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int64 0
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stringlengths 10
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31,800 |
Baron Munchausen told a story. "There were a whole crowd of us. We reached a crossroads. Then half of our group turned left, a third turned right, and a fifth went straight." "But wait, the Duke remarked, the sum of half, a third, and a fifth isn't equal to one, so you are lying!" The Baron replied, "I'm not lying, I'm rounding. For example, there are 17 people. I say that a third turned. Should one person split in your opinion? No, with rounding, six people turned. From whole numbers, the closest to the fraction $17 / 3$ is 6. And if I say that half of the 17 people turned, it means 8 or 9 people." It is known that Baron Munchausen never lies. What is the largest number of people that could have been in the crowd? | 37 | 93.75 |
31,801 | One dimension of a cube is increased by $2$, another is decreased by $2$, and the third is increased by $3$. The volume of the new rectangular solid is $7$ less than the volume of the cube. Find the original volume of the cube. | 27 | 23.4375 |
31,802 | Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, calculate the value of $m$. | -5 | 17.1875 |
31,803 | Let $\triangle ABC$ be equilateral with integer side length. Point $X$ lies on $\overline{BC}$ strictly between $B$ and $C$ such that $BX<CX$ . Let $C'$ denote the reflection of $C$ over the midpoint of $\overline{AX}$ . If $BC'=30$ , find the sum of all possible side lengths of $\triangle ABC$ .
*Proposed by Connor Gordon* | 130 | 0 |
31,804 | In triangle \( ABC \), angle \( B \) is right. The midpoint \( M \) is marked on side \( BC \), and there is a point \( K \) on the hypotenuse such that \( AB = AK \) and \(\angle BKM = 45^{\circ}\). Additionally, there are points \( N \) and \( L \) on sides \( AB \) and \( AC \) respectively, such that \( BC = CL \) and \(\angle BLN = 45^{\circ}\). In what ratio does point \( N \) divide the side \( AB \)? | 1:2 | 0.78125 |
31,805 | If triangle $PQR$ has sides of length $PQ = 8,$ $PR = 7,$ and $QR = 5,$ then calculate
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\] | \frac{7}{4} | 27.34375 |
31,806 | What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? | 16.67\% | 90.625 |
31,807 | The diagram shows the miles traveled by cyclists Clara and David. After five hours, how many more miles has Clara cycled than David?
[asy]
/* Modified AMC8 1999 #4 Problem */
draw((0,0)--(6,0)--(6,4.5)--(0,4.5)--cycle);
for(int x=0; x <= 6; ++x) {
for(real y=0; y <=4.5; y+=0.9) {
dot((x, y));
}
}
draw((0,0)--(5,3.6)); // Clara's line
draw((0,0)--(5,2.7)); // David's line
label(rotate(36)*"David", (3,1.35));
label(rotate(36)*"Clara", (3,2.4));
label(scale(0.75)*rotate(90)*"MILES", (-1, 2.25));
label(scale(0.75)*"HOURS", (3, -1));
label(scale(0.85)*"90", (0, 4.5), W);
label(scale(0.85)*"72", (0, 3.6), W);
label(scale(0.85)*"54", (0, 2.7), W);
label(scale(0.85)*"36", (0, 1.8), W);
label(scale(0.85)*"18", (0, 0.9), W);
label(scale(0.86)*"1", (1, 0), S);
label(scale(0.86)*"2", (2, 0), S);
label(scale(0.86)*"3", (3, 0), S);
label(scale(0.86)*"4", (4, 0), S);
label(scale(0.86)*"5", (5, 0), S);
label(scale(0.86)*"6", (6, 0), S);
[/asy] | 18 | 47.65625 |
31,808 | If an integer $a$ ($a \neq 1$) makes the solution of the linear equation in one variable $ax-3=a^2+2a+x$ an integer, then the sum of all integer roots of this equation is. | 16 | 50 |
31,809 | Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with its upper vertex at $(0,2)$ and an eccentricity of $\frac{\sqrt{5}}{3}$.
(1) Find the equation of ellipse $C$;
(2) From a point $P$ on the ellipse $C$, draw two tangent lines to the circle $x^{2}+y^{2}=1$, with the tangent points being $A$ and $B$. When the line $AB$ intersects the $x$-axis and $y$-axis at points $N$ and $M$, respectively, find the minimum value of $|MN|$. | \frac{5}{6} | 14.84375 |
31,810 | A set of 10 distinct integers $S$ is chosen. Let $M$ be the number of nonempty subsets of $S$ whose elements have an even sum. What is the minimum possible value of $M$ ?
<details><summary>Clarifications</summary>
- $S$ is the ``set of 10 distinct integers'' from the first sentence.
</details>
*Ray Li* | 511 | 79.6875 |
31,811 | Given the function $f(x)=\cos ωx \cdot \sin (ωx- \frac {π}{3})+ \sqrt {3}\cos ^{2}ωx- \frac{ \sqrt {3}}{4}(ω > 0,x∈R)$, and the distance from one symmetry center of the graph of the function $y=f(x)$ to the nearest symmetry axis is $\frac {π}{4}$.
(I) Find the value of $ω$ and the equation of the symmetry axis of $f(x)$;
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $f(A)= \frac { \sqrt {3}}{4}, \sin C= \frac {1}{3}, a= \sqrt {3}$, find the value of $b$. | \frac {3+2 \sqrt {6}}{3} | 0 |
31,812 | Find the constant term in the expansion of $\left(1+x+\frac{1}{x^{2}}\right)^{10}$. | 4351 | 59.375 |
31,813 | Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, determine the value of $m$. | -5 | 19.53125 |
31,814 | Let $a_{1}$, $a_{2}$, $a_{3}$, $\ldots$, $a_{n}$ be a geometric sequence with the first term $3$ and common ratio $3\sqrt{3}$. Find the smallest positive integer $n$ that satisfies the inequality $\log _{3}a_{1}-\log _{3}a_{2}+\log _{3}a_{3}-\log _{3}a_{4}+\ldots +(-1)^{n+1}\log _{3}a_{n} \gt 18$. | 25 | 17.96875 |
31,815 | Given the broadcast time of the "Midday News" program is from 12:00 to 12:30 and the news report lasts 5 minutes, calculate the probability that Xiao Zhang can watch the entire news report if he turns on the TV at 12:20. | \frac{1}{6} | 23.4375 |
31,816 | Given the set $\{1,2,3,5,8,13,21,34,55\}$, calculate the number of integers between 3 and 89 that cannot be written as the sum of exactly two elements of the set. | 53 | 0.78125 |
31,817 | If the function $f(x)=\tan (\omega x+ \frac {\pi}{4})$ ($\omega > 0$) has a minimum positive period of $2\pi$, then $\omega=$ ______ ; $f( \frac {\pi}{6})=$ ______ . | \sqrt {3} | 0 |
31,818 | Find the maximum value of the function $y=\sin^2x+3\sin x\cos x+4\cos^2x$ for $0 \leqslant x \leqslant \frac{\pi}{2}$ and the corresponding value of $x$. | \frac{\pi}{8} | 25 |
31,819 | Susan wants to determine the average and median number of candies in a carton. She buys 9 cartons of candies, opens them, and counts the number of candies in each one. She finds that the cartons contain 5, 7, 8, 10, 12, 14, 16, 18, and 20 candies. What are the average and median number of candies per carton? | 12 | 3.125 |
31,820 | In the Nanjing area, the weather in July and August is relatively hot. Xiaohua collected the highest temperature for ten consecutive days, obtaining the following set of data in sequence: 34, 35, 36, 34, 36, 37, 37, 36, 37, 37 (unit: ℃). The mode of this set of data is ▲, and the median is ▲. | 36 | 5.46875 |
31,821 | The Hoopers, coached by Coach Loud, have 15 players. George and Alex are the two players who refuse to play together in the same lineup. Additionally, if George plays, another player named Sam refuses to play. How many starting lineups of 6 players can Coach Loud create, provided the lineup does not include both George and Alex? | 3795 | 0.78125 |
31,822 | From the numbers 1, 2, 3, 5, 7, 8, two numbers are randomly selected and added together. Among the different sums that can be obtained, let the number of sums that are multiples of 2 be $a$, and the number of sums that are multiples of 3 be $b$. Then, the median of the sample 6, $a$, $b$, 9 is ____. | 5.5 | 1.5625 |
31,823 | Given $\alpha \in (0, \pi)$, if $\sin \alpha + \cos \alpha = \frac{\sqrt{3}}{3}$, calculate the value of $\cos^2 \alpha - \sin^2 \alpha$. | -\frac{\sqrt{5}}{3} | 24.21875 |
31,824 | In the Cartesian coordinate system $xOy$, point $P\left( \frac{1}{2},\cos^2\theta\right)$ is on the terminal side of angle $\alpha$, and point $Q(\sin^2\theta,-1)$ is on the terminal side of angle $\beta$, and $\overrightarrow{OP}\cdot \overrightarrow{OQ}=-\frac{1}{2}$.
$(1)$ Find $\cos 2\theta$;
$(2)$ Find the value of $\sin(\alpha+\beta)$. | -\frac{\sqrt{10}}{10} | 21.875 |
31,825 | Let \\(f(x)\\) be defined on \\((-∞,+∞)\\) and satisfy \\(f(2-x)=f(2+x)\\) and \\(f(7-x)=f(7+x)\\). If in the closed interval \\([0,7]\\), only \\(f(1)=f(3)=0\\), then the number of roots of the equation \\(f(x)=0\\) in the closed interval \\([-2005,2005]\\) is . | 802 | 64.0625 |
31,826 | What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? | 16.67\% | 86.71875 |
31,827 | Given the function $f(x) = \sin^2(wx) - \sin^2(wx - \frac{\pi}{6})$ ($x \in \mathbb{R}$, $w$ is a constant and $\frac{1}{2} < w < 1$), the graph of function $f(x)$ is symmetric about the line $x = \pi$.
(I) Find the smallest positive period of the function $f(x)$;
(II) In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a = 1$ and $f(\frac{3}{5}A) = \frac{1}{4}$, find the maximum area of $\triangle ABC$. | \frac{\sqrt{3}}{4} | 7.03125 |
31,828 | How many positive four-digit integers are divisible by both 7 and 6? | 215 | 54.6875 |
31,829 | There are 10 questions available, among which 6 are Type A questions and 4 are Type B questions. Student Xiao Ming randomly selects 3 questions to solve.
(1) Calculate the probability that Xiao Ming selects at least 1 Type B question.
(2) Given that among the 3 selected questions, there are 2 Type A questions and 1 Type B question, and the probability of Xiao Ming correctly answering each Type A question is $\dfrac{3}{5}$, and the probability of correctly answering each Type B question is $\dfrac{4}{5}$. Assuming the correctness of answers to different questions is independent, calculate the probability that Xiao Ming correctly answers at least 2 questions. | \dfrac{93}{125} | 23.4375 |
31,830 | Given that $F$ is the right focus of the hyperbola $C: x^{2}- \frac {y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \sqrt {6})$, when the perimeter of $\triangle APF$ is minimized, the ordinate of point $P$ is ______. | 2 \sqrt {6} | 0 |
31,831 | How many solutions does the equation
\[
\frac{(x-1)(x-2)(x-3)\dotsm(x-200)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-10^2)(x-11^3)(x-12^3)\dotsm(x-13^3)}
\]
have for \(x\)? | 190 | 31.25 |
31,832 | A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimum value of $n$ . | 17 | 46.09375 |
31,833 | A laptop is originally priced at $\$1200$. It is on sale for $15\%$ off. John applies two additional coupons: one gives $10\%$ off the discounted price, and another gives $5\%$ off the subsequent price. What single percent discount would give the same final price as these three successive discounts? | 27.325\% | 41.40625 |
31,834 | In triangle ABC below, find the length of side AB.
[asy]
unitsize(1inch);
pair A,B,C;
A = (0,0);
B = (1,0);
C = (0,1);
draw (A--B--C--A,linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$18\sqrt{2}$",C/2,W);
label("$45^\circ$",(0.7,0),N);
[/asy] | 18\sqrt{2} | 33.59375 |
31,835 | Convert the complex number \(1 + i \sqrt{3}\) into its exponential form \(re^{i \theta}\) and find \(\theta\). | \frac{\pi}{3} | 84.375 |
31,836 | In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam. | 13 | 18.75 |
31,837 | Arrange the digits \(1, 2, 3, 4, 5, 6, 7, 8, 9\) in some order to form a nine-digit number \(\overline{\text{abcdefghi}}\). If \(A = \overline{\text{abc}} + \overline{\text{bcd}} + \overline{\text{cde}} + \overline{\text{def}} + \overline{\text{efg}} + \overline{\text{fgh}} + \overline{\text{ghi}}\), find the maximum possible value of \(A\). | 4648 | 45.3125 |
31,838 | A circle with center O is tangent to the coordinate axes and to the hypotenuse of a $45^\circ$-$45^\circ$-$90^\circ$ triangle ABC, where AB = 2. Determine the exact radius of the circle. | 2 + \sqrt{2} | 14.0625 |
31,839 | If set $A=\{-4, 2a-1, a^2\}$, $B=\{a-5, 1-a, 9\}$, and $A \cap B = \{9\}$, then the value of $a$ is. | -3 | 23.4375 |
31,840 | Rolling a die twice, let the points shown the first and second times be \(a\) and \(b\) respectively. Find the probability that the quadratic equation \(x^{2} + a x + b = 0\) has two distinct real roots both less than -1. (Answer with a number). | 1/12 | 22.65625 |
31,841 | Given the functions $f(x)=x-\frac{1}{x}$ and $g(x)=2a\ln x$.
(1) When $a\geqslant -1$, find the monotonically increasing interval of $F(x)=f(x)-g(x)$;
(2) Let $h(x)=f(x)+g(x)$, and $h(x)$ has two extreme values $({{x}_{1}},{{x}_{2}})$, where ${{x}_{1}}\in (0,\frac{1}{3}]$, find the minimum value of $h({{x}_{1}})-h({{x}_{2}})$. | \frac{20\ln 3-16}{3} | 1.5625 |
31,842 | The value of $1.000 + 0.101 + 0.011 + 0.001$ is: | 1.113 | 75 |
31,843 | What is the sum of all the integers between -25.4 and 15.8, excluding the integer zero? | -200 | 3.90625 |
31,844 | Rhombus $ABCD$ has side length $3$ and $\angle B = 110$°. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?
**A)** $0.81$
**B)** $1.62$
**C)** $2.43$
**D)** $2.16$
**E)** $3.24$ | 2.16 | 17.1875 |
31,845 | The average yield per unit area of a rice variety for five consecutive years was 9.4, 9.7, 9.8, 10.3, and 10.8 (unit: t/hm²). Calculate the variance of this sample data. | 0.244 | 0 |
31,846 | Along the school corridor hangs a Christmas garland consisting of red and blue bulbs. Next to each red bulb, there must be a blue bulb. What is the maximum number of red bulbs that can be in this garland if there are a total of 50 bulbs? | 33 | 0 |
31,847 | Evaluate the following product of sequences: $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{2187} \cdot \frac{6561}{1}$. | 81 | 43.75 |
31,848 | In order to cultivate students' financial management skills, Class 1 of the second grade founded a "mini bank". Wang Hua planned to withdraw all the money from a deposit slip. In a hurry, the "bank teller" mistakenly swapped the integer part (the amount in yuan) with the decimal part (the amount in cents) when paying Wang Hua. Without counting, Wang Hua went home. On his way home, he spent 3.50 yuan on shopping and was surprised to find that the remaining amount of money was twice the amount he was supposed to withdraw. He immediately contacted the teller. How much money was Wang Hua supposed to withdraw? | 14.32 | 0 |
31,849 | A 9x9 chessboard has its squares labeled such that the label of the square in the ith row and jth column is given by $\frac{1}{2 \times (i + j - 1)}$. We need to select one square from each row and each column. Find the minimum sum of the labels of the nine chosen squares. | \frac{1}{2} | 34.375 |
31,850 | Let square $WXYZ$ have sides of length $8$. An equilateral triangle is drawn such that no point of the triangle lies outside $WXYZ$. Determine the maximum possible area of such a triangle. | 16\sqrt{3} | 20.3125 |
31,851 | Consider equilateral triangle $ABC$ with side length $1$ . Suppose that a point $P$ in the plane of the triangle satisfies \[2AP=3BP=3CP=\kappa\] for some constant $\kappa$ . Compute the sum of all possible values of $\kappa$ .
*2018 CCA Math Bonanza Lightning Round #3.4* | \frac{18\sqrt{3}}{5} | 0 |
31,852 | Given the hyperbola $C\_1$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively, and hyperbola $C\_2$: $\frac{x^2}{16} - \frac{y^2}{4} = 1$, determine the length of the major axis of hyperbola $C\_1$ given that point $M$ lies on one of the asymptotes of hyperbola $C\_1$, $OM \perp MF\_2$, and the area of $\triangle OMF\_2$ is $16$. | 16 | 42.1875 |
31,853 | As shown in the diagram, three circles intersect to create seven regions. Fill the integers $0 \sim 6$ into the seven regions such that the sum of the four numbers within each circle is the same. What is the maximum possible value of this sum? | 15 | 64.84375 |
31,854 | There are two types of electronic toy cars, Type I and Type II, each running on the same two circular tracks. Type I completes a lap every 5 minutes, while Type II completes a lap every 3 minutes. At a certain moment, both Type I and Type II cars start their 19th lap simultaneously. How many minutes earlier did the Type I car start running compared to the Type II car? | 36 | 69.53125 |
31,855 | In the polar coordinate system, circle $C$ is centered at point $C\left(2, -\frac{\pi}{6}\right)$ with a radius of $2$.
$(1)$ Find the polar equation of circle $C$;
$(2)$ Find the length of the chord cut from circle $C$ by the line $l$: $\theta = -\frac{5\pi}{12} (\rho \in \mathbb{R})$. | 2\sqrt{2} | 7.8125 |
31,856 | Xibing is a local specialty in Haiyang, with a unique flavor, symbolizing joy and reunion. Person A and person B went to the market to purchase the same kind of gift box filled with Xibing at the same price. Person A bought $2400$ yuan worth of Xibing, which was $10$ boxes less than what person B bought for $3000$ yuan.<br/>$(1)$ Using fractional equations, find the quantity of Xibing that person A purchased;<br/>$(2)$ When person A and person B went to purchase the same kind of gift box filled with Xibing again, they coincidentally encountered a store promotion where the unit price was $20$ yuan less per box compared to the previous purchase. Person A spent the same total amount on Xibing as before, while person B bought the same quantity as before. Then, the average unit price of Xibing for person A over the two purchases is ______ yuan per box, and for person B is ______ yuan per box (write down the answers directly). | 50 | 55.46875 |
31,857 | $30$ same balls are put into four boxes $A$, $B$, $C$, $D$ in such a way that sum of number of balls in $A$ and $B$ is greater than sum of in $C$ and $D$. How many possible ways are there? | 2600 | 0 |
31,858 | Perform the calculations:
3.21 - 1.05 - 1.95
15 - (2.95 + 8.37)
14.6 × 2 - 0.6 × 2
0.25 × 1.25 × 32 | 10 | 13.28125 |
31,859 | Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for all $n\ge 2$ . Define : $P_n=\left(1+\frac{1}{a_1}\right)...\left(1+\frac{1}{a_n}\right)$ Compute $\lim_{n\to \infty} P_n$ | e | 67.1875 |
31,860 | A sweater costs 160 yuan, it was first marked up by 10% and then marked down by 10%. Calculate the current price compared to the original. | 0.99 | 0.78125 |
31,861 | Determine the probability that two edges selected at random from the twelve edges of a cube with side length 1 are skew lines (i.e., non-intersecting and not in the same plane). | \frac{4}{11} | 20.3125 |
31,862 | Let the internal angles $A$, $B$, $C$ of $\triangle ABC$ be opposite to the sides $a$, $b$, $c$ respectively, and $c\cos B= \sqrt {3}b\sin C$.
$(1)$ If $a^{2}\sin C=4 \sqrt {3}\sin A$, find the area of $\triangle ABC$;
$(2)$ If $a=2 \sqrt {3}$, $b= \sqrt {7}$, and $c > b$, the midpoint of side $BC$ is $D$, find the length of $AD$. | \sqrt {13} | 0 |
31,863 | Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron each of whose edges measures 2 meters. A bug, starting from vertex $A$, follows the rule that at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. What is the probability that the bug is at vertex $A$ after crawling exactly 10 meters? | \frac{20}{81} | 0 |
31,864 | In $\triangle ABC$, it is known that $AB=2$, $AC=3$, and $A=60^{\circ}$.
$(1)$ Find the length of $BC$;
$(2)$ Find the value of $\sin 2C$. | \frac{4\sqrt{3}}{7} | 73.4375 |
31,865 | Given vectors $\overrightarrow{a}=(\sin x, \sqrt{3}\cos x)$, $\overrightarrow{b}=(-1,1)$, and $\overrightarrow{c}=(1,1)$, where $x \in [0, \pi]$.
(1) If $(\overrightarrow{a} + \overrightarrow{b}) \parallel \overrightarrow{c}$, find the value of $x$;
(2) If $\overrightarrow{a} \cdot \overrightarrow{b} = \frac{1}{2}$, find the value of the function $\sin \left(x + \frac{\pi}{6}\right)$. | \frac{\sqrt{15}}{4} | 57.03125 |
31,866 | Given a geometric sequence \(\{a_n\}\) with the sum of the first \(n\) terms \(S_n\) such that \(S_n = 2^n + r\) (where \(r\) is a constant), let \(b_n = 2(1 + \log_2 a_n)\) for \(n \in \mathbb{N}^*\).
1. Find the sum of the first \(n\) terms of the sequence \(\{a_n b_n\}\), denoted as \(T_n\).
2. If for any positive integer \(n\), the inequality \(\frac{1 + b_1}{b_1} \cdot \frac{1 + b_2}{b_2} \cdots \cdot \frac{1 + b_n}{b_n} \geq k \sqrt{n + 1}\) holds, determine \(k\). | \frac{3}{4} \sqrt{2} | 0 |
31,867 | In right triangle $XYZ$, we have $\angle Y = \angle Z$ and $XY = 8\sqrt{2}$. What is the area of $\triangle XYZ$? | 64 | 59.375 |
31,868 | Given the function $f(x)= \frac{x}{4} + \frac{a}{x} - \ln x - \frac{3}{2}$, where $a \in \mathbb{R}$, and the curve $y=f(x)$ has a tangent at the point $(1,f(1))$ which is perpendicular to the line $y=\frac{1}{2}x$.
(i) Find the value of $a$;
(ii) Determine the intervals of monotonicity and the extreme values for the function $f(x)$. | -\ln 5 | 28.125 |
31,869 | The line $2x+ay-2=0$ is parallel to the line $ax+(a+4)y-4=0$. Find the value of $a$. | -2 | 17.1875 |
31,870 | Given the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ passing through the point $E(\sqrt{3}, 1)$, with an eccentricity of $\frac{\sqrt{6}}{3}$, and $O$ as the coordinate origin.
(I) Find the equation of the ellipse $C$;
(II) If point $P$ is a moving point on the ellipse $C$, and the perpendicular bisector of segment $AP$, where $A(3, 0)$, intersects the $y$-axis at point $B$, find the minimum value of $|OB|$. | \sqrt{6} | 74.21875 |
31,871 | If the tangent line of the curve $y=\ln x$ at point $P(x_{1}, y_{1})$ is tangent to the curve $y=e^{x}$ at point $Q(x_{2}, y_{2})$, then $\frac{2}{{x_1}-1}+x_{2}=$____. | -1 | 21.875 |
31,872 | For any number $y$, define the operations $\&y = 2(7-y)$ and $\&y = 2(y-7)$. What is the value of $\&(-13\&)$? | 66 | 51.5625 |
31,873 | Given that the function $f(x) = x^3 + ax^2 + bx + a^2$ has an extreme value of 10 at $x = 1$, find the slope of the tangent to the function at $x = 2$. | 17 | 38.28125 |
31,874 | Given two vectors in space, $\overrightarrow{a} = (x - 1, 1, -x)$ and $\overrightarrow{b} = (-x, 3, -1)$. If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$, find the value of $x$. | -1 | 7.03125 |
31,875 | 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 | 19.53125 |
31,876 | The number of students in James' graduating class is greater than 100 but fewer than 200 and is 1 less than a multiple of 4, 2 less than a multiple of 5, and 3 less than a multiple of 6. How many students are in James' graduating class? | 183 | 4.6875 |
31,877 | Find the product of the three smallest prime factors of 180. | 30 | 55.46875 |
31,878 | A total of 6 letters are used to spell the English word "theer". Calculate the probability that the person spells this English word incorrectly. | \frac{59}{60} | 2.34375 |
31,879 | The number of elements in a finite set $P$ is denoted as $\text{card}(P)$. It is known that $\text{card}(M) = 10$, $A \subseteq M$, $B \subseteq M$, $A \cap B = \emptyset$, and $\text{card}(A) = 2$, $\text{card}(B) = 3$. If the set $X$ satisfies $A \subseteq X \subseteq M$, then the number of such sets $X$ is ____. (Answer with a number) | 256 | 78.125 |
31,880 | Given that out of 6 products, 2 are defective and the rest are qualified, calculate the probability of selecting exactly one defective product from these 6 products. | \frac{8}{15} | 92.1875 |
31,881 | Given the geometric sequence $\{a_{n}\}$, $3a_{5}-a_{3}a_{7}=0$. If $\{b_{n}\}$ is an arithmetic sequence where $b_{5}=a_{5}$, find the sum of the first 9 terms, $S_{9}$, of $\{b_{n}\}$. | 27 | 98.4375 |
31,882 | A sequence $(c_n)$ is defined as follows: $c_1 = 1$, $c_2 = \frac{1}{3}$, and
\[c_n = \frac{2 - c_{n-1}}{3c_{n-2}}\] for all $n \ge 3$. Find $c_{100}$. | \frac{1}{3} | 85.15625 |
31,883 | Cagney can frost a cupcake every 25 seconds and Lacey can frost a cupcake every 35 seconds. If Lacey spends the first minute exclusively preparing frosting and then both work together to frost, determine the number of cupcakes they can frost in 10 minutes. | 37 | 21.875 |
31,884 | According to Moor's Law, the number of shoes in Moor's room doubles every year. In 2013, Moor's room starts out having exactly one pair of shoes. If shoes always come in unique, matching pairs, what is the earliest year when Moor has the ability to wear at least 500 mismatches pairs of shoes? Note that left and right shoes are distinct, and Moor must always wear one of each. | 2018 | 30.46875 |
31,885 | The distance traveled by the center \( P \) of a circle with radius 1 as it rolls inside a triangle with side lengths 6, 8, and 10, returning to its initial position. | 12 | 5.46875 |
31,886 | Given the hyperbola with the equation $\frac{x^{2}}{4} - \frac{y^{2}}{9} = 1$, where $F\_1$ and $F\_2$ are its foci, and point $M$ lies on the hyperbola.
(1) If $\angle F\_1 M F\_2 = 90^{\circ}$, find the area of $\triangle F\_1 M F\_2$.
(2) If $\angle F\_1 M F\_2 = 60^{\circ}$, what is the area of $\triangle F\_1 M F\_2$? If $\angle F\_1 M F\_2 = 120^{\circ}$, what is the area of $\triangle F\_1 M F\_2$? | 3 \sqrt{3} | 24.21875 |
31,887 | What is the smallest positive integer $k$ such that the number $\textstyle\binom{2k}k$ ends in two zeros? | 13 | 99.21875 |
31,888 | Given a sequence $\{a_n\}$, where $a_{n+1} + (-1)^n a_n = 2n - 1$, calculate the sum of the first 12 terms of $\{a_n\}$. | 78 | 36.71875 |
31,889 | What is the product of the solutions of the equation $45 = -x^2 - 4x?$ | -45 | 59.375 |
31,890 | A certain clothing wholesale market sells a type of shirt, with a purchase price of $50$ yuan per shirt. It is stipulated that the selling price of each shirt is not lower than the purchase price. According to a market survey, the monthly sales volume $y$ (in units) and the selling price $x$ (in yuan) per unit satisfy a linear function relationship. Some of the data is shown in the table below:
| Selling Price $x$ (yuan/unit) | $60$ | $65$ | $70$ |
|-------------------------------|------|------|------|
| Sales Volume $y$ (units) | $1400$ | $1300$ | $1200$ |
$(1)$ Find the functional expression between $y$ and $x$; (no need to find the range of the independent variable $x$)<br/>$(2)$ The wholesale market wants to make a profit of $24000$ yuan from the sales of this type of shirt each month, while also providing customers with affordable prices. How should they price this type of shirt?<br/>$(3)$ The price department stipulates that the profit per shirt should not exceed $30\%$ of the purchase price. If the total monthly profit from this type of shirt is $w$ yuan, what price should be set to maximize the profit? What is the maximum profit? | 19500 | 57.03125 |
31,891 | Two identical cylindrical containers are connected at the bottom by a small tube with a tap. While the tap was closed, water was poured into the first container, and oil was poured into the second one, so that the liquid levels were the same and equal to $h = 40$ cm. At what level will the water in the first container settle if the tap is opened? The density of water is 1000 kg/$\mathrm{m}^3$, and the density of oil is 700 kg/$\mathrm{m}^3$. Neglect the volume of the connecting tube. Give the answer in centimeters. | 16.47 | 17.96875 |
31,892 | Consider the 100th, 101st, and 102nd rows of Pascal's triangle, denoted as sequences $(p_i)$, $(q_i)$, and $(r_i)$ respectively. Calculate:
\[
\sum_{i = 0}^{100} \frac{q_i}{r_i} - \sum_{i = 0}^{99} \frac{p_i}{q_i}.
\] | \frac{1}{2} | 50 |
31,893 | Given that $f(x)$ is a function defined on $\mathbb{R}$ with a period of $2$, in the interval $[1,3]$, $f(x)= \begin{cases}x+ \frac {a}{x}, & 1\leqslant x < 2 \\ bx-3, & 2\leqslant x\leqslant 3\end{cases}$, and $f( \frac {7}{2})=f(- \frac {7}{2})$, find the value of $15b-2a$. | 41 | 3.125 |
31,894 | Several points were marked on a line, and then two additional points were placed between each pair of neighboring points. This procedure was repeated once more with the entire set of points. Could there have been 82 points on the line as a result? | 10 | 0 |
31,895 | Given a circle O with a radius of 6, the length of chord AB is 6.
(1) Find the size of the central angle α corresponding to chord AB;
(2) Find the arc length l and the area S of the sector where α is located. | 6\pi | 89.0625 |
31,896 | In the diagram, $\mathrm{ABCD}$ is a right trapezoid with $\angle \mathrm{DAB} = \angle \mathrm{ABC} = 90^\circ$. A rectangle $\mathrm{ADEF}$ is constructed externally along $\mathrm{AD}$, with an area of 6.36 square centimeters. Line $\mathrm{BE}$ intersects $\mathrm{AD}$ at point $\mathrm{P}$, and line $\mathrm{PC}$ is then connected. The area of the shaded region in the diagram is: | 3.18 | 60.15625 |
31,897 | Consider the matrix
\[\mathbf{N} = \begin{pmatrix} 2x & -y & z \\ y & x & -2z \\ y & -x & z \end{pmatrix}\]
and it is known that $\mathbf{N}^T \mathbf{N} = \mathbf{I}$. Find $x^2 + y^2 + z^2$. | \frac{2}{3} | 10.9375 |
31,898 | Let $f$ be a quadratic function which satisfies the following condition. Find the value of $\frac{f(8)-f(2)}{f(2)-f(1)}$ .
For two distinct real numbers $a,b$ , if $f(a)=f(b)$ , then $f(a^2-6b-1)=f(b^2+8)$ . | 13 | 3.90625 |
31,899 | Given that $\binom{20}{13} = 77520$, $\binom{20}{14} = 38760$ and $\binom{18}{12} = 18564$, find $\binom{19}{13}$. | 27132 | 28.90625 |
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