Unnamed: 0
int64 0
40.3k
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stringlengths 10
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|---|---|---|---|
28,000 | Dave's sister Amy baked $4$ dozen pies. Among these:
- $5/8$ of them contained chocolate.
- $3/4$ of them contained marshmallows.
- $2/3$ of them contained cayenne.
- $1/4$ of them contained salted soy nuts.
Additionally, all pies with salted soy nuts also contained marshmallows. How many pies at most did not contain any of these ingredients? | 16 | 2.34375 |
28,001 | Four friends make cookies from the same amount of dough with the same thickness. Art's cookies are circles with a radius of 2 inches, and Trisha's cookies are squares with a side length of 4 inches. If Art can make 18 cookies in his batch, determine the number of cookies Trisha will make in one batch. | 14 | 96.875 |
28,002 | Given that two children, A and B, and three adults, 甲, 乙, and 丙, are standing in a line, A is not at either end, and exactly two of the three adults are standing next to each other. The number of different arrangements is $\boxed{\text{answer}}$. | 48 | 39.84375 |
28,003 | In the diagram, each of \( \triangle W X Z \) and \( \triangle X Y Z \) is an isosceles right-angled triangle. The length of \( W X \) is \( 6 \sqrt{2} \). The perimeter of quadrilateral \( W X Y Z \) is closest to | 23 | 12.5 |
28,004 | Find \( x_{1000} \) if \( x_{1} = 4 \), \( x_{2} = 6 \), and for any natural \( n \geq 3 \), \( x_{n} \) is the smallest composite number greater than \( 2 x_{n-1} - x_{n-2} \). | 2002 | 11.71875 |
28,005 | Given cos($$α+ \frac {π}{6}$$)= $$\frac {1}{3}$$, find the value of sin($$ \frac {5π}{6}+2α$$). | -$$\frac {7}{9}$$ | 0 |
28,006 | When $\frac{1}{909}$ is expressed as a decimal, what is the sum of the first 30 digits after the decimal point? | 14 | 3.125 |
28,007 | Let $a_1$, $a_2$, $a_3$, $d_1$, $d_2$, and $d_3$ be real numbers such that for every real number $x$, we have
\[
x^8 - x^6 + x^4 - x^2 + 1 = (x^2 + a_1 x + d_1)(x^2 + a_2 x + d_2)(x^2 + a_3 x + d_3)(x^2 + 1).
\]
Compute $a_1 d_1 + a_2 d_2 + a_3 d_3$. | -1 | 7.8125 |
28,008 | There is a parking lot with $10$ empty spaces. Three different cars, A, B, and C, are going to park in such a way that each car has empty spaces on both sides, and car A must be parked between cars B and C. How many different parking arrangements are there? | 40 | 3.90625 |
28,009 | Let $\overrightarrow{m} = (\sin(x - \frac{\pi}{3}), 1)$ and $\overrightarrow{n} = (\cos x, 1)$.
(1) If $\overrightarrow{m} \parallel \overrightarrow{n}$, find the value of $\tan x$.
(2) If $f(x) = \overrightarrow{m} \cdot \overrightarrow{n}$, where $x \in [0, \frac{\pi}{2}]$, find the maximum and minimum values of $f(x)$. | 1 - \frac{\sqrt{3}}{2} | 6.25 |
28,010 | How many distinct sequences of four letters can be made from the letters in "EXAMPLE" if each letter can be used only once and each sequence must begin with X and not end with E? | 80 | 5.46875 |
28,011 | Given that \( f \) is a mapping from the set \( M = \{a, b, c\} \) to the set \( N = \{-3, -2, \cdots, 3\} \). Determine the number of mappings \( f \) that satisfy
$$
f(a) + f(b) + f(c) = 0
$$ | 37 | 100 |
28,012 | Given a cone and a cylinder made of rubber, the cone has a base radius of $5$ and a height of $4$, while the cylinder has a base radius of $2$ and a height of $8$. If they are remade into a new cone and a new cylinder with the same base radius, while keeping the total volume and height unchanged, find the new base radius. | r = \sqrt{7} | 62.5 |
28,013 | No two people stand next to each other, find the probability that, out of the ten people sitting around a circular table, no two adjacent people will stand after flipping their fair coins. | \dfrac{123}{1024} | 0 |
28,014 | Suppose $ABC$ is a scalene right triangle, and $P$ is the point on hypotenuse $\overline{AC}$ such that $\angle{ABP} = 30^{\circ}$. Given that $AP = 3$ and $CP = 1$, compute the area of triangle $ABC$. | \frac{12}{5} | 7.8125 |
28,015 | Given $a\in R$, $b \gt 0$, $a+b=2$, then the minimum value of $\frac{1}{2|a|}+\frac{|a|}{b}$ is ______. | \frac{3}{4} | 28.125 |
28,016 | Given $\cos \left( \frac {π}{4}-α \right) = \frac {3}{5}$, and $\sin \left( \frac {5π}{4}+β \right) = - \frac {12}{13}$, with $α \in \left( \frac {π}{4}, \frac {3π}{4} \right)$ and $β \in (0, \frac {π}{4})$, find the value of $\sin (α+β)$. | \frac {56}{65} | 16.40625 |
28,017 | Define
\[ A' = \frac{1}{1^2} + \frac{1}{7^2} - \frac{1}{11^2} - \frac{1}{13^2} + \frac{1}{19^2} + \frac{1}{23^2} - \dotsb, \]
which omits all terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 5, and
\[ B' = \frac{1}{5^2} - \frac{1}{25^2} + \frac{1}{35^2} - \frac{1}{55^2} + \frac{1}{65^2} - \frac{1}{85^2} + \dotsb, \]
which includes only terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 5.
Determine $\frac{A'}{B'}.$ | 26 | 0.78125 |
28,018 | 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 $\triangle PQR$. | 1818 | 0 |
28,019 | Given that point $M$ lies on the circle $C:x^{2}+y^{2}-4x-14y+45=0$, and point $Q(-2,3)$.
(1) If $P(a,a+1)$ is on circle $C$, find the length of segment $PQ$ and the slope of line $PQ$;
(2) Find the maximum and minimum values of $|MQ|$;
(3) If $M(m,n)$, find the maximum and minimum values of $\frac{n-{3}}{m+{2}}$. | 2- \sqrt {3} | 0 |
28,020 | A large cube is made up of 27 small cubes. A plane is perpendicular to one of the diagonals of this large cube and bisects the diagonal. How many small cubes are intersected by this plane? | 19 | 11.71875 |
28,021 | Given the function $f(x) = \frac{bx}{\ln x} - ax$, where $e$ is the base of the natural logarithm.
(1) If the equation of the tangent line to the graph of the function $f(x)$ at the point $({e}^{2}, f({e}^{2}))$ is $3x + 4y - e^{2} = 0$, find the values of the real numbers $a$ and $b$.
(2) When $b = 1$, if there exist $x_{1}, x_{2} \in [e, e^{2}]$ such that $f(x_{1}) \leq f'(x_{2}) + a$ holds, find the minimum value of the real number $a$. | \frac{1}{2} - \frac{1}{4e^{2}} | 0 |
28,022 | Given that the sides opposite to the internal angles A, B, and C of triangle ABC are a, b, and c respectively, if -c cosB is the arithmetic mean of $\sqrt {2}$a cosB and $\sqrt {2}$b cosA, find the maximum value of sin2A•tan²C. | 3 - 2\sqrt{2} | 0 |
28,023 | Let $x, y,$ and $z$ be positive real numbers. Find the minimum value of:
\[
\frac{(x^2 + 4x + 2)(y^2 + 4y + 2)(z^2 + 4z + 2)}{xyz}.
\] | 216 | 7.03125 |
28,024 | Given that real numbers \( a \) and \( b \) are such that the equation \( a x^{3} - x^{2} + b x - 1 = 0 \) has three positive real roots, find the minimum value of \( P = \frac{5 a^{2} - 3 a b + 2}{a^{2}(b - a)} \) for all such \( a \) and \( b \). | 12 \sqrt{3} | 2.34375 |
28,025 | Given a moving large circle $\odot O$ tangent externally to a fixed small circle $\odot O_{1}$ with radius 3 at point $P$, $AB$ is the common external tangent of the two circles with $A$ and $B$ as the points of tangency. A line $l$ parallel to $AB$ is tangent to $\odot O_{1}$ at point $C$ and intersects $\odot O$ at points $D$ and $E$. Find $C D \cdot C E = \quad$. | 36 | 3.125 |
28,026 | Given the numbers 1, 2, 3, 4, find the probability that $\frac{a}{b}$ is not an integer, where $a$ and $b$ are randomly selected numbers from the set $\{1, 2, 3, 4\}$. | \frac{2}{3} | 0 |
28,027 | Solve the application problem by setting up equations:<br/>A gift manufacturing factory receives an order for a batch of teddy bears and plans to produce them in a certain number of days. If they produce $20$ teddy bears per day, they will be $100$ short of the order. If they produce $23$ teddy bears per day, they will exceed the order by $20$. Find out how many teddy bears were ordered and how many days were originally planned to complete the task. | 40 | 50 |
28,028 | Let \( P \) be a point inside regular pentagon \( ABCDE \) such that \( \angle PAB = 48^\circ \) and \( \angle PDC = 42^\circ \). Find \( \angle BPC \), in degrees. | 84 | 6.25 |
28,029 | In a pile of apples, the ratio of large apples to small apples is $9:1$. Now, a fruit sorting machine is used for screening, with a probability of $5\%$ that a large apple is sorted as a small apple and a probability of $2\%$ that a small apple is sorted as a large apple. Calculate the probability that a "large apple" selected from the sorted apples is indeed a large apple. | \frac{855}{857} | 59.375 |
28,030 | Consider a central regular hexagon surrounded by six regular hexagons, each of side length $\sqrt{2}$. Three of these surrounding hexagons are selected at random, and their centers are connected to form a triangle. Calculate the area of this triangle. | 2\sqrt{3} | 5.46875 |
28,031 | In the Cartesian coordinate system \(xOy\), the set of points \(K=\{(x, y) \mid x, y=-1,0,1\}\). Three points are randomly selected from \(K\). What is the probability that the distance between any two of these three points does not exceed 2? | 5/14 | 67.96875 |
28,032 | Let $a,b,c,d$ be positive integers such that $a+c=20$ and $\frac{a}{b}+\frac{c}{d}<1$ . Find the maximum possible value of $\frac{a}{b}+\frac{c}{d}$ . | 20/21 | 3.125 |
28,033 | Let \( M \) be a set of \( n \) points in the plane such that:
1. There are 7 points in \( M \) that form the vertices of a convex heptagon.
2. For any 5 points in \( M \), if these 5 points form a convex pentagon, then the interior of this convex pentagon contains at least one point from \( M \).
Find the minimum value of \( n \). | 11 | 5.46875 |
28,034 | Given the function $f(x) = 2\sin\omega x \cdot \cos(\omega x) + (\omega > 0)$ has the smallest positive period of $4\pi$.
(1) Find the value of the positive real number $\omega$;
(2) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and it satisfies $2b\cos A = a\cos C + c\cos A$. Find the value of $f(A)$. | \frac{\sqrt{3}}{2} | 0 |
28,035 | A traffic light cycles repeatedly in the following order: green for 45 seconds, then yellow for 5 seconds, and red for 50 seconds. Cody picks a random five-second time interval to observe the light. What is the probability that the color changes while he is watching? | \frac{3}{20} | 45.3125 |
28,036 | In the diagram, three circles each with a radius of 5 units intersect at exactly one common point, which is the origin. Calculate the total area in square units of the shaded region formed within the triangular intersection of the three circles. Express your answer in terms of $\pi$.
[asy]
import olympiad; import geometry; size(100); defaultpen(linewidth(0.8));
filldraw(Circle((0,0),5));
filldraw(Circle((4,-1),5));
filldraw(Circle((-4,-1),5));
[/asy] | \frac{150\pi - 75\sqrt{3}}{12} | 0 |
28,037 | A ticket contains six digits \(a, b, c, d, e, f\). This ticket is said to be "lucky" if \(a + b + c = d + e + f\). How many lucky tickets are there (including the ticket 000000)? | 55252 | 97.65625 |
28,038 | Given the function $y=\cos (2x+\frac{\pi }{6})$, a point $P(\frac{\pi }{4},t)$ on its graph is moved right by $m (m>0)$ units to a new point $P'$, where $P'$ lies on the graph of the function $y=\cos 2x$. Determine the value of $t$ and the minimum value of $m$. | \frac{\pi}{12} | 43.75 |
28,039 | Given a hyperbola with asymptotes $2x \pm y=0$, that passes through the intersection of the lines $x+y-3=0$ and $2x-y+3t=0$, where $-2 \leq t \leq 5$. Find the maximum possible length of the real axis of the hyperbola. | 4\sqrt{3} | 3.125 |
28,040 | Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $(2\overrightarrow{a}+\overrightarrow{b})\bot \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{2\pi}{3} | 96.09375 |
28,041 | Given that $f(x) = \frac{1}{3^x + \sqrt{3}}$, find the value of $f(-12) + f(-11) + f(-10) + ... + f(0) + ... + f(11) + f(12) + f(13)$. | \frac{13\sqrt{3}}{3} | 13.28125 |
28,042 | Find the length of side $XY$ in the triangle below.
[asy]
unitsize(1inch);
pair X,Y,Z;
X = (0,0);
Y= (2,0);
Z = (0,sqrt(3));
draw (X--Y--Z--X,linewidth(0.9));
draw(rightanglemark(Y,X,Z,3));
label("$X$",X,S);
label("$Y$",Y,S);
label("$Z$",Z,N);
label("$12$",Z/2,W);
label("$60^\circ$",(1.2,0),N);
[/asy] | 24 | 9.375 |
28,043 | In a city, a newspaper stand buys the "Evening News" from the publisher at a price of $0.20 per copy and sells it at $0.30 per copy. Unsold newspapers can be returned to the publisher at $0.05 per copy. In a month (considered as 30 days), there are 20 days when 400 copies can be sold each day, and for the remaining 10 days, only 250 copies can be sold each day. However, the number of copies bought from the publisher must be the same every day. How many copies should the stand owner buy from the publisher each day to maximize the monthly profit? And calculate the maximum profit he can earn in a month? | 825 | 10.9375 |
28,044 | Dr. Math's house number $WXYZ$ is a four-digit number where each digit $W$, $X$, $Y$, and $Z$ is non-zero, and the two portions of the house number, $WX$ and $YZ$, form two-digit primes. Every prime number selected must be less than 50. Additionally, the sum of the digits in $YZ$ must be even, and $WX$ and $YZ$ must be different. How many such house numbers $WXYZ$ are possible? | 30 | 0 |
28,045 | In a bag are all natural numbers less than or equal to $999$ whose digits sum to $6$ . What is the probability of drawing a number from the bag that is divisible by $11$ ? | 1/7 | 35.9375 |
28,046 | A line \( l \), parallel to the diagonal \( AC_1 \) of a unit cube \( ABCDA_1B_1C_1D_1 \), is equidistant from the lines \( BD \), \( A_1D_1 \), and \( CB_1 \). Find the distances from the line \( l \) to these lines. | \frac{\sqrt{2}}{6} | 0 |
28,047 | Find the smallest positive integer \( n \) such that every \( n \)-element subset of \( S = \{1, 2, \ldots, 150\} \) contains 4 numbers that are pairwise coprime (it is known that there are 35 prime numbers in \( S \)). | 111 | 10.9375 |
28,048 | Given $\overrightarrow{a}=(2\sin x,1)$ and $\overrightarrow{b}=(2\cos (x-\frac{\pi }{3}),\sqrt{3})$, let $f(x)=\overrightarrow{a}\bullet \overrightarrow{b}-2\sqrt{3}$.
(I) Find the smallest positive period and the zeros of $f(x)$;
(II) Find the maximum and minimum values of $f(x)$ on the interval $[\frac{\pi }{24},\frac{3\pi }{4}]$. | -\sqrt{2} | 52.34375 |
28,049 | Sasha and Masha each picked a natural number and communicated them to Vasya. Vasya wrote the sum of these numbers on one piece of paper and their product on another piece, then hid one of the pieces and showed the other (on which the number 2002 was written) to Sasha and Masha. Seeing this number, Sasha said he did not know the number Masha had picked. Upon hearing this, Masha said she did not know the number Sasha had picked. What number did Masha pick? | 1001 | 20.3125 |
28,050 | A box contains 5 white balls and 6 black balls. You draw them out of the box, one at a time. What is the probability that the first four draws alternate in colors, starting with a black ball? | \frac{2}{33} | 1.5625 |
28,051 | Consider a rectangle $ABCD$, and inside it are four squares with non-overlapping interiors. Two squares have the same size and an area of 4 square inches each, located at corners $A$ and $C$ respectively. There is another small square with an area of 1 square inch, and a larger square, twice the side length of the smaller one, both adjacent to each other and located centrally from $B$ to $D$. Calculate the area of rectangle $ABCD$. | 12 | 2.34375 |
28,052 | Given the numbers \( x, y, z \in \left[0, \frac{\pi}{2}\right] \), find the maximum value of the expression
\[ A = \sin(x-y) + \sin(y-z) + \sin(z-x). \] | \sqrt{2} - 1 | 61.71875 |
28,053 | The dimensions of a part on a drawing are $7{}_{-0.02}^{+0.05}$ (unit: $mm$), indicating that the maximum requirement for processing this part should not exceed ______, and the minimum should not be less than ______. | 6.98 | 85.9375 |
28,054 | Natural numbers are arranged according to the following pattern:
\begin{tabular}{cccc}
$1-2$ & 5 & 10 & 17 \\
$\mid$ & $\mid$ & $\mid$ & $\mid$ \\
$4-3$ & 6 & 11 & 18 \\
& $\mid$ & $\mid$ & $\mid$ \\
$9-8-7$ & 12 & 19 \\
& $\mid$ & $\mid$ \\
$16-15-14-13$ & 20 \\
$25-24-23-22-21$
\end{tabular}
Then the number at the 2002nd row from the top and the 2003rd column from the left is ( ). | 2002 \times 2003 | 0 |
28,055 | A shape is created by aligning five unit cubes in a straight line. Then, one additional unit cube is attached to the top of the second cube in the line and another is attached beneath the fourth cube in the line. Calculate the ratio of the volume to the surface area. | \frac{1}{4} | 1.5625 |
28,056 | How many positive odd integers greater than 1 and less than $200$ are square-free? | 82 | 0 |
28,057 | Given positive real numbers \(a\) and \(b\) that satisfy \(ab(a+b) = 4\), find the minimum value of \(2a + b\). | 2\sqrt{3} | 31.25 |
28,058 | Determine the sum of all integer values $n$ for which $\binom{25}{n} + \binom{25}{12} = \binom{26}{13}$. | 25 | 46.09375 |
28,059 |
Let \( M \) be a set composed of a finite number of positive integers,
\[
M = \bigcup_{i=1}^{20} A_i = \bigcup_{i=1}^{20} B_i, \text{ where}
\]
\[
A_i \neq \varnothing, B_i \neq \varnothing \ (i=1,2, \cdots, 20)
\]
satisfying the following conditions:
1. For any \( 1 \leqslant i < j \leqslant 20 \),
\[
A_i \cap A_j = \varnothing, \ B_i \cap B_j = \varnothing;
\]
2. For any \( 1 \leqslant i \leqslant 20, \ 1 \leqslant j \leqslant 20 \), if \( A_i \cap B_j = \varnothing \), then \( \left|A_i \cup B_j\right| \geqslant 18 \).
Find the minimum number of elements in the set \( M \) (denoted as \( |X| \) representing the number of elements in set \( X \)). | 180 | 60.15625 |
28,060 | Cube $ABCDEFGH,$ labeled as shown below, has edge length $2$ and is cut by a plane passing through vertex $D$ and the midpoints $M$ and $N$ of $\overline{AB}$ and $\overline{CG}$ respectively. The plane divides the cube into two solids. Find the volume of the smaller of the two solids. | \frac{1}{6} | 0.78125 |
28,061 | Pat wrote a strange example on the board:
$$
550+460+359+340=2012 .
$$
Mat wanted to correct it, so he searched for an unknown number to add to each of the five numbers listed, so that the example would be numerically correct. What was that number?
Hint: How many numbers does Mat add to the left side and how many to the right side of the equation? | 75.75 | 30.46875 |
28,062 | The first three stages of a pattern are shown below, where each line segment represents a straw. If the pattern continues such that at each successive stage, four straws are added to the previous arrangement, how many straws are necessary to create the arrangement for the 100th stage? | 400 | 39.0625 |
28,063 | Compute the integer $k > 3$ for which
\[\log_{10} (k - 3)! + \log_{10} (k - 2)! + 3 = 2 \log_{10} k!.\] | 10 | 2.34375 |
28,064 | Find the smallest possible value of $x$ in the simplified form $x=\frac{a+b\sqrt{c}}{d}$ if $\frac{7x}{5}-2=\frac{4}{x}$, where $a, b, c,$ and $d$ are integers. What is $\frac{acd}{b}$? | -5775 | 85.15625 |
28,065 | In the 2011 Zhejiang Province Pilot Module Examination, there were a total of 18 questions. Each examinee was required to choose 6 questions to answer. Examinee A would definitely not choose questions 1, 2, 9, 15, 16, 17, and 18, while Examinee B would definitely not choose questions 3, 9, 15, 16, 17, and 18. Moreover, the 6 questions chosen by A and B had none in common. The total number of ways to select questions that meet these conditions is ______. | 462 | 33.59375 |
28,066 | Veronica has 6 marks on her report card.
The mean of the 6 marks is 74.
The mode of the 6 marks is 76.
The median of the 6 marks is 76.
The lowest mark is 50.
The highest mark is 94.
Only one mark appears twice, and no mark appears more than twice.
Assuming all of her marks are integers, the number of possibilities for her second lowest mark is: | 17 | 1.5625 |
28,067 | Consider the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$. Its right focus and one endpoint of its conjugate axis are $F$ and $A$, respectively. Let $P$ be a point on the left branch of the hyperbola $C$. If the minimum value of the perimeter of $\triangle APF$ is $6b$, then find the eccentricity of the hyperbola $C$. | \frac{\sqrt{85}}{7} | 9.375 |
28,068 | Given the function $f(x)=\sin \left(2x+\frac{\pi }{3}\right)+\sin \left(2x-\frac{\pi }{3}\right)+2\cos ^{2}(x)-1$, where $x\in R$.
(1) Simplify the function $f(x)$ in the form of $A\sin (\omega x+\phi )$ $(A,\omega > 0,0 < \phi < \frac{\pi }{2})$
(2) Find the maximum and minimum values of the function $f(x)$ in the interval $\left[-\frac{\pi }{4},\frac{\pi }{4}\right]$. | -1 | 55.46875 |
28,069 | Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=10$. A beam of light is emitted from vertex $A$ and reflects off face $BCFG$ at point $P$, which is 3 units from $\overline{BG}$ and 4 units from $\overline{BC}$. The beam continues its path, reflecting off the faces of the cube. The length of the light path from when it leaves point $A$ until it reaches another vertex of the cube for the first time is expressed as $r\sqrt{s}$, where $r$ and $s$ are integers and $s$ is not divisible by the square of any prime. Determine $r+s$. | 55 | 7.03125 |
28,070 | Rhombus $ABCD$ is inscribed in rectangle $WXYZ$ such that vertices $A$, $B$, $C$, and $D$ are on sides $\overline{WX}$, $\overline{XY}$, $\overline{YZ}$, and $\overline{ZW}$, respectively. It is given that $WA=12$, $XB=9$, $BD=15$, and the diagonal $AC$ of rhombus equals side $XY$ of the rectangle. Calculate the perimeter of rectangle $WXYZ$. | 66 | 0.78125 |
28,071 | If two circles $(x-m)^2+y^2=4$ and $(x+1)^2+(y-2m)^2=9$ are tangent internally, then the real number $m=$ ______ . | -\frac{2}{5} | 24.21875 |
28,072 | Fifteen square tiles with side 10 units long are arranged as shown. An ant walks along the edges of the tiles, always keeping a black tile on its left. Find the shortest distance that the ant would walk in going from point \( P \) to point \( Q \). | 80 | 6.25 |
28,073 | Let \( x \) and \( y \) be real numbers such that
\[
1 < \frac{x - y}{x + y} < 3.
\]
If \( \frac{x}{y} \) is an integer, what is its value? | -2 | 31.25 |
28,074 | Arrange the 9 numbers 12, 13, ..., 20 in a row such that the sum of every three consecutive numbers is a multiple of 3. How many such arrangements are there? | 216 | 0 |
28,075 | Determine the integer $x$ that satisfies the following set of congruences:
\begin{align*}
4+x &\equiv 3^2 \pmod{2^3} \\
6+x &\equiv 4^2 \pmod{3^3} \\
8+x &\equiv 6^2 \pmod{5^3}
\end{align*}
Find the remainder when $x$ is divided by $120$. | 37 | 0.78125 |
28,076 | Let $g$ be defined by \[g(x) = \left\{
\begin{array}{cl}
x+3 & \text{ if } x \leq 2, \\
x^2 - 4x + 5 & \text{ if } x > 2.
\end{array}
\right.\]Calculate $g^{-1}(1)+g^{-1}(6)+g^{-1}(11)$. | 2 + \sqrt{5} + \sqrt{10} | 94.53125 |
28,077 | Given an angle $α$ whose terminal side passes through the point $(-1, \sqrt{2})$, find the values of $\tan α$ and $\cos 2α$. | -\frac{1}{3} | 64.0625 |
28,078 | The union of sets \( A \) and \( B \), \( A \cup B = \{a_1, a_2, a_3\} \), and \( A \neq B \). When \( (A, B) \) and \( (B, A) \) are considered as different pairs, how many such pairs \( (A, B) \) are there? | 27 | 1.5625 |
28,079 | In the Cartesian coordinate system $xOy$, the parametric equation of line $l_{1}$ is $\begin{cases} x=t- \sqrt {3} \\ y=kt\end{cases}$ (where $t$ is the parameter), and the parametric equation of line $l_{2}$ is $\begin{cases} x= \sqrt {3}-m \\ y= \frac {m}{3k}\end{cases}$ (where $m$ is the parameter). Let $p$ be the intersection point of $l_{1}$ and $l_{2}$. When $k$ varies, the trajectory of $p$ is curve $C_{1}$
(Ⅰ) Write the general equation and parametric equation of $C_{1}$;
(Ⅱ) Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis. Suppose the polar equation of curve $C_{2}$ is $p\sin (\theta+ \frac {\pi}{4})=4 \sqrt {2}$. Let $Q$ be a moving point on curve $C_{1}$, find the minimum distance from point $Q$ to $C_{2}$. | 3 \sqrt {2} | 0 |
28,080 | Suppose \(a\), \(b\), and \(c\) are real numbers such that:
\[
\frac{ac}{a + b} + \frac{ba}{b + c} + \frac{cb}{c + a} = -12
\]
and
\[
\frac{bc}{a + b} + \frac{ca}{b + c} + \frac{ab}{c + a} = 15.
\]
Compute the value of:
\[
\frac{a}{a + b} + \frac{b}{b + c} + \frac{c}{c + a}.
\] | -12 | 0 |
28,081 | Compute the following expression:
\[ 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4)))))))) \] | 1398100 | 3.125 |
28,082 | The axis cross-section $SAB$ of a cone with an equal base triangle side length of 2, $O$ as the center of the base, and $M$ as the midpoint of $SO$. A moving point $P$ is on the base of the cone (including the circumference). If $AM \perp MP$, then the length of the trajectory formed by point $P$ is ( ). | $\frac{\sqrt{7}}{2}$ | 0 |
28,083 | Find the maximum value of the expression \( x + y \) if \( (2 \sin x - 1)(2 \cos y - \sqrt{3}) = 0 \), \( x \in [0, \frac{3\pi}{2}] \), \( y \in [\pi, 2\pi] \). | \frac{8\pi}{3} | 57.8125 |
28,084 | A finite set $\{a_1, a_2, ... a_k\}$ of positive integers with $a_1 < a_2 < a_3 < ... < a_k$ is named *alternating* if $i+a$ for $i = 1, 2, 3, ..., k$ is even. The empty set is also considered to be alternating. The number of alternating subsets of $\{1, 2, 3,..., n\}$ is denoted by $A(n)$ .
Develop a method to determine $A(n)$ for every $n \in N$ and calculate hence $A(33)$ . | 5702887 | 3.125 |
28,085 | Let $M$ denote the number of $9$-digit positive integers in which the digits are in increasing order, given that repeated digits are allowed and the digit ‘0’ is permissible. Determine the remainder when $M$ is divided by $1000$. | 620 | 91.40625 |
28,086 | Suppose $X$ and $Y$ are digits in base $d > 8$ such that $\overline{XY}_d + \overline{XX}_d = 234_d$. Find $X_d - Y_d$ in base $d$. | -2 | 2.34375 |
28,087 | A list of seven positive integers has a median of 5 and a mean of 15. What is the maximum possible value of the list's largest element? | 87 | 57.03125 |
28,088 | In a cube \(A B C D-A_{1} B_{1} C_{1} D_{1}\) with edge length 1, \(P, Q, R\) are the midpoints of edges \(A B, A D,\) and \(A A_1\) respectively. A right triangular prism is constructed with \(\triangle PQR\) as its base, such that the vertices of the other base also lie on the surface of the cube. Find the volume of this right triangular prism. | \frac{3}{16} | 0.78125 |
28,089 | A certain middle school assigns numbers to each student, where the last digit indicates the gender of the student: 1 for male and 2 for female. If 028432 represents "a female student who is number 43 in class 8 and enrolled in the year 2002," then the number for a male student who is number 23 in class 6 and enrolled in the year 2008 is. | 086231 | 8.59375 |
28,090 | Square the numbers \(a=1001\) and \(b=1001001\). Extract the square root of the number \(c=1002003004005004003002001\). | 1001001001001 | 78.90625 |
28,091 | In triangle $XYZ$ where $XY=60$ and $XZ=15$, the area of the triangle is given as $225$. Let $W$ be the midpoint of $\overline{XY}$, and $V$ be the midpoint of $\overline{XZ}$. The angle bisector of $\angle YXZ$ intersects $\overline{WV}$ and $\overline{YZ}$ at points $P$ and $Q$, respectively. Determine the area of quadrilateral $PQYW$.
A) $110$
B) $120$
C) $123.75$
D) $130$ | 123.75 | 18.75 |
28,092 | In the diagram, \(O\) is the center of a circle with radii \(OA=OB=7\). A quarter circle arc from \(A\) to \(B\) is removed, creating a shaded region. What is the perimeter of the shaded region? | 14 + 10.5\pi | 0 |
28,093 | Two thirds of a pitcher is filled with orange juice and the remaining part is filled with apple juice. The pitcher is emptied by pouring an equal amount of the mixture into each of 6 cups. Calculate the percentage of the total capacity of the pitcher that each cup receives. | 16.67\% | 96.875 |
28,094 | In $\triangle ABC$, we have $AC = BC = 10$, and $AB = 8$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 12$. What is $BD$? | 2\sqrt{15} | 2.34375 |
28,095 | Given that $m$ and $n$ are two non-coincident lines, and $\alpha$, $\beta$, $\gamma$ are three pairwise non-coincident planes, consider the following four propositions:
$(1)$ If $m \perp \alpha$ and $m \perp \beta$, then $\alpha \parallel \beta$;
$(2)$ If $\alpha \perp \gamma$ and $\beta \perp \gamma$, then $\alpha \parallel \beta$;
$(3)$ If $m \subset \alpha$ and $n \subset \beta$, then $\alpha \parallel \beta$;
$(4)$ If $m \nparallel \beta$ and $\beta \nparallel \gamma$, then $m \nparallel \gamma$.
Among these propositions, the incorrect ones are __________. (Fill in all the correct proposition numbers) | (2)(3)(4) | 0 |
28,096 | Jia and his four friends each have a private car. The last digit of Jia's license plate is 0, and the last digits of his four friends' license plates are 0, 2, 1, 5, respectively. To comply with the local traffic restrictions from April 1st to 5th (cars with odd-numbered last digits are allowed on odd days, and cars with even-numbered last digits are allowed on even days), the five people discussed carpooling, choosing any car that meets the requirements each day. However, Jia's car can only be used for one day at most. The total number of different car use plans is \_\_\_\_\_\_. | 64 | 1.5625 |
28,097 | Let \(x_{1}, x_{2}\) be the roots of the quadratic equation \(ax^{2} + bx + c = 0\) with real coefficients. If \(x_{1}\) is an imaginary number and \(\frac{x_{1}^{2}}{x_{2}}\) is a real number, determine the value of \(S = 1 + \frac{x_{1}}{x_{2}} + \left(\frac{x_{1}}{x_{2}}\right)^{2} + \left(\frac{x_{1}}{x_{2}}\right)^{4} + \cdots + \left(\frac{x_{1}}{x_{2}}\right)^{1999}\). | -1 | 3.125 |
28,098 | Determine the maximum number of elements in the set \( S \) that satisfy the following conditions:
1. Each element in \( S \) is a positive integer not exceeding 100;
2. For any two different elements \( a \) and \( b \) in \( S \), there exists another element \( c \) in \( S \) such that the greatest common divisor of \( a + b \) and \( c \) is 1;
3. For any two different elements \( a \) and \( b \) in \( S \), there exists another element \( c \) in \( S \) such that the greatest common divisor of \( a + b \) and \( c \) is greater than 1. | 50 | 60.15625 |
28,099 | Let $S = {1, 2, \cdots, 100}.$ $X$ is a subset of $S$ such that no two distinct elements in $X$ multiply to an element in $X.$ Find the maximum number of elements of $X$ .
*2022 CCA Math Bonanza Individual Round #3* | 91 | 59.375 |
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