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40.3k
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100
|
|---|---|---|---|
9,700 |
Let \(ABC\) be a triangle such that \(\frac{|BC|}{|AB| - |BC|} = \frac{|AB| + |BC|}{|AC|}\). Determine the ratio \(\angle A : \angle C\).
|
1 : 2
| 18.75 |
9,701 |
Given a square \(ABCD\) with side length 1, determine the maximum value of \(PA \cdot PB \cdot PC \cdot PD\) where point \(P\) lies inside or on the boundary of the square.
|
\frac{5}{16}
| 0.78125 |
9,702 |
Determine the value of the expression $\sin 410^{\circ}\sin 550^{\circ}-\sin 680^{\circ}\cos 370^{\circ}$.
|
\frac{1}{2}
| 40.625 |
9,703 |
Let \( x, y, \) and \( z \) be real numbers such that \(\frac{4}{x} + \frac{2}{y} + \frac{1}{z} = 1\). Find the minimum of \( x + 8y + 4z \).
|
64
| 42.1875 |
9,704 |
Given a triangle \( ABC \), \( X \) and \( Y \) are points on side \( AB \), with \( X \) closer to \( A \) than \( Y \), and \( Z \) is a point on side \( AC \) such that \( XZ \) is parallel to \( YC \) and \( YZ \) is parallel to \( BC \). Suppose \( AX = 16 \) and \( XY = 12 \). Determine the length of \( YB \).
|
21
| 71.09375 |
9,705 |
There are 5 different types of books, with at least 3 copies of each. If we want to buy 3 books to gift to 3 students, with each student receiving one book, how many different ways are there to do this?
|
125
| 24.21875 |
9,706 |
Grisha has 5000 rubles. Chocolate bunnies are sold in a store at a price of 45 rubles each. To carry the bunnies home, Grisha will have to buy several bags at 30 rubles each. One bag can hold no more than 30 chocolate bunnies. Grisha bought the maximum possible number of bunnies and enough bags to carry all the bunnies. How much money does Grisha have left?
|
20
| 8.59375 |
9,707 |
A $1 \times 3$ rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?
A) $\frac{9\pi}{8}$
B) $\frac{12\pi}{8}$
C) $\frac{13\pi}{8}$
D) $\frac{15\pi}{8}$
E) $\frac{16\pi}{8}$
|
\frac{13\pi}{8}
| 26.5625 |
9,708 |
There are $10$ seats in each of $10$ rows of a theatre and all the seats are numbered. What is the probablity that two friends buying tickets independently will occupy adjacent seats?
|
\dfrac{1}{55}
| 28.90625 |
9,709 |
Let \( x, y \in \mathbf{R}^{+} \), and \(\frac{19}{x}+\frac{98}{y}=1\). Find the minimum value of \( x + y \).
|
117 + 14 \sqrt{38}
| 3.90625 |
9,710 |
Mrs. Thompson gives extra points on tests to her students with test grades that exceed the class median. Given that 101 students take the same test, what is the largest number of students who can be awarded extra points?
|
50
| 78.90625 |
9,711 |
In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( BC \), and \( N \) is the midpoint of line segment \( BM \). Given that \( \angle A = \frac{\pi}{3} \) and the area of \( \triangle ABC \) is \( \sqrt{3} \), find the minimum value of \( \overrightarrow{AM} \cdot \overrightarrow{AN} \).
|
\sqrt{3} + 1
| 12.5 |
9,712 |
In the arithmetic sequence $\{a_n\}$, $S_{10} = 4$, $S_{20} = 20$. What is $S_{30}$?
|
48
| 54.6875 |
9,713 |
A plane intersects a right circular cylinder of radius $2$ forming an ellipse. If the major axis of the ellipse is $30\%$ longer than the minor axis, what is the length of the major axis?
|
5.2
| 86.71875 |
9,714 |
Anton writes down all positive integers that are divisible by 2. Berta writes down all positive integers that are divisible by 3. Clara writes down all positive integers that are divisible by 4. The orderly Dora notes the numbers written by the others. She arranges these numbers in ascending order and does not write any number more than once. What is the 2017th number in her list?
|
3026
| 65.625 |
9,715 |
The numbers \(a_{1}, a_{2}, \ldots, a_{n}\) are such that the sum of any seven consecutive numbers is negative, and the sum of any eleven consecutive numbers is positive. What is the largest possible \(n\) for which this is true?
|
16
| 51.5625 |
9,716 |
Given real numbers $a$ and $b$ satisfying $ab=1$, and $a>b\geq \frac{2}{3}$, the maximum value of $\frac{a-b}{a^{2}+b^{2}}$ is \_\_\_\_\_\_.
|
\frac{30}{97}
| 3.125 |
9,717 |
In the number $2016^{* * * *} 02 * *$, each of the six asterisks must be replaced with any of the digits $0, 2, 4, 5, 7, 9$ (digits may be repeated) so that the resulting 12-digit number is divisible by 15. How many ways can this be done?
|
5184
| 37.5 |
9,718 |
How many three-digit multiples of 9 consist only of odd digits?
|
11
| 91.40625 |
9,719 |
A factory produces a certain type of component that undergoes two processes. The probability of producing a defective component in the first and second processes is 0.01 and 0.03, respectively. The production of defective components in each process is independent. What is the probability that the component is of acceptable quality after undergoing both processes? (Round to three decimal places)
|
0.960
| 60.15625 |
9,720 |
Angelica wants to choose a three-digit code for her suitcase lock. To make it easier to remember, Angelica wants all the digits in her code to be in non-decreasing order. How many different possible codes does Angelica have to choose from?
|
220
| 96.09375 |
9,721 |
Given a 3x3 matrix where each row and each column forms an arithmetic sequence, and the middle element $a_{22} = 5$, find the sum of all nine elements.
|
45
| 88.28125 |
9,722 |
Three friends, Rowan, Sara, and Tim, are playing a monetary game. Each starts with $3. A bell rings every 20 seconds, and with each ring, any player with money chooses one of the other two players independently at random and gives them $1. The game continues for 2020 rounds. What is the probability that at the end of the game, each player has $3?
A) $\frac{1}{8}$
B) $\frac{1}{4}$
C) $\frac{1}{3}$
D) $\frac{1}{2}$
|
\frac{1}{4}
| 43.75 |
9,723 |
During the process of choosing trial points using the 0.618 method, if the trial interval is $[3, 6]$ and the first trial point is better than the second, then the third trial point should be at _____.
|
5.292
| 0 |
9,724 |
Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$. Point $X$ is chosen on edge $A_{1} D_{1}$, and point $Y$ is chosen on edge $B C$. It is known that $A_{1} X=5$, $B Y=3$, and $B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects ray $D A$ at point $Z$. Find $D Z$.
|
20
| 1.5625 |
9,725 |
Given vectors $\overset{→}{a}=(\cos x,-1+\sin x)$ and $\overset{→}{b}=(2\cos x,\sin x)$,
(1) Express $\overset{→}{a}·\overset{→}{b}$ in terms of $\sin x$.
(2) Find the maximum value of $\overset{→}{a}·\overset{→}{b}$ and the corresponding value of $x$.
|
\frac{9}{4}
| 3.90625 |
9,726 |
A right trapezoid has an upper base that is 60% of the lower base. If the upper base is increased by 24 meters, it becomes a square. What was the original area of the right trapezoid in square meters?
|
2880
| 70.3125 |
9,727 |
Calculate $\cos \frac{\pi}{9} \cdot \cos \frac{2\pi}{9} \cdot \cos \frac{4\pi}{9} = $ ______.
|
\frac{1}{8}
| 77.34375 |
9,728 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$, respectively, and $C= \frac{3}{4}\pi$, $\sin A= \frac{\sqrt{5}}{5}$.
(I) Find the value of $\sin B$;
(II) If $c-a=5-\sqrt{10}$, find the area of $\triangle ABC$.
|
\frac{5}{2}
| 39.84375 |
9,729 |
In $\triangle ABC$, let $AB=6$, $BC=7$, $AC=4$, and $O$ be the incenter of $\triangle ABC$. If $\overrightarrow{AO} = p\overrightarrow{AB} + q\overrightarrow{AC}$, determine the value of $\frac{p}{q}$.
|
\frac{2}{3}
| 59.375 |
9,730 |
Let \( x, y, z \) be positive numbers that satisfy the following system of equations:
$$
\left\{\begin{array}{l}
x^{2}+x y+y^{2}=12 \\
y^{2}+y z+z^{2}=16 \\
z^{2}+x z+x^{2}=28
\end{array}\right.
$$
Find the value of the expression \( x y + y z + x z \).
|
16
| 90.625 |
9,731 |
The number \( x \) is such that \( \log _{2}\left(\log _{4} x\right) + \log _{4}\left(\log _{8} x\right) + \log _{8}\left(\log _{2} x\right) = 1 \). Find the value of the expression \( \log _{4}\left(\log _{2} x\right) + \log _{8}\left(\log _{4} x\right) + \log _{2}\left(\log _{8} x\right) \). If necessary, round your answer to the nearest 0.01.
|
0.87
| 12.5 |
9,732 |
The diameter \( AB \) and the chord \( CD \) intersect at point \( M \). Given that \( \angle CMB = 73^\circ \) and the angular measure of arc \( BC \) is \( 110^\circ \). Find the measure of arc \( BD \).
|
144
| 18.75 |
9,733 |
Given that $\sin(\alpha + \frac{\pi}{5}) = \frac{1}{3}$ and $\alpha$ is an obtuse angle, find the value of $\cos(\alpha + \frac{9\pi}{20})$.
|
-\frac{\sqrt{2} + 4}{6}
| 0.78125 |
9,734 |
Alice wants to write down a list of prime numbers less than 100, using each of the digits 1, 2, 3, 4, and 5 once and no other digits. Which prime number must be in her list?
|
41
| 8.59375 |
9,735 |
How many even integers are there between \( \frac{12}{3} \) and \( \frac{50}{2} \)?
|
10
| 0 |
9,736 |
Simplify the expression \(\left(\frac{2-n}{n-1}+4 \cdot \frac{m-1}{m-2}\right):\left(n^{2} \cdot \frac{m-1}{n-1}+m^{2} \cdot \frac{2-n}{m-2}\right)\) given that \(m=\sqrt[4]{400}\) and \(n=\sqrt{5}\).
|
\frac{\sqrt{5}}{5}
| 28.90625 |
9,737 |
Let $F_k(a,b)=(a+b)^k-a^k-b^k$ and let $S={1,2,3,4,5,6,7,8,9,10}$ . For how many ordered pairs $(a,b)$ with $a,b\in S$ and $a\leq b$ is $\frac{F_5(a,b)}{F_3(a,b)}$ an integer?
|
22
| 99.21875 |
9,738 |
In an eight-digit number, each digit (except the last one) is greater than the following digit. How many such numbers are there?
|
45
| 60.9375 |
9,739 |
Convert the binary number $110101_{(2)}$ to decimal.
|
53
| 96.09375 |
9,740 |
A bundle of wire was used in the following sequence:
- The first time, more than half of the total length was used, plus an additional 3 meters.
- The second time, half of the remaining length was used, minus 10 meters.
- The third time, 15 meters were used.
- Finally, 7 meters were left.
How many meters of wire were there originally in the bundle?
|
54
| 42.1875 |
9,741 |
The line $x+2ay-1=0$ is parallel to $\left(a-1\right)x+ay+1=0$, find the value of $a$.
|
\frac{3}{2}
| 92.1875 |
9,742 |
Find the number of natural numbers \( k \), not exceeding 333300, such that \( k^{2} - 2k \) is exactly divisible by 303.
|
4400
| 46.875 |
9,743 |
Determine the integer \( x \) for which the following equation holds true:
\[ 1 \cdot 2023 + 2 \cdot 2022 + 3 \cdot 2021 + \dots + 2022 \cdot 2 + 2023 \cdot 1 = 2023 \cdot 1012 \cdot x. \]
|
675
| 76.5625 |
9,744 |
Find the smallest natural number \( N \) that is divisible by \( p \), ends with \( p \), and has a digit sum equal to \( p \), given that \( p \) is a prime number and \( 2p+1 \) is a cube of a natural number.
|
11713
| 18.75 |
9,745 |
John learned that Lisa scored exactly 85 on the American High School Mathematics Examination (AHSME). Due to this information, John was able to determine exactly how many problems Lisa solved correctly. If Lisa's score had been any lower but still over 85, John would not have been able to determine this. What was Lisa's score? Remember, the AHSME consists of 30 multiple choice questions, and the score, $s$, is given by $s = 30 + 4c - w$, where $c$ is the number of correct answers, and $w$ is the number of wrong answers (no penalty for unanswered questions).
|
85
| 23.4375 |
9,746 |
How can you cut 50 cm from a string that is $2 / 3$ meters long without any measuring tools?
|
50
| 30.46875 |
9,747 |
In a toy store, there are large and small plush kangaroos. In total, there are 100 of them. Some of the large kangaroos are female kangaroos with pouches. Each female kangaroo has three small kangaroos in her pouch, and the other kangaroos have empty pouches. Find out how many large kangaroos are in the store, given that there are 77 kangaroos with empty pouches.
|
31
| 39.84375 |
9,748 |
Find all the ways in which the number 1987 can be written in another base as a three-digit number where the sum of the digits is 25.
|
19
| 82.03125 |
9,749 |
In an arithmetic sequence $\{a_n\}$ with a non-zero common difference, it is known that $a_1=4$ and $a_7^2=a_1a_{10}$. The sum of the first $n$ terms is $S_n$.
1. Find the general formula for the sequence $\{a_n\}$.
2. Find the maximum value of $S_n$ and the value of $n$ when the maximum is achieved.
|
26
| 3.90625 |
9,750 |
Calculate $\left(\frac{1}{3}\right)^{6} \div \left(\frac{2}{5}\right)^{-4} + \frac{1}{2}$.
|
\frac{455657}{911250}
| 11.71875 |
9,751 |
Three regular heptagons share a common center, and their sides are parallel. The sides of two heptagons are 6 cm and 30 cm, respectively. The third heptagon divides the area between the first two heptagons in a ratio of $1:5$, starting from the smaller heptagon. Find the side of the third heptagon.
|
6\sqrt{5}
| 79.6875 |
9,752 |
How many integers can be expressed as a sum of three distinct numbers if chosen from the set $\{4, 7, 10, 13, \ldots, 46\}$?
|
37
| 21.875 |
9,753 |
Given the planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, with magnitudes $|\overrightarrow{a}| = 1$ and $|\overrightarrow{b}| = 2$, and their dot product $\overrightarrow{a} \cdot \overrightarrow{b} = 1$. If $\overrightarrow{e}$ is a unit vector in the plane, find the maximum value of $|\overrightarrow{a} \cdot \overrightarrow{e}| + |\overrightarrow{b} \cdot \overrightarrow{e}|$.
|
\sqrt{7}
| 17.96875 |
9,754 |
Given the function \( f(x)=\left(1-x^{3}\right)^{-1 / 3} \), find \( f(f(f \ldots f(2018) \ldots)) \) where the function \( f \) is applied 2019 times.
|
2018
| 60.15625 |
9,755 |
a) Calculate the number of triangles whose three vertices are vertices of the cube.
b) How many of these triangles are not contained in a face of the cube?
|
32
| 59.375 |
9,756 |
A line passing through the left focus $F_1$ of a hyperbola at an inclination of 30° intersects with the right branch of the hyperbola at point $P$. If the circle with the diameter $PF_1$ just passes through the right focus of the hyperbola, determine the eccentricity of the hyperbola.
|
\sqrt{3}
| 17.96875 |
9,757 |
The maximum value of the function $y=\sin x \cos x + \sin x + \cos x$ is __________.
|
\frac{1}{2} + \sqrt{2}
| 35.9375 |
9,758 |
Calculate:<br/>$(1)-4^2÷(-32)×(\frac{2}{3})^2$;<br/>$(2)(-1)^{10}÷2+(-\frac{1}{2})^3×16$;<br/>$(3)\frac{12}{7}×(\frac{1}{2}-\frac{2}{3})÷\frac{5}{14}×1\frac{1}{4}$;<br/>$(4)1\frac{1}{3}×[1-(-4)^2]-(-2)^3÷\frac{4}{5}$.
|
-10
| 28.125 |
9,759 |
Juca has fewer than 800 marbles. He likes to separate the marbles into groups of the same size. He noticed that if he forms groups of 3 marbles each, exactly 2 marbles are left over. If he forms groups of 4 marbles, 3 marbles are left over. If he forms groups of 5 marbles, 4 marbles are left over. Finally, if he forms groups of 7 marbles each, 6 marbles are left over.
(a) If Juca formed groups of 20 marbles each, how many marbles would be left over?
(b) How many marbles does Juca have?
|
419
| 74.21875 |
9,760 |
Joey has 30 thin sticks, each stick has a length that is an integer from 1 cm to 30 cm. Joey first places three sticks on the table with lengths of 3 cm, 7 cm, and 15 cm, and then selects a fourth stick such that it, along with the first three sticks, forms a convex quadrilateral. How many different ways are there for Joey to make this selection?
|
17
| 3.90625 |
9,761 |
Let n rational numbers $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ satisfy $|x_{i}| \lt 1$ for $i=1,2,\ldots,n$, and $|x_{1}|+|x_{2}|+\ldots+|x_{n}|=19+|x_{1}+x_{2}+\ldots+x_{n}|$. Calculate the minimum value of $n$.
|
20
| 76.5625 |
9,762 |
The lines tangent to a circle with center $O$ at points $A$ and $B$ intersect at point $M$. Find the chord $AB$ if the segment $MO$ is divided by it into segments equal to 2 and 18.
|
12
| 21.09375 |
9,763 |
Given that \( \text{rem} \left(\frac{5}{7}, \frac{3}{4}\right) \) must be calculated, determine the value of the remainder.
|
\frac{5}{7}
| 78.90625 |
9,764 |
Let \(\left(x^{2}+2x-2\right)^{6}=a_{0}+a_{1}(x+2)+a_{2}(x+2)^{2}+\cdots+a_{12}(x+2)^{12}\), where \(a_{i} (i=0,1,2,\ldots,12)\) are real constants. Determine the value of \(a_{0}+a_{1}+2a_{2}+3a_{3}+\cdots+12a_{12}\).
|
64
| 15.625 |
9,765 |
The sum of two nonzero natural numbers is 210, and their least common multiple is 1547. What is their product? $\qquad$
|
10829
| 49.21875 |
9,766 |
In how many different ways can 7 different prizes be awarded to 5 students such that each student has at least one prize?
|
16800
| 96.09375 |
9,767 |
What is the mean of the set $\{m, m + 6, m + 8, m + 11, m + 18, m + 20\}$ if the median of the set is 19?
|
20
| 83.59375 |
9,768 |
Given the function $f(x) = x + \sin(\pi x) - 3$, study its symmetry center $(a, b)$ and find the value of $f\left( \frac {1}{2016} \right) + f\left( \frac {2}{2016} \right) + f\left( \frac {3}{2016} \right) + \ldots + f\left( \frac {4030}{2016} \right) + f\left( \frac {4031}{2016} \right)$.
|
-8062
| 38.28125 |
9,769 |
Let point $P$ be a moving point on the ellipse $x^{2}+4y^{2}=36$, and let $F$ be the left focus of the ellipse. The maximum value of $|PF|$ is _________.
|
6 + 3\sqrt{3}
| 53.125 |
9,770 |
\( \Delta ABC \) is an isosceles triangle with \( AB = 2 \) and \( \angle ABC = 90^{\circ} \). Point \( D \) is the midpoint of \( BC \) and point \( E \) is on \( AC \) such that the area of quadrilateral \( AEDB \) is twice the area of triangle \( ECD \). Find the length of \( DE \).
|
\frac{\sqrt{17}}{3}
| 60.15625 |
9,771 |
On a backpacking trip with 10 people, in how many ways can I choose 2 cooks and 1 medical helper if any of the 10 people may fulfill these roles?
|
360
| 80.46875 |
9,772 |
The diagram shows a triangle joined to a square to form an irregular pentagon. The triangle has the same perimeter as the square.
What is the ratio of the perimeter of the pentagon to the perimeter of the square?
|
3:2
| 0 |
9,773 |
Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 65 islanders gathered for a meeting. Each of them made the following statement in turn: "Among the statements made earlier, the true ones are exactly 20 less than the false ones." How many knights were present at this meeting?
|
23
| 12.5 |
9,774 |
From point $A$ outside a circle, a tangent and a secant are drawn to the circle. The distance from point $A$ to the point of tangency is 16, and the distance from point $A$ to one of the intersection points of the secant with the circle is 32. Find the radius of the circle if the distance from its center to the secant is 5.
|
13
| 45.3125 |
9,775 |
Given the polynomial $f(x) = 2x^7 + x^6 + x^4 + x^2 + 1$, calculate the value of $V_2$ using Horner's method when $x=2$.
|
10
| 25 |
9,776 |
The physical education teacher lined up the class so that everyone was facing him. There are 12 people to the right of Kolya, 20 people to the left of Sasha, and 8 people to the right of Sasha. How many people are to the left of Kolya?
|
16
| 71.09375 |
9,777 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $a=2$, $b= \sqrt {3}$, $B= \frac {\pi}{3}$, then $A=$ \_\_\_\_\_\_.
|
\frac{\pi}{2}
| 99.21875 |
9,778 |
Alice plays a game where she rolls a fair eight-sided die each morning. If Alice rolls a number divisible by 3, she wears red; otherwise, she wears blue. If she rolls a perfect square (1, 4), then she rolls again. In a leap year, what is the expected number of times Alice will roll her die?
|
488
| 71.09375 |
9,779 |
A store received apples of the first grade worth 228 rubles and apples of the second grade worth 180 rubles. During unloading, the apples got mixed up. Calculations showed that if all the apples are now sold at a price 90 kopeks lower than the price per kilogram of first grade apples, the planned revenue will be achieved. How many kilograms of apples were delivered, given that there were 5 kg more second grade apples than first grade apples?
|
85
| 25 |
9,780 |
A square sheet of paper with sides of length $10$ cm is initially folded in half horizontally. The folded paper is then folded diagonally corner to corner, forming a triangular shape. If this shape is then cut along the diagonal fold, what is the ratio of the perimeter of one of the resulting triangles to the perimeter of the original square?
A) $\frac{15 + \sqrt{125}}{40}$
B) $\frac{15 + \sqrt{75}}{40}$
C) $\frac{10 + \sqrt{50}}{40}$
D) $\frac{20 + \sqrt{100}}{40}$
|
\frac{15 + \sqrt{125}}{40}
| 57.8125 |
9,781 |
Let $\Delta ABC$ be an acute-angled triangle and let $H$ be its orthocentre. Let $G_1, G_2$ and $G_3$ be the centroids of the triangles $\Delta HBC , \Delta HCA$ and $\Delta HAB$ respectively. If the area of $\Delta G_1G_2G_3$ is $7$ units, what is the area of $\Delta ABC $ ?
|
63
| 96.875 |
9,782 |
A train travelling at constant speed takes five seconds to pass completely through a tunnel which is $85 \mathrm{~m}$ long, and eight seconds to pass completely through a second tunnel which is $160 \mathrm{~m}$ long. What is the speed of the train?
|
25
| 92.96875 |
9,783 |
Sequence $(a_n)$ is defined as $a_{n+1}-2a_n+a_{n-1}=7$ for every $n\geq 2$ , where $a_1 = 1, a_2=5$ . What is $a_{17}$ ?
|
905
| 41.40625 |
9,784 |
The bases $AB$ and $CD$ of the trapezoid $ABCD$ are equal to 101 and 20, respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors $\overrightarrow{AD}$ and $\overrightarrow{BC}$.
|
2020
| 62.5 |
9,785 |
In the sequence $\{a_n\}$, $a_1=2$, $a_2=5$, $a_{n+1}=a_{n+2}+a_{n}$, calculate the value of $a_6$.
|
-3
| 79.6875 |
9,786 |
In the parallelogram \(ABCD\), points \(E\) and \(F\) are located on sides \(AB\) and \(BC\) respectively, and \(M\) is the point of intersection of lines \(AF\) and \(DE\). Given that \(AE = 2BE\) and \(BF = 3CF\), find the ratio \(AM : MF\).
|
4:5
| 46.09375 |
9,787 |
Let $a$, $b$, and $c$ be the three roots of the equation $$
5x^3 + 505x + 1010 = 0.
$$ Find $(a + b)^3 + (b + c)^3 + (c + a)^3.$
|
606
| 91.40625 |
9,788 |
Given the polynomial with integer coefficients:
\[ f(x) = x^5 + a_1 x^4 + a_2 x^3 + a_3 x^2 + a_4 x + a_5 \]
If \( f(\sqrt{3} + \sqrt{2}) = 0 \) and \( f(1) + f(3) = 0 \), find \( f(-1) \).
|
24
| 35.9375 |
9,789 |
The construction rule of the sequence $\{a_{n}\}$ is as follows: $a_{1}=1$, if $a_{n}-2$ is a natural number and has not appeared before, then use the recursive formula $a_{n+1}=a_{n}-2$. Otherwise, use the recursive formula $a_{n+1}=3a_{n}$, then $a_{6}=$ _____.
|
15
| 82.03125 |
9,790 |
Teacher Li plans to buy 25 souvenirs for students from a store that has four types of souvenirs: bookmarks, postcards, notebooks, and pens, with 10 pieces available for each type (souvenirs of the same type are identical). Teacher Li intends to buy at least one piece of each type. How many different purchasing plans are possible? (Answer in numeric form.).
|
592
| 0 |
9,791 |
Given the scores $93$, $89$, $92$, $95$, $93$, $94$, $93$ of seven referees, determine the average value of the remaining data after removing the highest and lowest scores.
|
93
| 3.90625 |
9,792 |
Given the ellipse $$C: \frac {x^{2}}{4}+ \frac {y^{2}}{b^{2}}=1(0＜b＜2)$$, a straight line with a slope angle of $$\frac {3π}{4}$$ intersects the ellipse C at points A and B. The midpoint of the line segment AB is M, and O is the coordinate origin. The angle between $$\overrightarrow {OM}$$ and $$\overrightarrow {MA}$$ is θ, and |tanθ|=3. Find the value of b.
|
\sqrt{2}
| 9.375 |
9,793 |
Consider the set $\{8, -7, 2, -4, 20\}$. Find the smallest sum that can be achieved by adding three different numbers from this set.
|
-9
| 61.71875 |
9,794 |
Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters are there?
|
86
| 64.0625 |
9,795 |
Through how many squares does the diagonal of a 1983 × 999 chessboard pass?
|
2979
| 98.4375 |
9,796 |
Let the set \( S = \{1, 2, \cdots, 15\} \). Define \( A = \{a_{1}, a_{2}, a_{3}\} \) as a subset of \( S \), such that \( (a_{1}, a_{2}, a_{3}) \) satisfies \( 1 \leq a_{1} < a_{2} < a_{3} \leq 15 \) and \( a_{3} - a_{2} \leq 6 \). Find the number of such subsets that satisfy these conditions.
|
371
| 99.21875 |
9,797 |
Exactly half of the population of the island of Misfortune are hares, and the rest are rabbits. If a resident of Misfortune makes a statement, he sincerely believes what he says. However, hares are faithfully mistaken on average in one out of every four cases, and rabbits are faithfully mistaken on average in one out of every three cases. One day, a creature came to the center of the island and shouted, "I am not a hare!" Then he thought and sadly said, "I am not a rabbit." What is the probability that he is actually a hare?
|
27/59
| 89.0625 |
9,798 |
Little kids were eating candies. Each ate 11 candies less than the rest combined but still more than one candy. How many candies were eaten in total?
|
33
| 71.09375 |
9,799 |
Given that P is a point on the hyperbola $\frac{x^2}{4} - \frac{y^2}{3} = 1$, and $F_1$, $F_2$ are the two foci of the hyperbola. If $\angle F_1PF_2 = 60^\circ$, calculate the area of $\triangle PF_1F_2$.
|
3\sqrt{3}
| 84.375 |
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