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 |
|---|---|---|---|---|---|---|
Given the line $y=ax$ intersects the circle $C:x^2+y^2-2ax-2y+2=0$ at points $A$ and $B$, and $\Delta ABC$ is an equilateral triangle, then the area of circle $C$ is __________. | 6\pi | 0.625 | 5,584.0625 | 4,019.3 | 8,192 | |
If $x$ is a real number and $k$ is a nonnegative integer, recall that the binomial coefficient $\binom{x}{k}$ is defined by the formula
\[
\binom{x}{k} = \frac{x(x - 1)(x - 2) \dots (x - k + 1)}{k!} \, .
\]Compute the value of
\[
\frac{\binom{1/2}{2014} \cdot 4^{2014}}{\binom{4028}{2014}} \, .
\] | -\frac{1} { 4027} | 0 | 7,889.1875 | -1 | 7,889.1875 | |
If $x > 0$, $y > 0$, and $\frac{1}{2x+y} + \frac{4}{x+y} = 2$, find the minimum value of $7x + 5y$. | 7 + 2\sqrt{6} | 0.1875 | 7,598.5625 | 5,027 | 8,192 | |
Determine the sum of all integral values of $c$ such that $c \leq 18$ for which the equation $y = x^2 - 5x - c$ has exactly two rational roots. | 10 | 0.8125 | 5,712.1875 | 5,139.923077 | 8,192 | |
In triangle $ABC$ the medians $\overline{AD}$ and $\overline{CE}$ have lengths $18$ and $27$, respectively, and $AB=24$. Extend $\overline{CE}$ to intersect the circumcircle of $ABC$ at $F$. The area of triangle $AFB$ is $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any ... | 63 | 0.125 | 7,930.3125 | 7,097.5 | 8,049.285714 | |
Compute
\[
\left( 1 - \sin \frac {\pi}{8} \right) \left( 1 - \sin \frac {3\pi}{8} \right) \left( 1 - \sin \frac {5\pi}{8} \right) \left( 1 - \sin \frac {7\pi}{8} \right).
\] | \frac{1}{4} | 0 | 7,982.125 | -1 | 7,982.125 | |
The graph of the function $f(x)$ is symmetric about the $y$-axis, and for any $x \in \mathbb{R}$, it holds that $f(x+3)=-f(x)$. If $f(x)=(\frac{1}{2})^{x}$ when $x \in \left( \frac{3}{2}, \frac{5}{2} \right)$, then find $f(2017)$. | -\frac{1}{4} | 0.8125 | 4,770.9375 | 3,981.461538 | 8,192 | |
How many non-similar regular 720-pointed stars are there, given that a regular $n$-pointed star requires its vertices to not all align with vertices of a smaller regular polygon due to common divisors other than 1 between the step size and $n$? | 96 | 0.125 | 6,105.375 | 5,239.5 | 6,229.071429 | |
Let $a$, $b$, $c$ represent the lengths of the sides of a triangle, and they are all natural numbers, where $a \leq b \leq c$. If $b = 2008$, then the total number of triangles that satisfy this condition is . | 2017036 | 0.8125 | 5,640.375 | 5,209.461538 | 7,507.666667 | |
If $|x| + x + y = 14$ and $x + |y| - y = 16,$ find $x + y.$ | -2 | 0 | 3,723.8125 | -1 | 3,723.8125 | |
A function $f(x, y)$ is linear in $x$ and in $y . f(x, y)=\frac{1}{x y}$ for $x, y \in\{3,4\}$. What is $f(5,5)$? | \frac{1}{36} | The main fact that we will use in solving this problem is that $f(x+2, y)-f(x+1, y)=f(x+1, y)-f(x, y)$ whenever $f$ is linear in $x$ and $y$. Suppose that $f(x, y)=a x y+b y+c x+d=x(a y+c)+(b y+d)$ for some constants $a, b, c$, and $d$. Then it is easy to see that $$\begin{aligned} f(x+2, y)-f(x+1, y) & =(x+2)(a y+c)+(... | 0.125 | 6,018.4375 | 5,668 | 6,068.5 |
A youth radio station, to attract listeners' attention, gives away gifts and grand prizes among them. Gifts are given away hourly over sixteen hours (one gift per hour), and grand prizes are given away during four evening hours (one grand prize per hour). The probability that listeners win a prize is 0.3, and a grand p... | 0.862 | 0 | 7,499.625 | -1 | 7,499.625 | |
The smaller square in the figure below has a perimeter of $4$ cm, and the larger square has an area of $16$ $\text{cm}^2$. What is the distance from point $A$ to point $B$? Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(12,0));
draw((2,0)--(2,10));
draw((0,0)--(0,2));
draw((0,2)--(2,2));
dr... | 5.8 | 0 | 6,990.875 | -1 | 6,990.875 | |
The function \( g \), defined on the set of ordered pairs of positive integers, satisfies the following properties:
\[
\begin{align*}
g(x, x) &= x, \\
g(x, y) &= g(y, x), \quad \text{and} \\
(x + 2y)g(x, y) &= yg(x, x + 2y).
\end{align*}
\]
Calculate \( g(18, 66) \). | 198 | 0 | 8,192 | -1 | 8,192 | |
The cost of four pencils and one pen is $\$2.60$, and the cost of one pencil and three pens is $\$2.15$. Find the cost of three pencils and two pens. | 2.63 | 0.75 | 6,754.375 | 6,275.166667 | 8,192 | |
Find the smallest prime number that can be represented as the sum of two, three, four, five, and six distinct prime numbers. | 61 | 0.125 | 8,055.75 | 7,306 | 8,162.857143 | |
The equation \( \sin^2 x + \sin^2 3x + \sin^2 5x + \sin^2 6x = \frac{81}{32} \) can be reduced to the equivalent equation
\[ \cos ax \cos bx \cos cx = 0, \] for some positive integers \( a, \) \( b, \) and \( c. \) Find \( a + b + c. \) | 14 | 0 | 8,192 | -1 | 8,192 | |
If $\log_{10}2=a$ and $\log_{10}3=b$, then $\log_{5}12=?$ | \frac{2a+b}{1-a} | 1. **Identify the given values and the target expression**: We are given $\log_{10}2=a$ and $\log_{10}3=b$. We need to find $\log_{5}12$.
2. **Use the change of base formula**: According to the change of base formula, for any logarithms $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ is any positive number. Applying... | 0.9375 | 2,101.625 | 2,080.333333 | 2,421 |
The population size (in number of animals) of a certain animal species is given by the equation $y=a\log_{2}(x+1)$. Suppose that the population size of this animal species in the first year was 100 animals. What will be the population size in the 15th year. | 400 | 1 | 2,382.1875 | 2,382.1875 | -1 | |
Compute the surface area of a cube inscribed in a sphere of surface area $\pi$. | 2 | The sphere's radius $r$ satisfies $4 \pi r^{2}=\pi \Rightarrow r=1 / 2$, so the cube has body diagonal 1 , hence side length $1 / \sqrt{3}$. So, its surface area is $6(1 / \sqrt{3})^{2}=2$. | 1 | 1,443.4375 | 1,443.4375 | -1 |
Simplify $((5p+1)-2p\cdot4)(3)+(4-1\div3)(6p-9)$ to a much simpler expression of the form $ap-b$ , where $a$ and $b$ are positive integers. | 13p-30 | 1 | 1,943 | 1,943 | -1 | |
What is the smallest positive number that is prime and $10$ less than a perfect square? | 71 | 1 | 4,404.25 | 4,404.25 | -1 | |
The average age of the three Wilson children is 7 years. If the two younger children are 4 years old and 7 years old, how many years old is the oldest child? | 10 | 1 | 719.0625 | 719.0625 | -1 | |
The diagram shows five circles of the same radius touching each other. A square is drawn so that its vertices are at the centres of the four outer circles. What is the ratio of the area of the shaded parts of the circles to the area of the unshaded parts of the circles? | 2:3 | 0 | 7,447.5 | -1 | 7,447.5 | |
Given that $a > 1$ and $b > 0$, and $a + 2b = 2$, find the minimum value of $\frac{2}{a - 1} + \frac{a}{b}$. | 4(1 + \sqrt{2}) | 0 | 7,912.375 | -1 | 7,912.375 | |
In product inspection, the method of sampling inspection is often used. Now, 4 products are randomly selected from 100 products (among which there are 3 defective products) for inspection. The number of ways to exactly select 2 defective products is ____. (Answer with a number) | 13968 | 1 | 2,178.375 | 2,178.375 | -1 | |
Two cells in a \(20 \times 20\) board are adjacent if they have a common edge (a cell is not considered adjacent to itself). What is the maximum number of cells that can be marked in a \(20 \times 20\) board such that every cell is adjacent to at most one marked cell? | 100 | 0 | 8,192 | -1 | 8,192 | |
Sarah's bowling score was 40 points more than Greg's, and the average of their two scores was 102. What was Sarah's score? (Recall that the average of two numbers is their sum divided by 2.) | 122 | 1 | 1,331.125 | 1,331.125 | -1 | |
Given an ellipse $(C)$: $\frac{x^{2}}{3m} + \frac{y^{2}}{m} = 1 (m > 0)$ with the length of its major axis being $2\sqrt{6}$, and $O$ is the coordinate origin.
(I) Find the equation and eccentricity of the ellipse $(C)$;
(II) Let moving line $(l)$ intersect with the $y$-axis at point $B$, and the symmetric point $P(3, ... | \sqrt{6} | 0.1875 | 7,733 | 5,744 | 8,192 | |
Consider the infinite series: $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \cdots$. Let $T$ be the limiting sum of this series. Find $T$.
**A)** $\frac{3}{26}$
**B)** $\frac{15}{26}$
**C)** $\frac{27}{26}$
**D)** $\frac{1}{26}$
**E)** $\frac{40}{26}$ | \frac{15}{26} | 0 | 7,256.375 | -1 | 7,256.375 | |
X is the point (1994p, 7·1994p), where p is a prime, and O is the point (0, 0). How many triangles XYZ have Y and Z at lattice points, incenter at O, and YXZ as a right-angle? | 36 | 0 | 8,192 | -1 | 8,192 | |
The base of a quadrilateral pyramid is a square \(ABCD\) with each side equal to 2. The lateral edge \(SA\) is perpendicular to the base plane and also equals 2. A plane is passed through the lateral edge \(SC\) and a point on side \(AB\) such that the resulting cross-section of the pyramid has the smallest perimeter. ... | \sqrt{6} | 0.8125 | 6,126.5 | 5,668.461538 | 8,111.333333 | |
In the diagram, each of four identical circles touch three others. The circumference of each circle is 48. Calculate the perimeter of the shaded region formed within the central area where all four circles touch. Assume the circles are arranged symmetrically like petals of a flower. | 48 | 0.6875 | 6,999.5625 | 6,460.909091 | 8,184.6 | |
Given is a isosceles triangle ABC so that AB=BC. Point K is in ABC, so that CK=AB=BC and <KAC=30°.Find <AKB=? | 150 | 0.0625 | 8,192 | 8,192 | 8,192 | |
The sum of three different numbers is 100. The two larger numbers differ by 8 and the two smaller numbers differ by 5. What is the value of the largest number? | \frac{121}{3} | 0.875 | 4,773.8125 | 4,285.5 | 8,192 | |
Given a basketball player has a probability of $a$ for scoring 3 points in a shot, $b$ for scoring 2 points, and $c$ for not scoring any points, where $a, b, c \in (0, 1)$, and the mathematical expectation for scoring points in one shot is 2, determine the minimum value of $\frac{2}{a} + \frac{1}{3b}$. | \frac{16}{3} | 1 | 4,077.6875 | 4,077.6875 | -1 | |
There are 8 identical balls in a box, consisting of three balls numbered 1, three balls numbered 2, and two balls numbered 3. A ball is randomly drawn from the box, returned, and then another ball is randomly drawn. The product of the numbers on the balls drawn first and second is denoted by $\xi$. Find the expected va... | 225/64 | 0.625 | 6,050.75 | 4,766 | 8,192 | |
What is $3.57 - 1.14 - 0.23$? | 2.20 | 1 | 1,077.625 | 1,077.625 | -1 | |
Let $m, n > 2$ be integers. One of the angles of a regular $n$-gon is dissected into $m$ angles of equal size by $(m-1)$ rays. If each of these rays intersects the polygon again at one of its vertices, we say $n$ is $m$-cut. Compute the smallest positive integer $n$ that is both 3-cut and 4-cut. | 14 | For the sake of simplicity, inscribe the regular polygon in a circle. Note that each interior angle of the regular $n$-gon will subtend $n-2$ of the $n$ arcs on the circle. Thus, if we dissect an interior angle into $m$ equal angles, then each must be represented by a total of $\frac{n-2}{m}$ arcs. However, since each ... | 0 | 7,112.4375 | -1 | 7,112.4375 |
Throw 6 dice at a time, find the probability, in the lowest form, such that there will be exactly four kinds of the outcome. | 325/648 | 0.4375 | 7,080 | 6,174.428571 | 7,784.333333 | |
What is the smallest positive value of $x$ such that $x + 4321$ results in a palindrome? | 13 | 1 | 4,977.875 | 4,977.875 | -1 | |
A store sells chalk in three types of packages: regular, unusual, and excellent. Initially, the quantitative ratio of the types was 2:3:6. After some packages of regular and unusual chalk—totaling no more than 100 packages—were delivered to the store, and $40\%$ of the excellent chalk packages were sold, the ratio chan... | 24 | 0.0625 | 7,249.6875 | 5,015 | 7,398.666667 | |
Let $P$ be the product of the first $100$ positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k .$ | 49 | We are obviously searching for multiples of three set S of odd numbers 1-199. Starting with 3, every number $\equiv 2 \pmod{3}$ in set S will be divisible by 3. In other words, every number $\equiv 3 \pmod{6}$. This is because the LCM must be divisible by 3, and 2, because the set is comprised of only odd numbers. Usin... | 0.5625 | 6,257.5625 | 4,753 | 8,192 |
The diagram shows a square and a regular decagon that share an edge. One side of the square is extended to meet an extended edge of the decagon. What is the value of \( x \)? | 18 | 0 | 6,787.125 | -1 | 6,787.125 | |
Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. Find the number of such distinct triangles whose area is a positive integer. | 600 | Let the two points $P$ and $Q$ be defined with coordinates; $P=(x_1,y_1)$ and $Q=(x_2,y_2)$
We can calculate the area of the parallelogram with the determinant of the matrix of the coordinates of the two points(shoelace theorem).
$\det \left(\begin{array}{c} P \\ Q\end{array}\right)=\det \left(\begin{array}{cc}x_1 &y_... | 0.75 | 5,756.75 | 4,945 | 8,192 |
Given real numbers $b$ and $c$, and the function $f(x) = x^2 + bx + c$, the equation $f(f(x)) = 0$ has exactly three different real roots. Find the maximum value of the sum of the roots of $f(x)$. | 1/2 | 0.1875 | 8,126.6875 | 7,843.666667 | 8,192 | |
How many odd integers are there between $\frac{17}{4}$ and $\frac{35}{2}?$ | 7 | 1 | 2,533.5625 | 2,533.5625 | -1 | |
Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 29x\]for all integers $x.$ Find the smallest possible value of $a.$ | 10 \pi - 29 | 0 | 7,612 | -1 | 7,612 | |
Given two circles $A:(x+4)^2+y^2=25$ and $B:(x-4)^2+y^2=1$, a moving circle $M$ is externally tangent to both fixed circles. Let the locus of the center of moving circle $M$ be curve $C$.
(I) Find the equation of curve $C$;
(II) If line $l$ intersects curve $C$ at points $P$ and $Q$, and $OP \perp OQ$. Is $\frac{1}{|... | \frac{1}{6} | 0.125 | 7,897.8125 | 7,156.5 | 8,003.714286 | |
\dfrac{1}{10} + \dfrac{9}{100} + \dfrac{9}{1000} + \dfrac{7}{10000} = | 0.1997 | 1. **Convert each fraction to its decimal form**:
- $\dfrac{1}{10} = 0.1$
- $\dfrac{9}{100} = 0.09$
- $\dfrac{9}{1000} = 0.009$
- $\dfrac{7}{10000} = 0.0007$
2. **Add the decimals**:
- Start by adding the largest decimal places first:
\[
0.1 + 0.09 = 0.19
\]
- Next, add the result to t... | 0.5625 | 564.5 | 493 | 656.428571 |
Find an $n$ such that $n!-(n-1)!+(n-2)!-(n-3)!+\cdots \pm 1$ ! is prime. Be prepared to justify your answer for $\left\{\begin{array}{c}n, \\ {\left[\frac{n+225}{10}\right],}\end{array} n \leq 25\right.$ points, where $[N]$ is the greatest integer less than $N$. | 3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160 | $3,4,5,6,7,8,10,15,19,41$ (26 points), 59, 61 (28 points), 105 (33 points), 160 (38 points) are the only ones less than or equal to 335. If anyone produces an answer larger than 335, then we ask for justification to call their bluff. It is not known whether or not there are infinitely many such $n$. | 0 | 8,192 | -1 | 8,192 |
The number of significant digits in the measurement of the side of a square whose computed area is $1.1025$ square inches to the nearest ten-thousandth of a square inch is: | 5 | To find the number of significant digits in the measurement of the side of a square, we start by considering the given area of the square, which is $1.1025$ square inches. This value is given to the nearest ten-thousandth of a square inch, indicating precision in the measurement.
1. **Identify the number of significan... | 0.375 | 7,073.25 | 5,906.666667 | 7,773.2 |
Find the largest prime divisor of $36^2 + 49^2$. | 13 | 0 | 5,995 | -1 | 5,995 | |
Every day, from Monday to Friday, an old man went to the blue sea and cast his net into the sea. Each day, the number of fish caught in the net was no greater than the number caught on the previous day. In total, over the five days, the old man caught exactly 100 fish. What is the minimum total number of fish he could ... | 50 | 0.0625 | 8,024.375 | 8,000 | 8,026 | |
Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads? | 56 | To solve this problem, we need to calculate the expected number of coins that show heads after up to three tosses. We will use the concept of probability and expected value to find the solution.
1. **Probability of a coin showing heads after each toss:**
- The probability of a coin landing heads on the first toss i... | 1 | 3,215.4375 | 3,215.4375 | -1 |
For any positive integer $n,$ let $\langle n \rangle$ denote the closest integer to $\sqrt{n}.$ Evaluate
\[\sum_{n = 1}^\infty \frac{2^{\langle n \rangle} + 2^{-\langle n \rangle}}{2^n}.\] | 3 | 0.5 | 7,600.6875 | 7,009.375 | 8,192 | |
There exist $r$ unique nonnegative integers $n_1 > n_2 > \cdots > n_r$ and $r$ unique integers $a_k$ ($1\le k\le r$) with each $a_k$ either $1$ or $- 1$ such that \[a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 1025.\]
Find $n_1 + n_2 + \cdots + n_r$. | 17 | 0 | 8,035.375 | -1 | 8,035.375 | |
For a certain weekend, the weatherman predicts that it will rain with a $40\%$ probability on Saturday and a $50\%$ probability on Sunday. Assuming these probabilities are independent, what is the probability that it rains over the weekend (that is, on at least one of the days)? Express your answer as a percentage. | 70\% | 1 | 1,903.625 | 1,903.625 | -1 | |
The mathematician John is having trouble remembering his girlfriend Alicia's 7-digit phone number. He remembers that the first four digits consist of one 1, one 2, and two 3s. He also remembers that the fifth digit is either a 4 or 5. While he has no memory of the sixth digit, he remembers that the seventh digit is 9 m... | 240 | There are $\frac{4!}{2!}=12$ possibilities for the first four digits. There are two possibilities for the fifth digit. There are 10 possibilities for the sixth digit, and this uniquely determines the seventh digit. So he has to dial $12 \cdot 2 \cdot 10=240$ numbers. | 0.6875 | 2,434.5 | 2,616.363636 | 2,034.4 |
A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\tfrac{m}{n},$ where $m$... | 173 | [asy] size(10cm); pair A, B, C, D, O, P, H, L, X, Y; A = (-1, 0); B = (1, 0); H = (0, 0); C = (5, 0); D = (9, 0); L = (7, 0); O = (0, sqrt(160/13 - 1)); P = (7, sqrt(160/13 - 4)); X = (0, sqrt(160/13 - 4)); Y = (O + P) / 2; draw(A -- O -- B -- cycle); draw(C -- P -- D -- cycle); draw(B -- C); draw(O -- P); draw(P -- X,... | 0.125 | 7,581.5625 | 5,906 | 7,820.928571 |
A triangle has sides of length 888, 925, and $x>0$. Find the value of $x$ that minimizes the area of the circle circumscribed about the triangle. | 259 | 259. | 0.25 | 7,389.25 | 5,340.75 | 8,072.083333 |
Consider the set of numbers $\{1, 10, 10^2, 10^3, \ldots, 10^{10}\}$. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer? | 9 | To solve the problem, we need to find the ratio of the largest element in the set $\{1, 10, 10^2, 10^3, \ldots, 10^{10}\}$ to the sum of all other elements in the set. The largest element in this set is $10^{10}$.
1. **Calculate the sum of the other elements:**
The sum of the other elements is $1 + 10 + 10^2 + 10^3... | 0.6875 | 6,823.3125 | 6,201.181818 | 8,192 |
Given $\tan \alpha = -\frac{1}{2}$, find the value of $\frac{1+2\sin \alpha \cos \alpha}{\sin^2 \alpha - \cos^2 \alpha}$. | -\frac{1}{3} | 0.875 | 4,441.5 | 3,905.714286 | 8,192 | |
Emma's telephone number is $548-1983$ and her apartment number contains different digits. The sum of the digits in her four-digit apartment number is the same as the sum of the digits in her phone number. What is the lowest possible value for Emma’s apartment number? | 9876 | 0 | 8,182.75 | -1 | 8,182.75 | |
Four cyclists. Four identical circles represent four tracks. The four cyclists start from the center at noon. Each moves along their track at speeds: the first at 6 km/h, the second at 9 km/h, the third at 12 km/h, and the fourth at 15 km/h. They agreed to ride until they all meet again in the center for the fourth tim... | 12:26:40 | 0.0625 | 7,230.875 | 6,908 | 7,252.4 | |
Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. | 15 | 1 | 1,926.9375 | 1,926.9375 | -1 | |
Determine all rational numbers \(a\) for which the matrix \(\left(\begin{array}{cccc} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a \\ 0 & 1 & a & -a \end{array}\right)\) is the square of a matrix with all rational entries. | a=0 | We will show that the only such number is \(a=0\). Let \(A=\left(\begin{array}{cccc} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a \\ 0 & 1 & a & -a \end{array}\right)\) and suppose that \(A=B^{2}\). It is easy to compute the characteristic polynomial of \(A\), which is \(p_{A}(x)=\operatorname{det}(A-x I)=\left... | 0 | 8,192 | -1 | 8,192 |
Given a hyperbola $C_{1}$ defined by $2x^{2}-y^{2}=1$, find the area of the triangle formed by a line parallel to one of the asymptotes of $C_{1}$, the other asymptote, and the x-axis. | \frac{\sqrt{2}}{8} | 0 | 8,192 | -1 | 8,192 | |
Let $A B C$ be a triangle with $A B=5, A C=4, B C=6$. The angle bisector of $C$ intersects side $A B$ at $X$. Points $M$ and $N$ are drawn on sides $B C$ and $A C$, respectively, such that $\overline{X M} \| \overline{A C}$ and $\overline{X N} \| \overline{B C}$. Compute the length $M N$. | \frac{3 \sqrt{14}}{5} | By Stewart's Theorem on the angle bisector, $$C X^{2}=A C \cdot B C\left(1-\frac{A B}{A C+B C}^{2}\right)$$ Thus, $$C X^{2}=4 \cdot 6\left(1-\frac{5}{10}^{2}\right)=18$$ Since $\overline{X M} \| \overline{A C}$ and $\overline{X N} \| \overline{B C}$, we produce equal angles. So, by similar triangles, $X M=X N=\frac{4 \... | 0 | 6,242.0625 | -1 | 6,242.0625 |
Find the sum of the roots of the equation $\tan^2x - 8\tan x + 2 = 0$ that are between $x = 0$ and $x = 2\pi$ radians. | 3\pi | 0 | 7,922.6875 | -1 | 7,922.6875 | |
How many distinct arrangements of the letters in the word "balloon" are there? | 1260 | 0.375 | 1,915.5 | 1,988.666667 | 1,871.6 | |
Two boys $A$ and $B$ start at the same time to ride from Port Jervis to Poughkeepsie, $60$ miles away. $A$ travels $4$ miles an hour slower than $B$. $B$ reaches Poughkeepsie and at once turns back meeting $A$ $12$ miles from Poughkeepsie. The rate of $A$ was: | 8 | 1. **Define Variables:**
Let the speed of boy $A$ be $a$ mph, and the speed of boy $B$ be $b$ mph. Given that $A$ travels $4$ mph slower than $B$, we have:
\[ b = a + 4 \]
2. **Set Up Distance Equations:**
- Boy $A$ travels until the meeting point, which is $12$ miles from Poughkeepsie, so he travels:
\[... | 0.8125 | 3,989.3125 | 3,235.153846 | 7,257.333333 |
If $M = 1! \times 2! \times 3! \times 4! \times 5! \times 6! \times 7! \times 8! \times 9!$, calculate the number of divisors of $M$ that are perfect squares. | 672 | 0.75 | 5,215.5625 | 5,142.583333 | 5,434.5 | |
A divisor of a number is a proper divisor if it is not equal to the number. What is the sum of the proper divisors of $432$? | 808 | 0.8125 | 5,173.375 | 4,656.692308 | 7,412.333333 | |
What is the area of the quadrilateral formed by the lines $y=8$, $y=x+3$, $y=-x+3$, and $x=5$? | 25 | 0.0625 | 6,755.5 | 7,464 | 6,708.266667 | |
How many squares of side at least $8$ have their four vertices in the set $H$, where $H$ is defined by the points $(x,y)$ with integer coordinates, $-8 \le x \le 8$ and $-8 \le y \le 8$? | 285 | 0 | 8,192 | -1 | 8,192 | |
Four students participate in a knowledge contest, each student must choose one of the two questions, A or B, to answer. Correctly answering question A earns 60 points, while an incorrect answer results in -60 points. Correctly answering question B earns 180 points, while an incorrect answer results in -180 points. The ... | 44 | 0 | 7,705.625 | -1 | 7,705.625 | |
In $\triangle PAT,$ $\angle P=36^{\circ},$ $\angle A=56^{\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG},$ respectively. What is the degree measure of the acute ang... | 80 | 1. **Identify Given Information and Setup:**
- In $\triangle PAT$, we have $\angle P = 36^\circ$, $\angle A = 56^\circ$, and $PA = 10$.
- Points $U$ and $G$ are on $\overline{TP}$ and $\overline{TA}$ respectively, such that $PU = AG = 1$.
- $M$ and $N$ are midpoints of $\overline{PA}$ and $\overline{UG}$ respe... | 0.625 | 7,461.9375 | 7,023.9 | 8,192 |
Find the product of the greatest common divisor and the least common multiple of $18$ and $42.$ | 756 | 1 | 2,108.6875 | 2,108.6875 | -1 | |
Given algebraic expressions $A=2m^{2}+3my+2y-1$ and $B=m^{2}-my$. Find:<br/>
$(1)$ Simplify $3A-2\left(A+B\right)$.<br/>
$(2)$ If $\left(m-1\right)^{2}+|y+2|=0$, find the value of $3A-2\left(A+B\right)$.<br/>
$(3)$ If the value of $3A-2\left(A+B\right)$ is independent of $y$, find the value of $m$. | -0.4 | 0 | 2,370.0625 | -1 | 2,370.0625 | |
Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjacent vertices consists of edges with at least two different colors. What is the smallest possible value of $N$? | 6 | Note that the condition is equivalent to having no edges of the same color sharing a vertex by just considering paths of length two. Consider a hexagon made out of six triangles. Six edges meet at the center, so $N \geq 6$. To prove $N=6$, simply use two colors for each of the three possible directions of an edge, and ... | 0 | 7,933.625 | -1 | 7,933.625 |
Given that the plane unit vectors $\overrightarrow{{e}_{1}}$ and $\overrightarrow{{e}_{2}}$ satisfy $|2\overrightarrow{{e}_{1}}-\overrightarrow{{e}_{2}}|\leqslant \sqrt{2}$. Let $\overrightarrow{a}=\overrightarrow{{e}_{1}}+\overrightarrow{{e}_{2}}$, $\overrightarrow{b}=3\overrightarrow{{e}_{1}}+\overrightarrow{{e}_{2}}... | \frac{28}{29} | 0.625 | 6,274 | 5,123.2 | 8,192 | |
There exists a scalar $k$ such that for any vectors $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ such that $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0},$ the equation
\[k (\mathbf{b} \times \mathbf{a}) + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} = \mathbf{0}\]holds. Find $k.$ | 2 | 0.75 | 5,182.5 | 4,179.333333 | 8,192 | |
In the final phase of a professional bowling competition, the top five players compete as follows: first, the fifth and fourth place players compete, and the loser gets the 5th place prize; the winner then competes with the third place player, and the loser gets the 4th place prize; the winner then competes with the se... | 16 | 0.375 | 7,749.8125 | 7,012.833333 | 8,192 | |
Given that a tetrahedron $ABCD$ is inscribed in a sphere $O$, and $AD$ is the diameter of the sphere $O$. If triangles $\triangle ABC$ and $\triangle BCD$ are equilateral triangles with side length 1, calculate the volume of tetrahedron $ABCD$. | \frac{\sqrt{3}}{12} | 0 | 8,192 | -1 | 8,192 | |
If $\sqrt{3\sqrt{s-3}} = \sqrt[4]{9 - s}$, then find $s$. | 3.6 | 0 | 2,717.0625 | -1 | 2,717.0625 | |
Given $\overrightarrow{a}=(1,-1)$ and $\overrightarrow{b}=(1,2)$, calculate the projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$. | -\frac{\sqrt{2}}{2} | 0 | 2,750.9375 | -1 | 2,750.9375 | |
Cut a 3-meter-long rope into 7 equal segments. Each segment accounts for \_\_\_\_\_\_ of the total length, and each segment is \_\_\_\_\_\_ meters long. | \frac{3}{7} | 0.375 | 354.25 | 360.666667 | 350.4 | |
Let \[g(x) = \left\{ \begin{aligned} 3x+6 & \quad \text{ if } x < 0 \\ 2x - 13 & \quad \text{ if } x \ge 0 \end{aligned} \right.\]Find all solutions to the equation $g(x) = 3.$ | -1, 8 | 0.4375 | 2,243.8125 | 2,832.142857 | 1,786.222222 | |
Consider the following data from a new season graph, showing the number of home runs hit in April by the top hitters in the baseball league:
- 5 players hit 6 home runs each.
- 6 players hit 8 home runs each.
- 4 players hit 10 home runs each.
Calculate the mean number of home runs hit by these players. | \frac{118}{15} | 0 | 524.3125 | -1 | 524.3125 | |
Given a sequence $a_1,$ $a_2,$ $a_3,$ $\dots,$ let $S_n$ denote the sum of the first $n$ terms of the sequence.
If $a_1 = 1$ and
\[a_n = \frac{2S_n^2}{2S_n - 1}\]for all $n \ge 2,$ then find $a_{100}.$ | -\frac{2}{39203} | 0.625 | 6,684.625 | 5,780.2 | 8,192 | |
Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that
\[ (2 + i)^n = a_n + b_ni \]for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is
\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\] | \frac{7}{16} | 1. **Express $(2+i)$ in polar form**:
We start by expressing the complex number $2+i$ in polar form. We calculate the modulus and the argument of $2+i$:
\[ |2+i| = \sqrt{2^2 + 1^2} = \sqrt{5}, \]
\[ \theta = \arctan\left(\frac{1}{2}\right). \]
Therefore, we can write:
\[ 2+i = \sqrt{5} \left(\cos \theta... | 0.25 | 7,252.8125 | 6,114 | 7,632.416667 |
Let $ABC$ be an equilateral triangle . Let point $D$ lie on side $AB,E$ lie on side $AC, D_1$ and $E_1$ lie on side BC such that $AB=DB+BD_1$ and $AC=CE+CE_1$ . Calculate the smallest angle between the lines $DE_1$ and $ED_1$ . | 60 | 0 | 8,192 | -1 | 8,192 | |
Given that a set of $n$ people participate in an online video soccer tournament, the statistics from the tournament reveal: The average number of complete teams wholly contained within randomly chosen subsets of $10$ members equals twice the average number of complete teams found within randomly chosen subsets of $7$ m... | 450 | 0 | 8,192 | -1 | 8,192 | |
The numbers $a_1,$ $a_2,$ $a_3,$ $b_1,$ $b_2,$ $b_3,$ $c_1,$ $c_2,$ $c_3$ are equal to the numbers $1,$ $2,$ $3,$ $\dots,$ $9$ in some order. Find the smallest possible value of
\[a_1 a_2 a_3 + b_1 b_2 b_3 + c_1 c_2 c_3.\] | 214 | 0 | 8,192 | -1 | 8,192 | |
Compute $\frac{x^6-16x^3+64}{x^3-8}$ when $x=6$. | 208 | 1 | 2,508.9375 | 2,508.9375 | -1 | |
The parabola $y = x^2+2$ and the hyperbola $y^2 - mx^2 = 1$ are tangent. Find $m.$ | 4+2\sqrt3 | 0.875 | 5,042.25 | 4,592.285714 | 8,192 | |
In the Cartesian coordinate system $xOy$, given the parabola $(E): y^2 = 2px (p > 0)$ with focus $F$, $P$ is an arbitrary point on the parabola $(E)$ in the first quadrant, and $Q$ is a point on the line segment $PF$ such that $\overrightarrow{OQ} = \frac{2}{3} \overrightarrow{OP} + \frac{1}{3} \overrightarrow{OF}$. De... | \sqrt{2} | 0.9375 | 5,094 | 4,887.466667 | 8,192 | |
Find the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. | 2600 | 0.1875 | 8,026.25 | 7,308 | 8,192 | |
The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is | 20 | To solve this problem, we need to determine the number of smaller cubes in the large cube that have at least one face shaded. We are given that the large cube is composed of $27$ smaller cubes and that the shading pattern on one face is mirrored exactly on the opposite face.
1. **Understanding the Cube Structure**: Th... | 0.0625 | 7,508.6875 | 6,108 | 7,602.066667 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.