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22,400 | On a lengthy, one-way, single-lane highway, cars travel at uniform speeds and maintain a safety distance determined by their speed: the separation distance from the back of one car to the front of another is one car length for each 10 kilometers per hour of speed or fraction thereof. Cars are exceptionally long, each 5 meters in this case. Assume vehicles can travel at any integer speed, and calculate $N$, the maximum total number of cars that can pass a sensor in one hour. Determine the result of $N$ divided by 100 when rounded down to the nearest integer. | 20 | 0.78125 |
22,401 | In parallelogram $EFGH$, $EF = 5z + 5$, $FG = 4k^2$, $GH = 40$, and $HE = k + 20$. Determine the values of $z$ and $k$ and find $z \times k$. | \frac{7 + 7\sqrt{321}}{8} | 0.78125 |
22,402 | If the function $f(x)=\sin \left( \frac{x+\varphi}{3}\right)$ is an even function, determine the value of $\varphi$. | \frac{3\pi}{2} | 78.90625 |
22,403 | Calculate $\sqrt[4]{\sqrt{\frac{32}{10000}}}$. | \frac{\sqrt[8]{2}}{\sqrt{5}} | 39.0625 |
22,404 | Given vectors $\overrightarrow{a}=(\cos x,\sin x)$, $\overrightarrow{b}=( \sqrt {3}\sin x,\sin x)$, where $x\in R$, define the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}- \dfrac {1}{2}$.
(1) Find the smallest positive period of the function $f(x)$;
(2) Find the maximum and minimum values of the function $f(x)$ on $\[0, \dfrac {\pi}{2}\]$. | -\dfrac{1}{2} | 86.71875 |
22,405 | "In a tree with black pearls hidden, this item is only available in May. Travelers who pass by taste one, with a mouthful of sweetness and sourness, never wanting to leave." The Dongkui waxberry is a sweet gift in summer. Each batch of Dongkui waxberries must undergo two rounds of testing before entering the market. They can only be sold if they pass both rounds of testing; otherwise, they cannot be sold. It is known that the probability of not passing the first round of testing is $\frac{1}{9}$, and the probability of not passing the second round of testing is $\frac{1}{10}$. The two rounds of testing are independent of each other.<br/>$(1)$ Find the probability that a batch of waxberries cannot be sold;<br/>$(2)$ If the waxberries can be sold, the profit for that batch is $400$ yuan; if the waxberries cannot be sold, the batch will incur a loss of $800$ yuan (i.e., a profit of $-800$ yuan). It is known that there are currently 4 batches of waxberries. Let $X$ represent the profit from the 4 batches of waxberries (the sales of waxberries in each batch are independent of each other). Find the probability distribution and mathematical expectation of $X$. | 640 | 35.15625 |
22,406 | A math conference is hosting a series of lectures by seven distinct lecturers. Dr. Smith's lecture depends on Dr. Jones’s lecture, and additionally, Dr. Brown's lecture depends on Dr. Green’s lecture. How many valid orders can these seven lecturers be scheduled, given these dependencies? | 1260 | 94.53125 |
22,407 | Find the constant term in the expansion of \\((x+ \frac {2}{x}+1)^{6}\\) (Answer with a numerical value) | 581 | 27.34375 |
22,408 | Calculate using factorization:<br/>$(1)\frac{2021×2023}{2022^2-1}$;<br/>$(2)2\times 101^{2}+2\times 101\times 98+2\times 49^{2}$. | 45000 | 55.46875 |
22,409 | A circle with radius \(5\) has its center on the \(x\)-axis, and the \(x\)-coordinate of the center is an integer. The circle is tangent to the line \(4x+3y-29=0\).
(Ⅰ) Find the equation of the circle;
(Ⅱ) Let the line \(ax-y+5=0\) (\(a > 0\)) intersect the circle at points \(A\) and \(B\), find the range of values for the real number \(a\);
(Ⅲ) Under the condition of (Ⅱ), determine if there exists a real number \(a\) such that the perpendicular bisector line \(l\) of chord \(AB\) passes through point \(P(-2,4)\), and if so, find the value of \(a\); if not, explain why. | a = \dfrac {3}{4} | 27.34375 |
22,410 | Given that the Green Park Middle School chess team consists of three boys and four girls, and a girl at each end and the three boys and one girl alternating in the middle, determine the number of possible arrangements. | 144 | 60.9375 |
22,411 | Let $\mathbf{u},$ $\mathbf{v},$ and $\mathbf{w}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = \frac{1}{4} \|\mathbf{v}\| \|\mathbf{w}\| \mathbf{u}.\]
Let $\phi$ be the angle between $\mathbf{v}$ and $\mathbf{w}.$ Find $\sin \phi.$ | \frac{\sqrt{15}}{4} | 91.40625 |
22,412 | In a club election, 10 officer positions are available. There are 20 candidates, of which 8 have previously served as officers. Determine how many different slates of officers include at least one of the past officers. | 184,690 | 0 |
22,413 | Let \(ABCD\) be a convex quadrilateral with \(\angle ABC = 90^\circ\), \(\angle BAD = \angle ADC = 80^\circ\). Let \(M\) and \(N\) be points on \([AD]\) and \([BC]\) such that \(\angle CDN = \angle ABM = 20^\circ\). Finally, assume \(MD = AB\). What is the measure of \(\angle MNB\)? | 70 | 17.96875 |
22,414 | Jackie and Alex have two fair coins and a third coin that comes up heads with probability $\frac{2}{5}$. Jackie flips the three coins, and then Alex flips the same three coins. Determine the probability that Jackie gets the same number of heads as Alex, where the probability is expressed as a reduced fraction $\frac{p}{q}$. Find the sum $p + q$. | 263 | 17.1875 |
22,415 | A box contains four balls, each with a unique number from 1 to 4, and all balls are identical in shape and size.
(1) If two balls are randomly drawn from the box, what is the probability that the sum of their numbers is greater than 5?
(2) If one ball is drawn from the box, its number is recorded as $a$, and then the ball is put back. Another ball is drawn, and its number is recorded as $b$. What is the probability that $|a-b| \geq 2$? | \frac{3}{8} | 73.4375 |
22,416 | Consider the set $S$ of permutations of $1, 2, \dots, 2022$ such that for all numbers $k$ in the
permutation, the number of numbers less than $k$ that follow $k$ is even.
For example, for $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$ If $|S| = (a!)^b$ where $a, b \in \mathbb{N}$ , then find the product $ab$ . | 2022 | 58.59375 |
22,417 | A cube with side length 2 is cut three times to form various polyhedral pieces. It is first cut through the diagonal planes intersecting at vertex W, then an additional cut is made that passes through the midpoint of one side of the cube and perpendicular to its adjacent face, effectively creating 16 pieces. Consider the piece containing vertex W, which now forms a pyramid with a triangular base. What is the volume of this pyramid? | \frac{4}{3} | 1.5625 |
22,418 | Given a right triangle \( ABC \) with hypotenuse \( AB \). One leg \( AC = 15 \) and the altitude from \( C \) to \( AB \) divides \( AB \) into segments \( AH \) and \( HB \) with \( HB = 16 \). What is the area of triangle \( ABC \)? | 150 | 56.25 |
22,419 | The sum of the first n terms of the sequence $-1, 4, -7, 10, \ldots, (-1)^{n}(3n-2)$ is given by $S_{n}$. Calculate $S_{11}+S_{20}$. | 14 | 36.71875 |
22,420 | Let the hyperbola $C: \frac{x^2}{a^2} - y^2 = 1$ ($a > 0$) intersect with the line $l: x + y = 1$ at two distinct points $A$ and $B$.
(Ⅰ) Find the range of values for the eccentricity $e$ of the hyperbola $C$.
(Ⅱ) Let the intersection of line $l$ with the y-axis be $P$, and $\overrightarrow{PA} = \frac{5}{12} \overrightarrow{PB}$. Find the value of $a$. | \frac{17}{13} | 34.375 |
22,421 | If \( \sqrt{\frac{3}{x} + 3} = \frac{5}{3} \), solve for \( x \). | -\frac{27}{2} | 77.34375 |
22,422 | If the Cesaro sum of a sequence with 99 terms is 1000, calculate the Cesaro sum of the sequence with 100 terms consisting of the numbers 1 and the first 99 terms of the original sequence. | 991 | 57.8125 |
22,423 | Distribute 5 students into three groups: A, B, and C. Group A must have at least two students, while groups B and C must have at least one student each. Determine the number of different distribution schemes. | 80 | 15.625 |
22,424 | If \( a \) and \( b \) are positive numbers such that \( a^b = b^a \) and \( b = 4a \), then find the value of \( a \). | \sqrt[3]{4} | 85.9375 |
22,425 | If $\sin \left(\frac{\pi }{3}+\alpha \right)=\frac{1}{3}$, then find the value of $\cos \left(\frac{\pi }{3}-2\alpha \right)$. | -\frac{7}{9} | 44.53125 |
22,426 | Given that in quadrilateral $ABCD$, $m\angle B = m \angle C = 120^{\circ}, AB=3, BC=4,$ and $CD=5$, calculate the area of $ABCD$. | 8\sqrt{3} | 85.9375 |
22,427 | What is the largest four-digit number whose digits add up to 20? | 9920 | 91.40625 |
22,428 | Given the function $y=\sin (\omega x+\frac{\pi }{3})+2$, its graph shifts to the right by $\frac{4\pi }{3}$ units and coincides with the original graph. Find the minimum value of $|\omega|$. | \frac {3}{2} | 96.09375 |
22,429 | In the sequence $1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,2, \cdots$ where the number of 2s between consecutive 1s increases by 1 each time, what is the sum of the first 1234 terms? | 2419 | 87.5 |
22,430 | A child spends their time drawing pictures of Native Americans (referred to as "Indians") and Eskimos. Each drawing depicts either a Native American with a teepee or an Eskimo with an igloo. However, the child sometimes makes mistakes and draws a Native American with an igloo.
A psychologist noticed the following:
1. The number of Native Americans drawn is twice the number of Eskimos.
2. The number of Eskimos with teepees is equal to the number of Native Americans with igloos.
3. Each teepee drawn with an Eskimo is matched with three igloos.
Based on this information, determine the proportion of Native Americans among the inhabitants of teepees. | 7/8 | 7.03125 |
22,431 | Compute
\[\prod_{n = 1}^{15} \frac{n + 4}{n}.\] | 11628 | 6.25 |
22,432 | Given that point O is the center of the regular octagon ABCDEFGH, and Y is the midpoint of the side CD, determine the fraction of the area of the octagon that is shaded if the shaded region includes triangles DEO, EFO, and half of triangle CEO. | \frac{5}{16} | 84.375 |
22,433 | Let $[ABCD]$ be a convex quadrilateral with area $2014$ , and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have area $1$ . Let $R$ be the intersection of $[AC]$ and $[PQ]$ . Determine $\frac{\overline{RC}}{\overline{RA}}$ . | 2013 | 33.59375 |
22,434 | Suppose \( x \), \( y \), and \( z \) are positive numbers satisfying:
\[
x^2 \cdot y = 2, \\
y^2 \cdot z = 4, \text{ and} \\
z^2 / x = 5.
\]
Find \( x \). | 5^{1/7} | 91.40625 |
22,435 | Given in the Cartesian coordinate system $xOy$, a line $l$ passing through a fixed point $P$ with an inclination angle of $\alpha$ has the parametric equation: $$\begin{cases} x=t\cos\alpha \\ y=-2+t\sin\alpha \end{cases}$$ (where $t$ is the parameter). In the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinates of the center of the circle are $(3, \frac{\pi}{2})$, and the circle $C$ with a radius of 3 intersects the line $l$ at points $A$ and $B$. Then, $|PA|\cdot|PB|=$ \_\_\_\_\_. | 16 | 96.09375 |
22,436 | Calculate:<br/>$(1)-9+5-\left(-12\right)+\left(-3\right)$;<br/>Calculate:<br/>$(2)-(+1.5)-(-4\frac{1}{4})+3.75-(-8\frac{1}{2})$;<br/>$(3)$Read the following solution process and answer the question:<br/>Calculate:$\left(-15\right)\div (-\frac{1}{2}×\frac{25}{3}$）$÷\frac{1}{6}$<br/>Solution: Original expression $=\left(-15\right)\div (-\frac{25}{6})\times 6(Step 1)$<br/>$=\left(-15\right)\div \left(-25\right)(Step 2)$<br/>$=\frac{3}{5}$(Step 3)<br/>①The error in the solution process above starts from step ______, and the reason for the error is ______;<br/>②Please write down the correct solution process. | \frac{108}{5} | 46.09375 |
22,437 | Let $m$ be the smallest integer whose cube root is of the form $n+s$, where $n$ is a positive integer and $s$ is a positive real number less than $1/2000$. Find $n$. | 26 | 34.375 |
22,438 | Given the numbers \( x, y, z \in [0, \pi] \), find the minimum value of the expression
$$
A = \cos (x - y) + \cos (y - z) + \cos (z - x)
$$ | -1 | 92.96875 |
22,439 | In a right-angled triangle $LMN$, suppose $\sin N = \frac{5}{13}$ with $LM = 10$. Calculate the length of $LN$. | 26 | 86.71875 |
22,440 | In the polar coordinate system and the Cartesian coordinate system xOy, which have the same unit of length, with the origin O as the pole and the positive half-axis of x as the polar axis. The parametric equation of line l is $$\begin{cases} x=2+ \frac {1}{2}t \\ y= \frac { \sqrt {3}}{2}t \end{cases}$$ (t is the parameter), and the polar equation of curve C is $\rho\sin^2\theta=4\cos\theta$.
(1) Find the Cartesian equation of curve C;
(2) Suppose line l intersects curve C at points A and B, find the length of chord |AB|. | \frac {8 \sqrt {7}}{3} | 0 |
22,441 | How many positive integers divide the number $10! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10$ ? | 270 | 96.875 |
22,442 | In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b\sin A = \frac{{\sqrt{3}}}{2}a$. Find:<br/>
$(Ⅰ)$ The measure of angle $B$;<br/>
$(Ⅱ)$ If triangle $\triangle ABC$ is an acute triangle and $a=2c$, $b=2\sqrt{6}$, find the area of $\triangle ABC$. | 4\sqrt{3} | 86.71875 |
22,443 | Given the function $f(x) = x^{2-m}$ is defined on the interval $[-3-m, m^2-m]$ and is an odd function, then $f(m) = $ ? | -1 | 21.09375 |
22,444 | A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is $1709$ . If $n$ is the last page number, what is the largest prime factor of $n$ ? | 17 | 39.84375 |
22,445 | A class has a total of 54 students. Now, using the systematic sampling method based on the students' ID numbers, a sample of 4 students is drawn. It is known that students with ID numbers 3, 29, and 42 are in the sample. What is the ID number of the fourth student in the sample? | 16 | 34.375 |
22,446 | Alice and Carol each have a rectangular sheet of paper. Alice has a sheet of paper measuring 10 inches by 12 inches and rolls it into a tube by taping the two 10-inch sides together. Carol rolls her sheet, which measures 8 inches by 15 inches, by taping the two 15-inch sides together. Calculate $\pi$ times the positive difference of the volumes of the two tubes. | 150 | 74.21875 |
22,447 | Gloria's grandmother cycled a total of $\frac{3}{6} + \frac{4}{4} + \frac{3}{3} + \frac{2}{8}$ hours at different speeds, and she cycled a total of $\frac{3}{5} + \frac{4}{5} + \frac{3}{5} + \frac{2}{5}$ hours at a speed of 5 miles per hour. Calculate the difference in the total time she spent cycling. | 21 | 0 |
22,448 | An equilateral triangle $ABC$ shares a side with a square $BCDE$ . If the resulting pentagon has a perimeter of $20$ , what is the area of the pentagon? (The triangle and square do not overlap). | 16 + 4\sqrt{3} | 44.53125 |
22,449 | Given the set $A=\{2,3,4,8,9,16\}$, if $a\in A$ and $b\in A$, the probability that the event "$\log_{a}b$ is not an integer but $\frac{b}{a}$ is an integer" occurs is $\_\_\_\_\_\_$. | \frac{1}{18} | 15.625 |
22,450 | The sequence $\lg 1000, \lg \left(1000 \cos \frac{\pi}{3}\right), \lg \left(1000 \cos ^{2} \frac{\pi}{3}\right), \cdots, \lg \left(1000 \cos ^{n-1} \frac{\pi}{3}\right), \cdots$, when the sum of the first $n$ terms is maximized, the value of $n$ is ( ). | 10 | 57.03125 |
22,451 | The constant term in the expansion of (1+x)(e^(-2x)-e^x)^9. | 84 | 50 |
22,452 | What is the area enclosed by the graph of \( |x| + |3y| = 9 \)? | 54 | 98.4375 |
22,453 | Given the function $f(x)=\cos x\cos \left( x+\dfrac{\pi}{3} \right)$.
(1) Find the smallest positive period of $f(x)$;
(2) In $\triangle ABC$, angles $A$, $B$, $C$ correspond to sides $a$, $b$, $c$, respectively. If $f(C)=-\dfrac{1}{4}$, $a=2$, and the area of $\triangle ABC$ is $2\sqrt{3}$, find the value of side length $c$. | 2 \sqrt {3} | 0 |
22,454 | Encrypt integers using the following method: each digit of the number becomes the units digit of its product with 7, then replace each digit $a$ with $10-a$. If a number is encrypted using the above method and the result is 473392, then the original number is ______. | 891134 | 41.40625 |
22,455 | In $\triangle ABC$, $BC= \sqrt {5}$, $AC=3$, $\sin C=2\sin A$. Find:
1. The value of $AB$.
2. The value of $\sin(A- \frac {\pi}{4})$. | - \frac { \sqrt {10}}{10} | 0 |
22,456 | Given the function $f(x)=3\sin x+4\cos x$, if for any $x\in R$ we have $f(x)\geqslant f(α)$, then the value of $\tan α$ is equal to ___. | \frac {3}{4} | 78.90625 |
22,457 | Given three points \(A, B, C\) forming a triangle with angles \(30^{\circ}\), \(45^{\circ}\), and \(105^{\circ}\). Two of these points are chosen, and the perpendicular bisector of the segment connecting them is drawn. The third point is then reflected across this perpendicular bisector to obtain a fourth point \(D\). This procedure is repeated with the resulting set of four points, where two points are chosen, the perpendicular bisector is drawn, and all points are reflected across it. What is the maximum number of distinct points that can be obtained as a result of repeatedly applying this procedure? | 12 | 0 |
22,458 | Find \(\cos 2 \alpha\), given that \(2 \operatorname{ctg}^{2} \alpha+7 \operatorname{ctg} \alpha+3=0\) and the value of \(\alpha\) satisfies the inequalities:
a) \(\frac{3 \pi}{2}<\alpha<\frac{7 \pi}{4}\);
b) \(\frac{7 \pi}{4}<\alpha<2 \pi\). | \frac{4}{5} | 43.75 |
22,459 | There are three balls of the same size but different colors in a pocket. One ball is drawn each time, the color is recorded, and then it is put back. The drawing stops when all three colors of balls have been drawn. If it stops after exactly 5 draws, the number of different ways to draw is \_\_\_\_\_\_\_. | 42 | 8.59375 |
22,460 | What is the product of all real numbers that are tripled when added to their reciprocals? | -\frac{1}{2} | 82.03125 |
22,461 | Given \( f(x) = x^2 + 3x + 2 \) and \( S = \{0, 1, 2, 3, \cdots, 100\} \), if \( a \in S \) and \( f(a) \) is divisible by 6, how many such \( a \) exist? | 67 | 86.71875 |
22,462 | Given that 28×15=420, directly write out the results of the following multiplications:
2.8×1.5=\_\_\_\_\_\_、0.28×1.5=\_\_\_\_\_\_、0.028×0.15=\_\_\_\_\_\_. | 0.0042 | 84.375 |
22,463 | A sequence of twelve \(0\)s and/or \(1\)s is randomly generated and must start with a '1'. If the probability that this sequence does not contain two consecutive \(1\)s can be written in the form \(\dfrac{m}{n}\), where \(m,n\) are relatively prime positive integers, find \(m+n\). | 2281 | 4.6875 |
22,464 | A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$ . She also has a box with dimensions $10\times11\times14$ . The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out.
Find all possible values of the total number of bricks that she can pack. | 24 | 62.5 |
22,465 | Given three points $A$, $B$, and $C$ in a plane such that $|\overrightarrow{AB}| = 3$, $|\overrightarrow{BC}| = 5$, and $|\overrightarrow{CA}| = 6$, find the value of $\overrightarrow{AB} \cdot \overrightarrow{BC} + \overrightarrow{BC} \cdot \overrightarrow{CA} + \overrightarrow{CA} \cdot \overrightarrow{AB}$. | -35 | 31.25 |
22,466 | If
\[
x + \sqrt{x^2 - 1} + \frac{1}{x + \sqrt{x^2 - 1}} = 12,
\]
then find the value of
\[
x^3 + \sqrt{x^6 - 1} + \frac{1}{x^3 + \sqrt{x^6 - 1}}.
\] | 432 | 67.1875 |
22,467 | Point A is the intersection of the unit circle and the positive half of the x-axis, and point B is in the second quadrant. Let θ be the angle formed by the rays OA and OB. Given sin(θ) = 4/5, calculate the value of sin(π + θ) + 2sin(π/2 - θ) divided by 2tan(π - θ). | -\frac{3}{4} | 72.65625 |
22,468 | \( S \) is a set of 5 coplanar points, no 3 of which are collinear. \( M(S) \) is the largest area of a triangle with vertices in \( S \). Similarly, \( m(S) \) is the smallest area of such a triangle. What is the smallest possible value of \( \frac{M(S)}{m(S)} \) as \( S \) varies? | \frac{1 + \sqrt{5}}{2} | 17.1875 |
22,469 | Five friends did gardening for their local community and earned $15, $22, $28, $35, and $50 respectively. They decide to share their total earnings equally. How much money must the friend who earned $50 contribute to the pool?
A) $10
B) $15
C) $20
D) $25
E) $35 | 20 | 56.25 |
22,470 | How many ways can a 5-person executive board be chosen from a club of 12 people? | 792 | 90.625 |
22,471 | Given $$\frac {1}{3}$$≤a≤1, if the function f(x)=ax<sup>2</sup>-2x+1 has its maximum value M(a) and minimum value N(a) in the interval [1,3], let g(a)=M(a)-N(a).
(1) Find the expression for g(a);
(2) Describe the intervals where g(a) is increasing and decreasing (no proof required), and find the minimum value of g(a). | \frac {1}{2} | 14.84375 |
22,472 | The line \(y = -\frac{2}{3}x + 10\) crosses the \(x\)-axis at point \(P\) and the \(y\)-axis at point \(Q\). Point \(T(r, s)\) is on the segment \(PQ\). If the area of \(\triangle POQ\) is four times the area of \(\triangle TOP\), find the value of \(r + s\). | 13.75 | 32.8125 |
22,473 | Let $a, b, c$ , and $d$ be real numbers such that $a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018$ . Evaluate $3b + 8c + 24d + 37a$ . | 1215 | 6.25 |
22,474 | For how many integers \( n \), with \( 2 \leq n \leq 80 \), is \( \frac{(n-1)(n)(n+1)}{8} \) equal to an integer? | 49 | 3.90625 |
22,475 | Define a function $g(z) = (3 + i)z^2 + \alpha z + \gamma$ for all complex $z$, where $\alpha$ and $\gamma$ are complex numbers. Assume that $g(1)$ and $g(i)$ both yield real numbers. Determine the smallest possible value of $|\alpha| + |\gamma|$. | \sqrt{2} | 58.59375 |
22,476 | Given $|\vec{a}|=1$, $|\vec{b}|=2$, and the angle between $\vec{a}$ and $\vec{b}$ is $60^{\circ}$.
$(1)$ Find $\vec{a}\cdot \vec{b}$, $(\vec{a}- \vec{b})\cdot(\vec{a}+ \vec{b})$;
$(2)$ Find $|\vec{a}- \vec{b}|$. | \sqrt{3} | 98.4375 |
22,477 | Determine the smallest natural number $n =>2$ with the property:
For every positive integers $a_1, a_2,. . . , a_n$ the product of all differences $a_j-a_i$ ,
$1 <=i <j <=n$ , is divisible by 2001. | 30 | 38.28125 |
22,478 | (1) Calculate: $$\lg 4 + 2\lg 5 + (0.25)^{-\frac{1}{2}} - 8^{\frac{2}{3}}$$;
(2) Given that $f(x)$ is an odd function on $\mathbb{R}$ and $f(x+2) = -f(x)$, when $x \in (0, 2)$, $f(x) = 2x^2$, find $f(2015)$. | -2 | 92.96875 |
22,479 | Mary and Pat play the following number game. Mary picks an initial integer greater than $2017$ . She then multiplies this number by $2017$ and adds $2$ to the result. Pat will add $2019$ to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by $2017$ and add $2$ to the result when it is her turn. Pat will always add $2019$ to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by $2018$ . Mary wants to prevent Pat from winning the game.
Determine, with proof, the smallest initial integer Mary could choose in order to achieve this. | 2022 | 3.125 |
22,480 | The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately what fraction. | \frac{1}{9} | 17.1875 |
22,481 | A set containing three real numbers can be represented as $\{a,\frac{b}{a},1\}$, or as $\{a^{2}, a+b, 0\}$. Find the value of $a^{2023}+b^{2024}$. | -1 | 42.96875 |
22,482 | Let $T$ be a triangle whose vertices have integer coordinates, such that each side of $T$ contains exactly $m$ points with integer coordinates. If the area of $T$ is less than $2020$ , determine the largest possible value of $m$ .
| 64 | 16.40625 |
22,483 | Given the function $f(x) = (m^2 - m - 1)x^{-5m-3}$ is a power function, and it is increasing on the interval $(0, +\infty)$, determine the value of $m$. | -1 | 52.34375 |
22,484 | The measure of angle $ACB$ is 45 degrees. If ray $CA$ is rotated 510 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees? | 75 | 35.15625 |
22,485 | Let $a$, $b$, $c$, and $d$ be positive integers with $a < 3b$, $b < 3c$, and $c < 4d$. Additionally, suppose $b + d = 200$. The largest possible value for $a$ is:
A) 438
B) 440
C) 445
D) 449
E) 455 | 449 | 32.8125 |
22,486 | In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, with $a=6$ and $b\sin\frac{B+C}{2}=a\sin B$. Find:
1. The measure of angle $A$.
2. Let $M$ be a point inside triangle $\triangle ABC$. Extend $AM$ to intersect $BC$ at point $D$. _______. Find the area of triangle $\triangle ABC$. Choose one of the following conditions to supplement the blank line to ensure the existence of triangle $\triangle ABC$ and solve the problem:
- $M$ is the circumcenter of triangle $\triangle ABC$ and $AM=4$.
- $M$ is the centroid of triangle $\triangle ABC$ and $AM=2\sqrt{3}$.
- $M$ is the incenter of triangle $\triangle ABC$ and $AD=3\sqrt{3}$.
(Note: The point of intersection of the perpendicular bisectors of the sides of a triangle is called the circumcenter, the point of intersection of the medians is called the centroid, and the point of intersection of the angle bisectors is called the incenter.) | 9\sqrt{3} | 46.09375 |
22,487 | In triangle $ABC$, $\angle C = 90^\circ$, $AC = 4$, and $AB = \sqrt{41}$. What is $\tan B$? | \frac{4}{5} | 90.625 |
22,488 | The national student loan is a credit loan subsidized by the finance department, aimed at helping college students from families with financial difficulties to pay for tuition, accommodation, and living expenses during their study period in college. The total amount applied for each year shall not exceed 6,000 yuan. A graduate from the class of 2010 at a certain university, Ling Xiao, applied for a total of 24,000 yuan in student loans during his undergraduate period and promised to pay it all back within 3 years after graduation (calculated as 36 months). The salary standard provided by the contracted unit is 1,500 yuan per month for the first year, and starting from the 13th month, the monthly salary increases by 5% until it reaches 4,000 yuan. Ling Xiao plans to repay 500 yuan each month for the first 12 months, and starting from the 13th month, the monthly repayment amount will increase by x yuan each month.
(Ⅰ) If Ling Xiao just pays off the loan in the 36th month (i.e., three years after graduation), find the value of x;
(Ⅱ) When x=50, in which month will Ling Xiao pay off the last installment of the loan? Will his monthly salary balance be enough to meet the basic living expenses of 3,000 yuan that month?
(Reference data: $1.05^{18}=2.406$, $1.05^{19}=2.526$, $1.05^{20}=2.653$, $1.05^{21}=2.786$) | 31 | 7.8125 |
22,489 | Given that $f(x)= \frac {4}{4^{x}+2}$, $S\_n$ is the sum of the first $n$ terms of the sequence $\{a\_n\}$, and $\{a\_n\}$ satisfies $a\_1=0$, and when $n \geqslant 2$, $a\_n=f( \frac {1}{n})+f( \frac {2}{n})+f( \frac {3}{n})+…+f( \frac {n-1}{n})$, find the maximum value of $\frac {a_{n+1}}{2S\_n+a\_6}$. | \frac {2}{7} | 24.21875 |
22,490 | Given the geometric series $6 - \frac{12}{5} + \frac{36}{25} - \dots$, calculate the limiting sum of this series as the number of terms increases without bound. | \frac{30}{7} | 96.875 |
22,491 | How many $3$-digit positive integers have digits whose product equals $30$? | 12 | 17.96875 |
22,492 | Given an ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \left( a > b > 0 \right)$ with eccentricity $\dfrac{1}{2}$, and it passes through the point $\left( 1,\dfrac{3}{2} \right)$. If point $M\left( x_{0},y_{0} \right)$ is on the ellipse $C$, then the point $N\left( \dfrac{x_{0}}{a},\dfrac{y_{0}}{b} \right)$ is called an "elliptic point" of point $M$.
$(1)$ Find the standard equation of the ellipse $C$;
$(2)$ If the line $l:y=kx+m$ intersects the ellipse $C$ at points $A$ and $B$, and the "elliptic points" of $A$ and $B$ are $P$ and $Q$ respectively, and the circle with diameter $PQ$ passes through the origin, determine whether the area of $\Delta AOB$ is a constant value? If it is a constant, find the value; if not, explain why. | \sqrt{3} | 15.625 |
22,493 | Given an angle α with its vertex at the origin of coordinates, its initial side coinciding with the non-negative half-axis of the x-axis, and two points on its terminal side A(1,a), B(2,b), and cos(2α) = 2/3, determine the value of |a-b|. | \dfrac{\sqrt{5}}{5} | 47.65625 |
22,494 | In the right circular cone $P-ABC$, $PA \perp$ plane $ABC$, $AC \perp AB$, $PA=AB=2$, $AC=1$. Find the volume of the circumscribed sphere of the cone $P-ABC$. | \frac{9}{2}\pi | 0.78125 |
22,495 | Suppose that $20^{21} = 2^a5^b = 4^c5^d = 8^e5^f$ for positive integers $a,b,c,d,e,$ and $f$ . Find $\frac{100bdf}{ace}$ .
*Proposed by Andrew Wu* | 75 | 67.1875 |
22,496 | Evaluate the sum $$\lceil\sqrt{10}\rceil + \lceil\sqrt{11}\rceil + \lceil\sqrt{12}\rceil + \cdots + \lceil\sqrt{40}\rceil$$ | 170 | 2.34375 |
22,497 | The positive integer $m$ is a multiple of 111, and the positive integer $n$ is a multiple of 31. Their sum is 2017. Find $n - m$ . | 463 | 90.625 |
22,498 | A circle intersects the $y$ -axis at two points $(0, a)$ and $(0, b)$ and is tangent to the line $x+100y = 100$ at $(100, 0)$ . Compute the sum of all possible values of $ab - a - b$ . | 10000 | 5.46875 |
22,499 | What is the sum of all the four-digit positive integers that end in 0? | 4945500 | 86.71875 |
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