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29,900 | A biologist sequentially placed 150 beetles into ten jars. In each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is no less than half the number of beetles in the tenth jar. How many beetles are in the sixth jar? | 16 | 7.03125 |
29,901 | Let $ABCD$ be a convex quadrilateral such that $AB + BC = 2021$ and $AD = CD$ . We are also given that $\angle ABC = \angle CDA = 90^o$ . Determine the length of the diagonal $BD$ . | \frac{2021 \sqrt{2}}{2} | 31.25 |
29,902 | A trauma hospital uses a rectangular piece of white cloth that is 60m long and 0.8m wide to make triangular bandages with both base and height of 0.4m. How many bandages can be made in total? | 600 | 63.28125 |
29,903 | Maria baked 60 cakes, of which one-third contained strawberries, half contained blueberries, three-fifths contained raspberries, and one-tenth contained coconut flakes. What is the largest possible number of cakes that had none of these ingredients? | 24 | 6.25 |
29,904 | In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $\cos(B-C) - 2\sin B\sin C = -\frac{1}{2}$.
1. Find the measure of angle $A$.
2. When $a = 5$ and $b = 4$, find the area of $\triangle ABC$. | 2\sqrt{3} + \sqrt{39} | 19.53125 |
29,905 | If 1000 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? | 38 | 42.96875 |
29,906 | A geometric sequence of positive integers starts with the first term as 5, and the fifth term of the sequence is 320. Determine the second term of this sequence. | 10 | 58.59375 |
29,907 | Let $x_1< x_2 < x_3$ be the three real roots of the equation $\sqrt{100} x^3 - 210x^2 + 3 = 0$. Find $x_2(x_1+x_3)$. | 21 | 14.84375 |
29,908 | Tatiana's teacher drew a $3 \times 3$ grid on the board, with zero in each cell. The students then took turns to pick a $2 \times 2$ square of four adjacent cells, and to add 1 to each of the numbers in the four cells. After a while, the grid looked like the diagram on the right (some of the numbers in the cells have been rubbed out.) What number should be in the cell with the question mark?
A) 9
B) 16
C) 21
D) 29
E) 34 | 16 | 7.8125 |
29,909 | Given five letters a, b, c, d, and e arranged in a row, find the number of arrangements where both a and b are not adjacent to c. | 36 | 15.625 |
29,910 | Let $x$ be the number of points scored by the Sharks and $y$ be the number of points scored by the Eagles. It is given that $x + y = 52$ and $x - y = 6$. | 23 | 46.09375 |
29,911 | The numbers from 1 to 9 are placed in the cells of a $3 \times 3$ table such that the sum of the numbers on one diagonal is 7, and on the other diagonal is 21. What is the sum of the numbers in the five shaded cells?
 | 25 | 43.75 |
29,912 | A certain township responded to the call of "green waters and green mountains are mountains of gold and silver" and tailored the town to be a "ecological fruit characteristic town" according to local conditions. After investigation, it was found that the yield $W$ (unit: kg) of a certain fruit tree is related to the amount of fertilizer $x$ (unit: kg) applied as follows: $W(x)=\left\{{\begin{array}{l}{5({{x^2}+3}),0≤x≤2,}\\{\frac{{50x}}{{1+x}},2＜x≤5,}\end{array}}\right.$. The cost of fertilizer input is $10x$ yuan, and other cost inputs (such as cultivation management, fertilization, and labor costs) are $20x$ yuan. It is known that the market price of this fruit is approximately $15$ yuan/kg, and the sales are smooth and in short supply. Let the profit per tree of this fruit be $f\left(x\right)$ (unit: yuan).<br/>$(1)$ Find the relationship formula of the profit per tree $f\left(x\right)$ (yuan) with respect to the amount of fertilizer applied $x$ (kg);<br/>$(2)$ When the cost of fertilizer input is how much yuan, the maximum profit per tree of this fruit is obtained? What is the maximum profit? | 40 | 14.0625 |
29,913 | A positive integer $n$ is *funny* if for all positive divisors $d$ of $n$ , $d+2$ is a prime number. Find all funny numbers with the largest possible number of divisors. | 135 | 15.625 |
29,914 | A number composed of ten million, three hundred thousand, and fifty is written as \_\_\_\_\_\_, and this number is read as \_\_\_\_\_\_. | 10300050 | 26.5625 |
29,915 | Given three people, A, B, and C, they stand in a row in any random order. What is the probability that A stands in front of B and C does not stand in front of A? | \frac{1}{3} | 80.46875 |
29,916 | In the diagram, points \(B\), \(C\), and \(D\) lie on a line. Also, \(\angle ABC = 90^\circ\) and \(\angle ACD = 150^\circ\). The value of \(x\) is: | 60 | 55.46875 |
29,917 | Given two curves $y=x^{2}-1$ and $y=1-x^{3}$ have parallel tangents at point $x_{0}$, find the value of $x_{0}$. | -\dfrac{2}{3} | 13.28125 |
29,918 | Parallelogram $PQRS$ has vertices $P(4,4)$, $Q(-2,-2)$, $R(-8,-2)$, and $S(2,4)$. If a point is selected at random from the region determined by the parallelogram, what is the probability that the point lies below the $x$-axis? | \frac{1}{3} | 27.34375 |
29,919 | Given that $a_1, a_2, a_3, . . . , a_{99}$ is a permutation of $1, 2, 3, . . . , 99,$ find the maximum possible value of $$ |a_1 - 1| + |a_2 - 2| + |a_3 - 3| + \dots + |a_{99} - 99|. $$ | 4900 | 57.03125 |
29,920 | Let the function \( f(x) = x^3 + 3x^2 + 6x + 14 \), and \( f(a) = 1 \), \( f(b) = 19 \). Then \( a + b = \quad \). | -2 | 85.9375 |
29,921 | Let \( S = \{1, 2, \ldots, 98\} \). Find the smallest natural number \( n \) such that in any \( n \)-element subset of \( S \), it is always possible to select 10 numbers, and no matter how these 10 numbers are divided into two groups of five, there will always be a number in one group that is coprime with the other four numbers in the same group, and a number in the other group that is not coprime with the other four numbers in that group. | 50 | 21.875 |
29,922 | Given that $α \in (0,π)$, if $\sin α + \cos α = \frac{\sqrt{3}}{3}$, find the value of $\cos^2 α - \sin^2 α$. | \frac{\sqrt{5}}{3} | 24.21875 |
29,923 | Given that $F(c,0)$ is the right focus of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > b > 0)$, a line perpendicular to one asymptote of the hyperbola passes through $F$ and intersects the two asymptotes at points $A$ and $B$. $O$ is the coordinate origin, and the area of $\triangle OAB$ is $\frac{12a^2}{7}$. Find the eccentricity of the hyperbola $e = \frac{c}{a}$. | \frac{5}{4} | 19.53125 |
29,924 | Two congruent right circular cones each with base radius $5$ and height $12$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $4$ from the base of each cone. Determine the maximum possible value of the radius $r$ of a sphere that lies within both cones. | \frac{40}{13} | 5.46875 |
29,925 | Given the function $f(x)={e}^{-x}+ \frac{nx}{mx+n}$.
(1) If $m=0$, $n=1$, find the minimum value of the function $f(x)$;
(2) If $m > 0$, $n > 0$, and the minimum value of $f(x)$ on $[0,+\infty)$ is $1$, find the maximum value of $\frac{m}{n}$. | \frac{1}{2} | 2.34375 |
29,926 | Calculate the probability that four randomly chosen vertices of a cube form a tetrahedron (triangular pyramid). | \frac{29}{35} | 30.46875 |
29,927 | Represent the number 1000 as a sum of the maximum possible number of natural numbers, the sums of the digits of which are pairwise distinct. | 19 | 1.5625 |
29,928 | A line with slope $2$ passes through the focus $F$ of the parabola $y^2 = 2px$ $(p > 0)$ and intersects the parabola at points $A$ and $B$. The projections of $A$ and $B$ on the $y$-axis are $D$ and $C$ respectively. If the area of trapezoid $\triangle BCD$ is $6\sqrt{5}$, then calculate the value of $p$. | 2\sqrt{2} | 14.0625 |
29,929 | Given $f(x)=\sin \omega x \cos \omega x - \sqrt{3} \cos^2 \omega x$, where $(\omega > 0)$, ${x}_{1}$ and ${x}_{2}$ are the two zeros of the function $y=f(x)+\frac{2+\sqrt{3}}{2}$, and $|x_{1}-x_{2}|_{\min }=\pi$. When $x\in[0,\frac{7\pi}{12}]$, the sum of the minimum and maximum values of $f(x)$ is ______. | \frac{2-3\sqrt{3}}{2} | 0 |
29,930 | What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? | 16.\overline{6}\% | 0 |
29,931 | The focus of the parabola $y^{2}=4x$ is $F$, and the equation of the line $l$ is $x=ty+7$. Line $l$ intersects the parabola at points $M$ and $N$, and $\overrightarrow{MF}⋅\overrightarrow{NF}=0$. The tangents to the parabola at points $M$ and $N$ intersect at point $P$. Find the area of $\triangle PMN$. | 108 | 10.15625 |
29,932 | For a complex number $z \neq 3$ , $4$ , let $F(z)$ denote the real part of $\tfrac{1}{(3-z)(4-z)}$ . If \[
\int_0^1 F \left( \frac{\cos 2 \pi t + i \sin 2 \pi t}{5} \right) \; dt = \frac mn
\] for relatively prime positive integers $m$ and $n$ , find $100m+n$ .
*Proposed by Evan Chen* | 100 | 0.78125 |
29,933 | Let $PQRS$ be a convex pentagon, and let $H_P,$ $H_Q,$ $H_R,$ $H_S,$ and $H_T$ denote the centroids of the triangles $QRS,$ $RSP,$ $SPQ,$ $PQR,$ and $QRP$, respectively. Find $\frac{[H_PH_QH_RH_S]}{[PQRS]}$. | \frac{1}{9} | 82.03125 |
29,934 | Determine the largest value of \(x\) that satisfies the equation \(\sqrt{3x} = 5x^2\). Express your answer in simplest fractional form. | \left(\frac{3}{25}\right)^{\frac{1}{3}} | 7.8125 |
29,935 | Let $ABCD$ be a trapezoid of bases $AB$ and $CD$ . Let $O$ be the intersection point of the diagonals $AC$ and $BD$ . If the area of the triangle $ABC$ is $150$ and the area of the triangle $ACD$ is $120$ , calculate the area of the triangle $BCO$ . | \frac{200}{3} | 21.875 |
29,936 | In the diagram, \( PR \) and \( QS \) meet at \( X \). Also, \(\triangle PQX\) is right-angled at \(Q\) with \(\angle QPX = 62^\circ\) and \(\triangle RXS\) is isosceles with \( RX = SX \) and \(\angle XSR = y^\circ\). The value of \( y \) is: | 76 | 31.25 |
29,937 | In the ellipse $\dfrac {x^{2}}{36}+ \dfrac {y^{2}}{9}=1$, there are two moving points $M$ and $N$, and $K(2,0)$ is a fixed point. If $\overrightarrow{KM} \cdot \overrightarrow{KN} = 0$, find the minimum value of $\overrightarrow{KM} \cdot \overrightarrow{NM}$. | \dfrac{23}{3} | 23.4375 |
29,938 | What percentage of a seven-by-seven grid is shaded?
In a seven-by-seven grid, alternate squares are shaded starting with the top left square similar to a checkered pattern. However, an entire row (the fourth row from the top) and an entire column (the fourth column from the left) are left completely unshaded. | 73.47\% | 43.75 |
29,939 | Sally has six red cards numbered 1 through 6 and seven blue cards numbered 2 through 8. She attempts to create a stack where the colors alternate, and the number on each red card divides evenly into the number on each neighboring blue card. Can you determine the sum of the numbers on the middle three cards of this configuration?
A) 16
B) 17
C) 18
D) 19
E) 20 | 17 | 7.8125 |
29,940 | Leticia has a $9\times 9$ board. She says that two squares are *friends* is they share a side, if they are at opposite ends of the same row or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue or red. In each square a number will be written based on the following rules:
- If the square is green, write the number of red friends plus twice the number of blue friends.
- If the square is red, write the number of blue friends plus twice the number of green friends.
- If the square is blue, write the number of green friends plus twice the number of red friends.
Considering that Leticia can choose the coloring of the squares on the board, find the maximum possible value she can obtain when she sums the numbers in all the squares. | 486 | 24.21875 |
29,941 | The graph of $xy = 4$ is a hyperbola. Find the distance between the foci of this hyperbola. | 4\sqrt{2} | 32.03125 |
29,942 | A kindergarten received cards for learning to read: some are labeled "МА", and the rest are labeled "НЯ".
Each child took three cards and started to form words from them. It turned out that 20 children could form the word "МАМА" from their cards, 30 children could form the word "НЯНЯ", and 40 children could form the word "МАНЯ". How many children have all three cards the same? | 10 | 50 |
29,943 | A right rectangular prism, with edge lengths $\log_{5}x, \log_{6}x,$ and $\log_{8}x,$ must satisfy the condition that the sum of the squares of its face diagonals is numerically equal to 8 times the volume. What is $x?$
A) $24$
B) $36$
C) $120$
D) $\sqrt{240}$
E) $240$ | \sqrt{240} | 1.5625 |
29,944 | A chord $AB$ that makes an angle of $\frac{\pi}{6}$ with the horizontal passes through the left focus $F_1$ of the hyperbola $x^{2}- \frac{y^{2}}{3}=1$.
$(1)$ Find $|AB|$;
$(2)$ Find the perimeter of $\triangle F_{2}AB$ ($F_{2}$ is the right focus). | 3+3\sqrt{3} | 0 |
29,945 | Let $a$ be a real number. Find the minimum value of $\int_0^1 |ax-x^3|dx$ .
How many solutions (including University Mathematics )are there for the problem?
Any advice would be appreciated. :) | 1/8 | 61.71875 |
29,946 | The volume of a rectangular prism is 360 cubic units where \(a\), \(b\), and \(c\) are integers with \(1 < c < b < a\). Determine the largest possible value of \(b\). | 12 | 0 |
29,947 | Bill can buy mags, migs, and mogs for $\$3$, $\$4$, and $\$8$ each, respectively. What is the largest number of mogs he can purchase if he must buy at least one of each item and will spend exactly $\$100$? | 10 | 23.4375 |
29,948 | Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=10,$ find the smallest possible value of $|w^4+z^4|$. | 82 | 10.9375 |
29,949 | Given that $S_{n}$ is the sum of the first $n$ terms of the sequence ${a_{n}}$, and $S_{n}=n^{2}-4n+4$.
(1) Find the general term formula of the sequence ${a_{n}}$;
(2) Let ${c_{n}}$ be a sequence where all $c_{n}$ are non-zero, and the number of positive integers $k$ satisfying $c_{k}⋅c_{k+1} < 0$ is called the signum change number of this sequence. If $c_{n}=1- \frac {4}{a_{n}}$ ($n$ is a positive integer), find the signum change number of the sequence ${c_{n}}$;
(3) Let $T_{n}$ be the sum of the first $n$ terms of the sequence ${\frac {1}{a_{n}}}$. If $T_{2n+1}-T_{n}\leqslant \frac {m}{15}$ holds for all positive integers $n$, find the minimum value of the positive integer $m$. | 23 | 19.53125 |
29,950 | Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.303 | 0 |
29,951 | Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(0,3)$, the projection of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$ is ______. | \frac{6\sqrt{5}}{5} | 0 |
29,952 | Given the graph of $y = mx + 2$ passes through no lattice point with $0 < x \le 100$ for all $m$ such that $\frac{1}{2} < m < a$, find the maximum possible value of $a$. | \frac{50}{99} | 6.25 |
29,953 | There are 7 balls of each of the three colors: red, blue, and yellow. When randomly selecting 3 balls with different numbers, determine the total number of ways to pick such that the 3 balls are of different colors and their numbers are not consecutive. | 60 | 8.59375 |
29,954 | The intercept of the line $5x - 2y - 10 = 0$ on the x-axis is $a$, and on the y-axis is $b$. Find the values of $a$ and $b$. | -5 | 21.09375 |
29,955 | In rectangle ABCD, AB=2, BC=3, and points E, F, and G are midpoints of BC, CD, and AD, respectively. Point H is the midpoint of EF. What is the area of the quadrilateral formed by the points A, E, H, and G? | 1.5 | 0.78125 |
29,956 | If $S$, $H$, and $E$ are all distinct non-zero digits (each less than $6$) and the following is true, find the sum of the three values $S$, $H$, and $E$, expressing your answer in base $6$:
$$\begin{array}{c@{}c@{}c@{}c}
&S&H&E_6\\
&+&H&E_6\\
\cline{2-4}
&S&E&S_6\\
\end{array}$$ | 11_6 | 54.6875 |
29,957 | Given the function $f(x) = \sqrt{3}\sin x\cos x + \cos^2 x + a$.
(1) Find the smallest positive period and the monotonically increasing interval of $f(x)$;
(2) If the sum of the maximum and minimum values of $f(x)$ in the interval $[-\frac{\pi}{6}, \frac{\pi}{3}]$ is $1$, find the value of $a$. | a = -\frac{1}{4} | 17.96875 |
29,958 | Given a quadratic function $y=ax^{2}-4ax+3+b\left(a\neq 0\right)$.
$(1)$ Find the axis of symmetry of the graph of the quadratic function;
$(2)$ If the graph of the quadratic function passes through the point $\left(1,3\right)$, and the integers $a$ and $b$ satisfy $4 \lt a+|b| \lt 9$, find the expression of the quadratic function;
$(3)$ Under the conditions of $(2)$ and $a \gt 0$, when $t\leqslant x\leqslant t+1$ the function has a minimum value of $\frac{3}{2}$, find the value of $t$. | t = \frac{5}{2} | 3.125 |
29,959 | Given an isosceles triangle $PQR$ with $PQ=QR$ and the angle at the vertex $108^\circ$. Point $O$ is located inside the triangle $PQR$ such that $\angle ORP=30^\circ$ and $\angle OPR=24^\circ$. Find the measure of the angle $\angle QOR$. | 126 | 5.46875 |
29,960 | An isosceles right triangle with side lengths in the ratio 1:1:\(\sqrt{2}\) is inscribed in a circle with a radius of \(\sqrt{2}\). What is the area of the triangle and the circumference of the circle? | 2\pi\sqrt{2} | 21.875 |
29,961 | A machine can operate at different speeds, and some of the items it produces will have defects. The number of defective items produced per hour varies with the machine's operating speed. Let $x$ represent the speed (in revolutions per second), and $y$ represent the number of defective items produced per hour. Four sets of observations for $(x, y)$ are obtained as follows: $(8, 5)$, $(12, 8)$, $(14, 9)$, and $(16, 11)$. It is known that $y$ is strongly linearly correlated with $x$. If the number of defective items produced per hour is not allowed to exceed 10 in actual production, what is the maximum speed (in revolutions per second) the machine can operate at? (Round to the nearest integer)
Reference formula:
If $(x_1, y_1), \ldots, (x_n, y_n)$ are sample points, $\hat{y} = \hat{b}x + \hat{a}$,
$\overline{x} = \frac{1}{n} \sum\limits_{i=1}^{n}x_i$, $\overline{y} = \frac{1}{n} \sum\limits_{i=1}^{n}y_i$, $\hat{b} = \frac{\sum\limits_{i=1}^{n}(x_i - \overline{x})(y_i - \overline{y})}{\sum\limits_{i=1}^{n}(x_i - \overline{x})^2} = \frac{\sum\limits_{i=1}^{n}x_iy_i - n\overline{x}\overline{y}}{\sum\limits_{i=1}^{n}x_i^2 - n\overline{x}^2}$, $\hat{a} = \overline{y} - \hat{b}\overline{x}$. | 15 | 86.71875 |
29,962 | Given a fair die is thrown twice, and let the numbers obtained be denoted as a and b respectively, find the probability that the equation ax^2 + bx + 1 = 0 has real solutions. | \dfrac{19}{36} | 66.40625 |
29,963 | There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$ . | 3012 | 43.75 |
29,964 | All the complex roots of $(z + 1)^4 = 16z^4,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. | \frac{2}{3} | 5.46875 |
29,965 | The expression \( 29 \cdot 13 \) is a product of two integers greater than 1. When this product is divided by any integer \( k \) where \( 2 \leqslant k \leqslant 11 \), the remainder is 1. What is the difference between the two smallest such integers?
(29th Annual American High School Mathematics Examination, 1978) | 27720 | 59.375 |
29,966 | The sum of the first fifty positive odd integers subtracted from the sum of the first fifty positive even integers, each decreased by 3, calculate the result. | -100 | 0 |
29,967 | In triangle $ABC$ , $\angle ACB=50^{\circ}$ and $\angle CBA=70^{\circ}$ . Let $D$ be a foot of perpendicular from point $A$ to side $BC$ , $O$ circumcenter of $ABC$ and $E$ antipode of $A$ in circumcircle $ABC$ . Find $\angle DAE$ | 30 | 45.3125 |
29,968 | Knowing the horizontal component of the tension force, find the minimum force acting on a buoy from the current.
Consider the forces acting on the buoy along the horizontal axis: the force from the current acts in one direction, and the horizontal component of the tension force acts in the opposite direction.
\[M_5 \cdot a_{\sigma_{-} x} = T \cdot \sin(a) - F_{\text{tey_rop}}\]
where \(M_{\text{Б}}\) is the mass of the buoy, \(a_x\) is its acceleration along the horizontal axis, and \(\mathrm{F}_{\text{теч_го}}\) is the force acting on the buoy from the current in the horizontal direction.
Therefore, at the moment the anchor is lifted, the following equality is satisfied:
\[\mathrm{F}_{\text{Tey_rop}} = T \cdot \sin(a)\]
Hence, the minimum value of the force acting on the buoy from the current is equal to its horizontal projection, which is
\[\mathrm{F}_{\text{теч}} = \mathrm{F}_{\text{теч_гор}} = 400 \text{ N}\] | 400 | 49.21875 |
29,969 | When a swing is stationary, with the footboard one foot off the ground, pushing it forward two steps (in ancient times, one step was considered as five feet) equals 10 feet, making the footboard of the swing the same height as a person who is five feet tall, determine the length of the rope when pulled straight at this point. | 14.5 | 0 |
29,970 | Find \(\cos \frac{\alpha - \beta}{2}\), given \(\sin \alpha + \sin \beta = -\frac{27}{65}\), \(\tan \frac{\alpha + \beta}{2} = \frac{7}{9}\), \(\frac{5}{2} \pi < \alpha < 3 \pi\) and \(-\frac{\pi}{2} < \beta < 0\). | \frac{27}{7 \sqrt{130}} | 0 |
29,971 | Given the function $y=f(x)$, for any $x \in \mathbb{R}$, it satisfies $f(x+2)=\frac{1}{f(x)}$. When $x \in (0, 2]$, $f(x)=x$.
(1) Find the analytical expression of $f(x)$ when $x \in (2, 4]$;
(2) If $f(m)=1$, find the value of $m$;
(3) Calculate the sum: $f(1)+f(2)+f(3)+...+f(2015)$. | \frac{4535}{2} | 7.03125 |
29,972 | Three squares, with side-lengths 2, are placed together edge-to-edge to make an L-shape. The L-shape is placed inside a rectangle so that all five vertices of the L-shape lie on the rectangle, one of them at the midpoint of an edge, as shown.
What is the area of the rectangle?
A 16
B 18
C 20
D 22
E 24 | 24 | 67.96875 |
29,973 | In a corridor 100 meters long, 20 carpet strips with a total length of 1000 meters are laid down. What could be the maximum number of uncovered sections (the width of the carpet strip is equal to the width of the corridor)? | 10 | 3.125 |
29,974 | Given:
\\((1)y=x+ \\frac {4}{x}\\)
\\((2)y=\\sin x+ \\frac {4}{\\sin x}(0 < x < π)\\)
\\((3)y= \\frac {x^{2}+13}{ \\sqrt {x^{2}+9}}\\)
\\((4)y=4⋅2^{x}+2^{-x}\\)
\\((5)y=\\log \_{3}x+4\\log \_{x}3(0 < x < 1)\\)
Find the function(s) with a minimum value of $4$. (Fill in the correct question number) | (4) | 2.34375 |
29,975 | Suppose point P is on the curve $y=x^2+1$ (where $x \geq 0$), and point Q is on the curve $y=\sqrt{x-1}$ (where $x \geq 1$). Then the minimum value of the distance $|PQ|$ is ______. | \frac{3\sqrt{2}}{4} | 17.96875 |
29,976 | Consider all non-empty subsets of the set \( S = \{1, 2, \cdots, 10\} \). A subset is called a "good subset" if the number of even numbers in the subset is not less than the number of odd numbers. How many "good subsets" are there? | 637 | 0 |
29,977 | The diagonal $KM$ of trapezoid $KLMN$ is 3 times the length of segment $KP$ on this diagonal. The base $KN$ of the trapezoid is 3 times the length of the base $LM$. Find the ratio of the area of trapezoid $KLMN$ to the area of triangle $KPR$, where $R$ is the point of intersection of line $PN$ and side $KL$. | 32/3 | 0 |
29,978 | Given that $\{a_n\}$ is an arithmetic sequence, if $a_3 + a_5 + a_{12} - a_2 = 12$, calculate the value of $a_7 + a_{11}$. | 12 | 65.625 |
29,979 | Two subsets of the set $T = \{w, x, y, z, v\}$ need to be chosen so that their union is $T$ and their intersection contains exactly three elements. How many ways can this be accomplished, assuming the subsets are chosen without considering the order? | 20 | 54.6875 |
29,980 | Determine the number of ways to arrange the letters of the word "PERCEPTION". | 907,200 | 0 |
29,981 | The bases \( AB \) and \( CD \) of the trapezoid \( ABCD \) are 41 and 24 respectively, and its diagonals are mutually perpendicular. Find the scalar (dot) product of the vectors \( \overrightarrow{AD} \) and \( \overrightarrow{BC} \). | 984 | 51.5625 |
29,982 | The device consists of three independently operating elements. The probabilities of failure-free operation of the elements (over time $t$) are respectively: $p_{1}=0.7$, $p_{2}=0.8$, $p_{3}=0.9$. Find the probabilities that over time $t$, the following will occur:
a) All elements operate without failure;
b) Two elements operate without failure;
c) One element operates without failure;
d) None of the elements operate without failure. | 0.006 | 86.71875 |
29,983 | Given a hyperbola with eccentricity $2$ and equation $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, the right focus $F_2$ of the hyperbola is the focus of the parabola $y^2 = 8x$. A line $l$ passing through point $F_2$ intersects the right branch of the hyperbola at two points $P$ and $Q$. $F_1$ is the left focus of the hyperbola. If $PF_1 \perp QF_1$, then find the slope of line $l$. | \dfrac{3\sqrt{7}}{7} | 1.5625 |
29,984 | How many positive 3-digit numbers are multiples of 30, but not of 75? | 24 | 32.03125 |
29,985 | After the implementation of the "double reduction" policy, schools have attached importance to extended services and increased the intensity of sports activities in these services. A sports equipment store seized the opportunity and planned to purchase 300 sets of table tennis rackets and badminton rackets for sale. The number of table tennis rackets purchased does not exceed 150 sets. The purchase price and selling price are as follows:
| | Purchase Price (元/set) | Selling Price (元/set) |
|------------|-------------------------|------------------------|
| Table Tennis Racket | $a$ | $50$ |
| Badminton Racket | $b$ | $60$ |
It is known that purchasing 2 sets of table tennis rackets and 1 set of badminton rackets costs $110$ yuan, and purchasing 4 sets of table tennis rackets and 3 sets of badminton rackets costs $260$ yuan.
$(1)$ Find the values of $a$ and $b$.
$(2)$ Based on past sales experience, the store decided to purchase a number of table tennis rackets not less than half the number of badminton rackets. Let $x$ be the number of table tennis rackets purchased, and $y$ be the profit obtained after selling these sports equipment.
① Find the functional relationship between $y$ and $x$, and write down the range of values for $x$.
② When the store actually makes purchases, coinciding with the "Double Eleven" shopping festival, the purchase price of table tennis rackets is reduced by $a$ yuan per set $(0 < a < 10)$, while the purchase price of badminton rackets remains the same. Given that the selling price remains unchanged, and all the sports equipment can be sold out, how should the store make purchases to maximize profit? | 150 | 23.4375 |
29,986 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and form a geometric sequence with common ratio $r$. Additionally, it is given that $2c - 4a = 0$. Express $\cos B$ in terms of $a$ and $r$. | \dfrac {3}{4} | 91.40625 |
29,987 | Walking is a form of exercise that falls between walking and racewalking. It is a simple and safe aerobic exercise that can enhance lung capacity and promote heart health. A sports physiologist conducted a large number of surveys on the body fat percentage ($X$) of people engaged in walking activities and found that the body fat percentage of the survey respondents follows a normal distribution $N(0.2, \sigma^2)$. It is defined that people with a body fat percentage less than or equal to $0.17$ have a good body shape. If $16\%$ of the people participating in walking have a good body shape, then the value of $\sigma$ is approximately ____.<br/>Reference data: $P(\mu -\sigma \leq X \leq \mu + \sigma) \approx 0.6827$, $P(\mu -2\sigma \leq X \leq \mu + 2\sigma) \approx 0.9545$. | 0.03 | 7.8125 |
29,988 | Consider the curve $y=x^{n+1}$ (where $n$ is a positive integer) and its tangent at the point (1,1). Let the x-coordinate of the intersection point between this tangent and the x-axis be $x_n$.
(Ⅰ) Let $a_n = \log{x_n}$. Find the value of $a_1 + a_2 + \ldots + a_9$.
(Ⅱ) Define $nf(n) = x_n$. Determine whether there exists a largest positive integer $m$ such that the inequality $f(n) + f(n+1) + \ldots + f(2n-1) > \frac{m}{24}$ holds for all positive integers $n$. If such an $m$ exists, find its value; if not, explain why. | 11 | 30.46875 |
29,989 | In a right triangle $\triangle STU$, where $\angle S = 90^\circ$, suppose $\sin T = \frac{3}{5}$. If the length of $SU$ is 15, find the length of $ST$. | 12 | 0.78125 |
29,990 | Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\angle Q$ and $\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\triangle PQR$. | 40 | 0.78125 |
29,991 | Translate the graph of $y= \sqrt{2}\sin(2x+ \frac{\pi}{3})$ to the right by $\varphi$ ($0<\varphi<\pi$) units to obtain the graph of the function $y=2\sin x(\sin x-\cos x)-1$. Find the value of $\varphi$. | \frac{13\pi}{24} | 70.3125 |
29,992 | Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} | 97.65625 |
29,993 | It is known that there is a four-person team consisting of boys A and B and girls C and D participating in a quiz activity organized by a TV station. The activity consists of four rounds. Let the probabilities for the boys to pass rounds one to four be $\frac{5}{6}, \frac{4}{5}, \frac{3}{4}, \frac{2}{3}$, respectively, and the probabilities for the girls to pass rounds one to four be $\frac{4}{5}, \frac{3}{4}, $\frac{2}{3}, \frac{1}{2}$, respectively.
(1) Find the probability that a boy passes all four rounds;
(2) Let $\xi$ represent the number of people in the four-person team who pass all four rounds. Find the distribution and the expectation of the random variable $\xi$. | \frac{16}{15} | 12.5 |
29,994 | Suppose the quadratic function $f(x)=ax^2+bx+c$ has a maximum value $M$ and a minimum value $m$ in the interval $[-2,2]$, and the set $A={x|f(x)=x}$.
(1) If $A={1,2}$ and $f(0)=2$, find the values of $M$ and $m$.
(2) If $A={2}$ and $a\geqslant 1$, let $g(a)=M+m$, find the minimum value of $g(a)$. | \frac{63}{4} | 30.46875 |
29,995 | Given that $f(x)$ is an odd function on $\mathbb{R}$, when $x\geqslant 0$, $f(x)= \begin{cases} \log _{\frac {1}{2}}(x+1),0\leqslant x < 1 \\ 1-|x-3|,x\geqslant 1\end{cases}$. Find the sum of all the zeros of the function $y=f(x)+\frac {1}{2}$. | \sqrt {2}-1 | 0 |
29,996 | Given the hyperbola $C: \frac{x^{2}}{4} - \frac{y^{2}}{3} = 1$, with its right vertex at $P$.
(1) Find the standard equation of the circle centered at $P$ and tangent to both asymptotes of the hyperbola $C$;
(2) Let line $l$ pass through point $P$ with normal vector $\overrightarrow{n}=(1,-1)$. If there are exactly three points $P_{1}$, $P_{2}$, and $P_{3}$ on hyperbola $C$ with the same distance $d$ to line $l$, find the value of $d$. | \frac{3\sqrt{2}}{2} | 3.125 |
29,997 | In $\triangle ABC$, $AB = AC = 3.6$, point $D$ lies on $AB$ with $AD = 1.2$, and point $E$ lies on the extension of $AC$. The area of $\triangle AED$ is equal to the area of $\triangle ABC$. Find the length of $AE$. | 10.8 | 23.4375 |
29,998 | Cyclic pentagon \( A B C D E \) has a right angle \( \angle A B C = 90^\circ \) and side lengths \( A B = 15 \) and \( B C = 20 \). Supposing that \( A B = D E = E A \), find \( C D \). | 20 | 29.6875 |
29,999 | Given two lines $l_1: y=a$ and $l_2: y= \frac {18}{2a+1}$ (where $a>0$), $l_1$ intersects the graph of the function $y=|\log_{4}x|$ from left to right at points A and B, and $l_2$ intersects the graph of the function $y=|\log_{4}x|$ from left to right at points C and D. Let the projection lengths of line segments AC and BD on the x-axis be $m$ and $n$ respectively. When $a= \_\_\_\_\_\_$, $\frac {n}{m}$ reaches its minimum value. | \frac {5}{2} | 12.5 |
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