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int64 0
40.3k
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stringlengths 10
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stringlengths 1
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100
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|---|---|---|---|
27,100 | In triangle \(ABC\), side \(BC = 28\). The angle bisector \(BL\) is divided by the intersection point of the angle bisectors of the triangle in the ratio \(4:3\) from the vertex. Find the radius of the circumscribed circle around triangle \(ABC\) if the radius of the inscribed circle is 12. | 50 | 0.78125 |
27,101 | Find a positive integer that is divisible by 21 and has a square root between 30 and 30.5. | 903 | 0 |
27,102 | Out of 8 shots, 3 hit the target. The total number of ways in which exactly 2 hits are consecutive is: | 30 | 13.28125 |
27,103 | Given a quadrilateral $ABCD$ where the internal angles form an arithmetic progression, the angles are such that $\angle ADB$ is double the $\angle DBA$. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Furthermore, the angles within triangle $ABD$ form an arithmetic progression as well. Find the sum of the two smallest angles in $ABCD$. | 90 | 17.96875 |
27,104 | Eastbound traffic flows at 80 miles per hour and westbound traffic flows at 60 miles per hour. An eastbound driver observes 30 westbound vehicles in a 10-minute period. Calculate the number of westbound vehicles in a 150-mile section of the highway. | 193 | 25.78125 |
27,105 | If \(\frac{5+6+7+8}{4} = \frac{2014+2015+2016+2017}{N}\), calculate the value of \(N\). | 1240 | 83.59375 |
27,106 | A spherical decoration was suspended in a cylindrical container when the water inside it froze. The decoration was removed (without breaking the ice), leaving a hole 30 cm across at the top and 10 cm deep. If the water froze up to a height of 5 cm from the top of the sphere, what was the radius of the sphere (in centimeters)? | 5\sqrt{13} | 0 |
27,107 | In the number \(2 * 0 * 1 * 6 * 0 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0,1,2,3,4,5,6,7,8\) (digits can repeat) such that the resulting 10-digit number is divisible by 18. How many ways can this be done? | 32805 | 0 |
27,108 | Given $x_{1}=4$, $x_{2}=5$, $x_{3}=6$, calculate the standard deviation of this sample. | \frac{\sqrt{6}}{3} | 3.125 |
27,109 | Eight positive integers are written on the faces of a square prism (two bases and a lateral surface consisting of four faces). Each vertex is labeled with the product of the three numbers on the faces sharing that vertex (two from the lateral faces and one from the bases). If the sum of the numbers at the vertices equals $1176$, determine the sum of the numbers written on the faces. | 33 | 11.71875 |
27,110 | Determine the length of side $PQ$ in the right-angled triangle $PQR$, where $PR = 15$ units and $\angle PQR = 45^\circ$. | 15 | 0.78125 |
27,111 | To make a rectangular frame with lengths of 3cm, 4cm, and 5cm, the total length of wire needed is cm. If paper is then glued around the outside (seams not considered), the total area of paper needed is cm<sup>2</sup>. | 94 | 55.46875 |
27,112 | Yu Semo and Yu Sejmo have created sequences of symbols $\mathcal{U} = (\text{U}_1, \ldots, \text{U}_6)$ and $\mathcal{J} = (\text{J}_1, \ldots, \text{J}_6)$ . These sequences satisfy the following properties.
- Each of the twelve symbols must be $\Sigma$ , $\#$ , $\triangle$ , or $\mathbb{Z}$ .
- In each of the sets $\{\text{U}_1, \text{U}_2, \text{U}_4, \text{U}_5\}$ , $\{\text{J}_1, \text{J}_2, \text{J}_4, \text{J}_5\}$ , $\{\text{U}_1, \text{U}_2, \text{U}_3\}$ , $\{\text{U}_4, \text{U}_5, \text{U}_6\}$ , $\{\text{J}_1, \text{J}_2, \text{J}_3\}$ , $\{\text{J}_4, \text{J}_5, \text{J}_6\}$ , no two symbols may be the same.
- If integers $d \in \{0, 1\}$ and $i, j \in \{1, 2, 3\}$ satisfy $\text{U}_{i + 3d} = \text{J}_{j + 3d}$ , then $i < j$ .
How many possible values are there for the pair $(\mathcal{U}, \mathcal{J})$ ? | 24 | 0 |
27,113 | Given that \( x - \frac{1}{x} = 5 \), find the value of \( x^4 - \frac{1}{x^4} \). | 727 | 5.46875 |
27,114 | For each integer $i=0,1,2, \dots$ , there are eight balls each weighing $2^i$ grams. We may place balls as much as we desire into given $n$ boxes. If the total weight of balls in each box is same, what is the largest possible value of $n$ ? | 15 | 2.34375 |
27,115 | Let $a = \pi/4032$. Find the smallest positive integer $n$ such that \[2[\cos(a)\sin(a) + \cos(9a)\sin(3a) + \cos(25a)\sin(5a) + \cdots + \cos(n^2a)\sin(na)]\] is an integer, where $n$ is odd. | 4031 | 3.90625 |
27,116 | If a $5\times 5$ chess board exists, in how many ways can five distinct pawns be placed on the board such that each column and row contains no more than one pawn? | 14400 | 0.78125 |
27,117 | A doctor told Mikael to take a pill every 75 minutes. He took his first pill at 11:05. At what time did he take his fourth pill? | 14:50 | 17.96875 |
27,118 | For $n \geq 1$ , let $a_n$ be the number beginning with $n$ $9$ 's followed by $744$ ; eg., $a_4=9999744$ . Define $$ f(n)=\text{max}\{m\in \mathbb{N} \mid2^m ~ \text{divides} ~ a_n \} $$ , for $n\geq 1$ . Find $f(1)+f(2)+f(3)+ \cdots + f(10)$ . | 75 | 74.21875 |
27,119 | Solve the equation: $x^{2}-2x-8=0$. | -2 | 0.78125 |
27,120 | Determine the values of $x$ and $y$ given that 15 is the arithmetic mean of the set $\{8, x, 21, y, 14, 11\}$. | 36 | 0 |
27,121 | *How many odd numbers between 200 and 999 have distinct digits, and no digit greater than 7?* | 120 | 3.125 |
27,122 | Interior numbers begin in the third row of Pascal's Triangle. Calculate the sum of the squares of the interior numbers in the eighth row. | 3430 | 0 |
27,123 | Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | 2028 | 41.40625 |
27,124 | Right triangle $ABC$ is inscribed in circle $W$ . $\angle{CAB}=65$ degrees, and $\angle{CBA}=25$ degrees. The median from $C$ to $AB$ intersects $W$ and line $D$ . Line $l_1$ is drawn tangent to $W$ at $A$ . Line $l_2$ is drawn tangent to $W$ at $D$ . The lines $l_1$ and $l_2$ intersect at $P$ Determine $\angle{APD}$ | 50 | 9.375 |
27,125 | A semicircle is inscribed in a quarter circle. What fraction of the quarter circle is shaded? | $\frac{2}{3}$ | 0 |
27,126 | At a large gathering hosted by Benjamin Franklin, each man shakes hands with every other attendee except their own spouse and except Benjamin Franklin, who is also present. No handshakes take place between women, and Franklin only shakes hands with the women. If 15 married couples attended the gathering, calculate the total number of handshakes that occurred. | 225 | 5.46875 |
27,127 | Given a polynomial $(a+b+c+d+e+1)^N$, when expanded and like terms are combined, the expression contains exactly 2002 terms that include all five variables $a, b, c, d, e$, each to some positive power, determine the value of $N$. | 17 | 0 |
27,128 | Let $a$ and $b$ each be chosen at random from the set $\{1, 2, 3, \ldots, 40\}$. Additionally, let $c$ and $d$ also be chosen at random from the same set. Calculate the probability that the integer $2^c + 5^d + 3^a + 7^b$ has a units digit of $8$. | \frac{3}{16} | 10.9375 |
27,129 | Find the area of triangle $ABC$ below.
[asy]
unitsize(1inch);
pair A, B, C;
A = (0,0);
B= (sqrt(2),0);
C = (0,sqrt(2));
draw (A--B--C--A, linewidth(0.9));
draw(rightanglemark(B,A,C,3));
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,N);
label("$8$",(B+C)/2,NE);
label("$45^\circ$",(0,0.7),E);
[/asy] | 32 | 8.59375 |
27,130 | Jimmy finds that 8 bananas weigh the same as 6 oranges. If Jimmy has 40 oranges, and the weight of 4 oranges equals the weight of 5 apples, how many apples and bananas would Jimmy need to equal the weight of his 40 oranges? | 50 | 2.34375 |
27,131 | Compute
\[
\sin^2 3^\circ + \sin^2 6^\circ + \sin^2 9^\circ + \dots + \sin^2 177^\circ.
\] | 30 | 14.84375 |
27,132 | There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \cdots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \), circle \( C_{6} \) passes through exactly 6 points in \( M \), and so on, with circle \( C_{1} \) passing through exactly 1 point in \( M \). What is the minimum number of points in \( M \)? | 12 | 9.375 |
27,133 | The average of the data $x_1, x_2, \ldots, x_8$ is 6, and the standard deviation is 2. Then, the average and the variance of the data $2x_1-6, 2x_2-6, \ldots, 2x_8-6$ are | 16 | 53.90625 |
27,134 | Each of the numbers \( m \) and \( n \) is the square of an integer. The difference \( m - n \) is a prime number. Which of the following could be \( n \)? | 900 | 0.78125 |
27,135 | A necklace consists of 50 blue beads and some quantity of red beads. It is known that on any segment of the necklace containing 8 blue beads, there are at least 4 red beads. What is the minimum number of red beads that can be in this necklace? (The beads in the necklace are arranged cyclically, meaning the last bead is adjacent to the first one.) | 29 | 0.78125 |
27,136 | Let $\alpha$ be an acute angle, and $\cos\left(\alpha+ \frac{\pi}{6}\right) = \frac{3 \sqrt{10}}{10}$, $\tan(\alpha+\beta) = \frac{2}{5}$.
(1) Find the value of $\sin\left(2\alpha+ \frac{\pi}{6}\right)$.
(2) Find the value of $\tan\left(2\beta- \frac{\pi}{3}\right)$. | \frac{17}{144} | 0 |
27,137 | In a cylinder with a base radius of 6, there are two spheres each with a radius of 6, and the distance between their centers is 13. If a plane is tangent to both spheres and intersects the cylindrical surface, forming an ellipse, what is the sum of the lengths of the major and minor axes of this ellipse? ( ). | 25 | 31.25 |
27,138 | Given that $y$ is a multiple of $45678$, what is the greatest common divisor of $g(y)=(3y+4)(8y+3)(14y+9)(y+14)$ and $y$? | 1512 | 0 |
27,139 | There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.
| 2500 | 2.34375 |
27,140 | The cube shown is divided into 64 small cubes. Exactly one of the cubes is grey, as shown in the diagram. Two cubes are said to be 'neighbours' if they have a common face.
On the first day, the white neighbours of the grey cube are changed to grey. On the second day, the white neighbours of all the grey cubes are changed to grey.
How many grey cubes are there at the end of the second day? Choices: A 11, B 13, C 15, D 16, E 17 | 17 | 39.84375 |
27,141 | Given the vectors $\overrightarrow{a} = (1, 1)$, $\overrightarrow{b} = (2, 0)$, the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______. | \frac{\pi}{4} | 82.8125 |
27,142 | Elon Musk's Starlink project belongs to his company SpaceX. He plans to use tens of thousands of satellites to provide internet services to every corner of the Earth. A domestic company also plans to increase its investment in the development of space satellite networks to develop space internet. It is known that the research and development department of this company originally had 100 people, with an average annual investment of $a$ (where $a \gt 0$) thousand yuan per person. Now the research and development department personnel are divided into two categories: technical personnel and research personnel. There are $x$ technical personnel, and after the adjustment, the annual average investment of technical personnel is adjusted to $a(m-\frac{2x}{25})$ thousand yuan, while the annual average investment of research personnel increases by $4x\%$.
$(1)$ To ensure that the total annual investment of the adjusted research personnel is not less than the total annual investment of the original 100 research personnel, what is the maximum number of technical personnel after the adjustment?
$(2)$ Now it is required that the total annual investment of the adjusted research personnel is always not less than the total annual investment of the adjusted technical personnel. Find the maximum value of $m$ and the number of technical personnel at that time. | 50 | 30.46875 |
27,143 | If I have a $5\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 14400 | 0 |
27,144 | Given a point P(3, 2) outside the circle $x^2+y^2-2x-2y+1=0$, find the cosine of the angle between the two tangents drawn from this point to the circle. | \frac{3}{5} | 24.21875 |
27,145 | If the system of equations
\begin{align*}
8x - 6y &= c, \\
10y - 15x &= d.
\end{align*}
has a solution $(x,y)$ where $x$ and $y$ are both nonzero, find $\frac{c}{d},$ assuming $d$ is nonzero. | -\frac{4}{5} | 6.25 |
27,146 |
In the parallelogram $\mathrm{ABCD}$, points $\mathrm{E}$ and $\mathrm{F}$ lie on $\mathrm{AD}$ and $\mathrm{AB}$ respectively. Given that the area of $S_{A F I E} = 49$, the area of $\triangle B G F = 13$, and the area of $\triangle D E H = 35$, find the area of $S_{G C H I}$. | 97 | 21.875 |
27,147 | How many solutions does the equation $\tan x = \tan(\tan x + x)$ have on the interval $0 \leq x \leq \tan^{-1} 500$? | 160 | 17.96875 |
27,148 | Two 5-digit positive integers are formed using each of the digits from 0 through 9 once. What is the smallest possible positive difference between the two integers? | 247 | 0 |
27,149 | If $\lceil{\sqrt{x}}\rceil=12$, how many possible integer values of $x$ are there? | 23 | 89.84375 |
27,150 | Given that the function $f(x)=2\cos x-3\sin x$ reaches its minimum value when $x=\theta$, calculate the value of $\tan \theta$. | \frac{3}{2} | 40.625 |
27,151 | The distance from city $A$ to city $B$ is $999$ km. Along the highway leading from $A$ to $B$, there are kilometer markers indicating the distances from the marker to $A$ and $B$ as shown:
How many of these markers use only two different digits to indicate both distances? | 40 | 96.09375 |
27,152 | For a $k$-element subset $T$ of the set $\{1,2,\cdots,242\}$, every pair of elements (which may be the same) in $T$ has a sum that is not an integer power of 3. Find the maximum value of $k$. | 121 | 63.28125 |
27,153 | A card is secretly removed from a standard deck of 52 cards. Then two cards are dealt at random from the now 51-card deck. What is the probability that both cards drawn are from the same suit and the second card is a face card? | \frac{3}{50} | 1.5625 |
27,154 | For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | 16 | 8.59375 |
27,155 | Given the following matrix $$ \begin{pmatrix}
11& 17 & 25& 19& 16
24 &10 &13 & 15&3
12 &5 &14& 2&18
23 &4 &1 &8 &22
6&20&7 &21&9
\end{pmatrix}, $$ choose five of these elements, no two from the same row or column, in such a way that the minimum of these elements is as large as possible. | 17 | 0 |
27,156 | Two \(1 \times 1\) squares are removed from a \(5 \times 5\) grid as shown. Determine the total number of squares of various sizes on the grid. | 55 | 5.46875 |
27,157 | Let $p,$ $q,$ $r,$ $s,$ $t,$ $u$ be positive real numbers such that $p + q + r + s + t + u = 8.$ Find the minimum value of
\[\frac{1}{p} + \frac{9}{q} + \frac{16}{r} + \frac{25}{s} + \frac{36}{t} + \frac{49}{u}.\] | 84.5 | 32.03125 |
27,158 | A line segment $AB$ with a fixed length of $4$ has its endpoints moving along the positive $x$-axis and the positive $y$-axis, respectively, and $P(x,y)$ is a point on the circumcircle of triangle $OAB$. Find the maximum value of $x+y$. | 2\sqrt{2} | 0.78125 |
27,159 | Given that $y$ is a multiple of $42522$, what is the greatest common divisor of $g(y)=(3y+4)(8y+3)(14y+9)(y+17)$ and $y$? | 102 | 2.34375 |
27,160 | A keen archaeologist is holding a competition where participants must guess the age of a rare artifact. The age of the artifact is formed using the six digits: 2, 2, 3, 3, 7, and 9, and it must begin with an odd digit.
How many different ages can be there for the artifact? | 180 | 17.96875 |
27,161 | Consider a decreasing arithmetic sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. If $a\_3a\_5=63$ and $a\_2+{a}\_{6} =16$,
(1) Find the general term formula of the sequence.
(2) For what value of $n$ does $S\_n$ reach its maximum value? Also, find the maximum value.
(3) Calculate $|a\_1|+|a\_2|+|a\_3|+…+|a\_n|$. | 66 | 4.6875 |
27,162 | In a large square, points $P$, $Q$, $R$, and $S$ are midpoints of the sides. Inside the square, a triangle is formed by connecting point $P$ to the center of the square and point $Q$. If the area of the larger square is 80, what are the areas of the smaller square and the triangle formed? | 10 | 44.53125 |
27,163 | The side lengths of a cyclic quadrilateral are 25, 39, 52, and 60. What is the diameter of the circle? | 65 | 75 |
27,164 | Moor has $2016$ white rabbit candies. He and his $n$ friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that $n$ is a positive integer (Moor has at least $1$ friend), how many possible values of $n$ exist? | 35 | 88.28125 |
27,165 | The sides of a triangle are all integers, and the longest side is 11. Calculate the number of such triangles. | 36 | 82.8125 |
27,166 | Given the set \( A = \{x \mid (x-2)(x-6) \geqslant 3, x \in \mathbf{Z}, 0 \leqslant x \leq 7\} \), find the number of non-empty subsets of \( A \). | 63 | 1.5625 |
27,167 | Given the power function $f(x) = (m^2 - m - 1)x^{m^2 + m - 3}$ on the interval $(0, +\infty)$, determine the value of $m$ that makes it a decreasing function. | -1 | 25.78125 |
27,168 | Find the total perimeter of the combined shape of the flower bed, given that the garden is shaped as a right triangle adjacent to a rectangle, the hypotenuse of the triangle coincides with one of the sides of the rectangle, the triangle has legs of lengths 3 meters and 4 meters, and the rectangle has the other side of length 10 meters. | 22 | 16.40625 |
27,169 | Let \( (a_1, a_2, \dots, a_{12}) \) be a list of the first 12 positive integers such that for each \( 2 \le i \le 12 \), either \( a_i+1 \) or \( a_i-1 \) or both appear somewhere before \( a_i \) in the list. Determine the number of such lists. | 2048 | 8.59375 |
27,170 | Observing the equations:<br/>$\frac{1}{1×2}=1-\frac{1}{2}$; $\frac{1}{2×3}=\frac{1}{2}-\frac{1}{3}$; $\frac{1}{3×4}=\frac{1}{3}-\frac{1}{4}$.<br/>By adding both sides of the above three equations, we get:<br/>$\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}=1-\frac{1}{4}=\frac{3}{4}$.<br/>$(1)$ Make a conjecture and write down: $\frac{1}{n(n+1)}=\_\_\_\_\_\_.$<br/>$(2)$ Calculate:<br/>$\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}+\ldots +\frac{1}{2021×2022}$.<br/>$(3)$ Investigate and calculate:<br/>$\frac{1}{1×4}+\frac{1}{4×7}+\frac{1}{7×10}+\ldots +\frac{1}{2020×2023}$. | \frac{674}{2023} | 94.53125 |
27,171 | Let $Q(x) = x^2 - 4x - 16$. A real number $x$ is chosen at random from the interval $6 \le x \le 20$. The probability that $\lfloor\sqrt{Q(x)}\rfloor = \sqrt{Q(\lfloor x \rfloor)}$ is equal to $\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}$, where $a$, $b$, $c$, $d$, and $e$ are positive integers. Find $a + b + c + d + e$. | 17 | 0 |
27,172 | A circle passes through the vertex of a rectangle $ABCD$ and touches its sides $AB$ and $AD$ at $M$ and $N$ respectively. If the distance from $C$ to the line segment $MN$ is equal to $5$ units, find the area of rectangle $ABCD$ . | 25 | 1.5625 |
27,173 | Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sums of the first $n$ terms are denoted as $S_n$ and $T_n$ respectively. If for any natural number $n$, it holds that $\dfrac{S_n}{T_n} = \dfrac{2n-3}{4n-3}$, calculate the value of $\dfrac{a_3+a_{15}}{2(b_3+b_9)}+ \dfrac{a_3}{b_2+b_{10}}$. | \dfrac{19}{41} | 21.875 |
27,174 | Let $ABCD$ be a trapezoid such that $|AC|=8$ , $|BD|=6$ , and $AD \parallel BC$ . Let $P$ and $S$ be the midpoints of $[AD]$ and $[BC]$ , respectively. If $|PS|=5$ , find the area of the trapezoid $ABCD$ . | 24 | 17.96875 |
27,175 | Let set $A=\{(x,y) \,|\, |x|+|y| \leq 2\}$, and $B=\{(x,y) \in A \,|\, y \leq x^2\}$. Calculate the probability that a randomly selected element $P(x,y)$ from set $A$ belongs to set $B$. | \frac {17}{24} | 0 |
27,176 | Star writes down the whole numbers $1$ through $40$. Emilio copies Star's numbers, but he replaces each occurrence of the digit $3$ by the digit $2$. Calculate the difference between Star's sum of the numbers and Emilio's sum of the numbers. | 104 | 1.5625 |
27,177 | A line passing through point P(1, 2) is tangent to the circle $x^2+y^2=4$ and perpendicular to the line $ax-y+1=0$. Find the value of the real number $a$. | -\frac{3}{4} | 0.78125 |
27,178 | Let the function $f(x) = 2\cos^2x + 2\sqrt{3}\sin x\cos x + m$.
(1) Find the smallest positive period of the function $f(x)$ and its intervals of monotonic decrease;
(2) If $x \in \left[0, \frac{\pi}{2}\right]$, does there exist a real number $m$ such that the range of the function $f(x)$ is exactly $\left[\frac{1}{2}, \frac{7}{2}\right]$? If it exists, find the value of $m$; if not, explain why. | \frac{1}{2} | 9.375 |
27,179 | Triangle $ABC$ has $AB=25$ , $AC=29$ , and $BC=36$ . Additionally, $\Omega$ and $\omega$ are the circumcircle and incircle of $\triangle ABC$ . Point $D$ is situated on $\Omega$ such that $AD$ is a diameter of $\Omega$ , and line $AD$ intersects $\omega$ in two distinct points $X$ and $Y$ . Compute $XY^2$ .
*Proposed by David Altizio* | 252 | 0 |
27,180 | Let $S$ be a subset of $\{1, 2, . . . , 500\}$ such that no two distinct elements of S have a
product that is a perfect square. Find, with proof, the maximum possible number of elements
in $S$ . | 306 | 75 |
27,181 | a) In Mexico City, there are regulations that restrict private cars from driving on city streets on two specified days of the week. A family needs to have at least ten cars available every day. What is the minimum number of cars the family needs if they can choose the restricted days for their cars?
b) In Mexico City, each private car is restricted from driving on city streets on one specified day of the week. A wealthy family of ten people has bribed the police, allowing them to suggest two days, from which the police choose one as the restricted day. What is the minimum number of cars the family needs to buy to ensure that each family member can drive every day, given that the assignment of restricted days is done sequentially? | 12 | 20.3125 |
27,182 | Natural numbers are written in sequence on the blackboard, skipping over any perfect squares. The sequence looks like this:
$$
2,3,5,6,7,8,10,11, \cdots
$$
The first number is 2, the fourth number is 6, the eighth number is 11, and so on. Following this pattern, what is the 1992nd number written on the blackboard?
(High School Mathematics Competition, Beijing, 1992) | 2036 | 2.34375 |
27,183 | Let $T$ be the set of points $(x, y)$ in the Cartesian plane that satisfy
\[\big|\big| |x|-3\big|-1\big|+\big|\big| |y|-3\big|-1\big|=2.\]
What is the total length of all the lines that make up $T$? | 32\sqrt{2} | 0 |
27,184 | Given $\sin \alpha + \cos \alpha = \frac{\sqrt{2}}{3}$ ($\frac{\pi}{2} < \alpha < \pi$), find the values of the following expressions:
$(1) \sin \alpha - \cos \alpha$;
$(2) \sin^2\left(\frac{\pi}{2} - \alpha\right) - \cos^2\left(\frac{\pi}{2} + \alpha\right)$. | -\frac{4\sqrt{2}}{9} | 74.21875 |
27,185 | Real numbers \(a, b, c\) and a positive number \(\lambda\) such that \(f(x)=x^{3}+a x^{2}+b x+c\) has three real roots \(x_{1}, x_{2}, x_{3}\), satisfying:
(1) \(x_{2}-x_{1}=\lambda\);
(2) \(x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)\).
Find the maximum value of \(\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}\). | \frac{3\sqrt{3}}{2} | 3.125 |
27,186 | Let $p$ and $q$ be constants. Suppose that the equation \[\frac{(x+p)(x+q)(x+20)}{(x+4)^2} = 0\] has exactly $3$ distinct roots, while the equation \[\frac{(x+3p)(x+4)(x+10)}{(x+q)(x+20)} = 0\] has exactly $1$ distinct root. Compute $100p + q.$ | \frac{430}{3} | 29.6875 |
27,187 | Lil writes one of the letters \( \text{P}, \text{Q}, \text{R}, \text{S} \) in each cell of a \( 2 \times 4 \) table. She does this in such a way that, in each row and in each \( 2 \times 2 \) square, all four letters appear. In how many ways can she do this?
A) 12
B) 24
C) 48
D) 96
E) 198 | 24 | 25.78125 |
27,188 | In a right triangle, one leg is 24 inches, and the other leg is 10 inches more than twice the shorter leg. Calculate the area of the triangle and find the length of the hypotenuse. | \sqrt{3940} | 0 |
27,189 | Half of the blue flowers are tulips, five sixths of the yellow flowers are daisies, and four fifths of the flowers are yellow. What percent of the flowers are tulips? | 23.3 | 0 |
27,190 | On the altitude \( BH \) of triangle \( ABC \) a certain point \( D \) is marked. Line \( AD \) intersects side \( BC \) at point \( E \), and line \( CD \) intersects side \( AB \) at point \( F \). Points \( G \) and \( J \) are the projections of points \( F \) and \( E \) onto side \( AC \) respectively. The area of triangle \( HEJ \) is twice the area of triangle \( HFG \). In what ratio does the altitude \( BH \) divide the segment \( FE \)? | \sqrt{2} : 1 | 9.375 |
27,191 | In rectangle $ABCD$, diagonal $DB$ is divided into three segments of lengths $1$, $2$, and $3$ by parallel lines $L$ and $L'$, which pass through vertices $A$ and $C$ and are perpendicular to diagonal $DB$. Calculate the area of rectangle $ABCD$.
A) $6\sqrt{5}$
B) $12$
C) $12\sqrt{5}$
D) $30$
E) $18\sqrt{3}$ | 6\sqrt{5} | 5.46875 |
27,192 | If the Cesaro sum of the 50-term sequence \((b_1,\dots,b_{50})\) is 500, what is the Cesaro sum of the 51-term sequence \((2,b_1,\dots,b_{50})\)? | 492 | 4.6875 |
27,193 | Triangle \(ABC\) is isosceles with \(AB = AC\) and \(BC = 65 \, \text{cm}\). \(P\) is a point on \(BC\) such that the perpendicular distances from \(P\) to \(AB\) and \(AC\) are \(24 \, \text{cm}\) and \(36 \, \text{cm}\), respectively. The area of \(\triangle ABC\), in \(\text{cm}^2\), is | 2535 | 8.59375 |
27,194 | Find $ \#\left\{ (x,y)\in\mathbb{N}^2\bigg| \frac{1}{\sqrt{x}} -\frac{1}{\sqrt{y}} =\frac{1}{2016}\right\} , $ where $ \# A $ is the cardinal of $ A . $ | 165 | 24.21875 |
27,195 | Given the areas of three squares in the diagram, find the area of the triangle formed. The triangle shares one side with each of two squares and the hypotenuse with the third square.
[asy]
/* Modified AMC8-like Problem */
draw((0,0)--(10,0)--(10,10)--cycle);
draw((10,0)--(20,0)--(20,10)--(10,10));
draw((0,0)--(0,-10)--(10,-10)--(10,0));
draw((0,0)--(-10,10)--(0,20)--(10,10));
draw((9,0)--(9,1)--(10,1));
label("100", (5, 5));
label("64", (15, 5));
label("100", (5, -5));
[/asy]
Assume the triangle is a right isosceles triangle. | 50 | 9.375 |
27,196 | In triangle $XYZ$, where $XY = 8$, $YZ = 12$, and $XZ = 14$. Points $D$ and $E$ are selected on $\overline{XY}$ and $\overline{XZ}$ respectively, such that $XD = 3$ and $XE = 9$. Calculate the area of triangle $XDE$. | \frac{405 \sqrt{17}}{112} | 0 |
27,197 | In order to obtain steel for a specific purpose, the golden section method was used to determine the optimal addition amount of a specific chemical element. After several experiments, a good point on the optimal range $[1000, m]$ is in the ratio of 1618, find $m$. | 2000 | 1.5625 |
27,198 | Let the function $f(x)= \sqrt{3}\cos^2\omega x+\sin \omega x\cos \omega x+a$ where $\omega > 0$, $a\in\mathbb{R}$, and the graph of $f(x)$ has its first highest point on the right side of the y-axis at the x-coordinate $\dfrac{\pi}{6}$.
(Ⅰ) Find the smallest positive period of $f(x)$; (Ⅱ) If the minimum value of $f(x)$ in the interval $\left[-\dfrac{\pi}{3}, \dfrac{5\pi}{6}\right]$ is $\sqrt{3}$, find the value of $a$. | \dfrac{ \sqrt{3}+1}{2} | 1.5625 |
27,199 | George has 10 different colors available to paint his room and he must choose exactly 2 of them. However, George prefers to include the color blue as one of his choices if possible. In how many ways can he choose the colors? | 45 | 50 |
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