problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
In an isosceles triangle \(ABC\) (\(AB = BC\)), the angle bisectors \(AM\) and \(BK\) intersect at point \(O\). The areas of triangles \(BOM\) and \(COM\) are 25 and 30, respectively. Find the area of triangle \(ABC\). | 110 | 0 | 7,966.125 | -1 | 7,966.125 | |
Anton has two species of ants, Species A and Species B, in his ant farm. The two species are identical in appearance, but Anton knows that every day, there are twice as many ants of Species A than before, while there are three times as many ants of Species B. On Day 0, Anton counts that there are 30 ants in his ant far... | 608 | 0.875 | 1,866 | 1,928.428571 | 1,429 | |
Marisa has a collection of $2^{8}-1=255$ distinct nonempty subsets of $\{1,2,3,4,5,6,7,8\}$. For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain ... | \frac{1024}{255} | It suffices to compute the probability of each number appearing in the final subset. For any given integer $n \in[1,8]$, there are $2^{7}=128$ subsets with $n$ and $2^{7}-1=127$ without. When we focus on only this element, each operation is equivalent to taking two random sets and discarding one of them randomly. There... | 0.0625 | 7,924.75 | 4,793 | 8,133.533333 |
Let $A$ be the set of all numbers which can be represented as the sum of three consecutive positive integers. What is the greatest common divisor of all numbers in $A$? | 3 | 1 | 2,641.625 | 2,641.625 | -1 | |
The Greenhill Soccer Club has 25 players, including 4 goalies. During an upcoming practice, the team plans to have a competition in which each goalie will try to stop penalty kicks from every other player, including the other goalies. How many penalty kicks are required for every player to have a chance to kick against... | 96 | 0.1875 | 4,721.4375 | 5,556 | 4,528.846154 | |
Let \( [x] \) denote the greatest integer not exceeding \( x \), e.g., \( [\pi]=3 \), \( [5.31]=5 \), and \( [2010]=2010 \). Given \( f(0)=0 \) and \( f(n)=f\left(\left[\frac{n}{2}\right]\right)+n-2\left[\frac{n}{2}\right] \) for any positive integer \( n \). If \( m \) is a positive integer not exceeding 2010, find th... | 10 | 0.1875 | 7,902.4375 | 7,393 | 8,020 | |
The sequence \(\left\{a_{n}\right\}\) is defined as follows:
\[
\begin{align*}
a_{1} &= 0, \\
a_{2} &= 1, \\
a_{n+1} &= \frac{\sqrt{6}-\sqrt{2}}{2} a_{n} - a_{n-1} \quad \text{for } n \geq 2.
\end{align*}
\]
Determine the value of \(a_{2019}\). | \frac{\sqrt{6} - \sqrt{2}}{2} | 0 | 8,192 | -1 | 8,192 | |
There are nine parts in a bag, including five different genuine parts and four different defective ones. These parts are being drawn and inspected one by one. If the last defective part is found exactly on the fifth draw, calculate the total number of different sequences of draws. | 480 | 0 | 7,754.3125 | -1 | 7,754.3125 | |
How much money does Roman give Dale if Roman wins a contest with a prize of $\$ 200$, gives $30 \%$ of the prize to Jackie, and then splits $15 \%$ of what remains equally between Dale and Natalia? | \$ 10.50 | To determine $30 \%$ of Roman's $\$ 200$ prize, we calculate $\$ 200 \times 30 \%=\$ 200 \times \frac{30}{100}=\$ 2 \times 30=\$ 60$. After Roman gives $\$ 60$ to Jackie, he has $\$ 200-\$ 60=\$ 140$ remaining. He splits $15 \%$ of this between Dale and Natalia. The total that he splits is $\$ 140 \times 15 \%=\$ 140 \... | 0.875 | 431.3125 | 425.5 | 472 |
Find the solutions to $z^4 = -4.$ Enter the solutions, separated by commas. | 1 + i, 1 - i, -1 + i, -1 - i | 0 | 3,508.0625 | -1 | 3,508.0625 | |
In $\triangle ABC$, $AB=5$, $BC=7$, $AC=9$, and $D$ is on $\overline{AC}$ with $BD=5$. Find the ratio of $AD:DC$. | 19/8 | 1. **Apply the Law of Cosines in $\triangle ABC$**:
\[
c^2 = a^2 + b^2 - 2ab \cos C
\]
Here, $c = BC = 7$, $a = AB = 5$, and $b = AC = 9$. Plugging in these values, we get:
\[
7^2 = 5^2 + 9^2 - 2 \cdot 5 \cdot 9 \cdot \cos A
\]
Simplifying, we find:
\[
49 = 25 + 81 - 90 \cos A \implies 49 ... | 0.8125 | 3,938.875 | 4,049.923077 | 3,457.666667 |
The vertices of Durer's favorite regular decagon in clockwise order: $D_1, D_2, D_3, . . . , D_{10}$ . What is the angle between the diagonals $D_1D_3$ and $D_2D_5$ ? | 90 | 0 | 7,735.6875 | -1 | 7,735.6875 | |
Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How many minutes would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time? | 120 | 1 | 1,398.9375 | 1,398.9375 | -1 | |
Let $a,$ $b,$ $c,$ $d$ be real numbers such that $a + b + c + d = 10$ and
\[ab + ac + ad + bc + bd + cd = 20.\]
Find the largest possible value of $d.$ | \frac{5 + \sqrt{105}}{2} | 0 | 5,684.625 | -1 | 5,684.625 | |
A truncated right circular cone has a large base radius of 10 cm and a small base radius of 5 cm. The height of the truncated cone is 10 cm. Calculate the volume of this solid. | 583.33\pi | 0 | 3,840.75 | -1 | 3,840.75 | |
A magician and their assistant plan to perform a trick. The spectator writes a sequence of $N$ digits on a board. The magician's assistant then covers two adjacent digits with a black dot. Next, the magician enters and has to guess both covered digits (including the order in which they are arranged). What is the smalle... | 101 | 0.125 | 7,803.375 | 5,083 | 8,192 | |
Given the function \( f(x) = \lg \frac{1 + x}{1 - x} \), if \( f\left(\frac{y + z}{1 + y z}\right) = 1 \) and \( f\left(\frac{y - z}{1 - y z}\right) = 2 \), where \( -1 < y, z < 1 \), find the value of \( f(y) \cdot f(z) \). | -3/4 | 0 | 8,116.5625 | -1 | 8,116.5625 | |
What is the greatest common factor of 84, 112 and 210? | 14 | 0.9375 | 3,331.3125 | 3,007.266667 | 8,192 | |
There are 10 sprinters in the Olympic 100-meter finals. Four of the sprinters are from Spain. The gold, silver, and bronze medals are awarded to the top three finishers. In how many ways can the medals be awarded if at most two Spaniards get medals? | 696 | 0.8125 | 4,635.3125 | 3,814.538462 | 8,192 | |
How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her? | 262144 | 0.1875 | 7,872.125 | 6,829.333333 | 8,112.769231 | |
Given the prime factorization of $215^7$, $p^7 \cdot q^6 \cdot r^6$, where $p$, $q$, and $r$ are prime numbers, determine the number of positive integer divisors of $215^7$ that are perfect squares or perfect cubes (or both). | 21 | 0.125 | 5,386.9375 | 3,840.5 | 5,607.857143 | |
Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6)$, and the product of the radii is $68$. The x-axis and the line $y = mx$, where $m > 0$, are tangent to both circles. It is given that $m$ can be written in the form $a\sqrt {b}/c$, where $a$, $b$, and $c$ are positive integ... | 282 | 0.3125 | 7,320.875 | 5,588.2 | 8,108.454545 | |
Triangle $ABC$ is isosceles with $AC = BC$ and $\angle ACB = 106^\circ.$ Point $M$ is in the interior of the triangle so that $\angle MAC = 7^\circ$ and $\angle MCA = 23^\circ.$ Find the number of degrees in $\angle CMB.$ | 83 | 0.375 | 7,310.25 | 5,840.666667 | 8,192 | |
Given that $a > 0$ and $b > 0$, they satisfy the equation $3a + b = a^2 + ab$. Find the minimum value of $2a + b$. | 3 + 2\sqrt{2} | 0.875 | 5,731.875 | 5,380.428571 | 8,192 | |
Given the function $f(x)=f'(1)e^{x-1}-f(0)x+\frac{1}{2}x^{2}(f′(x) \text{ is } f(x))$'s derivative, where $e$ is the base of the natural logarithm, and $g(x)=\frac{1}{2}x^{2}+ax+b(a\in\mathbb{R}, b\in\mathbb{R})$
(I) Find the analytical expression and extreme values of $f(x)$;
(II) If $f(x)\geqslant g(x)$, find the max... | \frac{e}{4} | 0.25 | 7,474.5 | 5,322 | 8,192 | |
Given that the polar coordinate equation of curve C is ρ - 4cosθ = 0, establish a rectangular coordinate system with the pole as the origin and the polar axis as the positive semi-axis. Line l passes through point M(3, 0) with a slope angle of $\frac{\pi}{6}$.
(I) Find the rectangular coordinate equation of curve C and... | \frac{\sqrt{15}}{3} | 0 | 6,457.4375 | -1 | 6,457.4375 | |
Given an infinite grid where each cell is either red or blue, such that in any \(2 \times 3\) rectangle exactly two cells are red, determine how many red cells are in a \(9 \times 11\) rectangle. | 33 | 0 | 8,192 | -1 | 8,192 | |
There are 15 married couples among 30 people. Calculate the total number of handshakes that occurred among these people. | 301 | 0 | 3,229.5625 | -1 | 3,229.5625 | |
What is the largest multiple of 7 less than 50? | 49 | 1 | 326.1875 | 326.1875 | -1 | |
Suppose that $x, y, z$ are real numbers such that $x=y+z+2$, $y=z+x+1$, and $z=x+y+4$. Compute $x+y+z$. | -7 | Adding all three equations gives $$x+y+z=2(x+y+z)+7$$ from which we find that $x+y+z=-7$. | 1 | 2,472.4375 | 2,472.4375 | -1 |
Given the function $f(x)=|\log_{4}x|$, and real numbers $m$, $n$ satisfy $0 < m < n$ and $f(m)=f(n)$. If the maximum value of $f(x)$ in the interval $[m^{2},n]$ is $2$, then $\frac{n}{m}=$ ______. | 16 | 0.6875 | 5,213.6875 | 4,199.545455 | 7,444.8 | |
The four points $A(-1,2), B(3,-4), C(5,-6),$ and $D(-2,8)$ lie in the coordinate plane. Compute the minimum possible value of $PA + PB + PC + PD$ over all points P . | 23 | 0 | 8,192 | -1 | 8,192 | |
A circle, whose center lies on the line \( y = b \), intersects the parabola \( y = \frac{12}{5} x^2 \) at least at three points; one of these points is the origin, and two of the remaining points lie on the line \( y = \frac{12}{5} x + b \). Find all values of \( b \) for which this configuration is possible. | 169/60 | 0.0625 | 8,190.5 | 8,168 | 8,192 | |
The sum of all three-digit numbers that, when divided by 7 give a remainder of 5, when divided by 5 give a remainder of 2, and when divided by 3 give a remainder of 1, calculate the sum of these numbers. | 4436 | 1 | 3,947.8125 | 3,947.8125 | -1 | |
Given the coordinates of the three vertices of $\triangle P_{1}P_{2}P_{3}$ are $P_{1}(1,2)$, $P_{2}(4,3)$, and $P_{3}(3,-1)$, the length of the longest edge is ________, and the length of the shortest edge is ________. | \sqrt {10} | 0 | 1,951 | -1 | 1,951 | |
A student used the "five-point method" to draw the graph of the function $f(x)=A\sin(\omega x+\varphi)$ ($\omega\ \gt 0$, $|\varphi|＜\frac{π}{2}$) within one period. The student listed and filled in some of the data in the table below:
| $\omega x+\varphi$ | $0$ | $\frac{π}{2}$ | $\pi$ | $\frac{{3π}}{2}$ | $2\pi$ |
|-... | \frac{7\pi}{24} | 0.0625 | 7,878.75 | 5,055 | 8,067 | |
Let $a,$ $b,$ and $c$ be complex numbers such that $|a| = |b| = |c| = 1$ and
\[\frac{a^2}{bc} + \frac{b^2}{ac} + \frac{c^2}{ab} = -1.\]Find all possible values of $|a + b + c|.$
Enter all the possible values, separated by commas. | 1,2 | 0 | 8,094.9375 | -1 | 8,094.9375 | |
The equation \( x^{2} + mx + 1 + 2i = 0 \) has real roots. Find the minimum value of the modulus of the complex number \( m \). | \sqrt{2 + 2\sqrt{5}} | 0 | 7,495.3125 | -1 | 7,495.3125 | |
A cat has found $432_{9}$ methods in which to extend each of her nine lives. How many methods are there in base 10? | 353 | 0.875 | 2,200 | 1,846.642857 | 4,673.5 | |
Given triangle $ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, if $\cos B= \frac{1}{4}, b=3$, and $\sin C=2\sin A$, find the area of triangle $ABC$. | \frac{9\sqrt{15}}{16} | 0 | 5,955.9375 | -1 | 5,955.9375 | |
The intercept of the line $5x - 2y - 10 = 0$ on the x-axis is $a$, and on the y-axis is $b$. Find the values of $a$ and $b$. | -5 | 1 | 1,558.5 | 1,558.5 | -1 | |
Let $a, b \in \mathbb{R}$. If the line $l: ax+y-7=0$ is transformed by the matrix $A= \begin{bmatrix} 3 & 0 \\ -1 & b\end{bmatrix}$, and the resulting line is $l′: 9x+y-91=0$. Find the values of the real numbers $a$ and $b$. | 13 | 0.625 | 5,140.125 | 3,841.7 | 7,304.166667 | |
Among 6 internists and 4 surgeons, there is one chief internist and one chief surgeon. Now, a 5-person medical team is to be formed to provide medical services in rural areas. How many ways are there to select the team under the following conditions?
(1) The team includes 3 internists and 2 surgeons;
(2) The team i... | 191 | 0 | 7,791.4375 | -1 | 7,791.4375 | |
Consider triangle $A B C$ with $\angle A=2 \angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{A B}$ at $E$. If $\frac{D E}{D C}=\frac{1}{3}$, compute $\frac{A B}{A C}$. | \frac{7}{9} | Let $A E=x$ and $B E=y$. Using angle-bisector theorem on $\triangle A C E$ we have $x: D E=A C: D C$, so $A C=3 x$. Using some angle chasing, it is simple to see that $\angle A D E=\angle A E D$, so $A D=A E=x$. Then, note that $\triangle C D A \sim \triangle C E B$, so $y:(D C+D E)=x: D C$, so $y: x=1+\frac{1}{3}=\fra... | 0.125 | 7,756.9375 | 6,348 | 7,958.214286 |
Given that $\sin A+\sin B=1$ and $\cos A+\cos B=3 / 2$, what is the value of $\cos (A-B)$? | 5/8 | Squaring both equations and add them together, one obtains $1+9 / 4=2+2(\cos (A) \cos (B)+\sin (A) \sin (B))=2+2 \cos (A-B)$. Thus $\cos A-B=5 / 8$. | 0.9375 | 3,188.1875 | 2,854.6 | 8,192 |
Define a function $g(x),$ for positive integer values of $x,$ by
\[
g(x) = \left\{
\begin{aligned}
\log_3 x & \quad \text{if } \log_3 x \text{is an integer} \\
1 + g(x + 1) & \quad \text{otherwise}.
\end{aligned}
\right.
\]
Compute $g(50)$. | 35 | 0.5625 | 6,557.875 | 5,286.888889 | 8,192 | |
Given points $M(4,0)$ and $N(1,0)$, any point $P$ on curve $C$ satisfies: $\overset{→}{MN} \cdot \overset{→}{MP} = 6|\overset{→}{PN}|$.
(I) Find the trajectory equation of point $P$;
(II) A line passing through point $N(1,0)$ intersects curve $C$ at points $A$ and $B$, and intersects the $y$-axis at point $H$. If $\o... | -\frac{8}{3} | 0.875 | 6,302.4375 | 6,032.5 | 8,192 | |
What is the period of $y = \sin x + \cos x$? | 2 \pi | 1 | 2,713.4375 | 2,713.4375 | -1 | |
A large equilateral triangle with side length $12$ is to be filled completely with non-overlapping equilateral triangles, each of side length $2$. Calculate the number of small triangles needed to fill the large triangle. | 36 | 0.875 | 4,970.25 | 4,510 | 8,192 | |
Find $A+B$ (in base 10), given the following addition problem \[ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}& & & 4 & A & B_{6}\\ &+& & & 4 & 1_{6}\\ \cline{2-6}& & & 5 & 3 & A_{6}\\ \end{array} \] | 9 | 1 | 3,310.8125 | 3,310.8125 | -1 | |
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$ . Determine $f (2014)$ . $N_0=\{0,1,2,...\}$ | 671 | 0.5625 | 7,079 | 6,213.333333 | 8,192 | |
A granary has collected 1536 shi of rice, and upon inspection, it is found that out of 224 grains, 28 are weeds. Determine the approximate amount of weeds in this batch of rice. | 192 | 0.875 | 929.625 | 336.857143 | 5,079 | |
Let $S$ be a subset of $\{1, 2, 3, \ldots, 100\}$ such that no pair of distinct elements in $S$ has a sum divisible by $5$. What is the maximum number of elements in $S$? | 40 | 0.0625 | 7,595.625 | 8,192 | 7,555.866667 | |
If the graph of the function $y=3\sin(2x+\phi)$ $(0 < \phi < \pi)$ is symmetric about the point $\left(\frac{\pi}{3},0\right)$, then $\phi=$ ______. | \frac{\pi}{3} | 0.875 | 4,743.3125 | 4,250.642857 | 8,192 | |
Calculate: $9-8+7\times6+5-4\times3+2-1$ | 37 | 1 | 489.9375 | 489.9375 | -1 | |
The value of \(0.9 + 0.99\) is: | 1.89 | 1 | 1,850.625 | 1,850.625 | -1 | |
There are 3 females and 3 males to be arranged in a sequence of 6 contestants, with the restriction that no two males can perform consecutively and the first contestant cannot be female contestant A. Calculate the number of different sequences of contestants. | 132 | 0.125 | 7,671.375 | 5,774 | 7,942.428571 | |
A fair coin is flipped 8 times. What is the probability that at least 6 consecutive flips come up heads? | \frac{3}{128} | 0 | 8,154.8125 | -1 | 8,154.8125 | |
Teacher Li and three students (Xiao Ma, Xiao Lu, and Xiao Zhou) depart from school one after another and walk the same route to a cinema. The walking speed of the three students is equal, and Teacher Li's walking speed is 1.5 times that of the students. Currently, Teacher Li is 235 meters from the school, Xiao Ma is 87... | 42 | 0.0625 | 7,079.6875 | 1,868 | 7,427.133333 | |
Among the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, draw one at random. The probability of drawing a prime number is ____, and the probability of drawing a composite number is ____. | \frac{4}{9} | 0.375 | 589.9375 | 570.833333 | 601.4 | |
A store owner purchases merchandise at a discount of 30% off the original list price. To ensure a profit, the owner wants to mark up the goods such that after offering a 15% discount on the new marked price, the final selling price still yields a 30% profit compared to the cost price. What percentage of the original li... | 107\% | 0.0625 | 5,898.125 | 2,137 | 6,148.866667 | |
The domain of the function $f(x)=\log_{\frac{1}{2}}(\log_4(\log_{\frac{1}{4}}(\log_{16}(\log_{\frac{1}{16}}x))))$ is an interval of length $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | 271 | To find the domain of the function
\[ f(x) = \log_{\frac{1}{2}}\left(\log_4\left(\log_{\frac{1}{4}}\left(\log_{16}\left(\log_{\frac{1}{16}}x\right)\right)\right)\right), \]
we need to ensure that each logarithm in the nested expression is defined and positive, as the logarithm function is only defined for positive arg... | 0.6875 | 6,156.375 | 5,245.545455 | 8,160.2 |
A triangle is divided into 1000 triangles. What is the maximum number of distinct points that can be vertices of these triangles? | 1002 | 0.0625 | 7,974.3125 | 5,721 | 8,124.533333 | |
Consider the following scenario where three stores offer different discounts on a television priced at $149.99$:
\begin{tabular}{|l|l|}
\hline
\textbf{Store} & \textbf{Sale Price for Television $Y$} \\
\hline
Value Market & $\$10$~off the list price~$\$149.99$ \\
Tech Bargains & $30\%$~off the list price~$\$149.99$ \\
... | 3500 | 0.75 | 4,613.25 | 3,998.916667 | 6,456.25 | |
For how many $n=2,3,4,\ldots,109,110$ is the base-$n$ number $432143_n$ a multiple of $11$? | 10 | 0.3125 | 7,388 | 5,688.8 | 8,160.363636 | |
An equilateral triangle \( ABC \) is inscribed in the ellipse \( \frac{x^2}{p^2} + \frac{y^2}{q^2} = 1 \), such that vertex \( B \) is at \( (0, q) \), and \( \overline{AC} \) is parallel to the \( x \)-axis. The foci \( F_1 \) and \( F_2 \) of the ellipse lie on sides \( \overline{BC} \) and \( \overline{AB} \), respe... | \frac{8}{5} | 0.0625 | 8,152.9375 | 7,567 | 8,192 | |
Consider a terminal with fifteen gates arranged in a straight line with exactly $90$ feet between adjacent gates. A passenger's departure gate is assigned at random. Later, the gate is changed to another randomly chosen gate. Calculate the probability that the passenger walks $360$ feet or less to the new gate. Express... | 31 | 0.0625 | 7,281.75 | 6,778 | 7,315.333333 | |
In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules:
1) The marksman first chooses a column from which a target is to be broken.
2) The marksman must then break the lowest... | 560 | 0.9375 | 2,781.4375 | 2,420.733333 | 8,192 | |
A square piece of paper has sides of length $100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at a distance $\sqrt{17}$ from the corner, and they meet on the diagonal at an angle of $60^{\circ}$ (see the figure below). The paper is then folded up along... | 871 | In the final pyramid, let $ABCD$ be the smaller square and let $A'B'C'D'$ be the larger square such that $AA'$, etc. are edges.
It is obvious from the diagram that $\angle A'AB = \angle A'AD = 105^\circ$.
Let $AB$ and $AD$ be the positive $x$ and $y$ axes in a 3-d coordinate system such that $A'$ has a positive $z$ c... | 0 | 8,192 | -1 | 8,192 |
Given real numbers \( x_{1}, x_{2}, \cdots, x_{1991} \) that satisfy the condition
\[
\sum_{i=1}^{1990} \left| x_{i} - x_{i+1} \right| = 1991,
\]
and \( y_{k} = \frac{1}{k} \sum_{i=1}^{k} x_{i} \) for \( k = 1, 2, \cdots, 1991 \), determine the maximum value of \( \sum_{i=1}^{1990} \left| y_{i} - y_{i+1} \right| \). | 1990 | 0 | 8,192 | -1 | 8,192 | |
What is the greatest common divisor of $8!$ and $10!$? | 40320 | 1 | 2,717.0625 | 2,717.0625 | -1 | |
Let $A$ be a set of positive integers satisfying the following : $a.)$ If $n \in A$ , then $n \le 2018$ . $b.)$ If $S \subset A$ such that $|S|=3$ , then there exists $m,n \in S$ such that $|n-m| \ge \sqrt{n}+\sqrt{m}$ What is the maximum cardinality of $A$ ? | 44 | 0 | 8,192 | -1 | 8,192 | |
Let ellipse M be defined by the equation $$\frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1$$ where $a>b>0$. The eccentricity of ellipse M and the eccentricity of the hyperbola defined by $x^{2}-y^{2}=1$ are reciprocals of each other, and ellipse M is inscribed in the circle defined by $x^{2}+y^{2}=4$.
(1) Find the equat... | \sqrt {2} | 0 | 7,933 | -1 | 7,933 | |
Let triangle $A B C$ have $A B=5, B C=6$, and $A C=7$, with circumcenter $O$. Extend ray $A B$ to point $D$ such that $B D=5$, and extend ray $B C$ to point $E$ such that $O D=O E$. Find $C E$. | \sqrt{59}-3 | Because $O D=O E, D$ and $E$ have equal power with respect to the circle, so $(E C)(E B)=(D B)(D A)=50$. Letting $E C=x$, we have $x(x+6)=50$, and taking the positive root gives $x=\sqrt{59}-3$. | 0.25 | 8,056.5 | 7,650 | 8,192 |
If $r_1$ and $r_2$ are the distinct real roots of $x^2+px+8=0$, then it must follow that: | $|r_1+r_2|>4\sqrt{2}$ | 1. **Identify the coefficients and the discriminant condition**:
Given the quadratic equation $x^2 + px + 8 = 0$, we know that the sum of the roots $r_1 + r_2 = -p$ and the product of the roots $r_1r_2 = 8$, according to Vieta's formulas. Since $r_1$ and $r_2$ are distinct real roots, the discriminant must be posit... | 0 | 4,833.6875 | -1 | 4,833.6875 |
In parallelogram $ABCD$ , the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$ . If the area of $XYZW$ is $10$ , find the area of $ABCD$ | 40 | 0 | 8,150.1875 | -1 | 8,150.1875 | |
Given a function f(x) defined on ℝ that satisfies f(x-2)=f(-2-x), and when x ≥ -2, f(x)=2^x-3. If the function f(x) has a zero point in the interval (k,k+1) (k ∈ ℤ), determine the value of k. | -6 | 0 | 7,832.625 | -1 | 7,832.625 | |
Given a random variable $X\sim N(2, \sigma ^{2})$, $P(X\leqslant 0)=0.15$, calculate $P(2\leqslant X\leqslant 4)$. | 0.35 | 0.25 | 6,825.8125 | 7,086 | 6,739.083333 | |
It is known that \( m, n, \) and \( k \) are distinct natural numbers greater than 1, the number \( \log_{m} n \) is rational, and additionally,
$$
k^{\sqrt{\log_{m} n}} = m^{\sqrt{\log_{n} k}}
$$
Find the minimum possible value of the sum \( k + 5m + n \). | 278 | 0.0625 | 8,118.625 | 7,018 | 8,192 | |
The product of the positive differences of the sum of eight integers (28) is 6. To what power will it always be divisible? | 6^7 | 0 | 7,213.625 | -1 | 7,213.625 | |
The Fibonacci numbers are defined recursively by $F_{0}=0, F_{1}=1$, and $F_{i}=F_{i-1}+F_{i-2}$ for $i \geq 2$. Given 15 wooden blocks of weights $F_{2}, F_{3}, \ldots, F_{16}$, compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the... | 32 | Partition the blocks into sets $$\left\{F_{2}, F_{3}, F_{4}\right\},\left\{F_{5}, F_{6}, F_{7}\right\}, \ldots,\left\{F_{14}, F_{15}, F_{16}\right\}$$ We can show by bounding that $F_{16}$ belongs on the opposite side as $F_{15}$ and $F_{14}$, and, in general, that $F_{3 k+1}$ is on the opposite side as $F_{3 k}$ and $... | 0 | 8,192 | -1 | 8,192 |
What is the greatest prime factor of $12! + 14!$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.) | 61 | 1 | 1,912.375 | 1,912.375 | -1 | |
How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 5, 5, 7, 7, 7\}$ if no digit may be used more times than it appears in the given set of available digits? | 43 | 0.125 | 7,790.125 | 5,172.5 | 8,164.071429 | |
A certain shopping mall purchased a batch of daily necessities. If they are sold at a price of $5$ yuan per item, they can sell $30,000$ items per month. If they are sold at a price of $6$ yuan per item, they can sell $20,000$ items per month. It is assumed that the monthly sales quantity $y$ (items) and the price $x$ ... | 40000 | 1 | 4,920.0625 | 4,920.0625 | -1 | |
What is the value of the expression \( 4 + \frac{3}{10} + \frac{9}{1000} \)? | 4.309 | Converting from fractions to decimals, \( 4 + \frac{3}{10} + \frac{9}{1000} = 4 + 0.3 + 0.009 = 4.309 \). | 1 | 1,578.25 | 1,578.25 | -1 |
The school store sells 7 pencils and 8 notebooks for $4.15. It also sells 5 pencils and 3 notebooks for $1.77. How much do 16 pencils and 10 notebooks cost? | $5.84 | 1. **Define Variables:**
Let $p$ be the cost of one pencil in dollars, and $n$ be the cost of one notebook in dollars.
2. **Set Up Equations:**
From the problem, we have two equations based on the given transactions:
\[
7p + 8n = 4.15 \quad \text{(Equation 1)}
\]
\[
5p + 3n = 1.77 \quad \text{(Equ... | 0 | 3,968.5625 | -1 | 3,968.5625 |
The graph of an equation \[\sqrt{(x-3)^2 + (y+4)^2} + \sqrt{(x+5)^2 + (y-8)^2} = 20.\]is an ellipse. What is the distance between its foci? | 4\sqrt{13} | 1 | 2,599.25 | 2,599.25 | -1 | |
Find $\sin \frac{11 \pi}{3}.$ | -\frac{\sqrt{3}}{2} | 0 | 2,875.5 | -1 | 2,875.5 | |
What is the remainder when $2^{87} +3$ is divided by $7$? | 4 | 1 | 2,946.625 | 2,946.625 | -1 | |
Consider the eighth-sphere $\left\{(x, y, z) \mid x, y, z \geq 0, x^{2}+y^{2}+z^{2}=1\right\}$. What is the area of its projection onto the plane $x+y+z=1$ ? | \frac{\pi \sqrt{3}}{4} | Consider the three flat faces of the eighth-ball. Each of these is a quarter-circle of radius 1, so each has area $\frac{\pi}{4}$. Furthermore, the projections of these faces cover the desired area without overlap. To find the projection factor one can find the cosine of the angle $\theta$ between the planes, which is ... | 0 | 8,192 | -1 | 8,192 |
Given points \(A=(8,15)\) and \(B=(16,9)\) are on a circle \(\omega\), and the tangent lines to \(\omega\) at \(A\) and \(B\) meet at a point \(P\) on the x-axis, calculate the area of the circle \(\omega\). | 250\pi | 0 | 5,607.25 | -1 | 5,607.25 | |
The function $f(x)$ satisfies
\[f(x) + f \left( \frac{x + 1}{1 - 3x} \right) = x\]for all $x \neq \frac{1}{3}.$ Find $f(2).$ | \frac{48}{35} | 0.6875 | 5,674.5625 | 4,530.272727 | 8,192 | |
How many interior intersection points are there on a 12 by 12 grid of squares? | 121 | A 12 by 12 grid of squares will have 11 interior vertical lines and 11 interior horizontal lines. Each of the 11 interior vertical lines intersects each of the 11 interior horizontal lines and creates an interior intersection point. Therefore, the number of interior intersection points is $11 imes 11=121$. | 0.875 | 4,365.3125 | 3,818.642857 | 8,192 |
Two people, A and B, start from the same point on a 300-meter circular track and run in opposite directions. A runs at 2 meters per second, and B runs at 4 meters per second. When they first meet, A turns around and runs back. When A and B meet again, B turns around and runs back. Following this pattern, after how many... | 250 | 0 | 7,886.0625 | -1 | 7,886.0625 | |
Let \(ABC\) be a non-degenerate triangle and \(I\) the center of its incircle. Suppose that \(\angle A I B = \angle C I A\) and \(\angle I C A = 2 \angle I A C\). What is the value of \(\angle A B C\)? | 60 | 0.0625 | 5,643.375 | 6,724 | 5,571.333333 | |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-zero vectors, $\overrightarrow{m} = \overrightarrow{a} + t \overrightarrow{b} (t \in \mathbb{R})$, $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = 2$, the minimum value of $|\overrightarrow{m}|$ is obtained only when $t = \frac{1}{4}$. Determine the a... | \frac{2\pi}{3} | 0.75 | 1,953.6875 | 2,011.666667 | 1,779.75 | |
Calculate to 3 decimal places the following expressions:
1.
$$
\frac{2 \sqrt{3}}{\sqrt{3}-\sqrt{2}}
$$
2.
$$
\frac{(3+\sqrt{3})(1+\sqrt{5})}{(5+\sqrt{5})(1+\sqrt{3})}
$$ | 0.775 | 0.1875 | 7,864 | 6,442.666667 | 8,192 | |
Suppose that $x, y$, and $z$ are non-negative real numbers such that $x+y+z=1$. What is the maximum possible value of $x+y^{2}+z^{3}$ ? | 1 | Since $0 \leq y, z \leq 1$, we have $y^{2} \leq y$ and $z^{3} \leq z$. Therefore $x+y^{2}+z^{3} \leq x+y+z=1$. We can get $x+y^{2}+z^{3}=1$ by setting $(x, y, z)=(1,0,0)$. | 0.375 | 8,051.0625 | 7,816.166667 | 8,192 |
The function $x^2+px+q$ with $p$ and $q$ greater than zero has its minimum value when: | $x=\frac{-p}{2}$ | 1. **Identify the form of the function**: The given function is $x^2 + px + q$, which is a quadratic function in the standard form $ax^2 + bx + c$ where $a = 1$, $b = p$, and $c = q$.
2. **Determine the vertex**: The vertex of a parabola represented by the quadratic function $ax^2 + bx + c$ is given by the formula for... | 0 | 2,331.75 | -1 | 2,331.75 |
Find the number of integers from 1 to 1000 inclusive that give the same remainder when divided by 11 and by 12. | 87 | 0.875 | 5,849.875 | 5,515.285714 | 8,192 |
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