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31,000 | Six equilateral triangles, each with side $4$, are arranged in a line such that the midpoint of the base of one triangle is the vertex of the next triangle. Calculate the area of the region of the plane that is covered by the union of the six triangular regions.
A) $16\sqrt{3}$
B) $19\sqrt{3}$
C) $24\sqrt{3}$
D) $18\sqrt{3}$
E) $20\sqrt{3}$ | 19\sqrt{3} | 46.09375 |
31,001 | Three positive reals $x , y , z $ satisfy $x^2 + y^2 = 3^2
y^2 + yz + z^2 = 4^2
x^2 + \sqrt{3}xz + z^2 = 5^2 .$
Find the value of $2xy + xz + \sqrt{3}yz$ | 24 | 65.625 |
31,002 | A positive integer $n$ not exceeding $100$ is chosen such that if $n\le 60$, then the probability of choosing $n$ is $q$, and if $n > 60$, then the probability of choosing $n$ is $2q$. Find the probability that a perfect square is chosen.
- **A)** $\frac{1}{35}$
- **B)** $\frac{2}{35}$
- **C)** $\frac{3}{35}$
- **D)** $\frac{4}{35}$
- **E)** $\frac{6}{35}$ | \frac{3}{35} | 56.25 |
31,003 | In the diagram, triangles $ABC$ and $CBD$ are isosceles with $\angle ABC = \angle BAC$ and $\angle CBD = \angle CDB$. The perimeter of $\triangle CBD$ is $18,$ the perimeter of $\triangle ABC$ is $24,$ and the length of $BD$ is $8.$ If $\angle ABC = \angle CBD$, find the length of $AB.$ | 14 | 0.78125 |
31,004 | Given an ellipse C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a＞b＞0)$$ with left and right foci $F_1$ and $F_2$, respectively. Point A is the upper vertex of the ellipse, $|F_{1}A|= \sqrt {2}$, and the area of △$F_{1}AF_{2}$ is 1.
(1) Find the standard equation of the ellipse.
(2) Let M and N be two moving points on the ellipse such that $|AM|^2+|AN|^2=|MN|^2$. Find the equation of line MN when the area of △AMN reaches its maximum value. | y=- \frac {1}{3} | 7.03125 |
31,005 | The shelf life $y$ (in hours) of a certain food product and its storage temperature $x$ (in °C) satisfy the function relationship $y=e^{kx+b}$ (where $e=2.718\ldots$ is the base of the natural logarithm, and $k$, $b$ are constants). It is known that the shelf life of this food product is 192 hours at 0°C, and 24 hours at 33°C.
(1) Find the value of $k$.
(2) Find the shelf life of this food product at 11°C and 22°C. | 48 | 89.84375 |
31,006 | Let $\sigma_1 : \mathbb{N} \to \mathbb{N}$ be a function that takes a natural number $n$ , and returns the sum of the positive integer divisors of $n$ . For example, $\sigma_1(6) = 1 + 2 + 3 + 6 = 12$ . What is the largest number n such that $\sigma_1(n) = 1854$ ?
| 1234 | 95.3125 |
31,007 | Given an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ (where $a>0$, $b>0$) with eccentricity $\frac{\sqrt{6}}{3}$, the distance from the origin O to the line passing through points A $(0, -b)$ and B $(a, 0)$ is $\frac{\sqrt{3}}{2}$. Further, the line $y=kx+m$ ($k \neq 0$, $m \neq 0$) intersects the ellipse at two distinct points C and D, and points C and D both lie on the same circle centered at A.
(1) Find the equation of the ellipse;
(2) When $k = \frac{\sqrt{6}}{3}$, find the value of $m$ and the area of triangle $\triangle ACD$. | \frac{5}{4} | 2.34375 |
31,008 | If $\frac{1}{4}$ of all ninth graders are paired with $\frac{1}{3}$ of all sixth graders, what fraction of the total number of sixth and ninth graders are paired? | \frac{2}{7} | 43.75 |
31,009 | A cuckoo clock is on the wall. At the beginning of every hour, the cuckoo makes a number of "cuckoo" sounds equal to the hour displayed by the hour hand (for example, at 19:00 the cuckoo makes 7 sounds). One morning, Maxim approached the clock when it showed 9:05. He started turning the minute hand until he moved the time forward by 7 hours. How many times did the cuckoo make a sound during this time? | 43 | 0.78125 |
31,010 | In a right triangle $PQR$ where $\angle R = 90^\circ$, the lengths of sides $PQ = 15$ and $PR = 9$. Find $\sin Q$ and $\cos Q$. | \frac{3}{5} | 0 |
31,011 | Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$ . A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers $m$ and $n$ , compute $100m+n$ .
*Proposed by Eugene Chen* | 106 | 11.71875 |
31,012 | Determine the number of ways to arrange the letters of the word "PERCEPTION". | 907200 | 3.90625 |
31,013 | Given the sequence $\{a_{n}\}$ satisfies $a_{1}=1$, $({{a}\_{n+1}}-{{a}\_{n}}={{(-1)}^{n+1}}\dfrac{1}{n(n+2)})$, find the sum of the first 40 terms of the sequence $\{(-1)^{n}a_{n}\}$. | \frac{20}{41} | 7.03125 |
31,014 | How many diagonals within a regular nine-sided polygon span an odd number of vertices between their endpoints? | 18 | 32.8125 |
31,015 | Given the hyperbola \( C_1: 2x^2 - y^2 = 1 \) and the ellipse \( C_2: 4x^2 + y^2 = 1 \). If \( M \) and \( N \) are moving points on the hyperbola \( C_1 \) and ellipse \( C_2 \) respectively, such that \( OM \perp ON \) and \( O \) is the origin, find the distance from the origin \( O \) to the line \( MN \). | \frac{\sqrt{3}}{3} | 1.5625 |
31,016 | Given $\sin 2α - 2 = 2\cos 2α$, find the value of $\sin^{2}α + \sin 2α$. | \frac{8}{5} | 20.3125 |
31,017 | If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 30 | 5.46875 |
31,018 | For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $12^{10}$? | 409 | 32.8125 |
31,019 | Given a fixed point A (3, 4), and point P is a moving point on the parabola $y^2=4x$, the distance from point P to the line $x=-1$ is denoted as $d$. Find the minimum value of $|PA|+d$. | 2\sqrt{5} | 0.78125 |
31,020 | In a certain book, there were 100 statements written as follows:
1) "In this book, there is exactly one false statement."
2) "In this book, there are exactly two false statements."
...
3) "In this book, there are exactly one hundred false statements."
Which of these statements is true? | 99 | 46.09375 |
31,021 | Samantha leaves her house at 7:15 a.m. to catch the school bus, starts her classes at 8:00 a.m., and has 8 classes that last 45 minutes each, a 40-minute lunch break, and spends an additional 90 minutes in extracurricular activities. If she takes the bus home and arrives back at 5:15 p.m., calculate the total time spent on the bus. | 110 | 47.65625 |
31,022 | For positive real numbers $a,$ $b,$ and $c,$ compute the maximum value of
\[\frac{abc(a + b + c)}{(a + b)^2 (b + c)^2}.\] | \frac{1}{4} | 2.34375 |
31,023 | Each segment with endpoints at the vertices of a regular 100-sided polygon is colored red if there is an even number of vertices between the endpoints, and blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices such that the sum of their squares equals 1, and the product of the numbers at the endpoints is allocated to each segment. Then, the sum of the numbers on the red segments is subtracted by the sum of the numbers on the blue segments. What is the maximum possible result? | 1/2 | 12.5 |
31,024 | Consider the number $99,\!999,\!999,\!999$ squared. Following a pattern observed in previous problems, determine how many zeros are in the decimal expansion of this number squared. | 10 | 7.03125 |
31,025 | In triangle $PQR$, $\angle Q=90^\circ$, $PQ=15$ and $QR=20$. Points $S$ and $T$ are on $\overline{PR}$ and $\overline{QR}$, respectively, and $\angle PTS=90^\circ$. If $ST=12$, then what is the length of $PS$? | 15 | 14.0625 |
31,026 | Find the smallest positive multiple of 9 that can be written using only the digits: (a) 0 and 1; (b) 1 and 2. | 12222 | 3.125 |
31,027 | Jason rolls four fair standard six-sided dice. He looks at the rolls and decides to either reroll all four dice or keep two and reroll the other two. After rerolling, he wins if and only if the sum of the numbers face up on the four dice is exactly $9.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
**A)** $\frac{7}{36}$
**B)** $\frac{1}{18}$
**C)** $\frac{2}{9}$
**D)** $\frac{1}{12}$
**E)** $\frac{1}{4}$ | \frac{1}{18} | 3.125 |
31,028 | Given the set \( A = \{0, 1, 2, 3, 4, 5, 6, 7\} \), how many mappings \( f \) from \( A \) to \( A \) satisfy the following conditions?
1. For all \( i, j \in A \) with \( i \neq j \), \( f(i) \neq f(j) \).
2. For all \( i, j \in A \) with \( i + j = 7 \), \( f(i) + f(j) = 7 \). | 384 | 32.03125 |
31,029 | In an organization with 200 employees, those over the age of 50 account for 20%, those aged 40-50 make up 30%, and those under 40 account for 50%. If 40 employees are to be sampled, and the systematic sampling method is used—where all employees are randomly numbered 1-200 and evenly divided into 40 groups (numbers 1-5, 6-10, ..., 196-200)—and if the number 22 is drawn from the 5th group, then the number drawn from the 8th group would be ①. If stratified sampling is employed, then the number of individuals to be drawn from the under-40 age group would be ②. The correct data for ① and ② are respectively ( ). | 20 | 73.4375 |
31,030 | Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$. | 172822 | 0 |
31,031 | Find the value of \( \cos (\angle OBC + \angle OCB) \) in triangle \( \triangle ABC \), where angle \( \angle A \) is an obtuse angle, \( O \) is the orthocenter, and \( AO = BC \). | -\frac{\sqrt{2}}{2} | 0 |
31,032 | In the final phase of a professional bowling competition, the top five players compete as follows: first, the fifth and fourth place players compete, and the loser gets the 5th place prize; the winner then competes with the third place player, and the loser gets the 4th place prize; the winner then competes with the second place player, and the loser gets the 3rd place prize; the winner then competes with the first place player, and the loser gets the 2nd place prize, and the winner gets the 1st place prize. How many different award sequences are possible? | 16 | 82.03125 |
31,033 | Below is a portion of the graph of a quadratic function, $y=p(x)=dx^2 + ex + f$:
The value of $p(12)$ is an integer. The graph's axis of symmetry is $x = 10.5$, and the graph passes through the point $(3, -5)$. Based on this, what is the value of $p(12)$? | -5 | 6.25 |
31,034 | Given a quadratic function in terms of \\(x\\), \\(f(x)=ax^{2}-4bx+1\\).
\\((1)\\) Let set \\(P=\\{1,2,3\\}\\) and \\(Q=\\{-1,1,2,3,4\\}\\), randomly pick a number from set \\(P\\) as \\(a\\) and from set \\(Q\\) as \\(b\\), calculate the probability that the function \\(y=f(x)\\) is increasing in the interval \\([1,+∞)\\).
\\((2)\\) Suppose point \\((a,b)\\) is a random point within the region defined by \\( \\begin{cases} x+y-8\\leqslant 0 \\\\ x > 0 \\\\ y > 0\\end{cases}\\), denote \\(A=\\{y=f(x)\\) has two zeros, one greater than \\(1\\) and the other less than \\(1\\}\\), calculate the probability of event \\(A\\) occurring. | \dfrac{961}{1280} | 0.78125 |
31,035 | Let $f(x) = |\lg(x+1)|$, where $a$ and $b$ are real numbers, and $a < b$ satisfies $f(a) = f(- \frac{b+1}{b+2})$ and $f(10a + 6b + 21) = 4\lg2$. Find the value of $a + b$. | - \frac{11}{15} | 19.53125 |
31,036 | Given a square \( PQRS \) with an area of \( 120 \, \text{cm}^2 \). Point \( T \) is the midpoint of \( PQ \). The ratios are given as \( QU: UR = 2:1 \), \( RV: VS = 3:1 \), and \( SW: WP = 4:1 \).
Find the area, in \(\text{cm}^2\), of quadrilateral \( TUVW \). | 67 | 5.46875 |
31,037 | Given a sequence $\{a_{n}\}$ that satisfies ${a}_{1}+3{a}_{2}+9{a}_{3}+⋯+{3}^{n-1}{a}_{n}=\frac{n+1}{3}$, where the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is denoted as $S_{n}$, find the minimum value of the real number $k$ such that $S_{n} \lt k$ holds for all $n$. | \frac{5}{6} | 10.9375 |
31,038 | Given that in $\triangle ABC$, $BD:DC = 3:2$ and $AE:EC = 3:4$, and the area of $\triangle ABC$ is 1, find the area of $\triangle BMD$. | \frac{4}{15} | 0 |
31,039 | In rhombus $ABCD$, $\angle BAD=60^{\circ}$, $\overrightarrow{DE}=\overrightarrow{EC}$, $AB=2$, calculate $\overrightarrow{AE}\cdot\overrightarrow{DB}$. | -1 | 58.59375 |
31,040 | In triangle $XYZ$, $E$ lies on $\overline{YZ}$ and $G$ lies on $\overline{XY}$. Let $\overline{XE}$ and $\overline{YG}$ intersect at $Q.$
If $XQ:QE = 5:2$ and $GQ:QY = 3:4$, find $\frac{XG}{GY}.$ | \frac{4}{3} | 24.21875 |
31,041 | Given that A and B can only take on the first three roles, and the other three volunteers (C, D, and E) can take on all four roles, calculate the total number of different selection schemes for four people from five volunteers. | 72 | 13.28125 |
31,042 | Given the parabola $C$: $y^{2}=2px$ with the focus at $F(2,0)$, and points $P(m,0)$ and $Q(-m,n)$, a line $l$ passing through $P$ with a slope of $k$ (where $k\neq 0$) intersects the parabola $C$ at points $A$ and $B$.
(Ⅰ) For $m=k=2$, if $\vec{QA} \cdot \vec{QB} = 0$, find the value of $n$.
(Ⅱ) If $O$ represents the origin and $m$ is constant, for any change in $k$ such that $\vec{OA} \cdot \vec{OB} = 0$ always holds, find the value of the constant $m$.
(Ⅲ) For $k=1$, $n=0$, and $m < 0$, find the maximum area of triangle $QAB$ as $m$ changes. | \frac{32\sqrt{3}}{9} | 13.28125 |
31,043 | If the average of a set of sample data 4, 5, 7, 9, $a$ is 6, then the variance $s^2$ of this set of data is \_\_\_\_\_\_. | \frac{16}{5} | 21.09375 |
31,044 | a) In how many ways can a rectangle $8 \times 2$ be divided into $1 \times 2$ rectangles?
b) Imagine and describe a shape that can be divided into $1 \times 2$ rectangles in exactly 555 ways. | 34 | 39.84375 |
31,045 | For any four-digit number $m$, if the digits of $m$ are all non-zero and distinct, and the sum of the units digit and the thousands digit is equal to the sum of the tens digit and the hundreds digit, then this number is called a "mirror number". If we swap the units digit and the thousands digit of a "mirror number" to get a new four-digit number $m_{1}$, and swap the tens digit and the hundreds digit to get another new four-digit number $m_{2}$, let $F_{(m)}=\frac{{m_{1}+m_{2}}}{{1111}}$. For example, if $m=1234$, swapping the units digit and the thousands digit gives $m_{1}=4231$, and swapping the tens digit and the hundreds digit gives $m_{2}=1324$, the sum of these two four-digit numbers is $m_{1}+m_{2}=4231+1324=5555$, so $F_{(1234)}=\frac{{m_{1}+m_{2}}}{{1111}}=\frac{{5555}}{{1111}}=5$. If $s$ and $t$ are both "mirror numbers", where $s=1000x+100y+32$ and $t=1500+10e+f$ ($1\leqslant x\leqslant 9$, $1\leqslant y\leqslant 9$, $1\leqslant e\leqslant 9$, $1\leqslant f\leqslant 9$, $x$, $y$, $e$, $f$ are all positive integers), define: $k=\frac{{F_{(s)}}}{{F_{(t)}}}$. When $F_{(s)}+F_{(t)}=19$, the maximum value of $k$ is ______. | \frac{{11}}{8} | 0 |
31,046 | The cube below has sides of length 5 feet. If a cylindrical section of radius 1 foot is removed from the solid at an angle of $45^\circ$ to the top face, what is the total remaining volume of the cube? Express your answer in cubic feet in terms of $\pi$. | 125 - 5\sqrt{2}\pi | 60.9375 |
31,047 | Let $N$ be the smallest positive integer $N$ such that $2008N$ is a perfect square and $2007N$ is a perfect cube. Find the remainder when $N$ is divided by $25$ . | 17 | 2.34375 |
31,048 | A city does not have electric lighting yet, so candles are used in houses at night. In João's house, one candle is used per night without burning it completely, and with four of these candle stubs, João makes a new candle. How many nights can João light up his house with 43 candles? | 57 | 47.65625 |
31,049 | Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit perfect square!”
Claire asks, “If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?”
Cat says, “Yes! Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn’t know my favorite number.”
Claire says, “Now I know your favorite number!” What is Cat’s favorite number? | 25 | 15.625 |
31,050 | Given that $\overset{→}{a\_n}=\left(\cos \frac{nπ}{6},\sin \frac{nπ}{6}\right)$, $n∈ℕ^∗$, $\overset{→}{b}=\left( \frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, calculate the value of $y={\left| \overset{→}{{a\_1}}+ \overset{→}{b}\right|}^{2}+{\left| \overset{→}{{a\_2}}+ \overset{→}{b}\right|}^{2}+···+{\left| \overset{→}{{a\_2015}}+ \overset{→}{b}\right|}^{2}$. | 4029 | 17.96875 |
31,051 | Solve for $x$: $0.04x + 0.05(25 + x) = 13.5$. | 136.\overline{1} | 0 |
31,052 | Two chess players, A and B, are in the midst of a match. Player A needs to win 2 more games to be the final winner, while player B needs to win 3 more games. If each player has a probability of $\frac{1}{2}$ to win any given game, then calculate the probability of player A becoming the final winner. | \frac{11}{16} | 14.84375 |
31,053 | Given points $A(-2, -3)$ and $B(5, 3)$ on the $xy$-plane, find the point $C(x, n)$ such that $AC + CB$ is minimized, where $x = 2$. Find the value of $n$.
A) $\frac{6}{7}$
B) $\frac{12}{7}$
C) $6.5$
D) $\frac{25}{6}$
E) $\frac{13}{2}$ | \frac{13}{2} | 3.90625 |
31,054 | A traffic light cycles as follows: green for 45 seconds, yellow for 5 seconds, then red for 50 seconds. Felix chooses a random five-second interval to observe the light. What is the probability that the color changes while he is observing? | \frac{3}{20} | 52.34375 |
31,055 | Given that the function $F(x) = f(x) + x^2$ is an odd function, and $f(2) = 1$, find $f(-2) = ( \ )$. | -9 | 100 |
31,056 | A circle passes through the midpoints of the hypotenuse $AB$ and the leg $BC$ of the right triangle $ABC$ and touches the leg $AC$. In what ratio does the point of tangency divide the leg $AC$?
| 1 : 3 | 21.09375 |
31,057 | From the $8$ vertices of a cube, choose any $4$ vertices. The probability that these $4$ points lie in the same plane is ______ (express the result as a simplified fraction). | \frac{6}{35} | 25.78125 |
31,058 |
A batch of disaster relief supplies is loaded into 26 trucks. The trucks travel at a constant speed of \( v \) kilometers per hour directly to the disaster area. If the distance between the two locations is 400 kilometers and the distance between every two trucks must be at least \( \left(\frac{v}{20}\right)^{2} \) kilometers, how many hours will it take to transport all the supplies to the disaster area? | 10 | 8.59375 |
31,059 | Suppose that the number $\sqrt{7200} - 61$ can be expressed in the form $(\sqrt a - b)^3,$ where $a$ and $b$ are positive integers. Find $a+b.$ | 21 | 1.5625 |
31,060 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$ and $|2\overrightarrow{a}+\overrightarrow{b}|=2\sqrt{3}$, find the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} | 92.96875 |
31,061 | Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he wrote the sum of the numbers written next to its row and its column ("addition table"). What is the maximum number of sums in this table that could end up being rational numbers? | 1250 | 7.03125 |
31,062 | The price of a bottle of "Komfort" fabric softener used to be 13.70 Ft, and half a capful was needed for 15 liters of water. The new composition of "Komfort" now costs 49 Ft, and 1 capful is needed for 8 liters of water. By what percentage has the price of the fabric softener increased? | 1240 | 0 |
31,063 | Given that $\{a_{n}\}$ is an arithmetic progression, $\{b_{n}\}$ is a geometric progression, and $a_{2}+a_{5}=a_{3}+9=8b_{1}=b_{4}=16$.
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$.
$(2)$ Arrange the terms of $\{a_{n}\}$ and $\{b_{n}\}$ in ascending order to form a new sequence $\{c_{n}\}$. Let the sum of the first $n$ terms of $\{c_{n}\}$ be denoted as $S_{n}$. If $c_{k}=101$, find the value of $k$ and determine $S_{k}$. | 2726 | 7.03125 |
31,064 | The distance from point P(1, -1) to the line $ax+3y+2a-6=0$ is maximized when the line passing through P is perpendicular to the given line. | 3\sqrt{2} | 13.28125 |
31,065 | Let's consider two fictional states, Sunland and Moonland, which have different license plate formats. Sunland license plates have the format LLDDLLL (where 'L' stands for a letter and 'D' for a digit), while Moonland license plates have the format LLLDDD. Assuming all 10 digits and all 26 letters are equally likely to appear in their respective positions, calculate how many more license plates can Sunland issue than Moonland. | 1170561600 | 2.34375 |
31,066 | In a positive term geometric sequence ${a_n}$, ${a_5 a_6 =81}$, calculate the value of ${\log_{3}{a_1} + \log_{3}{a_5} +...+\log_{3}{a_{10}}}$. | 20 | 3.90625 |
31,067 | 15. Let $a_{n}$ denote the number of ternary strings of length $n$ so that there does not exist a $k<n$ such that the first $k$ digits of the string equals the last $k$ digits. What is the largest integer $m$ such that $3^{m} \mid a_{2023}$ ? | 2022 | 0 |
31,068 | Let $f(x) = 2^x + 3^x$ . For how many integers $1 \leq n \leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \dots, f(n-1)$ ? | 11 | 46.09375 |
31,069 | Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) and the circle $x^2 + y^2 = a^2 + b^2$ in the first quadrant, find the eccentricity of the hyperbola, where $|PF_1| = 3|PF_2|$. | \frac{\sqrt{10}}{2} | 22.65625 |
31,070 | In the diagram, three lines meet at the points \( A, B \), and \( C \). If \( \angle ABC = 50^\circ \) and \( \angle ACB = 30^\circ \), the value of \( x \) is: | 80 | 7.8125 |
31,071 | Given a sequence $\{a_n\}$ that satisfies $a_1=1$ and $a_{n+1}=2S_n+1$, where $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, and $n \in \mathbb{N}^*$.
$(1)$ Find $a_n$.
$(2)$ If the sequence $\{b_n\}$ satisfies $b_n=\frac{1}{(1+\log_3{a_n})(3+\log_3{a_n})}$, and the sum of the first $n$ terms of $\{b_n\}$ is $T_n$, and for any positive integer $n$, $T_n < m$, find the minimum value of $m$. | \frac{3}{4} | 86.71875 |
31,072 | What is $4512_6 - 2324_6 + 1432_6$? Express your answer in base 6. | 4020_6 | 23.4375 |
31,073 | The isosceles trapezoid has base lengths of 24 units (bottom) and 12 units (top), and the non-parallel sides are each 12 units long. How long is the diagonal of the trapezoid? | 12\sqrt{3} | 83.59375 |
31,074 | What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{2005}+1\right)^{11/8}$? | 375 | 3.90625 |
31,075 | Given that a full circle is 800 clerts on Venus and is 360 degrees, calculate the number of clerts in an angle of 60 degrees. | 133.\overline{3} | 0 |
31,076 | Compute the following expression:
\[ 2(1+2(1+2(1+2(1+2(1+2(1+2(1+2))))))) \] | 510 | 2.34375 |
31,077 | Given vectors $\overrightarrow{a} = (\cos x, -\sqrt{3}\cos x)$ and $\overrightarrow{b} = (\cos x, \sin x)$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} + 1$.
(Ⅰ) Find the interval of monotonic increase for the function $f(x)$;
(Ⅱ) If $f(\theta) = \frac{5}{6}$, where $\theta \in \left( \frac{\pi}{3}, \frac{2\pi}{3} \right)$, find the value of $\sin 2\theta$. | \frac{2\sqrt{3} - \sqrt{5}}{6} | 40.625 |
31,078 | Mary typed a six-digit number, but the two $1$ s she typed didn't show. What appeared was $2002$ . How many different six-digit numbers could she have typed? | 15 | 39.0625 |
31,079 | Kanga labelled the vertices of a square-based pyramid using \(1, 2, 3, 4,\) and \(5\) once each. For each face, Kanga calculated the sum of the numbers on its vertices. Four of these sums equaled \(7, 8, 9,\) and \(10\). What is the sum for the fifth face?
A) 11
B) 12
C) 13
D) 14
E) 15 | 13 | 17.1875 |
31,080 | A certain shopping mall sells a batch of brand-name shirts, with an average daily sales of 20 pieces and a profit of $40 per piece. In order to expand sales and reduce inventory quickly, the mall decides to take appropriate price reduction measures. After investigation, it was found that if the price of each shirt is reduced by $1, the mall can sell 2 more pieces on average per day. Find:<br/>$(1)$ If the price of each shirt is reduced by $3, the average daily sales quantity will be ______ pieces;<br/>$(2)$ If the mall needs to make an average daily profit of $1200, how much should each shirt be reduced by? | 20 | 87.5 |
31,081 | Determine the number of ways to arrange the letters of the word "PERCEPTION". | 907200 | 0.78125 |
31,082 | Let sets \( A \) and \( B \) satisfy:
\[
A \cup B = \{1, 2, \cdots, 10\}, \quad A \cap B = \varnothing
\]
If the number of elements in set \( A \) is not an element of \( A \) and the number of elements in set \( B \) is not an element of \( B \), find the total number of different sets \( A \) that satisfy the conditions. | 186 | 39.84375 |
31,083 | A club has 12 members - 6 boys and 6 girls. Each member is also categorized either as a senior or junior with equal distribution among genders. Two of the members are chosen at random. What is the probability that they are both girls where one girl is a senior and the other is a junior? | \frac{9}{66} | 0 |
31,084 | Complex numbers $p,$ $q,$ and $r$ are zeros of a polynomial $P(z) = z^3 + sz + t,$ and $|p|^2 + |q|^2 + |r|^2 = 325.$ The points corresponding to $p,$ $q,$ and $r$ in the complex plane form a right triangle with right angle at $q.$ Find the square of the hypotenuse, $h^2,$ of this triangle. | 487.5 | 3.90625 |
31,085 | Given that \(\alpha\) and \(\beta\) are both acute angles, and
$$(1+\tan \alpha)(1+\tan \beta)=2.$$
Find \(\alpha + \beta =\). | \frac{\pi}{4} | 90.625 |
31,086 | Given that
\[
2^{-\frac{5}{3} + \sin 2\theta} + 2 = 2^{\frac{1}{3} + \sin \theta},
\]
compute \(\cos 2\theta.\) | -1 | 10.9375 |
31,087 | Given the function $y=\cos \left( 2x+\dfrac{\pi}{3} \right)$, determine the horizontal shift required to obtain its graph from the graph of the function $y=\cos 2x$. | \dfrac{\pi}{6} | 16.40625 |
31,088 | In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} \overset{x=t}{y=1+t}\end{cases}$$ (t is the parameter), line m is parallel to line l and passes through the coordinate origin, and the parametric equation of circle C is $$\begin{cases} \overset{x=1+cos\phi }{y=2+sin\phi }\end{cases}$$ (φ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis.
1. Find the polar coordinate equations of line m and circle C.
2. Suppose line m and circle C intersect at points A and B. Find the perimeter of △ABC. | 2+ \sqrt {2} | 0 |
31,089 | Maria bakes a $24$-inch by $30$-inch pan of brownies, and the brownies are cut into pieces that measure $3$ inches by $4$ inches. Calculate the total number of pieces of brownies the pan contains. | 60 | 25 |
31,090 | Given that the vision data recorded by the five-point recording method satisfies the equation $L=5+\lg V$, and a student's vision test data using the decimal recording method is $0.8$, calculate the student's vision data using the five-point recording method. | 4.9 | 0 |
31,091 | There are 3 male students and 2 female students. Calculate the total number of different arrangement methods under the following different requirements:
(1) All are arranged in a row where "A" can only be in the middle or on the sides. There are \_\_\_\_\_\_ ways to arrange this;
(2) All are arranged in a row where the male students must be grouped together. There are \_\_\_\_\_\_ ways to arrange this;
(3) All are arranged in a row where the male students cannot be grouped together. There are \_\_\_\_\_\_ ways to arrange this;
(4) All are arranged in a row where the order of "A", "B", "C" (three males) from left to right remains unchanged. There are \_\_\_\_\_\_ ways to arrange this;
(5) All are arranged in a row where "A" is not on the far left and "B" is not on the far right. There are \_\_\_\_\_\_ ways to arrange this;
(6) If one more female student is added, all are arranged in a row where males and females are not next to each other. There are \_\_\_\_\_\_ ways to arrange this;
(7) They are arranged in two rows, with 3 people in the front row and 2 people in the back row. There are \_\_\_\_\_\_ ways to arrange this;
(8) All are arranged in a row where there must be 1 person between "A" and "B". There are \_\_\_\_\_\_ ways to arrange this. | 36 | 19.53125 |
31,092 | Given that Tian Ji's top horse is faster than the King of Qi's middle horse, and Tian Ji's middle horse is faster than the King of Qi's bottom horse, but Tian Ji's top horse is slower than the King of Qi's top horse, and Tian Ji's bottom horse is slower than the King of Qi's bottom horse, calculate the probability that the King of Qi's horse wins when one horse is randomly selected from each side for a race. | \frac{2}{3} | 38.28125 |
31,093 | How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\times 5$ square array of dots? | 100 | 17.1875 |
31,094 | Given the parabola $y^{2}=2px\left(p \gt 0\right)$ with focus $F$ and the intersection point $E$ of its directrix with the $x$-axis, let line $l$ pass through $E$.<br/>$(1)$ If line $l$ is tangent to the parabola at point $M$, then $\angle EMF=$____.<br/>$(2)$ Let $p=6$. If line $l$ intersects the parabola at points $A$ and $B$, and $AB\bot BF$, then $|AF|-|BF|=$____. | 12 | 17.96875 |
31,095 | Let $\left\{a_n\right\}$ be an arithmetic sequence, and $S_n$ be the sum of its first $n$ terms, with $S_{11}= \frac{11}{3}\pi$. Let $\left\{b_n\right\}$ be a geometric sequence, and $b_4, b_8$ be the two roots of the equation $4x^2+100x+{\pi}^2=0$. Find the value of $\sin \left(a_6+b_6\right)$. | -\frac{1}{2} | 18.75 |
31,096 | Given a parabola $y=x^2+bx+c$ intersects the y-axis at point Q(0, -3), and the sum of the squares of the x-coordinates of the two intersection points with the x-axis is 15, find the equation of the function and its axis of symmetry. | \frac{3}{2} | 10.15625 |
31,097 | A 24-hour digital clock shows the time in hours and minutes. How many times in one day will it display all four digits 2, 0, 1, and 9 in some order?
A) 6
B) 10
C) 12
D) 18
E) 24 | 10 | 92.96875 |
31,098 | Given \( a_{0}=1, a_{1}=2 \), and \( n(n+1) a_{n+1}=n(n-1) a_{n}-(n-2) a_{n-1} \) for \( n=1, 2, 3, \ldots \), find \( \frac{a_{0}}{a_{1}}+\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{50}}{a_{51}} \). | 51 | 2.34375 |
31,099 | In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given that $b= \sqrt {2}$, $c=3$, $B+C=3A$.
(1) Find the length of side $a$;
(2) Find the value of $\sin (B+ \frac {3π}{4})$. | \frac{\sqrt{10}}{10} | 65.625 |
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