \). | 6 \sqrt{2} | 2.34375 |
24,774 | A larger square is constructed, and another square is formed inside it by connecting the midpoints of each side of the larger square. If the area of the larger square is 100, what is the area of the smaller square formed inside? | 25 | 0 |
24,775 | Find the value of \( k \) such that, for all real numbers \( a, b, \) and \( c \),
$$
(a+b)(b+c)(c+a) = (a+b+c)(ab + bc + ca) + k \cdot abc
$$ | -2 | 0.78125 |
24,776 | 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 | 0.78125 |
24,777 | Let $\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$ . Let $I$ be the center of the inscribed circle of $\triangle{ABC}$ . If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$ , then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$ , $b$ , and $c$ are positive integers. Find $a+b+c$ .
<span style="color:#00f">Note that this problem is null because a diagram is impossible.</span>
*Proposed by Andy Xu* | 622 | 0 |
24,778 |
The wizard revealed the secret of wisdom. For those who wanted to know precisely where the secret is hidden, he left a clue in his magic book:
\[ 5 \cdot \text{BANK} = 6 \cdot \text{SAD} \]
Each letter in this clue represents a certain digit. Find these digits and substitute them into the GPS coordinates:
\[ (\mathrm{C} \cdot 6+1) . (\mathrm{C} \cdot 6)(\mathrm{6} \cdot 6) \, \text{DB(K:2)(H-1)} \]
\[ (K: 2)(H-1) \cdot (\text{D}-1)(H-1)(\text{C} \cdot 9+\text{D}) \text{A(K:2)} \]
Where exactly is the secret revealed by the wizard hidden?
Each expression in parentheses and each letter represents a number. There are no symbols between them other than those explicitly stated.
The coordinate format is a decimal fraction with a dot as the separator. | 55.543065317 | 0 |
24,779 | How many distinct equilateral triangles can be constructed by connecting three different vertices of a regular dodecahedron? | 60 | 0.78125 |
24,780 | Polly has three circles cut from three pieces of colored card. She originally places them on top of each other as shown. In this configuration, the area of the visible black region is seven times the area of the white circle.
Polly moves the circles to a new position, as shown, with each pair of circles touching each other. What is the ratio between the areas of the visible black regions before and after? | 7:6 | 0 |
24,781 | Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $7, n,$ and $n+2$ cents, $120$ cents is the greatest postage that cannot be formed. | 43 | 0 |
24,782 | The picture shows several circles connected by segments. Tanya chooses a natural number \( n \) and places different natural numbers in the circles so that the following property holds for all these numbers:
If numbers \( a \) and \( b \) are not connected by a segment, then the sum \( a^2 + b^2 \) must be coprime with \( n \). If they are connected, then the numbers \( a^2 + b^2 \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists? | 65 | 25.78125 |
24,783 | What is the coefficient of $x^3$ when
$$x^5 - 4x^3 + 3x^2 - 2x + 5$$
is multiplied by
$$3x^2 - 2x + 4$$
and further multiplied by
$$1 - x$$
and the like terms are combined? | -3 | 0.78125 |
24,784 | Let \( p \) and \( q \) be the two distinct solutions to the equation
\[ (x-6)(3x+10) = x^2 - 19x + 50. \]
What is \( (p + 2)(q + 2) \)? | 108 | 0 |
24,785 | The Grunters play the Screamers 6 times. The Grunters have a 60% chance of winning any given game. If a game goes to overtime, the probability of the Grunters winning changes to 50%. There is a 10% chance that any game will go into overtime. What is the probability that the Grunters will win all 6 games, considering the possibility of overtime? | \frac{823543}{10000000} | 0 |
24,786 | The hypotenuse of a right triangle is $10$ inches and the radius of the inscribed circle is $1$ inch. The perimeter of the triangle in inches is: | 24 | 3.125 |
24,787 | Given that the function $f(x) = \frac{1}{2}\sin(\omega x + \varphi)$ ($\omega > 0, 0 < \varphi < \pi$) is an even function, and points P and Q are the highest and lowest points respectively on the graph of $y = f(x)$ such that $| \overrightarrow{PQ} | = \sqrt{2}$,
(1) Find the explicit formula of the function $f(x)$;
(2) In triangle $\triangle ABC$, where $a, b, c$ are the sides opposite to angles $A, B, C$ respectively, given that $a=1, b= \sqrt{2}$, and $f\left(\frac{A}{\pi}\right) = \frac{\sqrt{3}}{4}$, determine the size of angle $C$. | \frac{\pi}{12} | 22.65625 |
24,788 | A line $l$ passes through two points $P(-1,2)$ and $Q(2,-2)$, and intersects the hyperbola $(y-2)^{2}-x^{2}=1$ at two points $A$ and $B$.
$(1)$ Write the parametric equation of $l$ as required by the question;
$(2)$ Find the distance between the midpoint $M$ of $AB$ and point $P$. | 5 \sqrt {65} | 0 |
24,789 | (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} | 36.71875 |
24,790 | A lucky integer is a positive integer which is divisible by the sum of its digits. What is the least positive multiple of 6 that is not a lucky integer? | 114 | 1.5625 |
24,791 | Each of the squares of an $8 \times 8$ board can be colored white or black. Find the number of colorings of the board such that every $2 \times 2$ square contains exactly 2 black squares and 2 white squares. | 8448 | 0 |
24,792 | How many whole numbers between 1 and 2000 do not contain the digit 2? | 6560 | 0 |
24,793 | Given 100 real numbers, with their sum equal to zero. What is the minimum number of pairs that can be selected from them such that the sum of the numbers in each pair is non-negative? | 99 | 0 |
24,794 | Each brick in the pyramid contains one number. Whenever possible, the number in each brick is the least common multiple of the numbers of the two bricks directly above it.
What number could be in the bottom brick? Determine all possible options.
(Hint: What is the least common multiple of three numbers, one of which is a divisor of another?) | 2730 | 0 |
24,795 | The points \( K, L, M, N \) are the centers of the circles inscribed in the faces \( S A B, S A C, S B C, \) and \( A B C \) of the tetrahedron \( S A B C \). It is known that \( A B = S C = 5 \), \( A C = S B = 7 \), \( B C = S A = 8 \). Find the volume of the tetrahedron \( K L M N \). If necessary, round your answer to two decimal places. | 0.66 | 5.46875 |
24,796 | The carbon dioxide emissions in a certain region reach a peak of a billion tons (a > 0) and then begin to decline. The relationship between the carbon dioxide emissions S (in billion tons) and time t (in years) satisfies the function S = a · b^t. If after 7 years, the carbon dioxide emissions are (4a)/5 billion tons, determine the time it takes to achieve carbon neutrality, where the region offsets its own carbon dioxide emissions by (a)/4 billion tons. | 42 | 3.90625 |
24,797 | Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$ . She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors? | 44 | 84.375 |
24,798 | Let $ K$ be the curved surface obtained by rotating the parabola $ y \equal{} \frac {3}{4} \minus{} x^2$ about the $ y$ -axis.Cut $ K$ by the plane $ H$ passing through the origin and forming angle $ 45^\circ$ for the axis. Find the volume of the solid surrounded by $ K$ and $ H.$
Note that you are not allowed to use Double Integral for the problem. | \frac{9\pi}{256} | 0 |
24,799 | How many solutions of the equation $\tan x = \tan 2x$ are in the interval $0 \le x \le \tan^{-1} 500$? Assume $\tan \theta > \theta$ for $0 < \theta < \frac{\pi}{2}$. | 159 | 0 |
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