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 function $f(x)=\sin \left( \frac {5\pi}{6}-2x\right)-2\sin \left(x- \frac {\pi}{4}\right)\cos \left(x+ \frac {3\pi}{4}\right).$
$(1)$ Find the minimum positive period and the intervals of monotonic increase for the function $f(x)$;
$(2)$ If $x_{0}\in\left[ \frac {\pi}{3}, \frac {7\pi}{12}\right]$ and $f(x... | - \frac {2 \sqrt {6}+1}{6} | 0 | 6,516.3125 | -1 | 6,516.3125 | |
The matrix for reflecting over a certain line $\ell,$ which passes through the origin, is given by
\[\begin{pmatrix} \frac{7}{25} & -\frac{24}{25} \\ -\frac{24}{25} & -\frac{7}{25} \end{pmatrix}.\]Find the direction vector of line $\ell.$ Enter your answer in the form $\begin{pmatrix} a \\ b \end{pmatrix},$ where $a,$... | \begin{pmatrix} 4 \\ -3 \end{pmatrix} | 0.8125 | 5,933 | 5,411.692308 | 8,192 | |
We have a calculator with two buttons that displays an integer $x$. Pressing the first button replaces $x$ by $\left\lfloor\frac{x}{2}\right\rfloor$, and pressing the second button replaces $x$ by $4 x+1$. Initially, the calculator displays 0. How many integers less than or equal to 2014 can be achieved through a seque... | 233 | We consider the integers from this process written in binary. The first operation truncates the rightmost digit, while the second operation appends 01 to the right. We cannot have a number with a substring 11. For simplicity, call a string valid if it has no consecutive $1^{\prime} s$. Note that any number generated by... | 0 | 8,192 | -1 | 8,192 |
Calculate the value of $\frac12\cdot\frac41\cdot\frac18\cdot\frac{16}{1} \dotsm \frac{1}{2048}\cdot\frac{4096}{1}$, and multiply the result by $\frac34$. | 1536 | 0 | 6,217.6875 | -1 | 6,217.6875 | |
Let $E(n)$ denote the sum of the even digits of $n$. For example, $E(5681) = 6+8 = 14$. Find $E(1)+E(2)+E(3)+\cdots+E(100)$ | 400 | 1. **Understanding the Problem**: We need to find the sum of the even digits in all numbers from 1 to 100. The function $E(n)$ represents the sum of even digits in the number $n$.
2. **Simplifying the Range**: We can consider the numbers from 00 to 99 instead of 1 to 100 for simplicity, as adding 00 and removing 100 d... | 0.0625 | 8,123.5 | 7,096 | 8,192 |
Given that $\tan\alpha=3$, calculate the following:
(1) $\frac{\sin\alpha+\cos\alpha}{2\sin\alpha-\cos\alpha}$
(2) $\sin^2\alpha+\sin\alpha\cos\alpha+3\cos^2\alpha$ | 15 | 0 | 3,141.9375 | -1 | 3,141.9375 | |
How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart? | 10 | 0 | 8,192 | -1 | 8,192 | |
Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?
$\textbf{(A)}\ \sqrt{15} \qquad \textbf{(B)}\ \sqrt{21} \qquad \textbf{(C)}\ 2\sqrt{6} \qquad \textbf{(D)}\ 5 \qquad \textbf{(... | 2\sqrt{6} | 0 | 5,893.375 | -1 | 5,893.375 | |
There are $100$ piles of $400$ stones each. At every move, Pete chooses two piles, removes one stone from each of them, and is awarded the number of points, equal to the non- negative difference between the numbers of stones in two new piles. Pete has to remove all stones. What is the greatest total score Pete can get,... | 3920000 | To solve this problem, we need to find the greatest total score Pete can get by removing all stones. Initially, we have 100 piles, each containing 400 stones.
### Strategy
To maximize the total score, Pete should aim to keep the piles as balanced as possible until they are empty. This involves making the difference be... | 0 | 8,192 | -1 | 8,192 |
If the final 5 contestants of "The Voice" season 4 must sign with one of the three companies A, B, and C, with each company signing at least 1 person and at most 2 people, calculate the total number of possible different signing schemes. | 90 | 0.3125 | 7,549.5 | 6,136 | 8,192 | |
Given the polynomial $f(x)=x^{6}-5x^{5}+6x^{4}+x^{2}+0.3x+2$, use Horner's method to calculate $f(-2)$ and find the value of $v_{1}$. | -7 | 0.8125 | 3,646.8125 | 3,016.846154 | 6,376.666667 | |
Triangle $ABC$ has side lengths $AB=120,BC=220$, and $AC=180$. Lines $\ell_A,\ell_B$, and $\ell_C$ are drawn parallel to $\overline{BC},\overline{AC}$, and $\overline{AB}$, respectively, such that the intersections of $\ell_A,\ell_B$, and $\ell_C$ with the interior of $\triangle ABC$ are segments of lengths $55,45$, an... | 715 | Notation shown on diagram. By similar triangles we have \[k_1 = \frac{EF}{BC} = \frac{AE}{AB} = \frac {AF}{AC} = \frac {1}{4},\] \[k_2 = \frac{F''E''}{AC} = \frac {BF''}{AB} = \frac{1}{4},\] \[k_3 = \frac{E'F'}{AB} = \frac{E'C }{AC} = \frac{1}{8}.\] So, \[\frac{ZE}{BC} = \frac{F''E}{AB} = \frac{AB - AE - BF''}{AB} = 1 ... | 0 | 8,192 | -1 | 8,192 |
What is the sum of all integer solutions to $4<(x-3)^2<64$? | 30 | 1 | 3,026 | 3,026 | -1 | |
The points $(2, 5), (10, 9)$, and $(6, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum? | 14 | 0.375 | 3,128.625 | 2,565.333333 | 3,466.6 | |
If $f(3)=1$ and $f(2x)=2f(x)$ for all $x$, find $f^{-1}(64)$. | 192 | 0.9375 | 4,095.0625 | 3,821.933333 | 8,192 | |
For a certain type of car, the purchase cost is $10$ ten thousand yuan, and the annual expenses for insurance, road maintenance, and car fees are about $0.9$ ten thousand yuan. The maintenance fee for the first year is $0.2$ ten thousand yuan, and it increases by $0.2$ ten thousand yuan each subsequent year. How many y... | 10 | 1 | 3,468.875 | 3,468.875 | -1 | |
Quantities $a$ and $b$ vary inversely. When $a$ is $800$, $b$ is $0.5$. If the product of $a$ and $b$ increases by $200$ when $a$ is doubled, what is $b$ when $a$ is $1600$? | 0.375 | 0.4375 | 5,140.1875 | 5,305 | 5,012 | |
Following the concept of a healthy, low-carbon lifestyle, an increasing number of people are renting bicycles for cycling tours. A particular bicycle rental point charges no fee for rentals that do not exceed two hours, and for rentals that exceed two hours, the charging standard is 2 yuan per hour (with fractions of a... | \frac{7}{2} | 0.3125 | 6,785.75 | 4,885.4 | 7,649.545455 | |
For an arithmetic sequence $b_1, b_2, b_3, \dots,$ let
\[S_n = b_1 + b_2 + b_3 + \dots + b_n,\]and let
\[T_n = S_1 + S_2 + S_3 + \dots + S_n.\]Given the value of $S_{2023},$ then you can uniquely determine the value of $T_n$ for some integer $n.$ What is this integer $n$? | 3034 | 0.625 | 6,570.6875 | 5,597.9 | 8,192 | |
How many four-digit numbers divisible by 17 are also even? | 265 | 0.875 | 3,819.6875 | 3,195.071429 | 8,192 | |
What weights can be measured using a balance scale with weights of $1, 3, 9, 27$ grams? Generalize the problem! | 40 | 0.375 | 7,507.1875 | 6,674 | 8,007.1 | |
$\tan 2\alpha = \frac{\cos \alpha}{2-\sin \alpha}$, where $0 < \alpha < \frac{\pi}{2}$, find the value of $\tan \alpha$. | \frac{\sqrt{15}}{15} | 0 | 3,478.0625 | -1 | 3,478.0625 | |
If \( 20 \times 21 \times 22 \times \ldots \times 2020 = 26^{k} \times m \), where \( m \) is an integer, what is the maximum value of \( k \)? | 165 | 1 | 4,807.125 | 4,807.125 | -1 | |
What is the median of the following list of $4040$ numbers?
\[1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2\] | 1976.5 | 1. **Identify the total number of terms and the position of the median**:
The list consists of $4040$ numbers, which includes all integers from $1$ to $2020$ and their squares. Since the list has an even number of terms, the median will be the average of the $2020$-th and $2021$-st terms.
2. **Determine the range ... | 0 | 7,897.1875 | -1 | 7,897.1875 |
It is known that the center C of a moving circle is on the parabola $x^2=2py$ ($p>0$), the circle passes through point A $(0, p)$, and intersects the x-axis at two points M and N. The maximum value of $\sin\angle MCN$ is. | \frac{1}{\sqrt{2}} | 0 | 7,216.125 | -1 | 7,216.125 | |
Tetrahedron \(ABCD\) has base \( \triangle ABC \). Point \( E \) is the midpoint of \( AB \). Point \( F \) is on \( AD \) so that \( FD = 2AF \), point \( G \) is on \( BD \) so that \( GD = 2BG \), and point \( H \) is on \( CD \) so that \( HD = 2CH \). Point \( M \) is the midpoint of \( FG \) and point \( P \) is... | 1/10 | 0.5 | 7,195.125 | 7,045 | 7,345.25 | |
Find $\begin{pmatrix} 3 & 0 \\ 1 & 2 \end{pmatrix} + \begin{pmatrix} -5 & -7 \\ 4 & -9 \end{pmatrix}.$ | \begin{pmatrix} -2 & -7 \\ 5 & -7 \end{pmatrix} | 1 | 1,645.75 | 1,645.75 | -1 | |
The first term of a sequence is 2, the second term is 3, and each subsequent term is formed such that each term is 1 less than the product of its two neighbors. What is the sum of the first 1095 terms of the sequence? | 1971 | 0.75 | 3,941.375 | 2,910.916667 | 7,032.75 | |
There are two rows of seats, with 4 seats in the front row and 5 seats in the back row. Now, we need to arrange seating for 2 people, and these 2 people cannot sit next to each other (sitting one in front and one behind is also considered as not adjacent). How many different seating arrangements are there? | 58 | 0.25 | 7,814.125 | 7,592.5 | 7,888 | |
What is the smallest three-digit positive integer which can be written in the form \( p q^{2} r \), where \( p, q \), and \( r \) are distinct primes? | 126 | 0.1875 | 7,805.25 | 6,581.333333 | 8,087.692308 | |
Determine the time the copy machine will finish all the paperwork if it starts at 9:00 AM and completes half the paperwork by 12:30 PM. | 4:00 | 0 | 477 | -1 | 477 | |
In a country there are $15$ cities, some pairs of which are connected by a single two-way airline of a company. There are $3$ companies and if any of them cancels all its flights, then it would still be possible to reach every city from every other city using the other two companies. At least how many two-way airli... | 21 | 0.1875 | 7,328.8125 | 6,757.333333 | 7,460.692308 | |
If the intended number is multiplied by 6, then 382 is added to the product, the result is the largest three-digit number written with two identical even digits and one odd digit. Find the intended number. | 101 | 0.5 | 6,218.1875 | 4,690 | 7,746.375 | |
Given vectors $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ -2 \\ 0 \end{pmatrix},$ determine the scalar $s$ such that
\[\begin{pmatrix} 5 \\ -4 \\ 1 \end{pmatrix} = s(\mathbf{a} \times \mathbf{b}) + p\mathbf{a} + q\mathbf{b},\]
where $p$ and $q$ are scalars. | -\frac{1}{45} | 0 | 4,539.6875 | -1 | 4,539.6875 | |
In the triangle $\triangle ABC$, $\angle A = 60^{\circ}$ and $\angle B = 45^{\circ}$. A line $DE$, with $D$ on $AB$ and $E$ on $BC$, such that $\angle ADE =75^{\circ}$, divides $\triangle ABC$ into two pieces of equal area. Determine the ratio $\frac{AD}{AB}$.
A) $\frac{1}{2}$
B) $\frac{1}{\sqrt{3}}$
C) $\frac{1}{\sqrt... | \frac{1}{\sqrt{6}} | 0 | 8,192 | -1 | 8,192 | |
Find the remainder when $3 \times 13 \times 23 \times 33 \times \ldots \times 183 \times 193$ is divided by $5$. | 1 | 1 | 3,062.6875 | 3,062.6875 | -1 | |
A subset $B$ of $\{1, 2, \dots, 2017\}$ is said to have property $T$ if any three elements of $B$ are the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property $T$ may contain. | 1009 | 0.125 | 8,010.375 | 6,739 | 8,192 | |
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, its left focus is $F$, left vertex is $A$, and point $B$ is a point on the ellipse in the first quadrant. The line $OB$ intersects the ellipse at another point $C$. If the line $BF$ bisects the line segment $AC$, find the eccentricity of the ellipse. | \frac{1}{3} | 0.75 | 4,921.5 | 3,831.333333 | 8,192 | |
Rachelle picks a positive integer \(a\) and writes it next to itself to obtain a new positive integer \(b\). For instance, if \(a=17\), then \(b=1717\). To her surprise, she finds that \(b\) is a multiple of \(a^{2}\). Find the product of all the possible values of \(\frac{b}{a^{2}}\). | 77 | Suppose \(a\) has \(k\) digits. Then \(b=a(10^{k}+1)\). Thus \(a\) divides \(10^{k}+1\). Since \(a \geq 10^{k-1}\), we have \(\frac{10^{k}+1}{a} \leq 11\). But since none of 2, 3, or 5 divide \(10^{k}+1\), the only possibilities are 7 and 11. These values are obtained when \(a=143\) and \(a=1\), respectively. | 0.0625 | 8,143.75 | 7,420 | 8,192 |
For any positive integer \( n \), the value of \( n! \) is the product of the first \( n \) positive integers. Calculate the greatest common divisor of \( 8! \) and \( 10! \). | 40320 | 1 | 2,738.9375 | 2,738.9375 | -1 | |
Let $f(x) = 5x^2 - 4$ and $g(f(x)) = x^2 + x + x/3 + 1$. Find the sum of all possible values of $g(49)$. | \frac{116}{5} | 0.8125 | 5,558 | 4,950.153846 | 8,192 | |
Let $ ABCD$ be a quadrilateral in which $ AB$ is parallel to $ CD$ and perpendicular to $ AD; AB \equal{} 3CD;$ and the area of the quadrilateral is $ 4$ . if a circle can be drawn touching all the four sides of the quadrilateral, find its radius. | \frac{\sqrt{3}}{2} | 0 | 4,016 | -1 | 4,016 | |
The function $f(x) = x(x - m)^2$ reaches its maximum value at $x = -2$. Determine the value of $m$. | -6 | 0 | 3,950.125 | -1 | 3,950.125 | |
Given $a\in \mathbb{R}$, $b\in \mathbb{R}$, if the set $\{a, \frac{b}{a}, 1\} = \{a^{2}, a-b, 0\}$, calculate the value of $a^{2019}+b^{2019}$. | -1 | 0.125 | 7,767.875 | 8,192 | 7,707.285714 | |
In writing the integers from 100 through 199 inclusive, how many times is the digit 7 written? | 20 | 0.4375 | 6,459.5625 | 4,590.571429 | 7,913.222222 | |
Let $x,$ $y,$ $z$ be real numbers such that
\begin{align*}
x + y + z &= 4, \\
x^2 + y^2 + z^2 &= 6.
\end{align*}Let $m$ and $M$ be the smallest and largest possible values of $x,$ respectively. Find $m + M.$ | \frac{8}{3} | 1 | 3,604.625 | 3,604.625 | -1 | |
Margie bought $3$ apples at a cost of $50$ cents per apple. She paid with a 5-dollar bill. How much change did Margie receive? | $3.50 | 1. **Convert the cost per apple to dollars:**
Since $50$ cents is equivalent to $\textdollar 0.50$, the cost per apple in dollars is $\textdollar 0.50$.
2. **Calculate the total cost for three apples:**
The total cost for three apples is calculated by multiplying the cost per apple by the number of apples:
... | 0 | 735.5 | -1 | 735.5 |
Let \( f(n) = \sum_{k=2}^{\infty} \frac{1}{k^n \cdot k!} \). Calculate \( \sum_{n=2}^{\infty} f(n) \). | 3 - e | 0 | 8,158.9375 | -1 | 8,158.9375 | |
Let $P$ be a cubic polynomial with $P(0) = k$, $P(1) = 2k$, and $P(-1) = 3k$. What is $P(2) + P(-2)$ ? | 14k | 0.875 | 4,188.25 | 3,616.285714 | 8,192 | |
Let \( T \) be a right triangle with sides having lengths 3, 4, and 5. A point \( P \) is called awesome if \( P \) is the center of a parallelogram whose vertices all lie on the boundary of \( T \). What is the area of the set of awesome points? | 3/2 | 0.0625 | 7,983.5625 | 8,192 | 7,969.666667 | |
Let $P_1^{}$ be a regular $r~\mbox{gon}$ and $P_2^{}$ be a regular $s~\mbox{gon}$ $(r\geq s\geq 3)$ such that each interior angle of $P_1^{}$ is $\frac{59}{58}$ as large as each interior angle of $P_2^{}$. What's the largest possible value of $s_{}^{}$? | 117 | As in above, we have $rs = 118r - 116s.$ This means that $rs + 116s - 118r = 0.$ Using SFFT we obtain $s(r+116) - 118(r+116) = -118 \cdot 116 \implies (s-118)(r+116) = -118 \cdot 116.$ Since $r+116$ is always positive, we know thta $s-118$ must be negative. Therefore the maximum value of $s$ must be $\boxed{117}$ which... | 0.25 | 7,993.5 | 7,398 | 8,192 |
A keen archaeologist is holding a competition where participants must guess the age of a rare artifact. The age of the artifact is formed using the six digits: 2, 2, 3, 3, 7, and 9, and it must begin with an odd digit.
How many different ages can be there for the artifact? | 180 | 0 | 5,335.4375 | -1 | 5,335.4375 | |
The roots of the equation $x^2 + kx + 8 = 0$ differ by 10. Find the greatest possible value of $k$. | 2\sqrt{33} | 1 | 3,901.375 | 3,901.375 | -1 | |
Three faces of a rectangular box meet at a corner, and the centers of these faces form the vertices of a triangle with side lengths of 4 cm, 5 cm, and 6 cm. What is the volume of the box, in cm^3? | 90 \sqrt{6} | 0.75 | 5,887.875 | 5,119.833333 | 8,192 | |
9. The real quartic $P x^{4}+U x^{3}+M x^{2}+A x+C$ has four different positive real roots. Find the square of the smallest real number $z$ for which the expression $M^{2}-2 U A+z P C$ is always positive, regardless of what the roots of the quartic are. | 16 | 0.0625 | 8,192 | 8,192 | 8,192 | |
A museum is organizing a quiz where participants must guess the year a certain artifact was created. Clues given: the year uses each of the digits: 1, 2, 2, 5, 5, 9 exactly once, and the year must start with a prime digit.
How many different possible years could be guessed based on these clues? | 120 | 0.5 | 6,720.8125 | 5,371.625 | 8,070 | |
If one-fourth of $2^{30}$ is equal to $2^x$, what is $x$? | 28 | 1 | 1,563.875 | 1,563.875 | -1 | |
Compute $(-64)\div (-32)$. | 2 | 1 | 976.1875 | 976.1875 | -1 | |
Determine the number of ways to arrange the letters of the word "BALLOONIST". | 907200 | 0.875 | 2,079.125 | 2,118 | 1,807 | |
Let \(A, B, C\), and \(D\) be four points that are not coplanar. A plane passes through the centroid of triangle \(ABC\) that is parallel to the lines \(AB\) and \(CD\). In what ratio does this plane divide the median drawn to the side \(CD\) of triangle \(ACD\)? | 1:2 | 0.625 | 6,865.1875 | 6,644.4 | 7,233.166667 | |
If the equation $x^{2}+(k^{2}-4)x+k-1=0$ has two roots that are opposite numbers, solve for $k$. | -2 | 0.5625 | 4,243.4375 | 3,656.111111 | 4,998.571429 | |
Determine the share of the Japanese yen in the currency structure of the NWF funds as of 01.07.2021 using one of the following methods:
First method:
a) Find the total amount of NWF funds placed in Japanese yen as of 01.07.2021:
\[ JPY_{22} = 1213.76 - 3.36 - 38.4 - 4.25 - 226.6 - 340.56 - 0.29 = 600.3 \text{ (billi... | -23.5 | 0.1875 | 3,927.375 | 3,556.333333 | 4,013 | |
If Greg rolls five fair eight-sided dice, what is the probability that he rolls more 1's than 8's? | \frac{10246}{32768} | 0 | 8,147.3125 | -1 | 8,147.3125 | |
Compute
\[\sum_{n = 2}^\infty \frac{4n^3 - n^2 - n + 1}{n^6 - n^5 + n^4 - n^3 + n^2 - n}.\] | 1 | 0 | 8,192 | -1 | 8,192 | |
How many times during a day does the angle between the hour and minute hands measure exactly $17^{\circ}$? | 44 | 0.1875 | 7,652.75 | 6,291.333333 | 7,966.923077 | |
James has 6 ounces of tea in a ten-ounce mug and 6 ounces of milk in a separate ten-ounce mug. He first pours one-third of the tea from the first mug into the second mug and stirs well. Then he pours one-fourth of the mixture from the second mug back into the first. What fraction of the liquid in the first mug is now m... | \frac{1}{4} | 0 | 4,003.75 | -1 | 4,003.75 | |
In the quadrilateral \(ABCD\), it is known that \(\angle BAC = \angle CAD = 60^\circ\), and \(AB + AD = AC\). Additionally, it is known that \(\angle ACD = 23^\circ\). What is the measure of angle \(ABC\) in degrees? | 83 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Let $\triangle ABC$ have side lengths $AB=30$, $BC=32$, and $AC=34$. Point $X$ lies in the interior of $\overline{BC}$, and points $I_1$ and $I_2$ are the incenters of $\triangle ABX$ and $\triangle ACX$, respectively. Find the minimum possible area of $\triangle AI_1I_2$ as $X$ varies along $\overline{BC}$.
| 126 | 0 | 8,192 | -1 | 8,192 | |
A laptop is originally priced at $800. The store offers a $15\%$ discount, followed by another $10\%$ discount on the discounted price. Tom also has a special membership card giving an additional $5\%$ discount on the second discounted price. What single percent discount would give the same final price as these three s... | 27.325\% | 0.75 | 6,691.0625 | 6,583.25 | 7,014.5 | |
Sarah stands at $(0,0)$ and Rachel stands at $(6,8)$ in the Euclidean plane. Sarah can only move 1 unit in the positive $x$ or $y$ direction, and Rachel can only move 1 unit in the negative $x$ or $y$ direction. Each second, Sarah and Rachel see each other, independently pick a direction to move at the same time, and m... | \[
\frac{63}{64}
\] | We make the following claim: In a game with $n \times m$ grid where $n \leq m$ and $n \equiv m(\bmod 2)$, the probability that Sarah wins is $\frac{1}{2^{n}}$ under optimal play. Proof: We induct on $n$. First consider the base case $n=0$. In this case Rachel is confined on a line, so Sarah is guaranteed to win. We the... | 0 | 7,825.25 | -1 | 7,825.25 |
For $p=1, 2, \cdots, 10$ let $S_p$ be the sum of the first $40$ terms of the arithmetic progression whose first term is $p$ and whose common difference is $2p-1$; then $S_1+S_2+\cdots+S_{10}$ is | 80200 | 1. **Identify the $40$th term of the sequence**:
For an arithmetic progression (AP) with first term $a = p$ and common difference $d = 2p - 1$, the $n$th term of the AP is given by:
\[
a_n = a + (n-1)d = p + (n-1)(2p-1).
\]
Substituting $n = 40$, we get:
\[
a_{40} = p + 39(2p - 1) = p + 78p - 39 =... | 0.9375 | 3,931.625 | 3,647.6 | 8,192 |
Find the number of positive integers that are divisors of at least one of $10^{10},15^7,18^{11}.$ | 435 | $10^{10} = 2^{10}\cdot 5^{10}$ so $10^{10}$ has $11\cdot11 = 121$ divisors.
$15^7 = 3^7\cdot5^7$ so $15^7$ has $8\cdot8 = 64$ divisors.
$18^{11} = 2^{11}\cdot3^{22}$ so $18^{11}$ has $12\cdot23 = 276$ divisors.
Now, we use the Principle of Inclusion-Exclusion. We have $121 + 64 + 276$ total potential divisors so far, ... | 0.5 | 5,746.3125 | 4,094.625 | 7,398 |
The parabola with equation $y=ax^2+bx+c$ is graphed below:
[asy]
xaxis(-3,7);
yaxis(-5,32);
real g(real x)
{
return 4(x-2)^2-4;
}
draw(graph(g,-1,5));
dot((2,-4));
label("Vertex: $(2,-4)$", (2,-4), SE);
dot((4,12));
label("$(4,12)$", (4,12), E);
[/asy]
The zeros of the quadratic $ax^2 + bx + c$ are at $x=m$ and ... | 2 | 1 | 2,029.25 | 2,029.25 | -1 | |
The first three stages of a pattern are shown below, where each line segment represents a straw. If the pattern continues such that at each successive stage, four straws are added to the previous arrangement, how many straws are necessary to create the arrangement for the 100th stage? | 400 | 0.3125 | 4,006.4375 | 5,555 | 3,302.545455 | |
A line passes through the distinct vectors $\mathbf{a}$ and $\mathbf{b}.$ Then for a certain value of $k,$ the vector
\[k \mathbf{a} + \frac{3}{4} \mathbf{b}\]must also lie on the line. Find $k.$ | \frac{1}{4} | 1 | 2,605.3125 | 2,605.3125 | -1 | |
Let P be any point on the curve $y=x^2-\ln x$. Find the minimum distance from point P to the line $y=x-4$. | 2\sqrt{2} | 0.9375 | 5,711.875 | 5,546.533333 | 8,192 | |
In a square piece of grid paper containing an integer number of cells, a hole in the shape of a square, also consisting of an integer number of cells, was cut out. How many cells did the large square contain if, after cutting out the hole, 209 cells remained? | 225 | 0.75 | 6,940.875 | 6,523.833333 | 8,192 | |
Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$ | 16 | 0 | 8,192 | -1 | 8,192 | |
Given that in quadrilateral ABCD, $\angle A : \angle B : \angle C : \angle D = 1 : 3 : 5 : 6$, express the degrees of $\angle A$ and $\angle D$ in terms of a common variable. | 144 | 0.6875 | 2,491.8125 | 2,824.454545 | 1,760 | |
John drove continuously from 8:30 a.m. until 2:15 p.m. of the same day and covered a distance of 246 miles. What was his average speed in miles per hour? | 42.78 | 0.3125 | 7,102.75 | 6,901.8 | 7,194.090909 | |
Given that positive real numbers $x$ and $y$ satisfy $e^{x}=y\ln x+y\ln y$, then the minimum value of $\frac{{e}^{x}}{x}-\ln y$ is ______. | e-1 | 0.25 | 7,983.3125 | 7,357.25 | 8,192 | |
Given that a four-digit integer $MMMM$, with all identical digits, is multiplied by the one-digit integer $M$, the result is the five-digit integer $NPMPP$. Assuming $M$ is the largest possible single-digit integer that maintains the units digit property of $M^2$, find the greatest possible value of $NPMPP$. | 89991 | 0.5 | 6,564.5625 | 5,570.75 | 7,558.375 | |
If \( k \) is the smallest positive integer such that \(\left(2^{k}\right)\left(5^{300}\right)\) has 303 digits when expanded, then the sum of the digits of the expanded number is | 11 | 0.9375 | 4,604.5625 | 4,365.4 | 8,192 | |
Evaluate the sum: 1 - 2 + 3 - 4 + $\cdots$ + 100 - 101 | -151 | 0.0625 | 4,868.625 | 8,192 | 4,647.066667 | |
The function $g(x),$ defined for $0 \le x \le 1,$ has the following properties:
(i) $g(0) = 0.$
(ii) If $0 \le x < y \le 1,$ then $g(x) \le g(y).$
(iii) $g(1 - x) = 1 - g(x)$ for all $0 \le x \le 1.$
(iv) $g\left(\frac{x}{4}\right) = \frac{g(x)}{3}$ for $0 \le x \le 1.$
(v) $g\left(\frac{1}{2}\right) = \frac{1}{3}.$
... | \frac{2}{9} | 0.1875 | 7,791.4375 | 6,055.666667 | 8,192 | |
There is a solution of table salt in a flask. From the flask, $\frac{1}{5}$ of the solution is poured into a test tube and evaporated until the salt concentration in the test tube doubles. After that, the evaporated solution is poured back into the flask. As a result, the salt concentration in the flask increases by $3... | 27 | 0.0625 | 8,071.0625 | 6,257 | 8,192 | |
A circle has an area of $M\text{ cm}^2$ and a circumference of $N\text{ cm}$. If $\dfrac{M}{N}=20$, what is the radius of the circle, in cm? | 40 | 1 | 1,303.6875 | 1,303.6875 | -1 | |
Given the parabola $C$: $x^2 = 2py (p > 0)$ and the line $2x-y+2=0$, they intersect at points $A$ and $B$. A vertical line is drawn from the midpoint of the line segment $AB$ to the $x$-axis, intersecting the parabola $C$ at point $Q$. If $\overrightarrow{QA} \cdot \overrightarrow{QB}=0$, calculate the value of $p$. | \frac{1}{4} | 1 | 4,722.3125 | 4,722.3125 | -1 | |
If $\dfrac {\cos (\pi-2\alpha)}{\sin (\alpha- \dfrac {\pi}{4})}=- \dfrac { \sqrt {2}}{2}$, then $\sin 2\alpha=$ \_\_\_\_\_\_ . | - \dfrac {3}{4} | 0.3125 | 7,716.8125 | 6,671.4 | 8,192 | |
Determine the area of the circle described by the equation \(3x^2 + 3y^2 - 15x + 9y + 27 = 0\) in terms of \(\pi\). | \frac{\pi}{2} | 0.0625 | 8,039.5 | 8,192 | 8,029.333333 | |
The letters of the alphabet are given numeric values based on the two conditions below.
$\bullet$ Only the numeric values of $-2,$ $-1,$ $0,$ $1$ and $2$ are used.
$\bullet$ Starting with A and going through Z, a numeric value is assigned to each letter according to the following pattern: $$
1, 2, 1, 0, -1, -2, -1,... | -1 | 0.625 | 5,301.625 | 5,074.4 | 5,680.333333 | |
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? | 500 | 1. **Define Variables:**
Let the length of the track be $x$ meters. Since Brenda and Sally start at diametrically opposite points, they start $\frac{x}{2}$ meters apart.
2. **Analyze the First Meeting:**
When Brenda and Sally first meet, Brenda has run 100 meters. Since they are running in opposite directions, t... | 0.3125 | 5,176.8125 | 4,104.4 | 5,664.272727 |
Triangle $ABC$ is an isosceles triangle with $AB=BC$. Point $D$ is the midpoint of both $\overline{BC}$ and $\overline{AE}$, and $\overline{CE}$ is 11 units long. What is the length of $\overline{BD}$? Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(3,112^.5)--(6,0)--cycle);
draw((6,0)--(9,1... | 5.5 | 0.875 | 5,408.375 | 5,010.714286 | 8,192 | |
There are three saline solutions with concentrations of 5%, 8%, and 9%, labeled A, B, and C, weighing 60g, 60g, and 47g respectively. We need to prepare 100g of a saline solution with a concentration of 7%. What is the maximum and minimum amount of solution A (5% concentration) that can be used? Please write down the s... | 84 | 1 | 3,528.8125 | 3,528.8125 | -1 | |
Let $ ABC$ be a triangle with $ AB \equal{} AC$ . The angle bisectors of $ \angle C AB$ and $ \angle AB C$ meet the sides $ B C$ and $ C A$ at $ D$ and $ E$ , respectively. Let $ K$ be the incentre of triangle $ ADC$. Suppose that $ \angle B E K \equal{} 45^\circ$ . Find all possible values of $ \angle C AB$ .
[i]Jan ... | 60^\circ \text{ and } 90^\circ |
Given a triangle \( ABC \) with \( AB = AC \) (isosceles triangle), we are tasked with finding all possible values of \( \angle CAB \) given the specific geometric conditions.
### Problem Setup
1. **Notation and Known Values:**
- Let \( ABC \) be an isosceles triangle with \( AB = AC \).
- The angle bisector o... | 0 | 8,192 | -1 | 8,192 |
The three medians of a triangle has lengths $3, 4, 5$ . What is the length of the shortest side of this triangle? | \frac{10}{3} | 0.1875 | 7,878.6875 | 6,576.333333 | 8,179.230769 | |
Al-Karhi's rule for approximating the square root. If \(a^{2}\) is the largest square contained in the given number \(N\), and \(r\) is the remainder, then
$$
\sqrt{N}=\sqrt{a^{2}+r}=a+\frac{r}{2a+1}, \text{ if } r<2a+1
$$
Explain how Al-Karhi might have derived this rule. Estimate the error by calculating \(\sqrt{41... | 20.366 | 0 | 7,974.25 | -1 | 7,974.25 | |
Determine the ratio $\frac{s}{r}$, where $r$ is the total number of rectangles and $s$ is the number of squares formed by the grid of a $7\times7$ checkerboard. Express $\frac{s}{r}$ in its simplest form and find the sum of the numerator and denominator. | 33 | 1 | 2,509.0625 | 2,509.0625 | -1 | |
When five students are lining up to take a photo, and two teachers join in, with the order of the five students being fixed, calculate the total number of ways for the two teachers to stand in line with the students for the photo. | 42 | 0.1875 | 6,120.5625 | 4,744 | 6,438.230769 | |
Solve in positive real numbers: $n+ \lfloor \sqrt{n} \rfloor+\lfloor \sqrt[3]{n} \rfloor=2014$ | 1958 |
We are asked to solve the equation \( n+ \lfloor \sqrt{n} \rfloor + \lfloor \sqrt[3]{n} \rfloor = 2014 \) for positive real numbers \( n \).
To begin, we denote:
- \( x = \lfloor \sqrt{n} \rfloor \),
- \( y = \lfloor \sqrt[3]{n} \rfloor \).
Thus, we have:
\[
x \leq \sqrt{n} < x+1
\]
\[
y \leq \sqrt[3]{n} < y+1
\]
T... | 0.5 | 7,440.5 | 6,689 | 8,192 |
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