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 that points A and B are on the x-axis, and the two circles with centers at A and B intersect at points M $(3a-b, 5)$ and N $(9, 2a+3b)$, find the value of $a^{b}$. | \frac{1}{8} | 0.625 | 6,199.1875 | 5,375.8 | 7,571.5 | |
Let $x$ and $y$ be real numbers such that $\frac{\sin x}{\sin y} = 3$ and $\frac{\cos x}{\cos y} = \frac12$. Find the value of
\[\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}.\] | \frac{49}{58} | 0.875 | 5,235.3125 | 4,812.928571 | 8,192 | |
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hi... | 48 | 1. Let $x$ be the number of acorns that both the chipmunk and the squirrel hid. According to the problem, the chipmunk hides 3 acorns per hole and the squirrel hides 4 acorns per hole.
2. Let $h_c$ be the number of holes dug by the chipmunk and $h_s$ be the number of holes dug by the squirrel. We know that:
\[
3... | 1 | 1,442.625 | 1,442.625 | -1 |
How many four-digit numbers have the property that the second digit is the average of the first and third digits, and the digits are all even? | 50 | 0.4375 | 6,892.125 | 6,272 | 7,374.444444 | |
A rectangular table with dimensions $x$ cm $\times 80$ cm is covered with identical sheets of paper measuring 5 cm $\times 8$ cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous sheet. The last sheet is positi... | 77 | 0.375 | 7,122.4375 | 5,339.833333 | 8,192 | |
If $\sqrt{\frac{2}{x} + 2} = \frac{3}{2}$, solve for $x$. | 8 | 1 | 2,144.25 | 2,144.25 | -1 | |
Given a line and a point at a distance of $1 \mathrm{~cm}$ from the line. What is the volume of a cube where the point is one of the vertices and the line is one of the body diagonals? | \frac{3 \sqrt{6}}{4} | 0 | 5,331.8125 | -1 | 5,331.8125 | |
Consider a fictional language with ten letters in its alphabet: A, B, C, D, F, G, H, J, L, M. Suppose license plates of six letters utilize only letters from this alphabet. How many license plates of six letters are possible that begin with either B or D, end with J, cannot contain any vowels (A), and have no letters t... | 1680 | 0.1875 | 5,357.625 | 4,603.666667 | 5,531.615385 | |
There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are labeled with the 1000 different integers $(2... | 650 | For each $i$th switch (designated by $x_{i},y_{i},z_{i}$), it advances itself only one time at the $i$th step; thereafter, only a switch with larger $x_{j},y_{j},z_{j}$ values will advance the $i$th switch by one step provided $d_{i}= 2^{x_{i}}3^{y_{i}}5^{z_{i}}$ divides $d_{j}= 2^{x_{j}}3^{y_{j}}5^{z_{j}}$. Let $N = 2... | 0.125 | 7,767.75 | 7,419.5 | 7,817.5 |
The square $BCDE$ is inscribed in circle $\omega$ with center $O$ . Point $A$ is the reflection of $O$ over $B$ . A "hook" is drawn consisting of segment $AB$ and the major arc $\widehat{BE}$ of $\omega$ (passing through $C$ and $D$ ). Assume $BCDE$ has area $200$ . To the nearest integer, what ... | 67 | 0.125 | 6,249.875 | 3,360 | 6,662.714286 | |
Three friends have a total of 6 identical pencils, and each one has at least one pencil. In how many ways can this happen? | 10 | 0.875 | 4,012.6875 | 3,415.642857 | 8,192 | |
For how many integers $n$, for $1 \leq n \leq 1000$, is the number $\frac{1}{2}\binom{2 n}{n}$ even? | 990 | In fact, the expression $\binom{2 n}{n}$ is always even, and it is not a multiple of four if and only if $n$ is a power of 2, and there are 10 powers of 2 between 1 and 1000. Let $f(N)$ denote the number of factors of 2 in $N$. Thus, $$f(n!)=\left\lfloor\frac{n}{2}\right\rfloor+\left\lfloor\frac{n}{4}\right\rfloor+\lef... | 0.1875 | 7,887.25 | 6,566.666667 | 8,192 |
Find the least odd prime factor of $2019^8+1$. | 97 | We know that $2019^8 \equiv -1 \pmod{p}$ for some prime $p$. We want to find the smallest odd possible value of $p$. By squaring both sides of the congruence, we find $2019^{16} \equiv 1 \pmod{p}$.
Since $2019^{16} \equiv 1 \pmod{p}$, the order of $2019$ modulo $p$ is a positive divisor of $16$.
However, if the order... | 0.875 | 5,211.375 | 4,785.571429 | 8,192 |
Determine the seventh element in Row 20 of Pascal's triangle. | 38760 | 0.9375 | 4,623.4375 | 4,385.533333 | 8,192 | |
In triangle $ΔABC$, $BC=a$, $AC=b$, where $a$ and $b$ are the two roots of the equation $x^2-2\sqrt{3}x+2=0$, and $2\cos(A+B)=1$.
(1) Find the angle $C$;
(2) Find the length of $AB$. | \sqrt{10} | 1 | 3,209.9375 | 3,209.9375 | -1 | |
Given the polar equation of circle $E$ is $\rho=4\sin \theta$, with the pole as the origin and the polar axis as the positive half of the $x$-axis, establish a Cartesian coordinate system with the same unit length (where $(\rho,\theta)$, $\rho \geqslant 0$, $\theta \in [0,2\pi)$).
$(1)$ Line $l$ passes through the or... | 2 \sqrt {2} | 0 | 6,808.3125 | -1 | 6,808.3125 | |
Zou and Chou are practicing their $100$-meter sprints by running $6$ races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\frac23$ if they won the previous race but only $\frac13$ if they lost the previous race. The probability that Zou will win exactly $5$... | 97 | Note that Zou wins one race. The probability that he wins the last race is $\left(\frac{2}{3}\right)^4\left(\frac{1}{3}\right)=\frac{16}{243}.$ Now, if he doesn't win the last race, then there must be two races where the winner of the previous race loses. We can choose any $4$ of the middle races for Zou to win. So the... | 0 | 8,192 | -1 | 8,192 |
$ABCDEFGH$ is a rectangular prism with $AB=CD=EF=GH=1$, $AD=BC=EH=FG=2$, and $AE=BF=CG=DH=3$. Find $\sin \angle GAC$. | \frac{3}{\sqrt{14}} | 0 | 4,619.625 | -1 | 4,619.625 | |
How many different primes are in the prime factorization of $87\cdot89\cdot91\cdot93$? | 6 | 1 | 1,549.375 | 1,549.375 | -1 | |
Juca has fewer than 800 marbles. He likes to separate the marbles into groups of the same size. He noticed that if he forms groups of 3 marbles each, exactly 2 marbles are left over. If he forms groups of 4 marbles, 3 marbles are left over. If he forms groups of 5 marbles, 4 marbles are left over. Finally, if he forms... | 419 | 0.9375 | 3,196 | 2,862.933333 | 8,192 | |
What is the largest number of integers that we can choose from the set $\{1, 2, 3, \ldots, 2017\}$ such that the difference between any two of them is not a prime number? | 505 | 0 | 8,192 | -1 | 8,192 | |
How many ways are there to make change for $55$ cents using any number of pennies, nickels, dimes, and quarters? | 60 | 0.1875 | 7,884.6875 | 6,553 | 8,192 | |
For the positive integer \( n \), if the expansion of \( (xy - 5x + 3y - 15)^n \) is combined and simplified, and \( x^i y^j \) (where \( i, j = 0, 1, \ldots, n \)) has at least 2021 terms, what is the minimum value of \( n \)? | 44 | 0.375 | 6,397.9375 | 4,532 | 7,517.5 | |
Given that the terminal side of angle $α$ rotates counterclockwise by $\dfrac{π}{6}$ and intersects the unit circle at the point $\left( \dfrac{3 \sqrt{10}}{10}, \dfrac{\sqrt{10}}{10} \right)$, and $\tan (α+β)= \dfrac{2}{5}$.
$(1)$ Find the value of $\sin (2α+ \dfrac{π}{6})$,
$(2)$ Find the value of $\tan (2β- \dfrac{π... | \dfrac{17}{144} | 0 | 7,965.375 | -1 | 7,965.375 | |
A, B, C, D, and E are five students who obtained the top 5 positions (no ties) in a math competition. When taking a photo, they stood in a line and each of them made the following statement:
A said: The two students next to me have rankings lower than mine;
B said: The two students next to me have rankings adjacent to... | 23514 | 0 | 8,100 | -1 | 8,100 | |
For a triangular piece of land with sides measuring 198 meters, 180 meters, and 210 meters, plant trees equidistantly along the three sides with 6 meters between each pair of consecutive trees, also requiring one tree planted at each of the three vertices. Determine the total number of trees planted. | 98 | 0.8125 | 5,278.1875 | 4,605.769231 | 8,192 | |
Find the positive integer that has three digits in both base-10 and base-8, and the sum of its digits in both bases is fourteen. | 455 | 0.5 | 7,157.9375 | 6,123.875 | 8,192 | |
Let $T = \{3^0, 3^1, 3^2, \ldots, 3^{10}\}$. Consider all possible positive differences of pairs of elements of $T$. Let $N$ be the sum of all these differences. Find $N$. | 783492 | 0 | 7,996.625 | -1 | 7,996.625 | |
What is the product of the coordinates of the midpoint of a line segment with endpoints at $(2,3)$ and $(-6,5)$? | -8 | 1 | 1,358.9375 | 1,358.9375 | -1 | |
In response to the national policy of expanding domestic demand, a manufacturer plans to hold a promotional event at the beginning of 2015. After investigation and estimation, the annual sales volume (i.e., the annual production volume $x$ in ten thousand units) and the annual promotional expenses $t$ (where $t > 0$) i... | 3 \sqrt {2} | 0 | 8,192 | -1 | 8,192 | |
Given $X \sim N(\mu, \sigma^2)$, $P(\mu-\sigma < X \leq \mu+\sigma) = 0.68$, $P(\mu-2\sigma < X \leq \mu+2\sigma) = 0.95$. In a city-wide exam with 20,000 participants, the math scores approximately follow a normal distribution $N(100, 100)$. How many students scored above 120? | 500 | 0.3125 | 5,025.0625 | 5,301.6 | 4,899.363636 | |
If \( \left\lfloor n^2/9 \right\rfloor - \lfloor n/3 \rfloor^2 = 5 \), then find all integer values of \( n \). | 14 | 0 | 7,857.625 | -1 | 7,857.625 | |
Four two-inch squares are placed with their bases on a line. The second square from the left is lifted out, rotated 45 degrees, then centered and lowered back until it touches its adjacent squares on both sides. Determine the distance, in inches, of point P, the top vertex of the rotated square, from the line on which ... | 1 + \sqrt{2} | 0.0625 | 8,025.625 | 7,311 | 8,073.266667 | |
In the extended Number Wall, numbers are added from adjacent blocks directly below to form the sum in the block above. What number will be in the block labeled '$n$' in this configuration?
```plaintext
__n__
__ __
__ __
8 5 3 2
``` | 34 | 0.6875 | 3,482.5625 | 3,236 | 4,025 | |
In the mystical mountain, there are only two types of legendary creatures: Nine-Headed Birds and Nine-Tailed Foxes. A Nine-Headed Bird has nine heads and one tail, while a Nine-Tailed Fox has nine tails and one head.
A Nine-Headed Bird discovers that, excluding itself, the total number of tails of the other creatures ... | 14 | 0.375 | 5,843.375 | 4,775 | 6,484.4 | |
The graph of the function $f(x)=\cos(\omega x+\frac{π}{4})$ $(\omega>0)$ is transformed to an odd function by shifting it to the left by $\frac{π}{3}$ units. Determine the minimum value of the real number $\omega$. | \frac{3}{4} | 0.5625 | 6,162.875 | 4,930 | 7,748 | |
The reciprocal of $\left( \frac{1}{2}+\frac{1}{3}\right)$ is | \frac{6}{5} | 1. **Identify the Expression to Simplify**:
The problem asks for the reciprocal of the expression $\left( \frac{1}{2}+\frac{1}{3}\right)$.
2. **Simplify the Expression**:
To simplify $\frac{1}{2} + \frac{1}{3}$, find a common denominator. The least common multiple of 2 and 3 is 6.
\[
\frac{1}{2} = \frac{3}... | 1 | 1,474.875 | 1,474.875 | -1 |
If $n$ is a real number, then the simultaneous system
$nx+y = 1$
$ny+z = 1$
$x+nz = 1$
has no solution if and only if $n$ is equal to | -1 |
To determine the values of $n$ for which the given system of equations has no solution, we analyze the system step-by-step.
**Step 1: Add all three equations.**
\[
\begin{align*}
nx + y &= 1, \\
ny + z &= 1, \\
x + nz &= 1.
\end{align*}
\]
Adding these equations, we get:
\[
nx + y + ny + z + x + nz = 3.
\]
Rearrangin... | 1 | 2,180.625 | 2,180.625 | -1 |
Given a student measures his steps from one sidewalk to another on a number line where each marking represents 3 meters, calculate the position marking $z$ after taking 5 steps from the starting point, given that the student counts 8 steps and the total distance covered is 48 meters. | 30 | 0.25 | 2,591.5625 | 370.25 | 3,332 | |
Jeff will pick a card at random from ten cards numbered 1 through 10. The number on this card will indicate his starting point on the number line shown below. He will then spin the fair spinner shown below (which has three congruent sectors) and follow the instruction indicated by his spin. From this new point he wi... | \frac{31}{90} | 0 | 8,058.8125 | -1 | 8,058.8125 | |
For how many different digits $n$ is the three-digit number $14n$ divisible by $n$?
Note: $14n$ refers to a three-digit number with the unit digit of $n,$ not the product of $14$ and $n.$ | 5 | 1 | 2,961.625 | 2,961.625 | -1 | |
Source: 1976 Euclid Part B Problem 1
-----
Triangle $ABC$ has $\angle{B}=30^{\circ}$ , $AB=150$ , and $AC=50\sqrt{3}$ . Determine the length of $BC$ . | 50\sqrt{3} | 0.1875 | 6,959.9375 | 6,378.666667 | 7,094.076923 | |
How many pairs $(m,n)$ of integers satisfy the equation $m+n=mn$? | 2 | To find the number of integer pairs $(m,n)$ that satisfy the equation $m+n=mn$, we can start by rearranging the equation:
\[ m+n = mn \]
Subtracting $m+n$ from both sides gives:
\[ mn - m - n = 0 \]
Adding 1 to both sides, we can factorize the left-hand side:
\[ mn - m - n + 1 = 1 \]
\[ (m-1)(n-1) = 1 \]
Now, we... | 1 | 4,500.5625 | 4,500.5625 | -1 |
Calculate the definite integral:
$$
\int_{0}^{2 \pi} \sin^{6} x \cos^{2} x \, dx
$$ | \frac{5\pi}{64} | 0.5625 | 6,624.5 | 5,740.222222 | 7,761.428571 | |
Let $x<0.1$ be a positive real number. Let the foury series be $4+4 x+4 x^{2}+4 x^{3}+\ldots$, and let the fourier series be $4+44 x+444 x^{2}+4444 x^{3}+\ldots$ Suppose that the sum of the fourier series is four times the sum of the foury series. Compute $x$. | \frac{3}{40} | The sum of the foury series can be expressed as \(\frac{4}{1-x}\) by geometric series. The fourier series can be expressed as $$ \begin{aligned} & \frac{4}{9}\left((10-1)+(100-1) x+(1000-1) x^{2}+\ldots\right) \\ & =\frac{4}{9}\left(\left(10+100 x+1000 x^{2}+\ldots\right)-\left(1+x+x^{2}+\ldots\right)\right) \\ & =\fra... | 0.5 | 4,652 | 4,381.625 | 4,922.375 |
Simplify $\sqrt[3]{1+8} \cdot \sqrt[3]{1+\sqrt[3]{8}}$. | 3 | 1 | 1,877.9375 | 1,877.9375 | -1 | |
It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it? | 40 | 1. **Define Variables:**
Let $s$ be the speed of the escalator and $c$ be the speed of Clea walking down the escalator. Let $d$ represent the distance of the escalator.
2. **Translate the Problem into Equations:**
- When the escalator is not operating, Clea walks the entire distance, so the equation is:
\[
... | 1 | 2,519.5 | 2,519.5 | -1 |
Given a random variable $\eta = 8 - \xi$, if $\xi \sim B(10, 0.6)$, then calculate $E\eta$ and $D\eta$. | 2.4 | 1 | 1,643.0625 | 1,643.0625 | -1 | |
As $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $11n+3$ and $6n+1$? | 7 | 1 | 3,160.625 | 3,160.625 | -1 | |
Given the regression line equation obtained from a certain sample data is $y=1.5x+45$, where $x\in\{1,7,10,13,19\}$, find the value of $\overline{y}$. | 60 | 1 | 2,055.875 | 2,055.875 | -1 | |
Let $s$ be a table composed of positive integers (the table may contain the same number). Among the numbers in $s$ there is the number 68. The arithmetic mean of all numbers in $s$ is 56. However, if 68 is removed, the arithmetic mean of the remaining numbers drops to 55. What is the largest possible number that can ap... | 649 | 0.6875 | 6,525.1875 | 5,767.545455 | 8,192 | |
How many multiples of 4 are between 100 and 350? | 62 | 0.875 | 4,599.5 | 4,491.928571 | 5,352.5 | |
Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$ . Find $\angle BCA$ . | 90 | 0 | 8,192 | -1 | 8,192 | |
Given complex number $z$, define sequence $z_{0}, z_{1}, z_{2}, \ldots$ as $z_{0}=z$ and $z_{n+1}=2 z_{n}^{2}+2 z_{n}$ for $n \geq 0$. Given that $z_{10}=2017$, find the minimum possible value of $|z|$. | \frac{\sqrt[1024]{4035}-1}{2} | Define $w_{n}=z_{n}+\frac{1}{2}$, so $z_{n}=w_{n}-\frac{1}{2}$, and the original equation becomes $$w_{n+1}-\frac{1}{2}=2\left(w_{n}-\frac{1}{2}\right)^{2}+2\left(w_{n}-\frac{1}{2}\right)=2 w_{n}^{2}-\frac{1}{2}$$ which reduces to $w_{n+1}=2 w_{n}^{2}$. it is not difficult to show that $$z_{10}+\frac{1}{2}=2017+\frac{1... | 0 | 8,192 | -1 | 8,192 |
There are 16 people standing in a circle: each of them is either truthful (always tells the truth) or a liar (always lies). Everyone said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle? | 10 | 0.125 | 7,731.9375 | 6,387.5 | 7,924 | |
In the Cartesian coordinate system $xOy$, it is known that $P$ is a moving point on the graph of the function $f(x)=\ln x$ ($x > 0$). The tangent line $l$ at point $P$ intersects the $x$-axis at point $E$. A perpendicular line to $l$ through point $P$ intersects the $x$-axis at point $F$. If the midpoint of the line se... | \dfrac {1}{2}(e+ \dfrac {1}{e}) | 0 | 4,933 | -1 | 4,933 | |
A company has 200 employees randomly assigned numbers from 1 to 200, and they are evenly divided into 40 groups. If employee number 22 is selected from the 5th group, find the number of the employee to be selected from the 10th group. | 47 | 0.4375 | 6,032.6875 | 4,933.571429 | 6,887.555556 | |
Calculate the limit of the function:
\[ \lim _{x \rightarrow \frac{\pi}{3}} \frac{1-2 \cos x}{\sin (\pi-3 x)} \] | -\frac{\sqrt{3}}{3} | 0 | 5,967.1875 | -1 | 5,967.1875 | |
On a straight road, there are an odd number of warehouses. The distance between adjacent warehouses is 1 kilometer, and each warehouse contains 8 tons of goods. A truck with a load capacity of 8 tons starts from the warehouse on the far right and needs to collect all the goods into the warehouse in the middle. It is kn... | 25 | 0 | 6,538.8125 | -1 | 6,538.8125 | |
Simplify the expression, then evaluate: $$(1- \frac {a}{a+1})\div \frac {1}{1-a^{2}}$$ where $a=-2$. | \frac {1}{3} | 0 | 1,275.8125 | -1 | 1,275.8125 | |
Two people, A and B, play a guessing game. First, A thinks of a number denoted as $a$, then B guesses the number A thought of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\{0,1,2,…,9\}$. If $|a-b|=1$, then A and B are said to have a "telepathic connection". If two people are randomly chosen to play t... | \dfrac {9}{50} | 0.9375 | 4,212 | 4,116.066667 | 5,651 | |
Every day, Xiaoming goes to school along a flat road \(AB\), an uphill road \(BC\), and a downhill road \(CD\) (as shown in the diagram). Given that \(AB : BC : CD = 1 : 2 : 1\) and that Xiaoming's speeds on flat, uphill, and downhill roads are in the ratio 3 : 2 : 4, respectively, find the ratio of the time Xiaoming t... | 19:16 | 0 | 5,385.4375 | -1 | 5,385.4375 | |
Let $A$ be the number of unordered pairs of ordered pairs of integers between 1 and 6 inclusive, and let $B$ be the number of ordered pairs of unordered pairs of integers between 1 and 6 inclusive. (Repetitions are allowed in both ordered and unordered pairs.) Find $A-B$. | 225 | There are $6 \cdot 6$ ordered pairs of integers between 1 and 6 inclusive and 21 unordered pairs of integers \left(\binom{6}{2}=15\right.$ different pairs and 6 doubles). Then, $A=\binom{36}{2}+36=666$ and $B=21 \cdot 21=441$. Therefore $A-B=225$. For general $n$, there are $n^{2}$ ordered pairs of integers and \frac{n... | 0.25 | 6,219.0625 | 4,443.5 | 6,810.916667 |
What is the sum of the six positive integer factors of 18? | 39 | 1 | 1,805.5 | 1,805.5 | -1 | |
If $\frac{3}{5}=\frac{M}{45}=\frac{60}{N}$, what is $M+N$? | 127 | Given the equation $\frac{3}{5}=\frac{M}{45}=\frac{60}{N}$, we can separate this into two equations and solve for $M$ and $N$ respectively.
1. **Solving for $M$:**
\[
\frac{3}{5} = \frac{M}{45}
\]
Cross-multiplying gives:
\[
3 \times 45 = 5 \times M \implies 135 = 5M \implies M = \frac{135}{5} = 27
... | 1 | 1,432.0625 | 1,432.0625 | -1 |
Let $n$ be the least positive integer for which $149^n-2^n$ is divisible by $3^3\cdot5^5\cdot7^7.$ Find the number of positive integer divisors of $n.$ | 270 | Lifting the Exponent shows that \[v_3(149^n-2^n) = v_3(n) + v_3(147) = v_3(n)+1\] so thus, $3^2$ divides $n$. It also shows that \[v_7(149^n-2^n) = v_7(n) + v_7(147) = v_7(n)+2\] so thus, $7^5$ divides $n$.
Now, setting $n = 4c$ (necessitated by $149^n \equiv 2^n \pmod 5$ in order to set up LTE), we see \[v_5(149^{4c}... | 0 | 7,746.875 | -1 | 7,746.875 |
Determine the number of digits in the value of $2^{15} \times 5^{12} - 10^5$. | 12 | 0 | 6,719.125 | -1 | 6,719.125 | |
Mary is $20\%$ older than Sally, and Sally is $40\%$ younger than Danielle. The sum of their ages is $23.2$ years. How old will Mary be on her next birthday? | 8 | 1. **Define the variables:**
Let $m$ be Mary's age, $s$ be Sally's age, and $d$ be Danielle's age.
2. **Set up the relationships:**
- Sally is $40\%$ younger than Danielle, which means Sally's age is $60\%$ of Danielle's age:
\[
s = 0.6d
\]
- Mary is $20\%$ older than Sally, so Mary's age is $1... | 1 | 1,949.875 | 1,949.875 | -1 |
Find the domain of the real-valued function $$f(x)=\sqrt{-10x^2-11x+6}.$$ Give the endpoints in your answer as common fractions, not mixed numbers or decimals. | \left[-\frac{3}{2}, \frac{2}{5}\right] | 0.9375 | 2,324.5625 | 2,169.466667 | 4,651 | |
In the diagram below, $AB = AC = 115,$ $AD = 38,$ and $CF = 77.$ Compute $\frac{[CEF]}{[DBE]}.$
[asy]
unitsize(0.025 cm);
pair A, B, C, D, E, F;
B = (0,0);
C = (80,0);
A = intersectionpoint(arc(B,115,0,180),arc(C,115,0,180));
D = interp(A,B,38/115);
F = interp(A,C,(115 + 77)/115);
E = extension(B,C,D,F);
draw(C--B... | \frac{19}{96} | 0 | 8,110.9375 | -1 | 8,110.9375 | |
As part of his effort to take over the world, Edward starts producing his own currency. As part of an effort to stop Edward, Alex works in the mint and produces 1 counterfeit coin for every 99 real ones. Alex isn't very good at this, so none of the counterfeit coins are the right weight. Since the mint is not perfect, ... | \frac{19}{28} | $5 \%$ of the coins are sent to the lab, and only $.95 \%$ of the coins are sent to the lab and counterfeit, so there is a $19 \%$ chance that a coin sent to the lab is counterfeit and an $81 \%$ chance that it is real. The lab could correctly detect a counterfeit coin or falsely accuse a real one of being counterfeit,... | 0 | 6,867.0625 | -1 | 6,867.0625 |
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 lowe... | 560 | Clearly, the marksman must shoot the left column three times, the middle column two times, and the right column three times.
From left to right, suppose that the columns are labeled $L,M,$ and $R,$ respectively. We consider the string $LLLMMRRR:$
Since the letter arrangements of $LLLMMRRR$ and the shooting orders hav... | 1 | 2,363.375 | 2,363.375 | -1 |
A tripod has three legs each of length $5$ feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top o... | 183 | [asy] size(200); import three; pointpen=black;pathpen=black+linewidth(0.65);pen ddash = dashed+linewidth(0.65); currentprojection = perspective(1,-10,3.3); triple O=(0,0,0),T=(0,0,5),C=(0,3,0),A=(-3*3^.5/2,-3/2,0),B=(3*3^.5/2,-3/2,0); triple M=(B+C)/2,S=(4*A+T)/5; draw(T--S--B--T--C--B--S--C);draw(B--A--C--A--S,ddash);... | 0 | 8,192 | -1 | 8,192 |
Calculate the sum $1 + 3 + 5 + \cdots + 15 + 17$. | 81 | 1 | 1,631.6875 | 1,631.6875 | -1 | |
There are 2011 positive numbers with both their sum and the sum of their reciprocals equal to 2012. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}.$ | \frac{8045}{2012} | 0.0625 | 7,847.375 | 6,139 | 7,961.266667 | |
Suppose we want to divide the 10 dogs into three groups, one with 3 dogs, one with 5 dogs, and one with 2 dogs. How many ways can we form the groups such that Fluffy is in the 3-dog group and Nipper is in the 5-dog group? | 420 | 0.4375 | 6,917.5625 | 5,984.571429 | 7,643.222222 | |
Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$ K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma} $$ obtains its minimum value. | 2/5 | 0 | 8,192 | -1 | 8,192 | |
Given that the sum of the first n terms of the sequence {a_n} is S_n, with a_1 = 2 and S_{n+1} = 2S_n - 1 for all positive integers n, find the value of a_{10}. | 256 | 1 | 3,711 | 3,711 | -1 | |
(12 points) Using the six digits 0, 1, 2, 3, 4, 5, complete the following three questions:
(1) If digits can be repeated, how many different five-digit even numbers can be formed?
(2) If digits cannot be repeated, how many different five-digit numbers divisible by 5, with the hundredth digit not being 3, can be formed?... | 20 | 0 | 8,046.5625 | -1 | 8,046.5625 | |
There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $\mid z \mid = 1$. These numbers have the form $z_{m} = \cos\theta_{m} + i\sin\theta_{m}$, where $0\leq\theta_{1} < \theta_{2} < \ldots < \theta_{2n} < 360$ and angles are measured in degrees. Find the value of $\theta_{2} + \theta_{4} + \ldo... | 840 | $z$ can be written in the form $\text{cis\,}\theta$. Rearranging, we find that $\text{cis\,}{28}\theta = \text{cis\,}{8}\theta+1$
Since the real part of $\text{cis\,}{28}\theta$ is one more than the real part of $\text{cis\,} {8}\theta$ and their imaginary parts are equal, it is clear that either $\text{cis\,}{28}\the... | 0 | 8,097.3125 | -1 | 8,097.3125 |
Given that \(1 \leqslant a_{1} \leqslant a_{2} \leqslant a_{3} \leqslant a_{4} \leqslant a_{5} \leqslant a_{6} \leqslant 64\), find the minimum value of \(Q = \frac{a_{1}}{a_{2}} + \frac{a_{3}}{a_{4}} + \frac{a_{5}}{a_{6}}\). | 3/2 | 0.0625 | 8,013.1875 | 8,192 | 8,001.266667 | |
Triangle $PQR$ has vertices $P(0, 6)$, $Q(3, 0)$, $R(9, 0)$. A line through $Q$ cuts the area of $\triangle PQR$ in half. Find the sum of the slope and $y$-intercept of this line. | -4 | 1 | 4,707 | 4,707 | -1 | |
If the roots of the quadratic equation $\frac32x^2+11x+c=0$ are $x=\frac{-11\pm\sqrt{7}}{3}$, then what is the value of $c$? | 19 | 1 | 2,212.5 | 2,212.5 | -1 | |
In parallelogram $ABCD$, point $M$ is on $\overline{AB}$ so that $\frac {AM}{AB} = \frac {17}{1000}$ and point $N$ is on $\overline{AD}$ so that $\frac {AN}{AD} = \frac {17}{2009}$. Let $P$ be the point of intersection of $\overline{AC}$ and $\overline{MN}$. Find $\frac {AC}{AP}$. | 177 | One of the ways to solve this problem is to make this parallelogram a straight line. So the whole length of the line is $APC$($AMC$ or $ANC$), and $ABC$ is $1000x+2009x=3009x.$
$AP$($AM$ or $AN$) is $17x.$
So the answer is $3009x/17x = \boxed{177}$ | 0.8125 | 4,835 | 4,060.307692 | 8,192 |
The congruent sides of an isosceles triangle are each 5 cm long, and the perimeter is 17 cm. In centimeters, what is the length of the base? | 7 | 1 | 1,406.625 | 1,406.625 | -1 | |
A function \( f(x) \) defined on the interval \([1,2017]\) satisfies \( f(1)=f(2017) \), and for any \( x, y \in [1,2017] \), \( |f(x) - f(y)| \leqslant 2|x - y| \). If the real number \( m \) satisfies \( |f(x) - f(y)| \leqslant m \) for any \( x, y \in [1,2017] \), find the minimum value of \( m \). | 2016 | 0.25 | 7,497.625 | 6,895 | 7,698.5 | |
Rearrange the digits of 124669 to form a different even number. | 240 | 0 | 557.75 | -1 | 557.75 | |
Find the minimum value of
\[\sin^4 x + \frac{3}{2} \cos^4 x,\]as $x$ varies over all real numbers. | \frac{3}{5} | 1 | 5,138.125 | 5,138.125 | -1 | |
If \( x, y \in \mathbf{R} \) and \(3x^{2} + 2y^{2} = 2x\), then the maximum value of \(x^{2} + y^{2}\) is? | \frac{4}{9} | 0.5 | 7,692.0625 | 7,192.125 | 8,192 | |
Burattino got on a train. After travelling half of the total distance, he fell asleep and slept until there was only half of the distance he slept left to travel. What fraction of the total journey did Burattino travel awake? | \frac{2}{3} | 0.1875 | 3,757.875 | 2,200.333333 | 4,117.307692 | |
A triangle $\triangle ABC$ satisfies $AB = 13$ , $BC = 14$ , and $AC = 15$ . Inside $\triangle ABC$ are three points $X$ , $Y$ , and $Z$ such that:
- $Y$ is the centroid of $\triangle ABX$
- $Z$ is the centroid of $\triangle BCY$
- $X$ is the centroid of $\triangle CAZ$
What is the area of $\... | 84/13 | 0.125 | 7,704.9375 | 6,842 | 7,828.214286 | |
Factorize the expression $27x^6 - 512y^6$ and find the sum of all integer coefficients in its factorized form. | 92 | 0.9375 | 4,550.0625 | 4,307.266667 | 8,192 | |
You are given the numbers 1, 2, 3, 4, 5, 6, 7, 8 to be placed at the eight vertices of a cube, such that the sum of any three numbers on each face of the cube is at least 10. Find the minimum possible sum of the four numbers on any face. | 16 | 0.0625 | 7,800.875 | 7,074 | 7,849.333333 | |
Find $b$ if $\log_{b}343=-\frac{3}{2}$. | b=\frac{1}{49} | 1 | 2,650.6875 | 2,650.6875 | -1 | |
$A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has travelled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, then the circumference of the track in yards i... | 480 | 1. **Define Variables:**
Let $x$ be half the circumference of the track. Therefore, the full circumference is $2x$.
2. **Analyze First Meeting:**
$A$ and $B$ start from diametrically opposite points on the track. When they first meet, $B$ has traveled $100$ yards. Since they start from opposite points, $A$ must ... | 0.75 | 4,485.3125 | 3,627.916667 | 7,057.5 |
Mrs. Johnson recorded the following scores for a test taken by her 120 students. Calculate the average percent score for these students.
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
95&10\\\hline
85&20\\\hline
75&40\\\hline
65&30\\\hline
55&15\\\hline
45&3... | 71.33 | 0.8125 | 494.1875 | 484.846154 | 534.666667 | |
Let $p,$ $q,$ $r$ be the roots of $x^3 - x^2 + x - 2 = 0.$ Find $p^3 + q^3 + r^3.$ | 4 | 0.875 | 4,597.375 | 4,083.857143 | 8,192 | |
Given any point $P$ on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\; \; (a > b > 0)$ with foci $F\_{1}$ and $F\_{2}$, if $\angle PF\_1F\_2=\alpha$, $\angle PF\_2F\_1=\beta$, $\cos \alpha= \frac{ \sqrt{5}}{5}$, and $\sin (\alpha+\beta)= \frac{3}{5}$, find the eccentricity of this ellipse. | \frac{\sqrt{5}}{7} | 0 | 7,670.125 | -1 | 7,670.125 | |
What is the largest $5$ digit integer congruent to $17 \pmod{26}$? | 99997 | 0 | 5,061.8125 | -1 | 5,061.8125 | |
Given an ellipse $C$ with its left and right foci at $F_{1}(-\sqrt{3},0)$ and $F_{2}(\sqrt{3},0)$, respectively, and the ellipse passes through the point $(-1, \frac{\sqrt{3}}{2})$.
(Ⅰ) Find the equation of the ellipse $C$;
(Ⅱ) Given a fixed point $A(1, \frac{1}{2})$, a line $l$ passing through the origin $O$ inters... | \sqrt{2} | 0.6875 | 7,612 | 7,577.363636 | 7,688.2 |
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