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 triangular pyramid $P-ABC$ where $PA\bot $ plane $ABC$, $PA=AB=2$, and $\angle ACB=30^{\circ}$, find the surface area of the circumscribed sphere of the triangular pyramid $P-ABC$. | 20\pi | 0.0625 | 7,846.625 | 5,779 | 7,984.466667 | |
Jeff rotates spinners $P$, $Q$ and $R$ and adds the resulting numbers. What is the probability that his sum is an odd number? | 1/3 | 1. **Identify the possible outcomes for each spinner:**
- Spinner $P$ has numbers 1, 2, 3. Thus, it has 1 even number (2) and 2 odd numbers (1, 3).
- Spinner $Q$ has numbers 2, 4, 6. All numbers are even.
- Spinner $R$ has numbers 1, 3, 5. All numbers are odd.
2. **Determine the conditions for an odd sum:**
... | 0 | 4,901.5625 | -1 | 4,901.5625 |
Paul wrote the list of all four-digit numbers such that the hundreds digit is $5$ and the tens digit is $7$ . For example, $1573$ and $7570$ are on Paul's list, but $2754$ and $571$ are not. Find the sum of all the numbers on Pablo's list. $Note$ . The numbers on Pablo's list cannot start with zero. | 501705 | 0.5 | 6,435.75 | 4,679.5 | 8,192 | |
If $q(x) = x^5 - 4x^3 + 5$, then find the coefficient of the $x^3$ term in the polynomial $(q(x))^2$. | 40 | 0 | 3,397.625 | -1 | 3,397.625 | |
A teacher asks her class what the value of $\left(5^2-4^2\right)^3$ is. What is the value of the expression? | 729 | 1 | 1,592.25 | 1,592.25 | -1 | |
Find the number of ways to pave a $1 \times 10$ block with tiles of sizes $1 \times 1, 1 \times 2$ and $1 \times 4$, assuming tiles of the same size are indistinguishable. It is not necessary to use all the three kinds of tiles. | 169 | 0.375 | 7,523.6875 | 7,027 | 7,821.7 | |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\sin ^{2}A+\cos ^{2}B+\cos ^{2}C=2+\sin B\sin C$.<br/>$(1)$ Find the measure of angle $A$;<br/>$(2)$ If $a=3$, the angle bisector of $\angle BAC$ intersects $BC$ at point $D$, find the maximum length of s... | \frac{\sqrt{3}}{2} | 0 | 8,111.875 | -1 | 8,111.875 | |
Let $a$ and $b$ be positive real numbers. Determine the minimum possible value of $$\sqrt{a^{2}+b^{2}}+\sqrt{(a-1)^{2}+b^{2}}+\sqrt{a^{2}+(b-1)^{2}}+\sqrt{(a-1)^{2}+(b-1)^{2}}$$ | 2 \sqrt{2} | Let $A B C D$ be a square with $A=(0,0), B=(1,0), C=(1,1), D=(0,1)$, and $P$ be a point in the same plane as $A B C D$. Then the desired expression is equivalent to $A P+B P+C P+D P$. By the triangle inequality, $A P+C P \geq A C$ and $B P+D P \geq B D$, so the minimum possible value is $A C+B D=2 \sqrt{2}$. This is ac... | 0.3125 | 7,424.375 | 5,735.6 | 8,192 |
Among the following 4 propositions, the correct one is:
(1) If a solid's three views are completely identical, then the solid is a cube;
(2) If a solid's front view and top view are both rectangles, then the solid is a cuboid;
(3) If a solid's three views are all rectangles, then the solid is a cuboid;
(4) If a... | (3) | 0.0625 | 6,441.1875 | 8,192 | 6,324.466667 | |
In a subject test, the average score of Xiaofang's four subjects: Chinese, Mathematics, English, and Science, is 88. The average score of the first two subjects is 93, and the average score of the last three subjects is 87. What is Xiaofang's English test score? | 95 | 0 | 623.6875 | -1 | 623.6875 | |
The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer.
For how many integer Fahren... | 539 | Examine $F - 32$ modulo 9.
If $F - 32 \equiv 0 \pmod{9}$, then we can define $9x = F - 32$. This shows that $F = \left[\frac{9}{5}\left[\frac{5}{9}(F-32)\right] + 32\right] \Longrightarrow F = \left[\frac{9}{5}(5x) + 32\right] \Longrightarrow F = 9x + 32$. This case works.
If $F - 32 \equiv 1 \pmod{9}$, then we can def... | 0 | 8,192 | -1 | 8,192 |
The opposite number of $-1 \frac{1}{2}$ is ______, its reciprocal is ______, and its absolute value is ______. | 1.5 | 0 | 519.25 | -1 | 519.25 | |
In the right triangle shown the sum of the distances $BM$ and $MA$ is equal to the sum of the distances $BC$ and $CA$.
If $MB = x, CB = h$, and $CA = d$, then $x$ equals: | \frac{hd}{2h+d} | 1. **Given Information and Equation Setup:**
We are given that the sum of the distances $BM$ and $MA$ is equal to the sum of the distances $BC$ and $CA$. This translates to the equation:
\[
BM + MA = BC + CA
\]
Substituting the given values, we have:
\[
x + \sqrt{(x+h)^2 + d^2} = h + d
\]
2. ... | 0 | 7,608.9375 | -1 | 7,608.9375 |
The length of the curve given by the parametric equations \(\left\{\begin{array}{l}x=2 \cos ^{2} \theta \\ y=3 \sin ^{2} \theta\end{array}\right.\) (where \(\theta\) is the parameter) is? | \sqrt{13} | 0.1875 | 7,445.625 | 5,312.333333 | 7,937.923077 | |
Let $a$, $b$, and $c$ be positive integers with $a \ge b \ge c$ such that
$a^2-b^2-c^2+ab=2011$ and
$a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$.
What is $a$? | 253 | 1. **Combine the given equations:**
\[
\begin{align*}
a^2 - b^2 - c^2 + ab &= 2011, \\
a^2 + 3b^2 + 3c^2 - 3ab - 2ac - 2bc &= -1997.
\end{align*}
\]
Adding these equations, we get:
\[
2a^2 + 2b^2 + 2c^2 - 4ab - 2ac - 2bc = 14.
\]
Simplifying, we have:
\[
a^2 + b^2 + c^2 - 2ab - ac... | 0.625 | 5,903.6875 | 4,530.7 | 8,192 |
How many positive, three-digit integers contain at least one $4$ as a digit but do not contain a $6$ as a digit? | 200 | 0.4375 | 6,655.125 | 4,679.142857 | 8,192 | |
Let S<sub>n</sub> be the sum of the first n terms of the arithmetic sequence {a<sub>n</sub>}, given that a<sub>7</sub> = 5 and S<sub>5</sub> = -55.
1. Find S<sub>n</sub>.
2. Let b<sub>n</sub> = $$\frac {S_{n}}{n}$$, find the sum of the first 19 terms, T<sub>19</sub>, of the sequence { $$\frac {1}{b_{n}b_{n+1}}$$}. | -\frac {1}{19} | 0.125 | 7,854.8125 | 6,031 | 8,115.357143 | |
You recently bought more than 100 eggs. The eggs are sorted in containers that can store exactly 12 eggs. However, upon inspecting the containers, you realize that two containers each hold only 11 eggs, while all the other containers hold 12 eggs. What is the smallest number of eggs you could have right now? | 106 | 1 | 3,222 | 3,222 | -1 | |
Quantities $r$ and $s$ vary inversely. When $r$ is $1200,$ $s$ is $0.35.$ What is the value of $s$ when $r$ is $2400$? Express your answer as a decimal to the nearest thousandths. | .175 | 1 | 1,509.9375 | 1,509.9375 | -1 | |
Sets $A$, $B$, and $C$, depicted in the Venn diagram, are such that the total number of elements in set $A$ is three times the total number of elements in set $B$. Their intersection has 1200 elements, and altogether, there are 4200 elements in the union of $A$, $B$, and $C$. If set $C$ intersects only with set $A$ add... | 3825 | 0.125 | 7,627.375 | 5,211.5 | 7,972.5 | |
Let $p, q, r,$ and $s$ be the roots of the polynomial $3x^4 - 8x^3 - 15x^2 + 10x - 2 = 0$. Find $pqrs$. | \frac{2}{3} | 0 | 6,360.625 | -1 | 6,360.625 | |
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of... | 238\pi | 1. **Identify the radius of the ball and adjust the radii of the arcs:**
The diameter of the ball is given as 4 inches, so the radius of the ball is half of that, which is 2 inches. When the ball rolls along the track, the center of the ball follows a path that is offset from the path defined by the edges of the tra... | 0 | 4,603 | -1 | 4,603 |
Let $A B C$ be an equilateral triangle with $A B=3$. Circle $\omega$ with diameter 1 is drawn inside the triangle such that it is tangent to sides $A B$ and $A C$. Let $P$ be a point on $\omega$ and $Q$ be a point on segment $B C$. Find the minimum possible length of the segment $P Q$. | \frac{3 \sqrt{3}-3}{2} | Let $P, Q$, be the points which minimize the distance. We see that we want both to lie on the altitude from $A$ to $B C$. Hence, $Q$ is the foot of the altitude from $A$ to $B C$ and $A Q=\frac{3 \sqrt{3}}{2}$. Let $O$, which must also lie on this line, be the center of $\omega$, and let $D$ be the point of tangency be... | 0 | 7,910.75 | -1 | 7,910.75 |
A malfunctioning thermometer shows a temperature of $+1^{\circ}$ in freezing water and $+105^{\circ}$ in the steam of boiling water. Currently, this thermometer shows $+17^{\circ}$; what is the true temperature? | 15.38 | 0.0625 | 4,811.8125 | 4,047 | 4,862.8 | |
Given that $a, b \in R^{+}$, and $a + b = 1$, find the maximum value of $- \frac{1}{2a} - \frac{2}{b}$. | -\frac{9}{2} | 1 | 5,483.5 | 5,483.5 | -1 | |
If \(0^{\circ} < \alpha < 30^{\circ}\), and \(\sin^6 \alpha + \cos^6 \alpha = \frac{7}{12}\), then \(1998 \cos \alpha = ?\) | 333 \sqrt{30} | 0.8125 | 5,303.1875 | 5,027.230769 | 6,499 | |
What is the total number of digits used when the first 2500 positive even integers are written? | 9448 | 0.625 | 5,372.3125 | 5,033 | 5,937.833333 | |
The left and right foci of the ellipse $\dfrac{x^{2}}{16} + \dfrac{y^{2}}{9} = 1$ are $F_{1}$ and $F_{2}$, respectively. There is a point $P$ on the ellipse such that $\angle F_{1}PF_{2} = 30^{\circ}$. Find the area of triangle $F_{1}PF_{2}$. | 18 - 9\sqrt{3} | 0.125 | 7,986.8125 | 7,292 | 8,086.071429 | |
Given that the sum of the first $n$ terms of a geometric sequence ${a_{n}}$ is $S_{n}=k+2( \frac {1}{3})^{n}$, find the value of the constant $k$. | -2 | 1 | 4,098 | 4,098 | -1 | |
Given that for reals $a_1,\cdots, a_{2004},$ equation $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$ has $2006$ positive real solution, find the maximum possible value of $a_1.$ | -2006 | 0.3125 | 7,698.9375 | 6,614.2 | 8,192 | |
In square \(ABCD\), an isosceles triangle \(AEF\) is inscribed such that point \(E\) lies on side \(BC\) and point \(F\) lies on side \(CD\), and \(AE = AF\). The tangent of angle \(AEF\) is 3. Find the cosine of angle \(FAD\). | \frac{2 \sqrt{5}}{5} | 0 | 4,858.125 | -1 | 4,858.125 | |
When $7$ fair standard $6$-sided dice are thrown, the probability that the sum of the numbers on the top faces is $10$ can be written as $\frac{n}{6^{7}}$, where $n$ is a positive integer. What is $n$? | 84 |
To find the number of ways to get a sum of $10$ when rolling $7$ fair $6$-sided dice, we can use the stars and bars method, considering the constraints that each die must show at least $1$ and at most $6$.
1. **Initial Setup**: Each die must show at least $1$. Therefore, we start by assigning $1$ to each die, reducin... | 0.8125 | 5,038.875 | 4,311.230769 | 8,192 |
Find the number of integers \( n \) that satisfy
\[ 15 < n^2 < 120. \] | 14 | 0.9375 | 3,498.875 | 3,487 | 3,677 | |
For a point $P=(x, y)$ in the Cartesian plane, let $f(P)=\left(x^{2}-y^{2}, 2 x y-y^{2}\right)$. If $S$ is the set of all $P$ so that the sequence $P, f(P), f(f(P)), f(f(f(P))), \ldots$ approaches $(0,0)$, then the area of $S$ can be expressed as $\pi \sqrt{r}$ for some positive real number $r$. Compute $\lfloor 100 r\... | 133 | For a point $P=(x, y)$, let $z(P)=x+y \omega$, where $\omega$ is a nontrivial third root of unity. Then $$\begin{aligned} z(f(P))=\left(x^{2}-y^{2}\right)+\left(2 x y-y^{2}\right) \omega=x^{2}+2 x y \omega+y^{2} & (-1-\omega) \\ & =x^{2}+2 x y \omega+y^{2} \omega^{2}=(x+y \omega)^{2}=z(P)^{2} \end{aligned}$$ Applying t... | 0 | 8,192 | -1 | 8,192 |
Given that $| \overrightarrow{a}|=5$, $| \overrightarrow{b}|=4$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-10$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ (denoted as $\langle \overrightarrow{a}, \overrightarrow{b} \rangle$). | \frac{2\pi}{3} | 0.0625 | 1,355.1875 | 1,390 | 1,352.866667 | |
Compute the sum of the squares of the first 10 base-6 numbers and express your answer in base 6. Specifically, find $1_6^2 + 2_6^2 + 3_6^2 + \cdots + 10_6^2$. | 231_6 | 0 | 6,130.375 | -1 | 6,130.375 | |
Let $A B C D$ be a quadrilateral, and let $E, F, G, H$ be the respective midpoints of $A B, B C, C D, D A$. If $E G=12$ and $F H=15$, what is the maximum possible area of $A B C D$? | 180 | The area of $E F G H$ is $E G \cdot F H \sin \theta / 2$, where $\theta$ is the angle between $E G$ and $F H$. This is at most 90. However, we claim the area of $A B C D$ is twice that of $E F G H$. To see this, notice that $E F=A C / 2=G H, F G=B D / 2=H E$, so $E F G H$ is a parallelogram. The half of this parallelog... | 0.375 | 5,993.75 | 4,789.833333 | 6,716.1 |
Suppose that $N = 3x + 4y + 5z$, where $x$ equals 1 or -1, and $y$ equals 1 or -1, and $z$ equals 1 or -1. How many of the following statements are true? - $N$ can equal 0. - $N$ is always odd. - $N$ cannot equal 4. - $N$ is always even. | 1 | When $N = 3x + 4y + 5z$ with each of $x, y$ and $z$ equal to either 1 or -1, there are 8 possible combinations of values for $x, y$ and $z$. From this information, $N$ cannot equal 0, $N$ is never odd, $N$ can equal 4, and $N$ is always even. Therefore, exactly one of the four given statements is true. | 0.625 | 2,620.4375 | 3,267.5 | 1,542 |
How many integers, $x$, satisfy $|5x - 3| \le 7$? | 3 | 1 | 3,605.75 | 3,605.75 | -1 | |
Find the remainder when $8\cdot10^{18}+1^{18}$ is divided by 9. | 0 | 0.875 | 3,821.0625 | 3,196.642857 | 8,192 | |
The mass of the first cast iron ball is $1462.5\%$ greater than the mass of the second ball. By what percentage less paint is needed to paint the second ball compared to the first ball? The volume of a ball with radius $R$ is $\frac{4}{3} \pi R^{3}$, and the surface area of a ball is $4 \pi R^{2}$. | 84 | 0.375 | 4,714.625 | 3,166 | 5,643.8 | |
Extend the definition of the binomial coefficient to $C_x^m = \frac{x(x-1)\dots(x-m+1)}{m!}$ where $x\in\mathbb{R}$ and $m$ is a positive integer, with $C_x^0=1$. This is a generalization of the binomial coefficient $C_n^m$ (where $n$ and $m$ are positive integers and $m\leq n$).
1. Calculate the value of $C_{-15}^3$.
... | \sqrt{2} | 0.5625 | 6,463.0625 | 6,005.222222 | 7,051.714286 | |
Ben rolls four fair 20-sided dice, and each of the dice has faces numbered from 1 to 20. What is the probability that exactly two of the dice show an even number? | \frac{3}{8} | 1 | 2,329.9375 | 2,329.9375 | -1 | |
It is now 3:15:20 PM, as read on a 12-hour digital clock. In 305 hours, 45 minutes, and 56 seconds, the time will be $X:Y:Z$. What is the value of $X + Y + Z$? | 26 | 0 | 8,116.625 | -1 | 8,116.625 | |
Given several numbers, one of them, $a$ , is chosen and replaced by the three numbers $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$ . This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A number $m$ is called *good* if there are $m... | 667 | 0.1875 | 7,723.6875 | 5,694.333333 | 8,192 | |
Consider the ellipse $\frac{x^{2}}{6} + \frac{y^{2}}{2} = 1$ and the hyperbola $\frac{x^{2}}{3} - y^{2} = 1$ with common foci $F_{1}$ and $F_{2}$. Let $P$ be one of the intersection points of the two curves. Find the value of $\cos \angle F_{1}PF_{2}$. | \frac{1}{3} | 0.875 | 4,836.0625 | 4,649.857143 | 6,139.5 | |
On an island of Liars and Knights, a circular arrangement is called correct if each person in the circle can say that among their two neighbors, there is at least one member of their tribe. One day, 2019 natives formed a correct circle. A liar approached them and said, "Now we too can form a correct circle." How many k... | 1346 | 0.0625 | 7,917.875 | 4,734 | 8,130.133333 | |
A cube with an edge length of 6 units has the same volume as a triangular-based pyramid with a base having equilateral triangle sides of 10 units and a height of $h$ units. What is the value of $h$? | \frac{216\sqrt{3}}{25} | 0 | 2,859.9375 | -1 | 2,859.9375 | |
A food factory regularly purchases flour. It is known that the factory needs 6 tons of flour per day, the price of each ton of flour is 1800 yuan, and the storage and other costs for flour are an average of 3 yuan per ton per day. Each time flour is purchased, a shipping fee of 900 yuan is required. How often should th... | 10 | 0.25 | 7,123.375 | 5,689 | 7,601.5 | |
In a tetrahedron \( PABC \), \(\angle APB = \angle BPC = \angle CPA = 90^\circ\). The dihedral angles formed between \(\triangle PBC, \triangle PCA, \triangle PAB\), and \(\triangle ABC\) are denoted as \(\alpha, \beta, \gamma\), respectively. Consider the following three propositions:
1. \(\cos \alpha \cdot \cos \beta... | (2) | 0 | 8,116.8125 | -1 | 8,116.8125 | |
Let \(a\), \(b\), and \(c\) be positive real numbers such that \(a + b + c = 5.\) Find the minimum value of
\[
\frac{9}{a} + \frac{16}{b} + \frac{25}{c}.
\] | 30 | 0 | 4,970.5625 | -1 | 4,970.5625 | |
Five people take a true-or-false test with five questions. Each person randomly guesses on every question. Given that, for each question, a majority of test-takers answered it correctly, let $p$ be the probability that every person answers exactly three questions correctly. Suppose that $p=\frac{a}{2^{b}}$ where $a$ is... | 25517 | There are a total of $16^{5}$ ways for the people to collectively ace the test. Consider groups of people who share the same problems that they got incorrect. We either have a group of 2 and a group of 3 , or a group 5 . In the first case, we can pick the group of two in $\binom{5}{2}$ ways, the problems they got wrong... | 0 | 7,110.625 | -1 | 7,110.625 |
A set \( \mathcal{S} \) of distinct positive integers has the property that for every integer \( x \) in \( \mathcal{S}, \) the arithmetic mean of the set of values obtained by deleting \( x \) from \( \mathcal{S} \) is an integer. Given that 1 belongs to \( \mathcal{S} \) and that 2310 is the largest element of \( \ma... | 20 | 0 | 8,101.5 | -1 | 8,101.5 | |
At Central Middle School the $108$ students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of $15$ cookies, lists these items:
$\bullet$ $1\frac{1}{2}$ cups of flou... | 11 | 1 | 519 | 519 | -1 | |
Points $A(-4,1), B(-1,4)$ and $C(-1,1)$ are the vertices of $\triangle ABC$. What will be the coordinates of the image of point A if $\triangle ABC$ is rotated 90 degrees clockwise about the origin? | (1, 4) | 0.9375 | 3,452.4375 | 3,136.466667 | 8,192 | |
If $11 = x^6 + \frac{1}{x^6}$, find the value of $x^3 + \frac{1}{x^3}$. | \sqrt{13} | 0.25 | 6,047.4375 | 3,965 | 6,741.583333 | |
Given that $F$ is the focus of the parabola $x^{2}=8y$, $P$ is a moving point on the parabola, and the coordinates of $A$ are $(0,-2)$, find the minimum value of $\frac{|PF|}{|PA|}$. | \frac{\sqrt{2}}{2} | 0 | 6,987.3125 | -1 | 6,987.3125 | |
Find $x$, such that $3^7 \cdot 3^x = 81$. | -3 | 1 | 1,709.8125 | 1,709.8125 | -1 | |
Consider the system of equations:
\[
8x - 6y = a,
\]
\[
12y - 18x = b.
\]
If there's a solution $(x, y)$ where both $x$ and $y$ are nonzero, determine $\frac{a}{b}$, assuming $b$ is nonzero. | -\frac{4}{9} | 0 | 8,064.9375 | -1 | 8,064.9375 | |
Determine the number of ways to select a positive number of squares on an $8 \times 8$ chessboard such that no two lie in the same row or the same column and no chosen square lies to the left of and below another chosen square. | 12869 | If $k$ is the number of squares chosen, then there are $\binom{8}{k}$ ways to choose $k$ columns, and $\binom{8}{k}$ ways to choose $k$ rows, and this would uniquely determine the set of squares selected. Thus the answer is $$\sum_{k=1}^{8}\binom{8}{k}\binom{8}{k}=-1+\sum_{k=0}^{8}\binom{8}{k}\binom{8}{k}=-1+\binom{16}... | 0 | 8,015.5625 | -1 | 8,015.5625 |
Given the ratio of length $AD$ to width $AB$ of the rectangle is $4:3$ and $AB$ is 40 inches, determine the ratio of the area of the rectangle to the combined area of the semicircles. | \frac{16}{3\pi} | 0.0625 | 6,941.9375 | 7,526 | 6,903 | |
In a closed right triangular prism ABC-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub> there is a sphere with volume $V$. If $AB \perp BC$, $AB=6$, $BC=8$, and $AA_{1}=3$, then the maximum value of $V$ is \_\_\_\_\_\_. | \frac{9\pi}{2} | 0 | 7,386.9375 | -1 | 7,386.9375 | |
In a game, \(N\) people are in a room. Each of them simultaneously writes down an integer between 0 and 100 inclusive. A person wins the game if their number is exactly two-thirds of the average of all the numbers written down. There can be multiple winners or no winners in this game. Let \(m\) be the maximum possible ... | 34 | Since the average of the numbers is at most 100, the winning number is an integer which is at most two-thirds of 100, or at most 66. This is achieved in a room with 34 people, in which 33 people pick 100 and one person picks 66, so the average number is 99. Furthermore, this cannot happen with less than 34 people. If t... | 0 | 8,192 | -1 | 8,192 |
What is the area enclosed by the graph of the equation $(x - 1)^2 + (y - 1)^2 = |x - 1| + |y - 1|$?
A) $\frac{\pi}{4}$
B) $\frac{\pi}{2}$
C) $\frac{\pi}{3}$
D) $\pi$ | \frac{\pi}{2} | 0 | 8,079.875 | -1 | 8,079.875 | |
A hotel has 5 distinct rooms, each with single beds for up to 2 people. The hotel has no other guests, and 5 friends want to stay there for the night. In how many ways can the 5 friends choose their rooms? | 2220 | 0.0625 | 7,777.625 | 8,192 | 7,750 | |
It is given that $x = -2272$ , $y = 10^3+10^2c+10b+a$ , and $z = 1$ satisfy the equation $ax + by + cz = 1$ , where $a, b, c$ are positive integers with $a < b < c$ . Find $y.$ | 1987 | 0.0625 | 8,110.125 | 6,882 | 8,192 | |
A circle inscribed in an isosceles trapezoid divides its lateral side into segments equal to 4 and 9. Find the area of the trapezoid. | 156 | 0.125 | 7,703.1875 | 4,281.5 | 8,192 | |
Let the function $f(x)=a\ln x-x- \frac {1}{2}x^{2}$.
(I) For $a=2$, find the extreme values of the function $f(x)$.
(II) Discuss the monotonicity of the function $f(x)$. | -\frac{3}{2} | 0.625 | 4,335.375 | 4,524.4 | 4,020.333333 | |
Jimmy notices $7$ oranges weigh the same as $5$ apples. If Jimmy has $28$ oranges, how many apples would Jimmy need to equal the weight of his $28$ oranges? | 20 | 1 | 899.125 | 899.125 | -1 | |
For positive integers $a$ and $b$, let $M(a, b)=\frac{\operatorname{lcm}(a, b)}{\operatorname{gcd}(a, b)}$, and for each positive integer $n \geq 2$, define $$x_{n}=M(1, M(2, M(3, \ldots, M(n-2, M(n-1, n)) \ldots)))$$ Compute the number of positive integers $n$ such that $2 \leq n \leq 2021$ and $5 x_{n}^{2}+5 x_{n+1}^... | 20 | The desired condition is that $x_{n}=5 x_{n+1}$ or $x_{n+1}=5 x_{n}$. Note that for any prime $p$, we have $\nu_{p}(M(a, b))=\left|\nu_{p}(a)-\nu_{p}(b)\right|$. Furthermore, $\nu_{p}(M(a, b)) \equiv \nu_{p}(a)+\nu_{p}(b) \bmod 2$. So, we have that $$\nu_{p}\left(x_{n}\right) \equiv \nu_{p}(1)+\nu_{p}(2)+\cdots+\nu_{p}... | 0 | 8,192 | -1 | 8,192 |
Given that the terminal side of angle $\alpha$ passes through point $P\left(\sin \frac{7\pi }{6},\cos \frac{11\pi }{6}\right)$, find the value of $\frac{1}{3\sin ^{2}\alpha -\cos ^{2}\alpha }=\_\_\_\_\_\_\_\_\_\_.$ | \frac{1}{2} | 1 | 3,873 | 3,873 | -1 | |
A monomial term $x_{i_{1}} x_{i_{2}} \ldots x_{i_{k}}$ in the variables $x_{1}, x_{2}, \ldots x_{8}$ is square-free if $i_{1}, i_{2}, \ldots i_{k}$ are distinct. (A constant term such as 1 is considered square-free.) What is the sum of the coefficients of the squarefree terms in the following product? $$\prod_{1 \leq i... | 764 | Let $a_{n}$ be the sum of the coefficients of the square-terms in the product $\prod_{1 \leq i<j \leq n}(1+$ $x_{i} x_{j}$ ). Square-free terms in this product come in two types: either they include $x_{n}$, or they do not. The sum of the coefficients of the terms that include $x_{n}$ is $(n-1) a_{n-2}$, since we can c... | 0 | 7,681.8125 | -1 | 7,681.8125 |
A ball is dropped from 10 feet high and always bounces back up half the distance it just fell. After how many bounces will the ball first reach a maximum height less than 1 foot? | 4 | 0.6875 | 5,084 | 3,671.272727 | 8,192 | |
In the diagram, points $A$, $B$, $C$, $D$, $E$, and $F$ lie on a straight line with $AB=BC=CD=DE=EF=3$. Semicircles with diameters $AF$, $AB$, $BC$, $CD$, $DE$, and $EF$ create a shape as depicted. What is the area of the shaded region underneath the largest semicircle that exceeds the areas of the other semicircles co... | \frac{45}{2}\pi | 0.0625 | 6,941.75 | 5,091 | 7,065.133333 | |
Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, the same letters to the same digits). The result was the word "ГВАТЕМАЛА". How many different numbers could Egor have originally written if his number was divisible by 30? | 21600 | 0 | 7,914.6875 | -1 | 7,914.6875 | |
Call a positive integer $N$ a 7-10 double if the digits of the base-$7$ representation of $N$ form a base-$10$ number that is twice $N$. For example, $51$ is a 7-10 double because its base-$7$ representation is $102$. What is the largest 7-10 double?
| 315 | 0 | 8,192 | -1 | 8,192 | |
In how many ways can four black balls, four white balls, and four blue balls be distributed into six different boxes? | 2000376 | 0.6875 | 4,488.375 | 3,673.818182 | 6,280.4 | |
If $x = (\log_82)^{(\log_28)}$, then $\log_3x$ equals: | -3 | 1. **Evaluate $x$:**
Given $x = (\log_8 2)^{(\log_2 8)}$. We start by simplifying each logarithm:
- $\log_8 2$ can be rewritten using the change of base formula:
\[
\log_8 2 = \frac{\log_2 2}{\log_2 8} = \frac{1}{3}
\]
since $\log_2 2 = 1$ and $\log_2 8 = 3$.
- $\log_2 8$ simplifies directl... | 1 | 2,728 | 2,728 | -1 |
Compute $\arccos (\sin 3).$ All functions are in radians. | 3 - \frac{\pi}{2} | 0.625 | 6,104.4375 | 4,851.9 | 8,192 | |
Eight people are sitting around a circular table for a meeting, and the recorder is sitting between the leader and the deputy leader. Calculate the total number of different seating arrangements possible, considering arrangements that can be made identical through rotation as the same. | 240 | 0.25 | 4,987.125 | 4,214.75 | 5,244.583333 | |
A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing it on top of the existi... | \frac{1}{8} | 0 | 8,176.125 | -1 | 8,176.125 | |
How many different positive integers can be represented as a difference of two distinct members of the set $\{1, 2, 3, 4, 5, 6 \}$? | 5 | 1 | 3,362.4375 | 3,362.4375 | -1 | |
Given the function $y=f(x)$ that satisfies $f(-x)=-f(x)$ and $f(1+x)=f(1-x)$ for $x \in [-1,1]$ with $f(x)=x^{3}$, find the value of $f(2015)$. | -1 | 0.6875 | 6,446 | 5,652.363636 | 8,192 | |
Find a positive integer $n$ with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of $n$ . | 35964 | 0.4375 | 7,431.75 | 6,454.285714 | 8,192 | |
Lucas chooses one, two or three different numbers from the list $2, 5, 7, 12, 19, 31, 50, 81$ and writes down the sum of these numbers. (If Lucas chooses only one number, this number is the sum.) How many different sums less than or equal to 100 are possible? | 41 | If Lucas chooses 1 number only, there are 8 possibilities for the sum, namely the 8 numbers themselves: $2, 5, 7, 12, 19, 31, 50, 81$. To count the number of additional sums to be included when Lucas chooses two numbers, we make a table, adding the number on left to the number on top when it is less than the number on ... | 0 | 8,192 | -1 | 8,192 |
Given an ellipse with $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), its left focal point is $F_1(-1, 0)$, and vertex P on the ellipse satisfies $\angle PF_1O = 45^\circ$ (where O is the origin).
(1) Determine the values of $a$ and $b$;
(2) Given that line $l_1: y = kx + m_1$ intersects the ellipse at points A ... | 2\sqrt{2} | 0.0625 | 8,192 | 8,192 | 8,192 | |
Given $y=f(x)+x^2$ is an odd function, and $f(1)=1$, then $f(-1)=$? | -3 | 1 | 1,980.375 | 1,980.375 | -1 | |
For how many positive integers $x$ is $(x-2)(x-4)(x-6) \cdots(x-2016)(x-2018) \leq 0$? | 1514 | We count the positive integers $x$ for which the product $(x-2)(x-4)(x-6) \cdots(x-2016)(x-2018)$ equals 0 and is less than 0 separately. The product equals 0 exactly when one of the factors equals 0. This occurs exactly when $x$ equals one of $2,4,6, \ldots, 2016,2018$. These are the even integers from 2 to 2018, incl... | 0 | 7,818 | -1 | 7,818 |
A certain high school has three mathematics teachers. For the convenience of the students, they arrange for a math teacher to be on duty every day from Monday to Friday, and two teachers are scheduled to be on duty on Monday. If each teacher is on duty for two days per week, there are ________ possible duty arrangement... | 36 | 0.125 | 7,392.25 | 4,162 | 7,853.714286 | |
Given \(w\) and \(z\) are complex numbers such that \(|w+z|=2\) and \(|w^2+z^2|=8,\) find the smallest possible value of \(|w^3+z^3|.\) | 20 | 0.375 | 7,285.875 | 6,109 | 7,992 | |
Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of odd integers among the 6 integers? | 1 | Let's denote the six integers as $a, b, c, d, e, f$. We are given the following conditions:
1. $a + b = 26$
2. $a + b + c + d = 41$
3. $a + b + c + d + e + f = 57$
From these conditions, we can derive the sums of the additional integers:
- From 1 and 2, $c + d = 41 - 26 = 15$
- From 2 and 3, $e + f = 57 - 41 = 16$
##... | 0.75 | 4,682.1875 | 4,026.416667 | 6,649.5 |
Let $G$ be the number of Google hits of "guts round" at 10:31PM on October 31, 2011. Let $B$ be the number of Bing hits of "guts round" at the same time. Determine $B / G$. Your score will be $$\max (0,\left\lfloor 20\left(1-\frac{20|a-k|}{k}\right)\right\rfloor)$$ where $k$ is the actual answer and $a$ is your answer. | .82721 | The number of Google hits was 7350. The number of Bing hits was 6080. The answer is thus $6080 / 7350=.82721$. | 0 | 6,030.6875 | -1 | 6,030.6875 |
An ant situated at point \( A \) decides to walk 1 foot east, then \( \frac{1}{2} \) foot northeast, then \( \frac{1}{4} \) foot east, then \( \frac{1}{8} \) foot northeast, then \( \frac{1}{16} \) foot east, and so on (that is, the ant travels alternately between east and northeast, and the distance travelled is decre... | \frac{2}{3} \sqrt{2 \sqrt{2}+5} | 0 | 6,556.3125 | -1 | 6,556.3125 | |
How many non- empty subsets $S$ of $\{1,2,3,\ldots ,15\}$ have the following two properties?
$(1)$ No two consecutive integers belong to $S$.
$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$. | 405 | 0.375 | 7,560.25 | 6,507.333333 | 8,192 | |
How many positive three-digit integers with a $7$ in the units place are divisible by $21$? | 39 | 0 | 7,184.3125 | -1 | 7,184.3125 | |
Given that $\{a_n\}$ is an arithmetic sequence, with the first term $a_1 > 0$, $a_5+a_6 > 0$, and $a_5a_6 < 0$, calculate the maximum natural number $n$ for which the sum of the first $n$ terms $S_n > 0$. | 10 | 0.1875 | 7,801.625 | 6,110 | 8,192 | |
Beginning with a $3 \mathrm{~cm}$ by $3 \mathrm{~cm}$ by $3 \mathrm{~cm}$ cube, a $1 \mathrm{~cm}$ by $1 \mathrm{~cm}$ by $1 \mathrm{~cm}$ cube is cut from one corner and a $2 \mathrm{~cm}$ by $2 \mathrm{~cm}$ by $2 \mathrm{~cm}$ cube is cut from the opposite corner. In $\mathrm{cm}^{2}$, what is the surface area of th... | 54 | 0.6875 | 6,435.75 | 5,807.181818 | 7,818.6 | |
The matrix
\[\begin{pmatrix} a & 3 \\ -8 & d \end{pmatrix}\]is its own inverse, for some real numbers $a$ and $d.$ Find the number of possible pairs $(a,d).$ | 2 | 1 | 2,325.4375 | 2,325.4375 | -1 | |
Given an ellipse $T$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\frac{\sqrt{3}}{2}$, a line passing through the right focus $F$ with slope $k (k > 0)$ intersects $T$ at points $A$ and $B$. If $\overline{AF} = 3\overline{FB}$, determine the value of $k$. | \sqrt{2} | 0.1875 | 7,893.3125 | 6,599 | 8,192 | |
Jenna is at a festival with six friends, making a total of seven people. They all want to participate in various group activities requiring groups of four or three people. How many different groups of four can be formed, and how many different groups of three can be formed from these seven people? | 35 | 0.5 | 680.75 | 774.5 | 587 |
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