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31,400 | Given 6 digits: \(0, 1, 2, 3, 4, 5\). Find the sum of all four-digit even numbers that can be written using these digits (the same digit can be repeated in a number). | 1769580 | 83.59375 |
31,401 | Given that the center of an ellipse is at the origin, the focus is on the $x$-axis, and the eccentricity $e= \frac { \sqrt {2}}{2}$, the area of the quadrilateral formed by connecting the four vertices of the ellipse in order is $2 \sqrt {2}$.
(1) Find the standard equation of the ellipse;
(2) Given that line $l$ intersects the ellipse at points $M$ and $N$, and $O$ is the origin. If point $O$ is on the circle with $MN$ as the diameter, find the distance from point $O$ to line $l$. | \frac{\sqrt{6}}{3} | 29.6875 |
31,402 | A research study group is investigating the traffic volume at a certain intersection near the school during the rush hour from 8:00 to 10:00. After long-term observation and statistics, they have established a simple function model between traffic volume and average vehicle speed. The model is as follows: Let the traffic volume be $y$ (thousand vehicles per hour) and the average vehicle speed be $v$ (kilometers per hour), then $y=\frac{25v}{{v}^{2}-5v+16}$ $(v>0)$.
$(1)$ If the traffic volume during the rush hour is required to be no less than 5 thousand vehicles per hour, in what range should the average vehicle speed be?
$(2)$ During the rush hour, at what average vehicle speed is the traffic volume maximum? What is the maximum traffic volume? | \frac{25}{3} | 57.03125 |
31,403 | A natural number is called a square if it can be written as the product of two identical numbers. For example, 9 is a square because \(9 = 3 \times 3\). The first squares are 1, 4, 9, 16, 25, ... A natural number is called a cube if it can be written as the product of three identical numbers. For example, 8 is a cube because \(8 = 2 \times 2 \times 2\). The first cubes are 1, 8, 27, 64, 125, ...
On a certain day, the square and cube numbers decided to go on strike. This caused the remaining natural numbers to take on new positions:
a) What is the number in the 12th position?
b) What numbers less than or equal to 2013 are both squares and cubes?
c) What is the new position occupied by the number 2013?
d) Find the number that is in the 2013th position. | 2067 | 75.78125 |
31,404 | Triangle \(ABC\) has sides \(AB = 14\), \(BC = 13\), and \(CA = 15\). It is inscribed in circle \(\Gamma\), which has center \(O\). Let \(M\) be the midpoint of \(AB\), let \(B'\) be the point on \(\Gamma\) diametrically opposite \(B\), and let \(X\) be the intersection of \(AO\) and \(MB'\). Find the length of \(AX\). | \frac{65}{12} | 2.34375 |
31,405 | The sultan gathered 300 court sages and proposed a trial. There are 25 different colors of hats, known in advance to the sages. The sultan informed them that each sage would be given one of these hats, and if they wrote down the number of hats for each color, all these numbers would be different. Each sage would see the hats of the other sages but not their own. Then all the sages would simultaneously announce the supposed color of their own hat. Can the sages agree in advance to act in such a way that at least 150 of them will correctly name their hat color? | 150 | 20.3125 |
31,406 | Three squares, $ABCD$, $EFGH$, and $GHIJ$, each have side length $s$. Point $C$ is located at the midpoint of side $HG$, and point $D$ is located at the midpoint of side $EF$. The line segment $AJ$ intersects the line segment $GH$ at point $X$. Determine the ratio of the area of the shaded region formed by triangle $AXD$ and trapezoid $JXCB$ to the total area of the three squares. | \frac{1}{3} | 17.1875 |
31,407 | Define $m(n)$ to be the greatest proper natural divisor of $n\in \mathbb{N}$ . Find all $n \in \mathbb{N} $ such that $n+m(n) $ is a power of $10$ .
*N. Agakhanov* | 75 | 13.28125 |
31,408 | If the function $f(x) = \frac{1}{2}(m-2)x^2 + (n-8)x + 1$ with $m \geq 0$ and $n \geq 0$ is monotonically decreasing in the interval $\left[\frac{1}{2}, 2\right]$, then the maximum value of $mn$ is __________. | 18 | 21.875 |
31,409 |
A finite arithmetic progression \( a_1, a_2, \ldots, a_n \) with a positive common difference has a sum of \( S \), and \( a_1 > 0 \). It is known that if the common difference of the progression is increased by 3 times while keeping the first term unchanged, the sum \( S \) doubles. By how many times will \( S \) increase if the common difference of the initial progression is increased by 4 times (keeping the first term unchanged)? | 5/2 | 75 |
31,410 | Gina's running app tracked her average rate in minutes per kilometre. After starting the app, Gina stood still for 15 seconds and then ran at a constant rate of 7 minutes per kilometre for the rest of the run. How many kilometres did Gina run between when her app showed her average rate as 7 minutes 30 seconds per kilometre and when it showed 7 minutes 5 seconds per kilometre? | 2.5 | 16.40625 |
31,411 | Given that the sum of the binomial coefficients in the expansion of {(5x-1/√x)^n} is 64, determine the constant term in its expansion. | 375 | 7.8125 |
31,412 | An integer is called a "good number" if it has 8 positive divisors and the sum of these 8 positive divisors is 3240. For example, 2006 is a good number because the sum of its divisors 1, 2, 17, 34, 59, 118, 1003, and 2006 is 3240. Find the smallest good number. | 1614 | 35.15625 |
31,413 | Given $2^{3x} = 128$, calculate the value of $2^{-x}$. | \frac{1}{2^{\frac{7}{3}}} | 9.375 |
31,414 | When the two-digit integer \( XX \), with equal digits, is multiplied by the one-digit integer \( X \), the result is the three-digit integer \( PXQ \). What is the greatest possible value of \( PXQ \) if \( PXQ \) must start with \( P \) and end with \( X \)? | 396 | 64.0625 |
31,415 | In $\triangle XYZ$, a triangle $\triangle MNO$ is inscribed such that vertices $M, N, O$ lie on sides $YZ, XZ, XY$, respectively. The circumcircles of $\triangle XMO$, $\triangle YNM$, and $\triangle ZNO$ have centers $P_1, P_2, P_3$, respectively. Given that $XY = 26, YZ = 28, XZ = 27$, and $\stackrel{\frown}{MO} = \stackrel{\frown}{YN}, \stackrel{\frown}{NO} = \stackrel{\frown}{XM}, \stackrel{\frown}{NM} = \stackrel{\frown}{ZO}$. The length of $ZO$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$. | 15 | 3.125 |
31,416 | Consider the sequence created by intermixing the following sets of numbers: the first $1000$ odd numbers, and the squares of the first $100$ integers. What is the median of the new list of $1100$ numbers?
- $1, 3, 5, \ldots, 1999$
- $1^2, 2^2, \ldots, 100^2$
A) $1089$
B) $1095$
C) $1100$
D) $1102$
E) $1105$ | 1100 | 45.3125 |
31,417 | Find the slope angle of the tangent line to the curve $f(x)=\frac{1}{3}{x}^{3}-{x}^{2}+5$ at $x=1$. | \frac{3\pi}{4} | 2.34375 |
31,418 | Find $d$, given that $\lfloor d\rfloor$ is a solution to \[3x^2 + 19x - 70 = 0\] and $\{d\} = d - \lfloor d\rfloor$ is a solution to \[4x^2 - 12x + 5 = 0.\] | -8.5 | 14.84375 |
31,419 | Define the derivative of the $(n-1)$th derivative as the $n$th derivative $(n \in N^{*}, n \geqslant 2)$, that is, $f^{(n)}(x)=[f^{(n-1)}(x)]'$. They are denoted as $f''(x)$, $f'''(x)$, $f^{(4)}(x)$, ..., $f^{(n)}(x)$. If $f(x) = xe^{x}$, then the $2023$rd derivative of the function $f(x)$ at the point $(0, f^{(2023)}(0))$ has a $y$-intercept on the $x$-axis of ______. | -\frac{2023}{2024} | 0 |
31,420 | Let $E(n)$ denote the sum of the even digits of $n$. Modify $E(n)$ such that if $n$ is prime, $E(n)$ is counted as zero, and if $n$ is not prime, $E(n)$ is counted twice. Calculate $E'(1)+E'(2)+E'(3)+\cdots+E'(200)$.
A) 1200
B) 1320
C) 1360
D) 1400
E) 1500 | 1360 | 39.0625 |
31,421 | How many different positive values of \( x \) will make this statement true: there are exactly 3 three-digit multiples of \( x \)? | 84 | 3.90625 |
31,422 | Consider the hyperbola $x^{2}-y^{2}=8$ with left and right foci denoted as $F_{1}$ and $F_{2}$, respectively. Let $P_{n}(x_{n},y_{n})$ be a sequence of points on its right branch such that $|P_{n+1}F_{2}|=|P_{n}F_{1}|$ and $P_{1}F_{2} \perp F_{1}F_{2}$. Determine the value of $x_{2016}$. | 8064 | 31.25 |
31,423 | The 2-digit integers from 31 to 75 are written consecutively to form the integer $M = 313233\cdots7475$. Suppose that $3^m$ is the highest power of 3 that is a factor of $M$. What is $m$?
A) 0
B) 1
C) 2
D) 3
E) more than 3 | \text{(A) } 0 | 0 |
31,424 | A regular octagon $ABCDEFGH$ has its sides' midpoints connected to form a smaller octagon inside it. Determine the fraction of the area of the larger octagon $ABCDEFGH$ that is enclosed by this smaller octagon. | \frac{1}{2} | 62.5 |
31,425 | Given a circle with 800 points labeled in sequence clockwise as \(1, 2, \ldots, 800\), dividing the circle into 800 arcs. Initially, one point is painted red, and subsequently, additional points are painted red according to the following rule: if the \(k\)-th point is already red, the next point to be painted red is found by moving clockwise \(k\) arcs from \(k\). What is the maximum number of red points that can be obtained on the circle? Explain the reasoning. | 25 | 45.3125 |
31,426 | Let $S$ be the set of positive real numbers. Let $g : S \to \mathbb{R}$ be a function such that
\[g(x) g(y) = g(xy) + 3003 \left( \frac{1}{x} + \frac{1}{y} + 3002 \right)\]for all $x,$ $y > 0.$
Let $m$ be the number of possible values of $g(2),$ and let $t$ be the sum of all possible values of $g(2).$ Find $m \times t.$ | \frac{6007}{2} | 7.8125 |
31,427 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $4b\sin A= \sqrt {7}a$.
(I) Find the value of $\sin B$;
(II) If $a$, $b$, and $c$ form an arithmetic sequence with a common difference greater than $0$, find the value of $\cos A-\cos C$. | \frac { \sqrt {7}}{2} | 0 |
31,428 | Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors, and $(2\overrightarrow{a}+ \overrightarrow{b})\cdot (\overrightarrow{a}-2\overrightarrow{b})=- \frac {3 \sqrt {3}}{2}$, calculate the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{6} | 98.4375 |
31,429 | Given that bag A contains 3 white balls and 5 black balls, and bag B contains 4 white balls and 6 black balls, calculate the probability that the number of white balls in bag A does not decrease after a ball is randomly taken from bag A and put into bag B, and a ball is then randomly taken from bag B and put back into bag A. | \frac{35}{44} | 7.8125 |
31,430 | How many integers $n$ are there subject to the constraint that $1 \leq n \leq 2020$ and $n^n$ is a perfect square? | 1032 | 48.4375 |
31,431 | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and satisfy $\frac{c}{\cos C}=\frac{a+b}{\cos A+\cos B}$. Point $D$ is the midpoint of side $BC$.
$(1)$ Find the measure of angle $C$.
$(2)$ If $AC=2$ and $AD=\sqrt{7}$, find the length of side $AB$. | 2\sqrt{7} | 15.625 |
31,432 | ABC is a triangle. D is the midpoint of AB, E is a point on the side BC such that BE = 2 EC and ∠ADC = ∠BAE. Find ∠BAC. | 30 | 0.78125 |
31,433 | Given the function $f(x)= \sqrt{3}\sin \omega x+\cos (\omega x+ \frac{\pi}{3})+\cos (\omega x- \frac{\pi}{3})-1$ ($\omega > 0$, $x\in\mathbb{R}$),
and the smallest positive period of the function $f(x)$ is $\pi$.
$(1)$ Find the analytical expression of the function $f(x)$;
$(2)$ In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $\alpha$ are $l$, $\alpha$, and $l$ respectively. If $\alpha$, $(\vec{BA}\cdot \vec{BC}= \frac{3}{2})$, and $a+c=4$, find the value of $b$. | \sqrt{7} | 30.46875 |
31,434 | Calculate the sum of all integers between 50 and 450 that end in 1 or 7. | 19920 | 32.8125 |
31,435 | For a four-digit natural number $M$, if the digit in the thousands place is $6$ more than the digit in the units place, and the digit in the hundreds place is $2$ more than the digit in the tens place, then $M$ is called a "naive number." For example, the four-digit number $7311$ is a "naive number" because $7-1=6$ and $3-1=2$. On the other hand, the four-digit number $8421$ is not a "naive number" because $8-1\neq 6$. Find the smallest "naive number" which is ______.
Let the digit in the thousands place of a "naive number" $M$ be $a$, the digit in the hundreds place be $b$, the digit in the tens place be $c$, and the digit in the units place be $d$. Define $P(M)=3(a+b)+c+d$ and $Q(M)=a-5$. If $\frac{{P(M)}}{{Q(M)}}$ is divisible by $10$, then the maximum value of $M$ that satisfies this condition is ______. | 9313 | 0.78125 |
31,436 | Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, the minimum value of the sum of the distance from point $P$ to point $Q$ and the distance from point $P$ to the directrix of the parabola is ______. | \sqrt{17}-1 | 3.125 |
31,437 | The function \( f(n) \) is an integer-valued function defined on the integers which satisfies \( f(m + f(f(n))) = -f(f(m+1)) - n \) for all integers \( m \) and \( n \). The polynomial \( g(n) \) has integer coefficients and satisfies \( g(n) = g(f(n)) \) for all \( n \). Find \( f(1991) \) and determine the most general form for \( g \). | -1992 | 7.03125 |
31,438 | Find the number of permutations \((b_1, b_2, b_3, b_4, b_5, b_6)\) of \((1,2,3,4,5,6)\) such that
\[
\frac{b_1 + 6}{2} \cdot \frac{b_2 + 5}{2} \cdot \frac{b_3 + 4}{2} \cdot \frac{b_4 + 3}{2} \cdot \frac{b_5 + 2}{2} \cdot \frac{b_6 + 1}{2} > 6!.
\] | 719 | 83.59375 |
31,439 | A parallelogram $ABCD$ is inscribed in the ellipse $\frac{x^{2}}{4}+y^{2}=1$, where the slope of the line $AB$ is $k_{1}=1$. Determine the slope of the line $AD$. | -\frac{1}{4} | 14.0625 |
31,440 | One TV was sold for a 12% profit and the other for a 12% loss at a selling price of 3080 yuan each. Determine the net profit or loss from these transactions. | -90 | 17.96875 |
31,441 | For rational numbers $x$, $y$, $a$, $t$, if $|x-a|+|y-a|=t$, then $x$ and $y$ are said to have a "beautiful association number" of $t$ with respect to $a$. For example, $|2-1|+|3-1|=3$, then the "beautiful association number" of $2$ and $3$ with respect to $1$ is $3$. <br/> $(1)$ The "beautiful association number" of $-1$ and $5$ with respect to $2$ is ______; <br/> $(2)$ If the "beautiful association number" of $x$ and $5$ with respect to $3$ is $4$, find the value of $x$; <br/> $(3)$ If the "beautiful association number" of $x_{0}$ and $x_{1}$ with respect to $1$ is $1$, the "beautiful association number" of $x_{1}$ and $x_{2}$ with respect to $2$ is $1$, the "beautiful association number" of $x_{2}$ and $x_{3}$ with respect to $3$ is $1$, ..., the "beautiful association number" of $x_{1999}$ and $x_{2000}$ with respect to $2000$ is $1$, ... <br/> ① The minimum value of $x_{0}+x_{1}$ is ______; <br/> ② What is the minimum value of $x_{1}+x_{2}+x_{3}+x_{4}+...+x_{2000}$? | 2001000 | 32.03125 |
31,442 | There are three flavors of chocolates in a jar: hazelnut, liquor, and milk. There are 12 chocolates that are not hazelnut, 18 chocolates that are not liquor, and 20 chocolates that are not milk. How many chocolates are there in total in the jar? | 50 | 0 |
31,443 | When 1524 shi of rice is mixed with an unknown amount of wheat, and in a sample of 254 grains, 28 are wheat grains, calculate the estimated amount of wheat mixed with this batch of rice. | 168 | 39.84375 |
31,444 | (1) Among the following 4 propositions:
① The converse of "If $a$, $G$, $b$ form a geometric sequence, then $G^2=ab$";
② The negation of "If $x^2+x-6\geqslant 0$, then $x > 2$";
③ In $\triangle ABC$, the contrapositive of "If $A > B$, then $\sin A > \sin B$";
④ When $0\leqslant \alpha \leqslant \pi$, if $8x^2-(8\sin \alpha)x+\cos 2\alpha\geqslant 0$ holds for $\forall x\in \mathbb{R}$, then the range of $\alpha$ is $0\leqslant \alpha \leqslant \frac{\pi}{6}$.
The numbers of the true propositions are ______.
(2) Given an odd function $f(x)$ whose graph is symmetric about the line $x=3$, and when $x\in [0,3]$, $f(x)=-x$, then $f(-16)=$ ______.
(3) The graph of the function $f(x)=a^{x-1}+4$ ($a > 0$ and $a\neq 1$) passes through a fixed point, then the coordinates of this point are ______.
(4) Given a point $P$ on the parabola $y^2=2x$, the minimum value of the sum of the distance from point $P$ to the point $(0,2)$ and the distance from $P$ to the directrix of the parabola is ______. | \frac{\sqrt{17}}{2} | 15.625 |
31,445 | The length of a rectangular yard exceeds twice its width by 30 feet, and the perimeter of the yard is 700 feet. What is the area of the yard in square feet? | 25955.56 | 3.125 |
31,446 | Given: $\cos\left(\alpha+ \frac{\pi}{4}\right) = \frac{3}{5}$, $\frac{\pi}{2} < \alpha < \frac{3\pi}{2}$, find $\cos\left(2\alpha+ \frac{\pi}{4}\right)$. | -\frac{31\sqrt{2}}{50} | 49.21875 |
31,447 | Determine the greatest number m such that the
system $x^2$ + $y^2$ = 1; | $x^3$ - $y^3$ |+|x-y|= $m^3$ has a solution. | \sqrt[3]{2} | 28.90625 |
31,448 | 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 area of this triangle is \_\_\_\_\_\_. | 12 \sqrt {6} | 0 |
31,449 | Consider the sum $$ S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|. $$ The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$ , satisfying $2c < d$ . Find the value of $c + d$ . | 3031 | 32.8125 |
31,450 | Screws are sold in packs of $10$ and $12$ . Harry and Sam independently go to the hardware store, and by coincidence each of them buys exactly $k$ screws. However, the number of packs of screws Harry buys is different than the number of packs Sam buys. What is the smallest possible value of $k$ ? | 60 | 96.875 |
31,451 | If the direction vectors of two skew lines $l_{1}$ and $l_{2}$ are $\overrightarrow{a}=(0,-1,-2)$ and $\overrightarrow{b}=(4,0,2)$, then the cosine value of the angle between the two skew lines $l_{1}$ and $l_{2}$ is ______. | \frac{2}{5} | 0 |
31,452 | If $a+b=1$, find the supremum of $$- \frac {1}{2a}- \frac {2}{b}.$$ | - \frac {9}{2} | 57.03125 |
31,453 | Four positive integers $p$, $q$, $r$, $s$ satisfy $p \cdot q \cdot r \cdot s = 9!$ and $p < q < r < s$. What is the smallest possible value of $s-p$? | 12 | 1.5625 |
31,454 | Determine the area, in square units, of triangle $PQR$, where the coordinates of the vertices are $P(-3, 4)$, $Q(4, 9)$, and $R(5, -3)$. | 44.5 | 8.59375 |
31,455 | A jacket was originally priced $\textdollar 100$ . The price was reduced by $10\%$ three times and increased by $10\%$ four times in some order. To the nearest cent, what was the final price? | 106.73 | 62.5 |
31,456 | Let the function $f(x)= \frac{ \sqrt{3}}{2}- \sqrt{3}\sin^2 \omega x-\sin \omega x\cos \omega x$ ($\omega > 0$) and the graph of $y=f(x)$ has a symmetry center whose distance to the nearest axis of symmetry is $\frac{\pi}{4}$.
$(1)$ Find the value of $\omega$; $(2)$ Find the maximum and minimum values of $f(x)$ in the interval $\left[\pi, \frac{3\pi}{2}\right]$ | -1 | 71.09375 |
31,457 | Find the number of addition problems in which a two digit number is added to a second two digit number to give a two digit answer, such as in the three examples: \[\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}2342\hline 65\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}3636\hline 72\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}4223\hline 65\end{tabular}\,.\] | 3240 | 81.25 |
31,458 | Consider a trapezoid field with base lengths of 120 meters and 180 meters and non-parallel sides each measuring 130 meters. The angles adjacent to the longer base are $60^\circ$. At harvest, the crops at any point in the field are brought to the nearest point on the field's perimeter. Determine the fraction of the crop that is brought to the longest base. | \frac{1}{2} | 7.03125 |
31,459 | The sum of the digits of the integer equal to \( 777777777777777^2 - 222222222222223^2 \) is | 74 | 1.5625 |
31,460 | Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ are perpendicular. If $DP= 15$ and $EQ = 20$, then what is ${DF}$? | \frac{20\sqrt{13}}{3} | 0.78125 |
31,461 | Find \( g(2021) \) if for any real numbers \( x \) and \( y \) the following equality holds:
\[ g(x-y) = 2021(g(x) + g(y)) - 2022xy \] | 2043231 | 22.65625 |
31,462 | On January 15 in the stormy town of Stormville, there is a $50\%$ chance of rain. Every day, the probability of it raining has a $50\%$ chance of being $\frac{2017}{2016}$ times that of the previous day (or $100\%$ if this new quantity is over $100\%$ ) and a $50\%$ chance of being $\frac{1007}{2016}$ times that of the previous day. What is the probability that it rains on January 20?
*2018 CCA Math Bonanza Lightning Round #3.3* | 243/2048 | 0.78125 |
31,463 | Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | 23 | 2.34375 |
31,464 | Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and the parabola $y^{2} = 4cx$, where $c = \sqrt{a^{2} + b^{2}}$, find the eccentricity of the hyperbola given that $|AB| = 4c$. | \sqrt{2} + 1 | 2.34375 |
31,465 | Rectangle ABCD has AB = 4 and BC = 3. Segment EF is constructed through B such that EF is perpendicular to DB, and A and C lie on DE and DF, respectively. Find the length of EF. | \frac{125}{12} | 20.3125 |
31,466 | A and B are playing a series of Go games, with the first to win 3 games declared the winner. Assuming in a single game, the probability of A winning is 0.6 and the probability of B winning is 0.4, with the results of each game being independent. It is known that in the first two games, A and B each won one game.
(1) Calculate the probability of A winning the match;
(2) Let $\xi$ represent the number of games played from the third game until the end of the match. Calculate the distribution and the mathematical expectation of $\xi$. | 2.48 | 16.40625 |
31,467 | Given $f(x) = \sin \left( \frac{\pi}{3}x \right)$, and the set $A = \{1, 2, 3, 4, 5, 6, 7, 8\}$. Now, choose any two distinct elements $s$ and $t$ from set $A$. Find out the number of possible pairs $(s, t)$ such that $f(s)\cdot f(t) = 0$. | 13 | 0.78125 |
31,468 | Xiao Ming, Xiao Hong, and Xiao Gang are three people whose ages are three consecutive even numbers. Their total age is 48 years old. What is the youngest age? What is the oldest age? | 18 | 78.125 |
31,469 | Given that the function $f(x)$ satisfies $f(x+y)=f(x)+f(y)$ for any $x, y \in \mathbb{R}$, and $f(x) < 0$ when $x > 0$, with $f(1)=-2$.
1. Determine the parity (odd or even) of the function $f(x)$.
2. When $x \in [-3, 3]$, does the function $f(x)$ have an extreme value (maximum or minimum)? If so, find the extreme value; if not, explain why. | -6 | 32.8125 |
31,470 | Let $P=\{1,2,\ldots,6\}$, and let $A$ and $B$ be two non-empty subsets of $P$. Find the number of pairs of sets $(A,B)$ such that the maximum number in $A$ is less than the minimum number in $B$. | 129 | 0 |
31,471 | Find the length of the diagonal and the area of a rectangle whose one corner is at (1, 1) and the opposite corner is at (9, 7). | 48 | 85.9375 |
31,472 | The sum of the three largest natural divisors of a natural number \( N \) is 10 times the sum of its three smallest natural divisors. Find all possible values of \( N \). | 40 | 27.34375 |
31,473 | A regular dodecagon \(Q_1 Q_2 \ldots Q_{12}\) is drawn in the coordinate plane with \(Q_1\) at \((2,0)\) and \(Q_7\) at \((4,0)\). If \(Q_n\) is the point \((x_n, y_n)\), compute the numerical value of the product:
\[
(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \ldots (x_{12} + y_{12} i).
\] | 531440 | 26.5625 |
31,474 | For all m and n satisfying \( 1 \leq n \leq m \leq 5 \), the polar equation \( \rho = \frac{1}{1 - C_{m}^{n} \cos \theta} \) represents how many different hyperbolas? | 10 | 32.03125 |
31,475 | Given that \( M \) is a subset of \(\{1, 2, 3, \cdots, 15\}\) such that the product of any 3 distinct elements of \( M \) is not a perfect square, determine the maximum possible number of elements in \( M \). | 11 | 8.59375 |
31,476 | Given points $A(\cos\alpha, \sin\alpha)$ and $B(\cos\beta, \sin\beta)$, where $\alpha, \beta$ are acute angles, and that $|AB| = \frac{\sqrt{10}}{5}$:
(1) Find the value of $\cos(\alpha - \beta)$;
(2) If $\tan \frac{\alpha}{2} = \frac{1}{2}$, find the values of $\cos\alpha$ and $\cos\beta$. | \frac{24}{25} | 57.8125 |
31,477 | If a positive integer \( n \) makes the equation \( x^{3} + y^{3} = z^{n} \) have a positive integer solution \( (x, y, z) \), then \( n \) is called a "good number." How many good numbers are there that do not exceed 2,019? | 1346 | 2.34375 |
31,478 | What is the coefficient of $x^3y^5$ in the expansion of $\left(\frac{4}{3}x - \frac{2y}{5}\right)^8$? | -\frac{114688}{84375} | 53.90625 |
31,479 | Circular arcs of radius 3 inches form a continuous pattern as shown. What is the area, in square inches, of the shaded region in a 2-foot length of this pattern? Each arc completes half of a circle. | 18\pi | 8.59375 |
31,480 | Given the equation $3x^{2}-4=-2x$, find the quadratic coefficient, linear coefficient, and constant term. | -4 | 1.5625 |
31,481 | Given the function $y=2\sin(2x+\frac{\pi}{3})$, its graph is symmetrical about the point $P(x\_0,0)$. If $x\_0\in[-\frac{\pi}{2},0]$, find the value of $x\_0$. | -\frac{\pi}{6} | 75.78125 |
31,482 | In an $n \times n$ matrix $\begin{pmatrix} 1 & 2 & 3 & … & n-2 & n-1 & n \\ 2 & 3 & 4 & … & n-1 & n & 1 \\ 3 & 4 & 5 & … & n & 1 & 2 \\ … & … & … & … & … & … & … \\ n & 1 & 2 & … & n-3 & n-2 & n-1\\end{pmatrix}$, if the number at the $i$-th row and $j$-th column is denoted as $a_{ij}(i,j=1,2,…,n)$, then the sum of all $a_{ij}$ that satisfy $2i < j$ when $n=9$ is _____ . | 88 | 38.28125 |
31,483 | Juan rolls a fair regular decagonal die marked with the numbers 1 through 10. Then Amal rolls a fair eight-sided die. What is the probability that the product of the two rolls is a multiple of 4? | \frac{2}{5} | 19.53125 |
31,484 | The surface of a 3 x 3 x 3 Rubik's Cube consists of 54 cells. What is the maximum number of cells you can mark such that the marked cells do not share any vertices? | 14 | 9.375 |
31,485 | Given: $\sqrt{23.6}=4.858$, $\sqrt{2.36}=1.536$, then calculate the value of $\sqrt{0.00236}$. | 0.04858 | 96.875 |
31,486 | There exist constants $b_1,$ $b_2,$ $b_3,$ $b_4,$ $b_5,$ $b_6,$ $b_7$ such that
\[\cos^7 \theta = b_1 \cos \theta + b_2 \cos 2 \theta + b_3 \cos 3 \theta + b_4 \cos 4 \theta + b_5 \cos 5 \theta + b_6 \cos 6 \theta + b_7 \cos 7 \theta\]for all angles $\theta.$ Find $b_1^2 + b_2^2 + b_3^2 + b_4^2 + b_5^2 + b_6^2 + b_7^2.$ | \frac{1555}{4096} | 0 |
31,487 | Given the function $f(x)=\sin(2x+\frac{π}{6})+2\sin^2x$.
$(1)$ Find the center of symmetry and the interval of monotonic decrease of the function $f(x)$;
$(2)$ If the graph of $f(x)$ is shifted to the right by $\frac{π}{12}$ units, resulting in the graph of the function $g(x)$, find the maximum and minimum values of the function $g(x)$ on the interval $[0,\frac{π}{2}]$. | -\frac{\sqrt{3}}{2}+1 | 2.34375 |
31,488 | From the set $\{1, 2, 3, \ldots, 10\}$, select 3 different elements such that the sum of these three numbers is a multiple of 3, and the three numbers cannot form an arithmetic sequence. Calculate the number of ways to do this. | 22 | 0 |
31,489 | Sets $A$, $B$, and $C$, depicted in the Venn diagram, are such that the total number of elements in set $A$ is three times the total number of elements in set $B$. Their intersection has 1200 elements, and altogether, there are 4200 elements in the union of $A$, $B$, and $C$. If set $C$ intersects only with set $A$ adding 300 more elements to the union, how many elements are in set $A$?
[asy]
label("$A$", (2,67));
label("$B$", (80,67));
label("$C$", (41,10));
draw(Circle((30,45), 22));
draw(Circle((58, 45), 22));
draw(Circle((44, 27), 22));
label("1200", (44, 45));
label("300", (44, 27));
[/asy] | 3825 | 9.375 |
31,490 | How many positive integers, not exceeding 200, are multiples of 3 or 5 but not 6? | 73 | 31.25 |
31,491 | Right triangle DEF has leg lengths DE = 18 and EF = 24. If the foot of the altitude from vertex E to hypotenuse DF is F', then find the number of line segments with integer length that can be drawn from vertex E to a point on hypotenuse DF. | 10 | 7.8125 |
31,492 | Archer Zhang Qiang has the probabilities of hitting the 10-ring, 9-ring, 8-ring, 7-ring, and below 7-ring in a shooting session as 0.24, 0.28, 0.19, 0.16, and 0.13, respectively. Calculate the probability that this archer in a single shot:
(1) Hits either the 10-ring or the 9-ring;
(2) Hits at least the 7-ring;
(3) Hits a ring count less than 8. | 0.29 | 41.40625 |
31,493 | Given an ellipse $T$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\frac{\sqrt{3}}{2}$, a line passing through the right focus $F$ with slope $k (k > 0)$ intersects $T$ at points $A$ and $B$. If $\overline{AF} = 3\overline{FB}$, determine the value of $k$. | \sqrt{2} | 21.09375 |
31,494 | In a particular country, the state of Sunland issues license plates with a format of one letter, followed by three digits, and then two letters (e.g., A123BC). Another state, Moonland, issues license plates where the format consists of two digits, followed by two letters, and then two more digits (e.g., 12AB34). Assuming all 10 digits and all 26 letters are equally likely to appear in their respective positions, calculate how many more license plates Sunland can issue compared to Moonland, given that Sunland always uses the letter 'S' as the starting letter in their license plates. | 6084000 | 57.8125 |
31,495 | Assume $x$, $y$, $z$, and $w$ are positive integers such that $x^3 = y^2$, $z^5 = w^4$, and $z - x = 31$. Determine $w - y$. | -2351 | 5.46875 |
31,496 | If $x^{2}+\left(m-1\right)x+9$ is a perfect square trinomial, then the value of $m$ is ____. | -5 | 69.53125 |
31,497 | Given the power function $y=(m^2-5m-5)x^{2m+1}$ is a decreasing function on $(0, +\infty)$, then the real number $m=$ . | -1 | 50.78125 |
31,498 | Given a circle with radius $4$, find the area of the region formed by all line segments of length $4$ that are tangent to this circle at their midpoints.
A) $2\pi$
B) $4\pi$
C) $8\pi$
D) $16\pi$ | 4\pi | 33.59375 |
31,499 | Suppose there are 3 counterfeit coins of equal weight mixed with 12 genuine coins. All counterfeit coins weigh differently from the genuine coins. A pair of coins is selected at random without replacement from the 15 coins, followed by selecting a second pair from the remaining 13 coins. The combined weight of the first pair equals the combined weight of the second pair. What is the probability that all four coins selected are genuine?
A) $\frac{7}{11}$
B) $\frac{9}{13}$
C) $\frac{11}{15}$
D) $\frac{15}{19}$
E) $\frac{15}{16}$ | \frac{15}{19} | 5.46875 |
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