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 |
|---|---|---|---|---|---|---|
Numbering the pages of an encyclopedia required 6869 digits. How many pages does it contain? | 1994 | 0.25 | 6,700.8125 | 3,491.5 | 7,770.583333 | |
Daniel worked for 50 hours per week for 10 weeks during the summer, earning \$6000. If he wishes to earn an additional \$8000 during the school year which lasts for 40 weeks, how many fewer hours per week must he work compared to the summer if he receives the same hourly wage? | 33.33 | 0.3125 | 692.8125 | 685.6 | 696.090909 | |
A circle of radius 2 is centered at $A$. An equilateral triangle with side 4 has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle? | $4(\pi - \sqrt{3})$ | 1. **Calculate the area of the circle:**
The formula for the area of a circle is $\pi r^2$. Given the radius $r = 2$, the area of the circle is:
\[
\text{Area of the circle} = \pi \times 2^2 = 4\pi
\]
2. **Calculate the area of the equilateral triangle:**
The formula for the area of an equilateral trian... | 0 | 5,442.1875 | -1 | 5,442.1875 |
What is the modular inverse of $11$, modulo $1000$?
Express your answer as an integer from $0$ to $999$, inclusive. | 91 | 0.9375 | 3,104.25 | 2,765.066667 | 8,192 | |
There are three buckets, X, Y, and Z. The average weight of the watermelons in bucket X is 60 kg, the average weight of the watermelons in bucket Y is 70 kg. The average weight of the watermelons in the combined buckets X and Y is 64 kg, and the average weight of the watermelons in the combined buckets X and Z is 66 kg... | 69 | 0 | 7,536.375 | -1 | 7,536.375 | |
Given an $8 \times 8$ checkerboard with alternating white and black squares, how many ways are there to choose four black squares and four white squares so that no two of the eight chosen squares are in the same row or column? | 20736 | Number both the rows and the columns from 1 to 8, and say that black squares are the ones where the rows and columns have the same parity. We will use, e.g. 'even rows' to refer to rows 2, 4, 6,8. Choosing 8 squares all in different rows and columns is equivalent to matching rows to columns. For each of the 8 rows, we ... | 0 | 8,144.25 | -1 | 8,144.25 |
Given \(1 \leq x^{2}+y^{2} \leq 4\), find the sum of the maximum and minimum values of \(x^{2}-xy+y^{2}\). | 6.5 | 0 | 6,950.9375 | -1 | 6,950.9375 | |
The reciprocal of $-2$ is equal to $\frac{1}{-2}$. | -\frac{1}{2} | 1 | 1,697.1875 | 1,697.1875 | -1 | |
Let vector $a = (\cos 25^\circ, \sin 25^\circ)$, $b = (\sin 20^\circ, \cos 20^\circ)$. If $t$ is a real number, and $u = a + tb$, then the minimum value of $|u|$ is \_\_\_\_\_\_\_\_. | \frac{\sqrt{2}}{2} | 0 | 5,695.0625 | -1 | 5,695.0625 | |
Compute $\frac{6! + 7!}{5!}$ | 48 | 1 | 1,821.6875 | 1,821.6875 | -1 | |
What is the smallest positive integer with exactly 16 positive divisors? | 216 | 0 | 4,963.1875 | -1 | 4,963.1875 | |
Find the result of $(1011101_2 + 1101_2) \times 101010_2 \div 110_2$. Express your answer in base 2. | 1110111100_2 | 0 | 8,192 | -1 | 8,192 | |
The equation $x^3 - 6x^2 - x + 3 = 0$ has three real roots $a$, $b$, $c$. Find $\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$. | \frac{37}{9} | 0.9375 | 5,019.8125 | 4,808.333333 | 8,192 | |
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between $0$ and $1$ will $20_{}^{}!$ be the resulting product? | 128 | If the fraction is in the form $\frac{a}{b}$, then $a < b$ and $\gcd(a,b) = 1$. There are 8 prime numbers less than 20 ($2, 3, 5, 7, 11, 13, 17, 19$), and each can only be a factor of one of $a$ or $b$. There are $2^8$ ways of selecting some combination of numbers for $a$; however, since $a<b$, only half of them will b... | 0.6875 | 6,093.5625 | 5,299.545455 | 7,840.4 |
What is the greatest integer less than 100 for which the greatest common factor of that integer and 18 is 3? | 93 | 0.875 | 5,271.4375 | 4,854.214286 | 8,192 | |
There is a magical tree with 63 fruits. On the first day, 1 fruit will fall from the tree. Starting from the second day, the number of fruits falling each day increases by 1 compared to the previous day. However, if the number of fruits on the tree is less than the number that should fall on that day, then the sequence... | 15 | 0.0625 | 6,041.625 | 5,681 | 6,065.666667 | |
Given the function $f(x)=\sin(\omega x+\varphi)$ is monotonically increasing on the interval ($\frac{π}{6}$,$\frac{{2π}}{3}$), and the lines $x=\frac{π}{6}$ and $x=\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, calculate the value of $f\left(-\frac{{5π}}{{12}}\right)$. | \frac{\sqrt{3}}{2} | 0 | 7,217.125 | -1 | 7,217.125 | |
For each positive integer $n$ , consider the highest common factor $h_n$ of the two numbers $n!+1$ and $(n+1)!$ . For $n<100$ , find the largest value of $h_n$ . | 97 | 0.875 | 5,166.375 | 4,734.142857 | 8,192 | |
Find the biggest real number $ k$ such that for each right-angled triangle with sides $ a$ , $ b$ , $ c$ , we have
\[ a^{3}\plus{}b^{3}\plus{}c^{3}\geq k\left(a\plus{}b\plus{}c\right)^{3}.\] | \frac{3\sqrt{2} - 4}{2} | 0 | 8,024.8125 | -1 | 8,024.8125 | |
Let sets \( A \) and \( B \) satisfy:
\[
A \cup B = \{1, 2, \cdots, 10\}, \quad A \cap B = \varnothing
\]
If the number of elements in set \( A \) is not an element of \( A \) and the number of elements in set \( B \) is not an element of \( B \), find the total number of different sets \( A \) that satisfy the condi... | 186 | 0.3125 | 6,956.0625 | 5,593.2 | 7,575.545455 | |
Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$ . Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$ . If $\angle MPN = 40^\circ$ , find the degree measure of $\angle BPC$ .
... | 80 | 0 | 7,802.6875 | -1 | 7,802.6875 | |
What is the probability of rolling eight standard, six-sided dice and getting exactly three pairs of identical numbers, while the other two numbers are distinct from each other and from those in the pairs? Express your answer as a common fraction. | \frac{525}{972} | 0 | 7,570.8125 | -1 | 7,570.8125 | |
Given $31.25 : x : 13.75 = 100 : 9.6 : y$, find the value of $x + y$. | 47 | 0.5 | 5,145.5 | 4,398.625 | 5,892.375 | |
The integer 48178 includes the block of digits 178. How many integers between 10000 and 100000 include the block of digits 178? | 280 | Since 100000 does not include the block of digits 178, each integer between 10000 and 100000 that includes the block of digits 178 has five digits. Such an integer can be of the form $178 x y$ or of the form $x 178 y$ or of the form $x y 178$ for some digits $x$ and $y$. The leading digit of a five-digit integer has 9 ... | 0.125 | 7,613 | 5,963.5 | 7,848.642857 |
In terms of $k$, for $k>0$ how likely is he to be back where he started after $2 k$ minutes? | \frac{1}{4}+\frac{3}{4}\left(\frac{1}{9}\right)^{k} | Again, Travis starts at $(0,0,0)$. At each step, exactly one of the three coordinates will change. The parity of the sum of the three coordinates will change at each step, so after $2 k$ steps, the sum of the coordinates must be even. There are only four possibilities for Travis's position: $(0,0,0),(1,1,0),(1,0,1)$, a... | 0 | 3,134.0625 | -1 | 3,134.0625 |
Given the binomial $(ax+b)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+…+a_{n}x^{n}(a，b∈R，n∈N^{*})$.
(1) When $a=b=1$, $n=6$, find:
① The value of $a_{1}+a_{2}+a_{3}+…+a_{n}$.
② The value of $a_{1}+2a_{2}+3a_{3}+…+na_{n}$.
(2) When $a=1$, $b=- \sqrt {3}$, $n=8$, find the value of $(a_{0}+a_{2}+a_{4}+a_{6}+a_{8})^{2}-(... | 256 | 0.8125 | 4,784.1875 | 3,997.769231 | 8,192 | |
The solutions to the equation $x^2 = x$ are $x=0$ and $x=1$. | 0 or 1 | 0 | 2,702.125 | -1 | 2,702.125 | |
What is the largest four-digit number that is divisible by 4? | 9996 | 0.875 | 867 | 916 | 524 | |
Given that $60\%$ of the high school students like ice skating, $50\%$ like skiing, and $70\%$ like either ice skating or skiing, calculate the probability that a randomly selected student who likes skiing also likes ice skating. | 0.8 | 0.9375 | 1,428.875 | 1,434.333333 | 1,347 | |
Let $a$ , $b$ , $c$ be positive reals for which
\begin{align*}
(a+b)(a+c) &= bc + 2
(b+c)(b+a) &= ca + 5
(c+a)(c+b) &= ab + 9
\end{align*}
If $abc = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , compute $100m+n$ .
*Proposed by Evan Chen* | 4532 | 0.875 | 4,083.8125 | 3,496.928571 | 8,192 | |
A mole has chewed a hole in a carpet in the shape of a rectangle with sides of 10 cm and 4 cm. Find the smallest size of a square patch that can cover this hole (a patch covers the hole if all points of the rectangle lie inside the square or on its boundary). | \sqrt{58} | 0.1875 | 7,386.625 | 6,290.666667 | 7,639.538462 | |
Given the function $f(x) = \sin^2x + 2\sqrt{3}\sin x\cos x + \sin(x + \frac{\pi}{4})\sin(x - \frac{\pi}{4})$, if $x = x_0 (0 \leq x_0 \leq \frac{\pi}{2})$ is a zero of the function $f(x)$, then $\cos 2x_0 = \_\_\_\_\_\_$. | \frac{3\sqrt{5} + 1}{8} | 0 | 6,512.0625 | -1 | 6,512.0625 | |
The equation $x^2 - kx - 24 = 0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$? | 40 | 0.9375 | 3,356.0625 | 3,275.533333 | 4,564 | |
The volume of a certain rectangular solidis $216\text{ cm}^3$, its total surface area is $288\text{ cm}^2$, and its three dimensions are in geometric progression. Find the sum of the lengths in cm of all the edges of this solid.
| 96 | 0.9375 | 4,301.25 | 4,349.533333 | 3,577 | |
Let $WXYZ$ be a rhombus with diagonals $WY = 20$ and $XZ = 24$. Let $M$ be a point on $\overline{WX}$, such that $WM = MX$. Let $R$ and $S$ be the feet of the perpendiculars from $M$ to $\overline{WY}$ and $\overline{XZ}$, respectively. Find the minimum possible value of $RS$. | \sqrt{244} | 0 | 7,885.9375 | -1 | 7,885.9375 | |
Of the final five contestants in a television show, three are female and two are male. If two are chosen randomly to be the final contestants, what is the probability that both are female? Express your answer as a common fraction. | \frac{3}{10} | 1 | 1,892.1875 | 1,892.1875 | -1 | |
In triangle $ABC$, $AB = 7$, $BC = 24$, and the area of triangle $ABC$ is 84 square units. Given that the length of median $AM$ from $A$ to $BC$ is 12.5, find $AC$. | 25 | 0 | 8,192 | -1 | 8,192 | |
From a large sheet of aluminum, triangular sheets (with each cell side equal to 1) are cut with vertices at marked points. What is the minimum area of the triangle that can be obtained? | $\frac{1}{2}$ | 0 | 6,301.125 | -1 | 6,301.125 | |
The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number.
For all real values of $t$ the expression $\sqrt{t^4+t^2}$ is equal to? | |t|\sqrt{1+t^2} | 1. **Identify the expression to simplify**: We start with the expression given in the problem:
\[
\sqrt{t^4 + t^2}
\]
2. **Factor out $t^2$ from the expression**: Recognizing that both terms inside the square root are powers of $t$, we can factor out the lowest power, $t^2$:
\[
\sqrt{t^4 + t^2} = \sqrt{... | 0 | 2,250.6875 | -1 | 2,250.6875 |
Given that $| \overrightarrow{a}|=6$, $| \overrightarrow{b}|=3$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-12$, find the projection of vector $\overrightarrow{a}$ onto vector $\overrightarrow{b}$. | -4 | 0.6875 | 4,639.5 | 3,602.545455 | 6,920.8 | |
A car travels due east at a speed of $\frac{5}{4}$ miles per minute on a straight road. Simultaneously, a circular storm with a 51-mile radius moves south at $\frac{1}{2}$ mile per minute. Initially, the center of the storm is 110 miles due north of the car. Calculate the average of the times, $t_1$ and $t_2$, when the... | \frac{880}{29} | 0 | 8,192 | -1 | 8,192 | |
Figure $ABCD$ is a trapezoid with $AB \parallel DC$, $AB=5$, $BC=3\sqrt{2}$, $\angle BCD=45^\circ$, and $\angle CDA=60^\circ$. The length of $DC$ is | 8 + \sqrt{3} | 1. **Identify the Trapezoid and Drop Perpendiculars**: Given that $ABCD$ is a trapezoid with $AB \parallel DC$, drop perpendiculars from $A$ and $B$ to line $CD$. Let these perpendiculars meet $CD$ at points $F$ and $E$, respectively.
2. **Analyze the Triangles Formed**: By dropping perpendiculars, we form two right t... | 0.125 | 7,161.4375 | 3,833 | 7,636.928571 |
Let $S$ be the set of all positive integers whose prime factorizations only contain powers of the primes 2 and 2017 (1, powers of 2, and powers of 2017 are thus contained in $S$). Compute $\sum_{s \in S} \frac{1}{s}$. | \frac{2017}{1008} | Since every $s$ can be written as $2^{i} \cdot 2017^{j}$ for non-negative integers $i$ and $j$, the given sum can be written as $\left(\sum_{i=0}^{\infty} \frac{1}{2^{i}}\right)\left(\sum_{j=0}^{\infty} \frac{1}{2017^{j}}\right)$. We can easily find the sum of these geometric series since they both have common ratio of... | 0.8125 | 3,845.9375 | 3,014.230769 | 7,450 |
Arrange numbers $ 1,\ 2,\ 3,\ 4,\ 5$ in a line. Any arrangements are equiprobable. Find the probability such that the sum of the numbers for the first, second and third equal to the sum of that of the third, fourth and fifth. Note that in each arrangement each number are used one time without overlapping. | 1/15 | 0 | 7,499.9375 | -1 | 7,499.9375 | |
In how many ways is it possible to arrange the digits of 11250 to get a five-digit multiple of 5? | 21 | 0.75 | 5,860.4375 | 5,083.25 | 8,192 | |
On the Saturday of a weekend softball tournament, Team A plays Team D, Team B plays Team E, and Team C gets a bye (no match). The winner of Team A vs. Team D plays against Team C in the afternoon, while the winner of Team B vs. Team E has no further matches on Saturday. On Sunday, the winners of Saturday's afternoon ma... | 48 | 0 | 8,192 | -1 | 8,192 | |
Convert 2 meters plus 3 centimeters plus 5 millimeters into meters. | 2.035 \text{ m} | Since there are 100 cm in 1 m, then 1 cm is 0.01 m. Thus, 3 cm equals 0.03 m. Since there are 1000 mm in 1 m, then 1 mm is 0.001 m. Thus, 5 mm equals 0.005 m. Therefore, 2 m plus 3 cm plus 5 mm equals $2+0.03+0.005=2.035 \mathrm{~m}$. | 0.3125 | 404.0625 | 391.8 | 409.636364 |
I have 6 shirts, 4 pairs of pants, and 6 hats. The pants come in tan, black, blue, and gray. The shirts and hats come in those colors, and also white and yellow. I refuse to wear an outfit in which all 3 items are the same color. How many choices for outfits, consisting of one shirt, one hat, and one pair of pants, do ... | 140 | 0.625 | 4,068.125 | 3,220.6 | 5,480.666667 | |
Given the polynomial $f(x) = 2x^7 + x^6 + x^4 + x^2 + 1$, calculate the value of $V_2$ using Horner's method when $x=2$. | 10 | 0.75 | 4,943.4375 | 3,860.583333 | 8,192 | |
9 seeds are divided among three pits, labeled A, B, and C, with each pit containing 3 seeds. Each seed has a 0.5 probability of germinating. If at least one seed in a pit germinates, then that pit does not need to be replanted; if no seeds in a pit germinate, then that pit needs to be replanted.
(Ⅰ) Calculate the pro... | 0.330 | 0.1875 | 5,555.0625 | 4,347.333333 | 5,833.769231 | |
The perimeter of quadrilateral PQRS, made from two similar right-angled triangles PQR and PRS, is given that the length of PQ is 3, the length of QR is 4, and ∠PRQ = ∠PSR. Find the perimeter of PQRS. | 22 | 0.25 | 5,852.125 | 3,702.25 | 6,568.75 | |
Suppose we flip four coins simultaneously: a penny, a nickel, a dime, and a quarter. What is the probability that at least 15 cents worth of coins come up heads? | \dfrac{5}{8} | 0.1875 | 7,405.4375 | 6,528 | 7,607.923077 | |
Let the internal angles $A$, $B$, $C$ of $\triangle ABC$ be opposite to the sides $a$, $b$, $c$ respectively, and $c\cos B= \sqrt {3}b\sin C$.
$(1)$ If $a^{2}\sin C=4 \sqrt {3}\sin A$, find the area of $\triangle ABC$;
$(2)$ If $a=2 \sqrt {3}$, $b= \sqrt {7}$, and $c > b$, the midpoint of side $BC$ is $D$, find the... | \sqrt {13} | 0 | 8,036.25 | -1 | 8,036.25 | |
Which of the following five values of \( n \) is a counterexample to the statement: For a positive integer \( n \), at least one of \( 6n-1 \) and \( 6n+1 \) is prime? | 20 | 0.5 | 6,277.5625 | 5,044.375 | 7,510.75 | |
Two painters are painting a fence that surrounds garden plots. They come every other day and paint one plot (there are 100 plots) in either red or green. The first painter is colorblind and mixes up the colors; he remembers which plots he painted, but cannot distinguish the color painted by the second painter. The firs... | 49 | 0 | 7,729.8125 | -1 | 7,729.8125 | |
Find the sum of $327_8$ and $73_8$ in base $8$. | 422_8 | 0.8125 | 4,312.0625 | 3,416.692308 | 8,192 | |
Among the nine fractions
$$
\frac{5}{4}, \frac{17}{6}, \frac{-5}{4}, \frac{10}{7}, \frac{2}{3}, \frac{14}{8}, \frac{-1}{3}, \frac{5}{3} \text { and } \frac{-3}{2}
$$
we have eight with the following properties:
- 2 fractions whose sum is $\frac{2}{5}$
- 2 fractions whose difference is $\frac{2}{5}$
- 2 fractions who... | -\frac{3}{2} | 0 | 8,192 | -1 | 8,192 | |
What is the value of the expression $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{1+2+3+4+5+6+7+8}$? | 1120 | 1. **Calculate the numerator**: The numerator of the given expression is the product of the first 8 positive integers:
\[
1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 = 8!
\]
Using the factorial of 8, we find:
\[
8! = 40320
\]
2. **Calculate the denominator**: The denominator is the s... | 1 | 1,940.25 | 1,940.25 | -1 |
A bag contains 4 blue marbles and 6 green marbles. Three marbles are drawn one after another without replacement. What is the probability that the first marble is blue, the second one is green, and the third one is also green? | \frac{1}{6} | 1 | 3,885.0625 | 3,885.0625 | -1 | |
A number $m$ is randomly selected from the set $\{12, 14, 16, 18, 20\}$, and a number $n$ is randomly selected from $\{2005, 2006, 2007, \ldots, 2024\}$. What is the probability that $m^n$ has a units digit of $6$?
A) $\frac{1}{5}$
B) $\frac{1}{4}$
C) $\frac{2}{5}$
D) $\frac{1}{2}$
E) $\frac{3}{5}$ | \frac{2}{5} | 0 | 5,730.75 | -1 | 5,730.75 | |
The numbers $1, 2, 3, 4, 5$ are to be arranged in a circle. An arrangement is $\textit{bad}$ if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the ... | 2 | To solve this problem, we need to determine how many arrangements of the numbers $1, 2, 3, 4, 5$ in a circle are "bad," meaning that it is not possible to find a subset of numbers that appear consecutively on the circle that sum to every integer from $1$ to $15$.
#### Step 1: Count the total number of distinct arrange... | 0 | 8,082.9375 | -1 | 8,082.9375 |
On an $8 \times 8$ grid, 64 points are marked at the center of each square. What is the minimum number of lines needed to separate all of these points from each other? | 14 | 0.0625 | 7,834.0625 | 6,633 | 7,914.133333 | |
Let \( x \in \mathbf{R} \). The algebraic expression
$$
(x+1)(x+2)(x+3)(x+4) + 2019
$$
has a minimum value of ( ). | 2018 | 0.5625 | 6,260.1875 | 4,920.222222 | 7,983 | |
There are two types of electronic toy cars, Type I and Type II, each running on the same two circular tracks. Type I completes a lap every 5 minutes, while Type II completes a lap every 3 minutes. At a certain moment, both Type I and Type II cars start their 19th lap simultaneously. How many minutes earlier did the Typ... | 36 | 0.125 | 3,466.75 | 1,637 | 3,728.142857 | |
A math conference is hosting a series of lectures by seven distinct lecturers. Dr. Smith's lecture depends on Dr. Jones’s lecture, and additionally, Dr. Brown's lecture depends on Dr. Green’s lecture. How many valid orders can these seven lecturers be scheduled, given these dependencies? | 1260 | 0.875 | 3,381.75 | 2,862.428571 | 7,017 | |
Ellie's drawer of hair clips contains 4 red clips, 5 blue clips, and 7 green clips. Each morning, she randomly chooses one hair clip to wear for the day. She returns this clip to the drawer each evening. One morning, Kyne removes $k$ hair clips before Ellie can make her daily selection. As a result, the probability tha... | 12 | Before Kyne removes hair clips, Ellie has 4 red clips and $4+5+7=16$ clips in total, so the probability that she randomly chooses a red clip is $ rac{4}{16}$ which equals $ rac{1}{4}$. After Kyne removes the clips, the probability that Ellie chooses a red clip is $2 imes rac{1}{4}$ or $ rac{1}{2}$. Since Ellie starts... | 0 | 6,192.5625 | -1 | 6,192.5625 |
Define the sequence $f_{1}, f_{2}, \ldots:[0,1) \rightarrow \mathbb{R}$ of continuously differentiable functions by the following recurrence: $$ f_{1}=1 ; \quad f_{n+1}^{\prime}=f_{n} f_{n+1} \quad \text { on }(0,1), \quad \text { and } \quad f_{n+1}(0)=1 $$ Show that \(\lim _{n \rightarrow \infty} f_{n}(x)\) exists fo... | \frac{1}{1-x} | First of all, the sequence $f_{n}$ is well defined and it holds that $$ f_{n+1}(x)=e^{\int_{0}^{x} f_{n}(t) \mathrm{d} t} \tag{2} $$ The mapping $\Phi: C([0,1)) \rightarrow C([0,1))$ given by $$ \Phi(g)(x)=e^{\int_{0}^{x} g(t) \mathrm{d} t} $$ is monotone, i.e. if $f<g$ on $(0,1)$ then $$ \Phi(f)(x)=e^{\int_{0}^{x} f(t... | 0.4375 | 7,524.875 | 6,667.142857 | 8,192 |
March 12th is Tree Planting Day. A school organizes 65 high school students and their parents to participate in the "Plant a tree, green a land" tree planting activity as a family unit. The activity divides the 65 families into two groups, A and B. Group A is responsible for planting 150 silver poplar seedlings, while ... | \frac{12}{5} | 0.25 | 5,261.375 | 4,629.75 | 5,471.916667 | |
Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point... | 273 | Clearly Freddy's $x$-coordinate is irrelevant, so we let $E(y)$ be the expected value of the number of jumps it will take him to reach the river from a given $y$-coordinate. Observe that $E(24)=0$, and \[E(y)=1+\frac{E(y+1)+E(y-1)+2E(y)}{4}\] for all $y$ such that $1\le y\le 23$. Also note that $E(0)=1+\frac{2E(0)+E(1)... | 0.0625 | 7,466.8125 | 6,213 | 7,550.4 |
Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $ | (0, 0, 2), (1, 1, 2), (2, 2, 3) |
The problem requires us to find all triples \((x, y, p)\) consisting of two non-negative integers \(x\) and \(y\), and a prime number \(p\), such that:
\[
p^x - y^p = 1
\]
To solve this problem, we'll analyze it case by case, beginning with small values for \(x\) and considering the nature of \(y^p\) and \(p^x\).
#... | 0 | 7,404.0625 | -1 | 7,404.0625 |
Let $ S $ be the set of all sides and diagonals of a regular hexagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? | \frac{33}{105} | 0 | 4,867.1875 | -1 | 4,867.1875 | |
Find the measure of the angle
$$
\delta=\arccos \left(\left(\sin 2905^{\circ}+\sin 2906^{\circ}+\cdots+\sin 6505^{\circ}\right)^{\cos } 2880^{\circ}+\cos 2881^{\circ}+\cdots+\cos 6480^{\circ}\right)
$$ | 65 | 0.8125 | 5,582.5 | 5,394.538462 | 6,397 | |
Calculate the expression $(-2)^4 + (-2)^3 + (-2)^2 + (-2)^1 + 2^1 + 2^2 + 2^3 + 2^4$. | 40 | 0.875 | 3,332.4375 | 3,714.785714 | 656 | |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 10.\] | \frac{131}{11} | 0.1875 | 8,027.875 | 7,316.666667 | 8,192 | |
Given that $a$ is a multiple of $456$, find the greatest common divisor of $3a^3+a^2+4a+57$ and $a$. | 57 | 1 | 3,256.75 | 3,256.75 | -1 | |
A grocer sets up a pyramid display for a store promotion, where the topmost row has three cans and each succeeding row below has three more cans than the row immediately above it. If the grocer uses 225 cans in total for this display, how many rows are there in the display? | 12 | 0 | 8,192 | -1 | 8,192 | |
The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence plus the square of the first term. What is the value of the first term which exceeds 10000? | 16384 | 0.9375 | 4,519.375 | 4,482.8 | 5,068 | |
Given that $|\overrightarrow{a}|=4, |\overrightarrow{b}|=8$, and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{2\pi}{3}$.
(I) Find $|\overrightarrow{a}+\overrightarrow{b}|$;
(II) Find the value of $k$ such that $(\overrightarrow{a}+2\overrightarrow{b}) \perp (k\overrightarrow{a}-\overright... | -7 | 0.9375 | 2,930.0625 | 2,970.466667 | 2,324 | |
Mr. Mendez awards extra credit on quizzes to his students with quiz grades that exceed the class mean. Given that 107 students take the same quiz, what is the largest number of students who can be awarded extra credit? | 106 | 0.375 | 7,324.1875 | 5,877.833333 | 8,192 | |
Find the largest three-digit integer that is divisible by each of its distinct, non-zero digits. | 936 | 0.125 | 8,192 | 8,192 | 8,192 | |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $S_7=3(a_1+a_9)$, then the value of $\frac{a_5}{a_4}$ is \_\_\_\_\_. | \frac{7}{6} | 1 | 2,594.0625 | 2,594.0625 | -1 | |
A translation of the plane takes $-3 + 2i$ to $-7 - i.$ Find the complex number that the translation takes $-4 + 5i$ to. | -8 + 2i | 1 | 1,265.75 | 1,265.75 | -1 | |
What is the value of $9^3 + 3(9^2) + 3(9) + 1$? | 1000 | 1 | 2,607.875 | 2,607.875 | -1 | |
The Boston weatherman says there is a 75 percent chance of rain for each day of the Labor Day four-day weekend. If it doesn't rain, then the weather will be sunny. Paul and Yuri want it to be sunny one of those days for a World War III preenactment, but if it's sunny for more than one day they won't know what to do wi... | \frac{27}{64} | 0.9375 | 2,845.4375 | 2,489 | 8,192 | |
Calculate the value of $V_3$ in Horner's method (also known as Qin Jiushao algorithm) for finding the value of the polynomial $f(x) = 4x^6 + 3x^5 + 4x^4 + 2x^3 + 5x^2 - 7x + 9$ when $x = 4$. | 80 | 0 | 3,741.125 | -1 | 3,741.125 | |
Compute
\[\frac{(10^4+400)(26^4+400)(42^4+400)(58^4+400)}{(2^4+400)(18^4+400)(34^4+400)(50^4+400)}.\] | 962 | 0 | 8,055.6875 | -1 | 8,055.6875 | |
On a quadrilateral piece of paper, there are a total of 10 points, and if the vertices of the quadrilateral are included, there are a total of 14 points. It is known that any three of these points are not collinear. According to the following rules, cut this piece of paper into some triangles:
(1) Each triangle's verti... | 22 | 0.3125 | 7,217.8125 | 6,838.6 | 7,390.181818 | |
A train moves along a straight track, and from the moment it starts braking to the moment it stops, the distance $S$ in meters that the train travels in $t$ seconds after braking is given by $S=27t-0.45t^2$. Find the time in seconds after braking when the train stops, and the distance in meters the train has traveled d... | 405 | 0.9375 | 2,648.8125 | 2,279.266667 | 8,192 | |
The coefficient of $x^7$ in the polynomial expansion of $(1+2x-x^2)^4$ is | -8 | 1. **Identify the Polynomial and the Target Term**: We are given the polynomial \((1+2x-x^2)^4\) and need to find the coefficient of \(x^7\).
2. **Expand the Polynomial**: The polynomial can be expanded using the multinomial theorem, which states that:
\[
(a+b+c)^n = \sum_{k_1+k_2+k_3=n} \binom{n}{k_1, k_2, k_3}... | 0.875 | 6,924.5625 | 6,743.5 | 8,192 |
Find the largest real number \(\lambda\) such that for the real-coefficient polynomial \(f(x) = x^3 + ax^2 + bx + c\) with all roots non-negative real numbers, we have \(f(x) \geqslant \lambda(x - a)^3\) for all \(x \geqslant 0\). Also, determine when the equality holds. | -\frac{1}{27} | 0 | 8,192 | -1 | 8,192 | |
From the set $\{1, 2, 3, 4, \ldots, 20\}$, select four different numbers $a, b, c, d$ such that $a+c=b+d$. If the order of $a, b, c, d$ does not matter, calculate the total number of ways to select these numbers. | 525 | 0 | 8,192 | -1 | 8,192 | |
Given a parabola $C: y^2 = 2px (p > 0)$, and a circle $M: (x-2)^2 + y^2 = 4$, the distance from the center $M$ of the circle to the directrix of the parabola is $3$. Point $P(x_0, y_0)(x_0 \geqslant 5)$ is a point on the parabola in the first quadrant. Through point $P$, two tangent lines to circle $M$ are drawn, inter... | \frac{25}{2} | 0.125 | 8,062.75 | 7,158 | 8,192 | |
There is a positive integer s such that there are s solutions to the equation $64sin^2(2x)+tan^2(x)+cot^2(x)=46$ in the interval $(0,\frac{\pi}{2})$ all of the form $\frac{m_k}{n_k}\pi$ where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, . . . , s$ . Find $(m_1 + n_1) + (m_2 + n_2)... | 100 | 0.125 | 8,004.3125 | 6,690.5 | 8,192 | |
At a crossroads, if vehicles are not allowed to turn back, calculate the total number of possible driving routes. | 12 | 0.0625 | 6,814.1875 | 7,121 | 6,793.733333 | |
In a right triangle where the ratio of the legs is 1:3, a perpendicular is dropped from the vertex of the right angle to the hypotenuse. Find the ratio of the segments created on the hypotenuse by this perpendicular. | 1:9 | 0.5 | 6,138.375 | 4,721 | 7,555.75 | |
Find the ratio of the volume of a regular hexagonal pyramid to the volume of a regular triangular pyramid, given that the sides of their bases are equal and their slant heights are twice the length of the sides of the base. | \frac{6 \sqrt{1833}}{47} | 0 | 6,380.6875 | -1 | 6,380.6875 | |
If $x_1=5, x_2=401$ , and
\[
x_n=x_{n-2}-\frac 1{x_{n-1}}
\]
for every $3\leq n \leq m$ , what is the largest value of $m$ ? | 2007 | 0.125 | 7,897.875 | 5,839 | 8,192 | |
Let $ABCD$ be a square with side length $16$ and center $O$ . Let $\mathcal S$ be the semicircle with diameter $AB$ that lies outside of $ABCD$ , and let $P$ be a point on $\mathcal S$ so that $OP = 12$ . Compute the area of triangle $CDP$ .
*Proposed by Brandon Wang* | 120 | 0 | 6,430.0625 | -1 | 6,430.0625 | |
Find the number of positive integers less than 1000000 that are divisible by some perfect cube greater than 1. | 168089 | Using the following code, we get the answer (denoted by the variable ans): ans $=0$ for $n$ in xrange $(1,1000000)$ : ``` divisible_by_cube = True for i in xrange(2,101): if n%(i*i*i)==0: divisible_by_cube = False break if divisible_by_cube: ans = ans + 1 print ans ``` This gives the output 168089 Alternatively, let $N... | 0 | 8,120.6875 | -1 | 8,120.6875 |
Consider a triangle with vertices at $(2, 1)$, $(4, 7)$, and $(7, 3)$. This triangle is reflected about the line $y=4$. Find the area of the union of the original and the reflected triangles. | 26 | 0 | 8,073.25 | -1 | 8,073.25 |
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