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 $\tan \alpha=-2$, the focus of the parabola $y^{2}=2px (p > 0)$ is $F(-\sin \alpha\cos \alpha,0)$, and line $l$ passes through point $F$ and intersects the parabola at points $A$ and $B$ with $|AB|=4$, find the distance from the midpoint of segment $AB$ to the line $x=-\frac{1}{2}$. | \frac{21}{10} | 0.5 | 7,215.75 | 6,239.5 | 8,192 | |
In solving the system of equations $y = 7$ and $x^2+ y^2= 100,$ what is the sum of the solutions for $x?$ | 0 | 1 | 1,311.9375 | 1,311.9375 | -1 | |
Three lines are drawn parallel to the sides of a triangle through a point inside it, dividing the triangle into six parts: three triangles and three quadrilaterals. The areas of all three inner triangles are equal. Determine the range within which the ratio of the area of each inner triangle to the area of the original... | 1/9 | 0 | 8,111.9375 | -1 | 8,111.9375 | |
Given vectors $a = (2, -1, 3)$, $b = (-1, 4, -2)$, and $c = (7, 5, \lambda)$, if vectors $a$, $b$, and $c$ are coplanar, the real number $\lambda$ equals ( ). | \frac{65}{9} | 0 | 3,748.8125 | -1 | 3,748.8125 | |
For any real number $x$, the symbol $\lfloor x \rfloor$ represents the largest integer not exceeding $x$. Evaluate the expression $\lfloor \log_{2}1 \rfloor + \lfloor \log_{2}2 \rfloor + \lfloor \log_{2}3 \rfloor + \ldots + \lfloor \log_{2}1023 \rfloor + \lfloor \log_{2}1024 \rfloor$. | 8204 | 0.4375 | 6,752.375 | 5,537.571429 | 7,697.222222 | |
Let $PROBLEMZ$ be a regular octagon inscribed in a circle of unit radius. Diagonals $MR$ , $OZ$ meet at $I$ . Compute $LI$ . | \sqrt{2} | 0.75 | 6,680.625 | 6,176.833333 | 8,192 | |
The sides of a triangle are all integers, and the longest side is 11. Calculate the number of such triangles. | 36 | 0.5 | 7,363.125 | 6,534.25 | 8,192 | |
There are four pairs of real numbers $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, and $(x_4,y_4)$ that satisfy both $x^3 - 3xy^2 = 2017$ and $y^3 - 3x^2y = 2016$. Compute the product $\left(1-\frac{x_1}{y_1}\right)\left(1-\frac{x_2}{y_2}\right)\left(1-\frac{x_3}{y_3}\right)\left(1-\frac{x_4}{y_4}\right)$. | \frac{-1}{1008} | 0 | 8,174.1875 | -1 | 8,174.1875 | |
Given $\sin (x-\frac{5π}{12})=\frac{1}{3}$, find $\cos (\frac{2021π}{6}-2x)$. | \frac{7}{9} | 0.9375 | 5,222.375 | 5,072.066667 | 7,477 | |
We wrote letters to ten of our friends and randomly placed the letters into addressed envelopes. What is the probability that exactly 5 letters will end up with their intended recipients? | 0.0031 | 0 | 4,463.125 | -1 | 4,463.125 | |
Let the function $f(x) = (\sin x + \cos x)^2 - \sqrt{3}\cos 2x$.
(Ⅰ) Find the smallest positive period of $f(x)$;
(Ⅱ) Find the maximum value of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$ and the corresponding value of $x$ when the maximum value is attained. | \frac{5\pi}{12} | 1 | 3,765.125 | 3,765.125 | -1 | |
Find the relationship between \(\arcsin \cos \arcsin x\) and \(\arccos \sin \arccos x\). | \frac{\pi}{2} | 0.5 | 5,811.0625 | 4,836.25 | 6,785.875 | |
Johan has a large number of identical cubes. He has made a structure by taking a single cube and then sticking another cube to each face. He wants to make an extended structure in the same way so that each face of the current structure will have a cube stuck to it. How many extra cubes will he need to complete his exte... | 18 | 0 | 7,771.5625 | -1 | 7,771.5625 | |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be unit vectors such that
\[\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \frac{\mathbf{b} + \mathbf{c}}{\sqrt{2}},\]and such that $\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}$ is a linearly independent set.
Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees. | 135^\circ | 0.8125 | 2,375.125 | 2,503.846154 | 1,817.333333 | |
For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\frac{n}{180}$ terminate? | 20 | 0.875 | 5,169.9375 | 4,738.214286 | 8,192 | |
In the parallelogram $\mathrm{ABCD}$, points $\mathrm{E}$ and $\mathrm{F}$ lie on $\mathrm{AD}$ and $\mathrm{AB}$ respectively. Given that the area of $S_{A F I E} = 49$, the area of $\triangle B G F = 13$, and the area of $\triangle D E H = 35$, find the area of $S_{G C H I}$. | 97 | 0 | 8,192 | -1 | 8,192 | |
Rhombus $ABCD$ is inscribed in rectangle $WXYZ$ such that vertices $A$, $B$, $C$, and $D$ are on sides $\overline{WX}$, $\overline{XY}$, $\overline{YZ}$, and $\overline{ZW}$, respectively. It is given that $WA=12$, $XB=9$, $BD=15$, and the diagonal $AC$ of rhombus equals side $XY$ of the rectangle. Calculate the perime... | 66 | 0 | 7,957.0625 | -1 | 7,957.0625 | |
Let $z$ and $w$ be complex numbers such that $|2z - w| = 25$, $|z + 2w| = 5$, and $|z + w| = 2$. Find $|z|$. | 9 | 0.8125 | 4,832.8125 | 4,525.692308 | 6,163.666667 | |
Find the number of ordered triples of sets $(T_1, T_2, T_3)$ such that
1. each of $T_1, T_2$ , and $T_3$ is a subset of $\{1, 2, 3, 4\}$ ,
2. $T_1 \subseteq T_2 \cup T_3$ ,
3. $T_2 \subseteq T_1 \cup T_3$ , and
4. $T_3\subseteq T_1 \cup T_2$ . | 625 | 0.4375 | 6,636.875 | 5,177.714286 | 7,771.777778 | |
Let $ABC$ be a right triangle with hypotenuse $AC$. Let $B^{\prime}$ be the reflection of point $B$ across $AC$, and let $C^{\prime}$ be the reflection of $C$ across $AB^{\prime}$. Find the ratio of $[BCB^{\prime}]$ to $[BC^{\prime}B^{\prime}]$. | 1 | Since $C, B^{\prime}$, and $C^{\prime}$ are collinear, it is evident that $[BCB^{\prime}]=\frac{1}{2}[BCC^{\prime}]$. It immediately follows that $[BCB^{\prime}]=[BC^{\prime}B^{\prime}]$. Thus, the ratio is 1. | 0.3125 | 7,632.4375 | 6,401.4 | 8,192 |
12 students are standing in two rows, with 4 in the front row and 8 in the back row. Now, 2 students from the back row are to be selected to stand in the front row. If the relative order of the other students remains unchanged, the number of different rearrangement methods is ______. | 28 | 0 | 6,264.3125 | -1 | 6,264.3125 | |
The real value of $x$ such that $\frac{81^{x-2}}{9^{x-2}} = 27^{3x+2}$. | -\frac{10}{7} | 1 | 1,915.0625 | 1,915.0625 | -1 | |
January 1st of a certain non-leap year fell on a Saturday. How many Fridays are there in this year? | 52 | 0.3125 | 7,672.25 | 6,528.8 | 8,192 | |
In a cylinder with a base radius of 6, there are two spheres, each with a radius of 6, and the distance between their centers is 13. If a plane is tangent to these two spheres and intersects the cylindrical surface forming an ellipse, what is the sum of the lengths of the major axis and minor axis of this ellipse? | 25 | 0.0625 | 8,155.625 | 7,610 | 8,192 | |
The four points $A(-4,0), B(0,-4), X(0,8),$ and $Y(14,k)$ are grouped on the Cartesian plane. If segment $AB$ is parallel to segment $XY$ what is the value of $k$? | -6 | 1 | 1,535.0625 | 1,535.0625 | -1 | |
In the geometric sequence ${a_{n}}$, $a_{n}=9$, $a_{5}=243$, find the sum of the first 4 terms. | 120 | 0.5 | 6,200.875 | 4,693.25 | 7,708.5 | |
Which of the following, when rounded to the nearest hundredth, does not round to 65.14?
A) 65.141
B) 65.138
C) 65.1339999
D) 65.1401
E) 65.14444
Your answer should be a letter: A, B, C, D, or E. | C | 0.8125 | 736 | 733.153846 | 748.333333 | |
What are the solutions for \(5x^2 + 6 = 2x - 15\)? Express your solutions in the form \(x = a \pm b i,\) and compute \(a + b^2\) presented as a fraction. | \frac{109}{25} | 0.75 | 2,324.5625 | 2,324.75 | 2,324 | |
Find all values of $z$ such that $z^4 - 4z^2 + 3 = 0$. Enter all the solutions, separated by commas. | -\sqrt{3},-1,1,\sqrt{3} | 0.0625 | 2,569.8125 | 2,074 | 2,602.866667 | |
The sequence $6075, 2025, 675 \ldots$, is made by repeatedly dividing by 3. How many integers are in this sequence? | 6 | 1 | 2,591.4375 | 2,591.4375 | -1 | |
Problem
Find all pairs of primes $(p,q)$ for which $p-q$ and $pq-q$ are both perfect squares. | \((p, q) = (3, 2)\) | We first consider the case where one of $p,q$ is even. If $p=2$ , $p-q=0$ and $pq-q=2$ which doesn't satisfy the problem restraints. If $q=2$ , we can set $p-2=x^2$ and $2p-2=y^2$ giving us $p=y^2-x^2=(y+x)(y-x)$ . This forces $y-x=1$ so $p=2x+1\rightarrow 2x+1=x^2+2 \rightarrow x=1$ giving us the solution $(p,q)=(3,2)... | 0 | 8,192 | -1 | 8,192 |
In the decimal representation of an even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) are used, and the digits may repeat. It is known that the sum of the digits of the number \( 2M \) equals 39, and the sum of the digits of the number \( M / 2 \) equals 30. What values can the sum of the digits of ... | 33 | 0 | 8,192 | -1 | 8,192 | |
18. Given the function $f(x)=x^3+ax^2+bx+5$, the equation of the tangent line to the curve $y=f(x)$ at the point $x=1$ is $3x-y+1=0$.
Ⅰ. Find the values of $a$ and $b$;
Ⅱ. Find the maximum and minimum values of $y=f(x)$ on the interval $[-3,1]$. | \frac{95}{27} | 0.75 | 3,648.8125 | 3,686.166667 | 3,536.75 | |
Given $f(x)=\cos x(\sqrt{3}\sin x-\cos x)+\frac{3}{2}$.
$(1)$ Find the interval on which $f(x)$ is monotonically decreasing on $[0,\pi]$.
$(2)$ If $f(\alpha)=\frac{2}{5}$ and $\alpha\in(\frac{\pi}{3},\frac{5\pi}{6})$, find the value of $\sin 2\alpha$. | \frac{-3\sqrt{3} - 4}{10} | 0 | 7,036.8125 | -1 | 7,036.8125 | |
The perimeter of a rectangle is 56 meters. The ratio of its length to its width is 4:3. What is the length in meters of a diagonal of the rectangle? | 20 | 1 | 1,172.8125 | 1,172.8125 | -1 | |
The scores (in points) of the 15 participants in the final round of a math competition are as follows: $56$, $70$, $91$, $98$, $79$, $80$, $81$, $83$, $84$, $86$, $88$, $90$, $72$, $94$, $78$. What is the $80$th percentile of these 15 scores? | 90.5 | 0 | 6,665.125 | -1 | 6,665.125 | |
Let the functions \( f(\alpha, x) \) and \( g(\alpha) \) be defined as
\[ f(\alpha, x)=\frac{\left(\frac{x}{2}\right)^{\alpha}}{x-1} \]
\[ g(\alpha)=\left.\frac{d^{4} f}{d x^{4}}\right|_{x=2} \]
Then \( g(\alpha) \) is a polynomial in \( \alpha \). Find the leading coefficient of \( g(\alpha) \). | 1/16 | 0 | 8,019.4375 | -1 | 8,019.4375 | |
Let $x$ , $y$ , $z$ be positive integers satisfying $x<y<z$ and $x+xy+xyz=37$ . Find the greatest possible value of $x+y+z$ . | 20 | 0.875 | 3,950.5 | 3,638.714286 | 6,133 | |
Find the area of the shaded region. | 6\dfrac{1}{2} | 1. **Identify the equations of the lines**:
- For the first line, using the points (0,4) and (8,1), the slope $m$ is calculated as:
\[
m = \frac{1-4}{8-0} = -\frac{3}{8}
\]
Thus, the equation of the line is:
\[
y_1 = -\frac{3}{8}x + 4
\]
- For the second line, using the points... | 0 | 7,914.375 | -1 | 7,914.375 |
If each of the variables represents a different digit, what is the value of $a+b+c+d$?
[asy]
label("$a$",(1,0),E);
label("$b$",(2,0),E);
label("$c$",(3,0),E);
label("$d$",(1,-1),E);
label("$c$",(2,-1),E);
label("$a$",(3,-1),E);
label("+",(-2,-1),E);
draw((-2.1,-1.4)--(4.1,-1.4),linewidth(0.5));
label("1",(0,-2),E);
fo... | 18 | 0.25 | 7,317.8125 | 5,575 | 7,898.75 | |
A high school has three math teachers. To facilitate students, math teachers are scheduled for duty from Monday to Friday, with two teachers on duty on Monday. If each teacher is on duty for two days a week, then there are ________ possible duty schedules for a week. | 36 | 0.0625 | 7,551.0625 | 6,327 | 7,632.666667 | |
90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99 = | 945 | We are given the sum of consecutive integers from $90$ to $99$. We can calculate this sum using the formula for the sum of an arithmetic series, or by simplifying the expression directly.
1. **Identify the series**: The series is $90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99$.
2. **Calculate the number of terms ($... | 1 | 435.5 | 435.5 | -1 |
Add $49.213$ to $27.569$, and round your answer to the nearest whole number. | 77 | 1 | 811.1875 | 811.1875 | -1 | |
How many of the twelve pentominoes pictured below have at least one line of reflectional symmetry?
[asy] unitsize(5mm); defaultpen(linewidth(1pt)); draw(shift(2,0)*unitsquare); draw(shift(2,1)*unitsquare); draw(shift(2,2)*unitsquare); draw(shift(1,2)*unitsquare); draw(shift(0,2)*unitsquare); draw(shift(2,4)*unitsquare)... | 6 | To solve this problem, we need to identify which of the twelve pentominoes have at least one line of reflectional symmetry. Reflectional symmetry in a shape means that there is at least one line (axis of symmetry) along which the shape can be folded or reflected onto itself perfectly.
1. **Identify each pentomino**: W... | 0.4375 | 5,134.4375 | 4,961.571429 | 5,268.888889 |
For real numbers $t,$ the point of intersection of the lines $tx - 2y - 3t = 0$ and $x - 2ty + 3 = 0$ is plotted. All the plotted points lie on what kind of curve?
(A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola
Enter the letter of the correct option. | \text{(E)} | 0 | 4,430 | -1 | 4,430 | |
The sides of this parallelogram measure 7,9, $8y-1$ and $2x+3$ units, consecutively. What is the value of $x+y$?
[asy]draw((0,0)--(21,0)--(30,25)--(9,25)--cycle);
label("$8y-1$",(10,0),S);
label("9",(25.5,12.5),E);
label("7",(19.5,25),N);
label("$2x+3$",(4.5,12.5),W);
[/asy] | 4 | 0.6875 | 1,555.5 | 1,873.636364 | 855.6 | |
A man bought a number of ping-pong balls where a 16% sales tax is added. If he did not have to pay tax, he could have bought 3 more balls for the same amount of money. If \( B \) is the total number of balls that he bought, find \( B \). | 18.75 | 0.3125 | 7,669.6875 | 6,913.4 | 8,013.454545 | |
Let \( a, b, c \) be real numbers such that \( 9a^2 + 4b^2 + 25c^2 = 1 \). Find the maximum value of
\[ 3a + 4b + 5c. \] | \sqrt{6} | 0.6875 | 5,865.25 | 4,807.636364 | 8,192 | |
The number of different arrangements possible for 6 acts, if the original sequence of the 4 acts remains unchanged. | 30 | 0.5625 | 4,988.6875 | 4,225.777778 | 5,969.571429 | |
In the tetrahedron $P-ABC$, $\Delta ABC$ is an equilateral triangle, and $PA=PB=PC=3$, $PA \perp PB$. The volume of the circumscribed sphere of the tetrahedron $P-ABC$ is __________. | \frac{27\sqrt{3}\pi}{2} | 0 | 6,618.1875 | -1 | 6,618.1875 | |
Find the roots of $z^2 - z = 5 - 5i.$
Enter the roots, separated by commas. | 3 - i, -2 + i | 0.125 | 2,802.9375 | 2,421 | 2,857.5 | |
$\frac{1-\frac{1}{3}}{1-\frac{1}{2}} =$ | $\frac{4}{3}$ | 1. **Simplify the numerator and the denominator separately:**
\[
1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}
\]
\[
1 - \frac{1}{2} = \frac{2}{2} - \frac{1}{2} = \frac{1}{2}
\]
2. **Form the fraction and simplify:**
\[
\frac{1-\frac{1}{3}}{1-\frac{1}{2}} = \frac{\frac{2}{3}}{\frac{... | 0 | 2,253.25 | -1 | 2,253.25 |
Compute $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}^{2018}.$ | \begin{pmatrix} 1 & 0 \\ 2018 & 1 \end{pmatrix} | 0.9375 | 4,495.25 | 4,248.8 | 8,192 | |
Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill'... | 4 | To solve this problem, we need to count the number of paths from Jack's house to Jill's house, avoiding the dangerous intersection. We can represent Jack's house as the origin (0,0) on a coordinate grid and Jill's house as the point (3,2). The dangerous intersection is at (1,1).
1. **Total Paths Without Restriction**:... | 0.5 | 6,617.25 | 5,176 | 8,058.5 |
Given that the sum of the first $n$ terms ($S_n$) of the sequence $\{a_n\}$ satisfies $S_n = 2a_n - 1$ ($n \in \mathbb{N}^*$).
(1) Find the general term formula of the sequence $\{a_n\}$;
(2) If the sequence $\{b_n\}$ satisfies $b_n = 1 + \log_2 a_n$,
(I) Find the sum of the first $n$ terms ($T_n$) of the sequence $... | \frac{13}{3} | 0.6875 | 6,004.9375 | 5,646.909091 | 6,792.6 | |
Determine the maximum value of the sum
\[S = \sum_{n=1}^\infty \frac{n}{2^n} (a_1 a_2 \cdots a_n)^{1/n}\]
over all sequences $a_1, a_2, a_3, \cdots$ of nonnegative real numbers satisfying
\[\sum_{k=1}^\infty a_k = 1.\] | 2/3 | The answer is $2/3$.
By AM-GM, we have
\begin{align*}
2^{n+1}(a_1\cdots a_n)^{1/n} &= \left((4a_1)(4^2a_2)\cdots (4^na_n)\right)^{1/n}\\
& \leq \frac{\sum_{k=1}^n (4^k a_k)}{n}.
\end{align*}
Thus
\begin{align*}
2S &\leq \sum_{n=1}^\infty \frac{\sum_{k=1}^n (4^k a_k)}{4^n} \\
&= \sum_{n=1}^\infty \sum_{k=1}^n (4^{k-n}... | 0 | 8,180.75 | -1 | 8,180.75 |
In the quadrilateral $ABCD$ , the angles $B$ and $D$ are right . The diagonal $AC$ forms with the side $AB$ the angle of $40^o$ , as well with side $AD$ an angle of $30^o$ . Find the acute angle between the diagonals $AC$ and $BD$ . | 80 | 0.0625 | 8,119.8125 | 8,192 | 8,115 | |
What is the probability that in a randomly chosen arrangement of the numbers and letters in "HMMT2005," one can read either "HMMT" or "2005" from left to right? | 23/144 | To read "HMMT," there are $\binom{8}{4}$ ways to place the letters, and $\frac{4!}{2}$ ways to place the numbers. Similarly, there are $\binom{8}{4} \frac{4!}{2}$ arrangements where one can read "2005." The number of arrangements in which one can read both is just $\binom{8}{4}$. The total number of arrangements is $\f... | 0 | 6,959.375 | -1 | 6,959.375 |
Bernardo randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Be... | \frac{37}{56} | We will solve this problem by considering two cases based on whether Bernardo picks the number 9 or not.
#### Case 1: Bernardo picks 9.
If Bernardo picks a 9, then any three-digit number he forms will be larger than any three-digit number Silvia can form, since Silvia's largest possible digit is 8. The probability tha... | 0 | 7,856.375 | -1 | 7,856.375 |
Solve for the largest value of $x$ such that $5(9x^2+9x+10) = x(9x-40).$ Express your answer as a simplified common fraction. | -\dfrac{10}{9} | 0.875 | 3,066.125 | 2,998.142857 | 3,542 | |
Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15}\dotsm\frac{3n+6}{3n}\dotsm\frac{3003}{2997}.\] | 1001 | 0.25 | 4,794.1875 | 2,658.5 | 5,506.083333 | |
Add $5_7 + 16_7.$ Express your answer in base $7.$ | 24_7 | 1 | 2,284.5625 | 2,284.5625 | -1 | |
A circular disc with diameter $D$ is placed on an $8 \times 8$ checkerboard with width $D$ so that the centers coincide. The number of checkerboard squares which are completely covered by the disc is | 32 | 1. **Understanding the Problem**: We are given a circular disc with diameter $D$ placed on an $8 \times 8$ checkerboard such that the centers of both the disc and the checkerboard coincide. We need to find the number of squares completely covered by the disc.
2. **Checkerboard and Disc Dimensions**: The checkerboard h... | 0 | 8,192 | -1 | 8,192 |
The four circles in the diagram intersect to divide the interior into 8 parts. Fill these 8 parts with the numbers 1 through 8 such that the sum of the 3 numbers within each circle is equal. Calculate the maximum possible sum and provide one possible configuration. | 15 | 0 | 7,889.8125 | -1 | 7,889.8125 | |
Given that all four roots of the equation $2x^4 + mx^2 + 8 = 0$ are integers, then $m = \ $, and the polynomial $2x^4 + mx^2 + 8$ can be factored into $\ $. | -10 | 0.25 | 5,476.5 | 5,989.5 | 5,305.5 | |
There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent. Find... | 310 | 0 | 7,956.25 | -1 | 7,956.25 | |
There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn. | 7 |
Given 7 arbitrary points in the plane, we need to determine the maximum number of circles that can be drawn through every 4 possible concyclic points.
To solve this, we consider the combinatorial aspect of selecting 4 points out of 7. The number of ways to choose 4 points from 7 is given by the binomial coefficient:
... | 0 | 7,861.5625 | -1 | 7,861.5625 |
Given that Liz had no money initially, and her friends gave her one-sixth, one-fifth, and one-fourth of their respective amounts, find the fractional part of the group's total money that Liz has. | \frac{1}{5} | 0.1875 | 5,626.75 | 5,557.666667 | 5,642.692308 | |
Given a tetrahedron $ABCD$, with $AD$ perpendicular to plane $BCD$, $BC$ perpendicular to $CD$, $AD=2$, $BD=4$, calculate the surface area of the circumscribed sphere of tetrahedron $ABCD$. | 20\pi | 0.625 | 6,382.375 | 5,376 | 8,059.666667 | |
A hemisphere-shaped bowl with radius 1 foot is filled full with chocolate. All of the chocolate is then evenly distributed between 36 cylindrical molds, each having a height equal to their diameter. What is the diameter of each cylinder? | \frac{2^{1/3}}{3} | 0 | 4,278.625 | -1 | 4,278.625 | |
A lieutenant is training recruits in marching drills. Upon arriving at the parade ground, he sees that all the recruits are arranged in several rows, with each row having the same number of soldiers, and that the number of soldiers in each row is 5 more than the number of rows. After finishing the drills, the lieutenan... | 24 | 0.5 | 7,853.5625 | 7,515.125 | 8,192 | |
An infinite geometric series has sum 2005. A new series, obtained by squaring each term of the original series, has 10 times the sum of the original series. The common ratio of the original series is $\frac mn$ where $m$ and $n$ are relatively prime integers. Find $m+n.$ | 802 | Let's call the first term of the original geometric series $a$ and the common ratio $r$, so $2005 = a + ar + ar^2 + \ldots$. Using the sum formula for infinite geometric series, we have $\;\;\frac a{1 -r} = 2005$. Then we form a new series, $a^2 + a^2 r^2 + a^2 r^4 + \ldots$. We know this series has sum $20050 = \frac{... | 0.8125 | 4,194.75 | 3,272.307692 | 8,192 |
Given that the length of the arc of a sector is $\pi$ and the radius is 3, the radian measure of the central angle of the sector is ______, and the area of the sector is ______. | \frac{3\pi}{2} | 1 | 1,546.75 | 1,546.75 | -1 | |
Given that the square of a number $y^2$ is the sum of squares of 11 consecutive integers, find the minimum value of $y^2$. | 121 | 0.25 | 7,311.1875 | 6,972 | 7,424.25 | |
Bully Vasya loves to run on the escalator in the subway, and he runs down twice as fast as up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is moving down, it takes Vasya 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down on an escal... | 324 | 0.125 | 7,937.25 | 7,700.5 | 7,971.071429 | |
Rhombus $PQRS$ is inscribed in rectangle $ABCD$ so that vertices $P$, $Q$, $R$, and $S$ are interior points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$, respectively. It is given that $PB=15$, $BQ=20$, $PR=30$, and $QS=40$. Let $m/n$, in lowest terms, denote the perimeter of $ABCD$. ... | 677 | 0.375 | 7,157.3125 | 5,432.833333 | 8,192 | |
A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $m=10$, what is the value of $n$?
[asy] size(200); defaultpen(linewidth(0.8)); draw(unitsquare); path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)... | 5 | 1. **Determine the interior angle of the decagon**:
The formula for the interior angle of a regular polygon with $m$ sides is given by:
\[
\text{Interior angle} = \frac{(m-2) \times 180^\circ}{m}
\]
For a decagon ($m=10$), the interior angle is:
\[
\text{Interior angle} = \frac{(10-2) \times 180^... | 0.1875 | 7,150.1875 | 4,093.333333 | 7,855.615385 |
Given that $|\cos\theta|= \frac {1}{5}$ and $\frac {5\pi}{2}<\theta<3\pi$, find the value of $\sin \frac {\theta}{2}$. | -\frac{\sqrt{15}}{5} | 0 | 5,288.6875 | -1 | 5,288.6875 | |
A convex polyhedron $Q$ has vertices $V_1,V_2,\ldots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\ldots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. ... | 300 |
#### Step-by-step Analysis:
1. **Understanding the Problem:**
- A convex polyhedron $Q$ has $n$ vertices and $100$ edges.
- Each vertex $V_k$ is associated with a plane $P_k$ that cuts all edges meeting at $V_k$.
- The cuts do not intersect each other inside or on $Q$.
- The result of these cuts is $n$ py... | 0.3125 | 7,700.0625 | 6,674.8 | 8,166.090909 |
For how many integers $n$ is $\frac n{20-n}$ the square of an integer? | 4 | We are given the expression $\frac{n}{20-n}$ and need to determine for how many integers $n$ this expression is a perfect square.
1. **Examine the domain of $n$:**
- If $n < 0$ or $n > 20$, the fraction $\frac{n}{20-n}$ is negative, and thus cannot be a perfect square since squares of real numbers are non-negative.... | 0.875 | 6,515.875 | 6,276.428571 | 8,192 |
The points $A$, $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$. It is given that $AB=13$, $BC=14$, $CA=15$, and that the distance from $O$ to $\triangle ABC$ is $\frac{m\sqrt{n}}k$, where $m$, $n$, and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by t... | 118 | Let $D$ be the foot of the perpendicular from $O$ to the plane of $ABC$. By the Pythagorean Theorem on triangles $\triangle OAD$, $\triangle OBD$ and $\triangle OCD$ we get:
\[DA^2=DB^2=DC^2=20^2-OD^2\]
It follows that $DA=DB=DC$, so $D$ is the circumcenter of $\triangle ABC$.
By Heron's Formula the area of $\triangl... | 1 | 3,477.3125 | 3,477.3125 | -1 |
Inside a circle \(\omega\) there is a circle \(\omega_{1}\) touching \(\omega\) at point \(K\). Circle \(\omega_{2}\) touches circle \(\omega_{1}\) at point \(L\) and intersects circle \(\omega\) at points \(M\) and \(N\). It turns out that the points \(K, L,\) and \(M\) are collinear. Find the radius of circle \(\omeg... | 11 | 0.0625 | 7,893.75 | 5,677 | 8,041.533333 | |
Let $a_{10} = 10$, and for each positive integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least positive $n > 10$ such that $a_n$ is a multiple of $99$. | 45 | We just notice that $100 \equiv 1 \pmod{99}$, so we are just trying to find $10 + 11 + 12 + \cdots + n$ modulo $99$, or $\dfrac{n(n+1)}{2} - 45$ modulo $99$. Also, the sum to $44$ is divisible by $99$, and is the first one that is. Thus, if we sum to $45$ the $45$ is cut off and thus is just a sum to $44$.
Without che... | 0.3125 | 7,426.5 | 6,259 | 7,957.181818 |
Given that line $l$ intersects circle $C$: $x^2+y^2+2x-4y+a=0$ at points $A$ and $B$, and the midpoint of chord $AB$ is $P(0,1)$.
(I) If the radius of circle $C$ is $\sqrt{3}$, find the value of the real number $a$;
(II) If the length of chord $AB$ is $6$, find the value of the real number $a$;
(III) When $a=1$, circle... | \sqrt{11} | 0.75 | 5,848.9375 | 5,067.916667 | 8,192 | |
The decimal representation of $m/n,$ where $m$ and $n$ are relatively prime positive integers and $m < n,$ contains the digits $2, 5$, and $1$ consecutively, and in that order. Find the smallest value of $n$ for which this is possible.
| 127 | 0 | 8,192 | -1 | 8,192 | |
The South China tiger is a first-class protected animal in our country. To save the species from the brink of extinction, the country has established a South China tiger breeding base. Due to scientific artificial cultivation, the relationship between the number of South China tigers $y$ (individuals) and the breeding ... | 46 | 0.5625 | 3,333.25 | 3,225.555556 | 3,471.714286 | |
Given that in triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Angle $B$ is obtuse. Let the area of $\triangle ABC$ be $S$. If $4bS=a(b^{2}+c^{2}-a^{2})$, then the maximum value of $\sin A + \sin C$ is ____. | \frac{9}{8} | 0.625 | 6,704.1875 | 5,811.5 | 8,192 | |
In response to the call of the commander, 55 soldiers came: archers and swordsmen. All of them were dressed either in golden or black armor. It is known that swordsmen tell the truth when wearing black armor and lie when wearing golden armor, while archers do the opposite.
- To the question "Are you wearing golden arm... | 22 | 0.125 | 7,601.375 | 6,930.5 | 7,697.214286 | |
A row consists of 10 chairs, but chair #5 is broken and cannot be used. Mary and James each sit in one of the available chairs, choosing their seats at random from the remaining chairs. What is the probability that they don't sit next to each other? | \frac{5}{6} | 0.0625 | 6,751.875 | 3,371 | 6,977.266667 | |
Given points P(-2, -2), Q(0, -1), and a point R(2, m) is chosen such that PR + PQ is minimized. What is the value of the real number $m$? | -2 | 0.125 | 7,465.5625 | 6,835 | 7,555.642857 | |
Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\tfrac{m}{n},$ where $m$ and $n$ are ... | 77 | 0.5625 | 7,227.8125 | 6,593.444444 | 8,043.428571 | |
Compute
\[\prod_{n = 1}^{15} \frac{n + 4}{n}.\] | 11628 | 0 | 5,488.8125 | -1 | 5,488.8125 | |
Let $f(x) = \frac{x + 6}{x}.$ The sequence $(f_n)$ of functions is defined by $f_1 = f$ and
\[f_n = f \circ f_{n - 1}\]for all $n \ge 2.$ For example,
\[f_2(x) = f(f(x)) = \frac{\frac{x + 6}{x} + 6}{\frac{x + 6}{x}} = \frac{7x + 6}{x + 6}\]and
\[f_3(x) = f(f_2(x)) = \frac{\frac{7x + 6}{x + 6} + 6}{\frac{7x + 6}{x + 6... | 2 | 0.4375 | 7,455.125 | 6,507.714286 | 8,192 | |
A light pulse starts at a corner of a reflective square. It bounces around inside the square, reflecting off of the square's perimeter $n$ times before ending in a different corner. The path of the light pulse, when traced, divides the square into exactly 2021 regions. Compute the smallest possible value of $n$. | 129 | The main claim is that if the light pulse reflects vertically (on the left/right edges) $a$ times and horizontally $b$ times, then $\operatorname{gcd}(a+1, b+1)=1$, and the number of regions is $\frac{(a+2)(b+2)}{2}$. This claim can be conjectured by looking at small values of $a$ and $b$; we give a full proof at the e... | 0 | 7,915.125 | -1 | 7,915.125 |
How many distinct three-digit numbers can be written with the digits $1$, $2$, $3$ and $4$ if no digit may be used more than once in a three-digit number? | 24 | 1 | 1,716.0625 | 1,716.0625 | -1 | |
Let $ABCD$ be a parallelogram and let $\overrightarrow{AA^\prime}$, $\overrightarrow{BB^\prime}$, $\overrightarrow{CC^\prime}$, and $\overrightarrow{DD^\prime}$ be parallel rays in space on the same side of the plane determined by $ABCD$. If $AA^{\prime} = 10$, $BB^{\prime}= 8$, $CC^\prime = 18$, and $DD^\prime = 22$ a... | 1 | 1. **Assume the Configuration**: Let $ABCD$ be a unit square with coordinates $A(0,0,0)$, $B(0,1,0)$, $C(1,1,0)$, and $D(1,0,0)$. Assume that the rays $\overrightarrow{AA'}$, $\overrightarrow{BB'}$, $\overrightarrow{CC'}$, and $\overrightarrow{DD'}$ extend in the positive $z$-direction from the plane of $ABCD$.
2. **D... | 0.375 | 7,078.6875 | 5,448.333333 | 8,056.9 |
Express $43210_{6}-3210_{7}$ in base 10. | 4776 | 0.875 | 4,245.1875 | 3,758.142857 | 7,654.5 | |
The value of
\[\frac{n}{2} + \frac{18}{n}\]is smallest for which positive integer $n$? | 6 | 1 | 2,298.5625 | 2,298.5625 | -1 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a > b$, $a=5$, $c=6$, and $\sin B=\dfrac{3}{5}$.
1. Find the values of $b$ and $\sin A$.
2. Find the value of $\sin (2A+\dfrac{\pi}{4})$. | \dfrac {7 \sqrt {2}}{26} | 0 | 4,863.5 | -1 | 4,863.5 | |
Let $f$ be a non-constant polynomial such that
\[f(x - 1) + f(x) + f(x + 1) = \frac{[f(x)]^2}{2013x}\]for all nonzero real numbers $x.$ Find the sum of all possible values of $f(1).$ | 6039 | 0.9375 | 4,664.75 | 4,429.6 | 8,192 |
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