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 |
|---|---|---|---|---|---|---|
How many positive four-digit integers less than 5000 have at least two digits that are the same? | 1984 | 0.5 | 6,382.6875 | 4,573.375 | 8,192 | |
James used a calculator to find the product $0.005 \times 3.24$. He forgot to enter the decimal points, and the calculator showed $1620$. If James had entered the decimal points correctly, what would the answer have been?
A) $0.00162$
B) $0.0162$
C) $0.162$
D) $0.01620$
E) $0.1620$ | 0.0162 | 0.125 | 3,594 | 648.5 | 4,014.785714 | |
For which values of \( a \) do the quadratic trinomials \( x^{2} + a x + 1 \) and \( x^{2} + x + a \) have a common root? | -2 | 0.3125 | 4,665.1875 | 4,159.2 | 4,895.181818 | |
Find the sum of $521_8$ and $146_8$ in base $8$. | 667_8 | 0.6875 | 4,288.6875 | 2,514.454545 | 8,192 | |
Gregor divides 2015 successively by 1, 2, 3, and so on up to and including 1000. He writes down the remainder for each division. What is the largest remainder he writes down? | 671 | 0.0625 | 8,006.375 | 8,192 | 7,994 | |
Let $S_i$ be the set of all integers $n$ such that $100i\leq n < 100(i + 1)$. For example, $S_4$ is the set ${400,401,402,\ldots,499}$. How many of the sets $S_0, S_1, S_2, \ldots, S_{999}$ do not contain a perfect square?
| 708 | 0 | 7,960.875 | -1 | 7,960.875 | |
Eight positive integers are written on the faces of a square prism (two bases and a lateral surface consisting of four faces). Each vertex is labeled with the product of the three numbers on the faces sharing that vertex (two from the lateral faces and one from the bases). If the sum of the numbers at the vertices equa... | 33 | 0 | 8,192 | -1 | 8,192 | |
Given a positive integer $n$, it can be shown that every complex number of the form $r+si$, where $r$ and $s$ are integers, can be uniquely expressed in the base $-n+i$ using the integers $0,1,2,\ldots,n^2$ as digits. That is, the equation
$r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0$
is true for a uniq... | 490 | First, we find the first three powers of $-3+i$:
$(-3+i)^1=-3+i ; (-3+i)^2=8-6i ; (-3+i)^3=-18+26i$
So we need to solve the diophantine equation $a_1-6a_2+26a_3=0 \Longrightarrow a_1-6a_2=-26a_3$.
The minimum the left hand side can go is -54, so $a_3\leq 2$, so we try cases:
Case 1: $a_3=2$
The only solution to that... | 0.4375 | 7,398.5 | 6,899 | 7,787 |
On a table near the sea, there are $N$ glass boxes where $N<2021$ , each containing exactly $2021$ balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wan... | 101 | 0.875 | 3,905.75 | 3,371.642857 | 7,644.5 | |
Let $a_1,$ $a_2,$ $\dots,$ $a_{12}$ be positive real numbers such that $a_1 + a_2 + \dots + a_{12} = 1.$ Find the minimum value of
\[\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_{12}}.\] | 144 | 1 | 3,138.5 | 3,138.5 | -1 | |
The sides of a right triangle are $a$ and $b$ and the hypotenuse is $c$. A perpendicular from the vertex divides $c$ into segments $r$ and $s$, adjacent respectively to $a$ and $b$. If $a : b = 1 : 3$, then the ratio of $r$ to $s$ is: | 1 : 9 | Given a right triangle with sides $a$, $b$, and hypotenuse $c$, where $a:b = 1:3$. Let's denote $a = k$ and $b = 3k$ for some positive constant $k$. By the Pythagorean theorem, we have:
\[ c^2 = a^2 + b^2 = k^2 + (3k)^2 = k^2 + 9k^2 = 10k^2 \]
Thus, the hypotenuse $c = \sqrt{10k^2} = k\sqrt{10}$.
Now, consider the per... | 0.1875 | 6,182 | 7,321 | 5,919.153846 |
The limit of $\frac {x^2-1}{x-1}$ as $x$ approaches $1$ as a limit is: | 2 | To find the limit of the function $\frac{x^2-1}{x-1}$ as $x$ approaches 1, we start by simplifying the expression.
1. **Factorize the Numerator**:
The numerator $x^2 - 1$ is a difference of squares, which can be factored as:
\[
x^2 - 1 = (x + 1)(x - 1)
\]
2. **Simplify the Expression**:
The expression... | 1 | 1,778.6875 | 1,778.6875 | -1 |
A regular tetrahedron has an edge length of $2$. Calculate the surface area of the sphere circumscribed around this tetrahedron. | 6\pi | 1 | 4,077.125 | 4,077.125 | -1 | |
Expand the following expression: $3(8x^2-2x+1)$. | 24x^2-6x+3 | 1 | 1,197.6875 | 1,197.6875 | -1 | |
Determine the value of $1 - 2 - 3 + 4 + 5 + 6 + 7 + 8 - 9 - 10 - \dots + 9801$, where the signs change after each perfect square and repeat every two perfect squares. | -9801 | 0 | 8,078.3125 | -1 | 8,078.3125 | |
Let \(x\), \(y\), and \(z\) be positive real numbers such that \(x + y + z = 3.\) Find the maximum value of \(x^3 y^3 z^2.\) | \frac{4782969}{390625} | 0 | 8,030.875 | -1 | 8,030.875 | |
Pentagon $S P E A K$ is inscribed in triangle $N O W$ such that $S$ and $P$ lie on segment $N O, K$ and $A$ lie on segment $N W$, and $E$ lies on segment $O W$. Suppose that $N S=S P=P O$ and $N K=K A=A W$. Given that $E P=E K=5$ and $E A=E S=6$, compute $O W$. | \frac{3 \sqrt{610}}{5} | Note that $[E S K]=[E P A]$, since one has half the base but double the height. Since the sides are the same, we must have $\sin \angle S E K=\sin \angle P E A$, so $\angle S E K+\angle P E A=180^{\circ}$. Let $O W=3 x$, so $S K=x$ and $P A=2 x$. Then by the law of cosines $$\begin{aligned} x^{2} & =61-60 \cos \angle S... | 0 | 8,192 | -1 | 8,192 |
The hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > 0, b > 0)$ has an asymptote that is perpendicular to the line $x + 2y + 1 = 0$. Let $F_1$ and $F_2$ be the foci of $C$, and let $A$ be a point on the hyperbola such that $|F_1A| = 2|F_2A|$. Find $\cos \angle AF_2F_1$. | \frac{\sqrt{5}}{5} | 0 | 6,665.5625 | -1 | 6,665.5625 | |
Given the function $y=\cos \left(x+ \frac {\pi}{3}\right)$, derive the horizontal shift of the graph of the function $y=\sin x$. | \frac {5\pi}{6} | 0.75 | 5,262.25 | 4,295.25 | 8,163.25 | |
Vasya thought of a four-digit number and wrote down the product of each pair of its adjacent digits on the board. After that, he erased one product, and the numbers 20 and 21 remained on the board. What is the smallest number Vasya could have in mind? | 3745 | 0 | 8,192 | -1 | 8,192 | |
Find the product of the solutions of: $|y|=2(|y|-1)$. | -4 | 1 | 1,808.6875 | 1,808.6875 | -1 | |
Compute $\lim _{n \rightarrow \infty} \frac{1}{\log \log n} \sum_{k=1}^{n}(-1)^{k}\binom{n}{k} \log k$. | 1 | Answer: 1. The idea is that if $f(k)=\int g^{k}$, then $\sum(-1)^{k}\binom{n}{k} f(k)=\int(1-g)^{n}$. To relate this to logarithm, we may use the Frullani integrals $\int_{0}^{\infty} \frac{e^{-x}-e^{-k x}}{x} d x=\lim _{c \rightarrow+0} \int_{c}^{\infty} \frac{e^{-x}}{x} d x-\int_{c}^{\infty} \frac{e^{-k x}}{x} d x=\l... | 0 | 8,165 | -1 | 8,165 |
Triangle $PQR$ has vertices at $P(7, 5)$, $Q(1, -3)$, and $R(4, 4)$. The point $S$ with coordinates $(x, y)$ is chosen inside the triangle so that the three smaller triangles $PQS$, $PRS$, and $QRS$ all have equal areas. What is the value of $12x + 3y$? | 54 | 0.875 | 3,359.3125 | 2,668.928571 | 8,192 | |
Let $s (n)$ denote the sum of digits of a positive integer $n$. Using six different digits, we formed three 2-digits $p, q, r$ such that $$p \cdot q \cdot s(r) = p\cdot s(q) \cdot r = s (p) \cdot q \cdot r.$$ Find all such numbers $p, q, r$. | (12, 36, 48), (21, 63, 84) |
To find the numbers \( p, q, r \) that satisfy the given conditions, we follow these steps:
Start by examining the conditions provided in the problem:
1. \( p \cdot q \cdot s(r) = p \cdot s(q) \cdot r \)
2. \( p \cdot s(q) \cdot r = s(p) \cdot q \cdot r \)
Since each of \( p, q, r \) is a two-digit number formed usi... | 0 | 8,192 | -1 | 8,192 |
Find the times between $8$ and $9$ o'clock, correct to the nearest minute, when the hands of a clock will form an angle of $120^{\circ}$. | 8:22 | 0.125 | 8,121.6875 | 7,629.5 | 8,192 | |
Let $f : [0, 1] \rightarrow \mathbb{R}$ be a monotonically increasing function such that $$ f\left(\frac{x}{3}\right) = \frac{f(x)}{2} $$ $$ f(1 0 x) = 2018 - f(x). $$ If $f(1) = 2018$ , find $f\left(\dfrac{12}{13}\right)$ .
| 2018 | 0 | 8,192 | -1 | 8,192 | |
In the geometric sequence ${a_n}$, $a_3$ and $a_{15}$ are the roots of the equation $x^2 + 6x + 2 = 0$, calculate the value of $$\frac{a_{2}a_{16}}{a_{9}}.$$ | \sqrt{2} | 0 | 7,558.6875 | -1 | 7,558.6875 | |
A container in the shape of a right circular cone is $12$ inches tall and its base has a $5$-inch radius. The liquid that is sealed inside is $9$ inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the height of the liquid is ... | 52 | The scale factor is uniform in all dimensions, so the volume of the liquid is $\left(\frac{3}{4}\right)^{3}$ of the container. The remaining section of the volume is $\frac{1-\left(\frac{3}{4}\right)^{3}}{1}$ of the volume, and therefore $\frac{\left(1-\left(\frac{3}{4}\right)^{3}\right)^{1/3}}{1}$ of the height when t... | 0.6875 | 6,365 | 5,534.545455 | 8,192 |
Let $p,$ $q,$ $r,$ $s$ be real numbers such that $p + q + r + s = 10$ and
\[ pq + pr + ps + qr + qs + rs = 20. \]
Find the largest possible value of $s$. | \frac{5 + \sqrt{105}}{2} | 0 | 6,003.6875 | -1 | 6,003.6875 | |
Given positive numbers $a$ and $b$ satisfying $\log _{6}(2a+3b)=\log _{3}b+\log _{6}9-1=\log _{2}a+\log _{6}9-\log _{2}3$, find $\lg \left(2a+3b\right)-\lg \left(10a\right)-\lg \left(10b\right)=\_\_\_\_\_\_$. | -2 | 0 | 8,192 | -1 | 8,192 | |
In a store, there are 21 white and 21 purple shirts hanging in a row. Find the smallest $k$ such that, regardless of the initial order of the shirts, it is possible to remove $k$ white and $k$ purple shirts, so that the remaining white shirts hang consecutively and the remaining purple shirts also hang consecutively. | 10 | 0.1875 | 7,721.6875 | 6,983.666667 | 7,892 | |
Given that Chelsea is ahead by 60 points halfway through a 120-shot archery contest, with each shot scoring 10, 8, 5, 3, or 0 points and Chelsea scoring at least 5 points on every shot, determine the smallest number of bullseyes (10 points) Chelsea needs to shoot in her next n attempts to ensure victory, assuming her o... | 49 | 0.125 | 7,691.625 | 7,329 | 7,743.428571 | |
Given the parametric equation of line $l$:
$$
\begin{cases}
x=t+1 \\
y= \sqrt {3}t
\end{cases}
$$
(where $t$ is the parameter), and the polar equation of curve $C$ is $\rho=2\cos\theta$, then the polar radius (taking the positive value) of the intersection point of line $l$ and curve $C$ is ______. | \sqrt{3} | 0.0625 | 8,169.1875 | 7,827 | 8,192 | |
A number $m$ is randomly selected from the set $\{11,13,15,17,19\}$, and a number $n$ is randomly selected from $\{1999,2000,2001,\ldots,2018\}$. What is the probability that $m^n$ has a units digit of $1$? | \frac{7}{20} | 1. **Reduction to Units Digits**:
Since we are interested in the units digit of $m^n$, we can reduce the set $\{11, 13, 15, 17, 19\}$ to their respective units digits, resulting in the set $A = \{1, 3, 5, 7, 9\}$.
2. **Case Analysis**:
- **Case $m = 1$**:
The units digit of $1^n$ is always $1$ for any $n$. ... | 0 | 4,694.5 | -1 | 4,694.5 |
Solve for $n$: $\frac{2-n}{n+1} + \frac{2n-4}{2-n} = 1$. | -\frac{1}{4} | 1 | 2,578.9375 | 2,578.9375 | -1 | |
The physical education teacher lined up the class so that everyone was facing him. There are 12 people to the right of Kolya, 20 people to the left of Sasha, and 8 people to the right of Sasha. How many people are to the left of Kolya? | 16 | 0.125 | 751 | 597.5 | 772.928571 | |
Given the parabola $y^2=2x$, find the equation of its directrix. | -\frac{1}{2} | 1 | 2,215.3125 | 2,215.3125 | -1 | |
There are 10 boys, each with a unique weight and height. For any two boys, $\mathbf{A}$ and $\mathbf{B}$, if $\mathbf{A}$ is heavier than $\mathbf{B}$, or if $\mathbf{A}$ is taller than $\mathbf{B}$, then $\mathbf{A}$ is not considered worse than $\mathbf{B}$. A boy who is not worse than the other 9 boys is called an "... | 10 | 0 | 7,757.1875 | -1 | 7,757.1875 | |
Given sets \( A = \{ x \mid 5x - a \leq 0 \} \) and \( B = \{ x \mid 6x - b > 0 \} \), where \( a, b \in \mathbf{N} \), and \( A \cap B \cap \mathbf{N} = \{ 2, 3, 4 \} \), determine the number of integer pairs \((a, b)\). | 30 | 0.0625 | 7,646.4375 | 6,052 | 7,752.733333 | |
How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x, y)$ satisfying $1 \leq x \leq 5$ and $1 \leq y \leq 5$? | 2164 | 0 | 8,192 | -1 | 8,192 | |
Given that cos(15°+α) = $\frac{3}{5}$, where α is an acute angle, find: $$\frac{tan(435° -α)+sin(α-165° )}{cos(195 ° +α )\times sin(105 ° +α )}$$. | \frac{5}{36} | 0.625 | 6,235.625 | 5,429.2 | 7,579.666667 | |
Given the formula $d=\sqrt[3]{\frac{16}{9}V}$, where $d$ is the diameter of a sphere and $V$ is its volume, find the volume of a sphere with a radius of $\frac{1}{3}$. | \frac{1}{6} | 1 | 4,516.9375 | 4,516.9375 | -1 | |
In the diagram, $ABCD$ is a trapezoid with an area of $18$. $CD$ is three times the length of $AB$. What is the area of $\triangle ABC$?
[asy]
draw((0,0)--(1,3)--(10,3)--(15,0)--cycle);
draw((10,3)--(0,0));
label("$D$",(0,0),W);
label("$A$",(1,3),NW);
label("$B$",(10,3),NE);
label("$C$",(15,0),E);
[/asy] | 4.5 | 0.0625 | 5,823.25 | 682 | 6,166 | |
Given $a = 1 + 2\binom{20}{1} + 2^2\binom{20}{2} + \ldots + 2^{20}\binom{20}{20}$, and $a \equiv b \pmod{10}$, determine the possible value(s) for $b$. | 2011 | 0 | 2,639.5 | -1 | 2,639.5 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $(2a-c)\cos B=b\cos C$.
(Ⅰ) Find the magnitude of angle $B$;
(Ⅱ) If $a=2$ and $c=3$, find the value of $\sin C$. | \frac {3 \sqrt {21}}{14} | 0 | 4,098 | -1 | 4,098 | |
Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^2 = s,$$where $r$ and $s$ are constants.
What is $r+s$? | 11 | 1 | 1,983.0625 | 1,983.0625 | -1 | |
Monica is renovating the floor of her 15-foot by 20-foot dining room. She plans to place two-foot by two-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. Additionally, there is a 1-foot by 1-foot column at the center of the room ... | 78 | 0 | 3,969.8125 | -1 | 3,969.8125 | |
Given a sector with a central angle of $\alpha$ and a radius of $r$.
$(1)$ If $\alpha = 60^{\circ}$ and $r = 3$, find the arc length of the sector.
$(2)$ If the perimeter of the sector is $16$, at what angle $\alpha$ will the area of the sector be maximized? Also, find the maximum area. | 16 | 0.9375 | 3,487.375 | 3,173.733333 | 8,192 | |
The distance between location A and location B originally required a utility pole to be installed every 45m, including the two poles at both ends, making a total of 53 poles. Now, the plan has been changed to install a pole every 60m. Excluding the two poles at both ends, how many poles in between do not need to be mov... | 12 | 0.4375 | 5,069.5625 | 5,242.857143 | 4,934.777778 | |
In triangle $ABC,$ angle bisectors $\overline{AD}$ and $\overline{BE}$ intersect at $P.$ If $AB = 7,$ $AC = 5,$ and $BC = 3,$ find $\frac{BP}{PE}.$ | 2 | 1 | 3,922.9375 | 3,922.9375 | -1 | |
Ten numbers are written around a circle with their sum equal to 100. It is known that the sum of each triplet of consecutive numbers is at least 29. Identify the smallest number \( A \) such that, in any such set of numbers, each number does not exceed \( A \). | 13 | 0 | 8,192 | -1 | 8,192 | |
How many positive integers $n$ with $n \leq 100$ can be expressed as the sum of four or more consecutive positive integers? | 63 | We consider first the integers that can be expressed as the sum of exactly 4 consecutive positive integers. The smallest such integer is $1+2+3+4=10$. The next smallest such integer is $2+3+4+5=14$. We note that when we move from $k+(k+1)+(k+2)+(k+3)$ to $(k+1)+(k+2)+(k+3)+(k+4)$, we add 4 to the total (this equals the... | 0 | 7,958.5625 | -1 | 7,958.5625 |
Last year, Isabella took 8 math tests and received 8 different scores, each an integer between 91 and 100, inclusive. After each test, she noted that the average of her test scores was an integer. Her score on the seventh test was 97. What was her score on the eighth test? | 96 | 0.0625 | 8,131.3125 | 7,221 | 8,192 | |
A pedestrian left city $A$ at noon heading towards city $B$. A cyclist left city $A$ at a later time and caught up with the pedestrian at 1 PM, then immediately turned back. After returning to city $A$, the cyclist turned around again and met the pedestrian at city $B$ at 4 PM, at the same time as the pedestrian.
By w... | 5/3 | 0.0625 | 7,651.625 | 6,401 | 7,735 | |
\[
\frac{\sin ^{2}\left(135^{\circ}-\alpha\right)-\sin ^{2}\left(210^{\circ}-\alpha\right)-\sin 195^{\circ} \cos \left(165^{\circ}-2 \alpha\right)}{\cos ^{2}\left(225^{\circ}+\alpha\right)-\cos ^{2}\left(210^{\circ}-\alpha\right)+\sin 15^{\circ} \sin \left(75^{\circ}-2 \alpha\right)}=-1
\] | -1 | 0.375 | 7,418.4375 | 6,241.333333 | 8,124.7 | |
If the line $l_{1}$: $x+my-2=0$ intersects the line $l_{2}$: $mx-y+2=0$ at point $P$, and a tangent line passing through point $P$ is drawn to the circle $C: (x+2)^{2} + (y+2)^{2} = 1$, with the point of tangency being $M$, then the maximum value of $|PM|$ is ____. | \sqrt{31} | 0.6875 | 6,862.8125 | 6,258.636364 | 8,192 | |
Find a necessary and sufficient condition on the natural number $ n$ for the equation
\[ x^n \plus{} (2 \plus{} x)^n \plus{} (2 \minus{} x)^n \equal{} 0
\]
to have a integral root. | n=1 |
To solve the problem and find the necessary and sufficient condition for the natural number \( n \) such that the equation
\[
x^n + (2 + x)^n + (2 - x)^n = 0
\]
has an integral root, we proceed as follows:
### Step 1: Analyze the Case \( n = 1 \)
Substitute \( n = 1 \) into the equation:
\[
x^1 + (2 + x)^1 + (2 -... | 0.1875 | 8,178.3125 | 8,119 | 8,192 |
Given that $0 < x < \frac{1}{2}$, find the minimum and maximum value of the function $x^{2}(1-2x)$. | \frac{1}{27} | 0.8125 | 6,295.625 | 5,858 | 8,192 | |
Simplify $\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{4}}{\sqrt{5}} \cdot \frac{\sqrt{6}}{\sqrt{7}}$ and rationalize the denominator of the resulting fraction. | \frac{4\sqrt{35}}{35} | 0 | 2,516.75 | -1 | 2,516.75 | |
Tommy takes a 25-question true-false test. He answers each question correctly with independent probability $\frac{1}{2}$ . Tommy earns bonus points for correct streaks: the first question in a streak is worth 1 point, the second question is worth 2 points, and so on. For instance, the sequence TFFTTTFT is worth 1 + ... | 50 | 0 | 7,955.0625 | -1 | 7,955.0625 | |
A company plans to invest in 3 different projects among 5 candidate cities around the Bohai Economic Rim, which are Dalian, Yingkou, Panjin, Jinzhou, and Huludao. The number of projects invested in the same city cannot exceed 2. How many different investment plans can the company have? (Answer with a number). | 120 | 0.4375 | 6,937.8125 | 5,325.285714 | 8,192 | |
Find the number of real solutions to the equation
\[
\frac{1}{x - 1} + \frac{2}{x - 2} + \frac{3}{x - 3} + \dots + \frac{50}{x - 50} = 2x.
\] | 51 | 0.125 | 6,823.75 | 6,033 | 6,936.714286 | |
Let $a,b,c,d,e,f,g,h,i$ be distinct integers from $1$ to $9.$ The minimum possible positive value of \[\dfrac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}\] can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 289 | To minimize a positive fraction, we minimize its numerator and maximize its denominator. It is clear that $\frac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i} \geq \frac{1}{7\cdot8\cdot9}.$
If we minimize the numerator, then $a \cdot b \cdot c - d \cdot e \cdot f = 1.$ Note that $a \cdot b \cdot c \cdot d ... | 0 | 8,090.9375 | -1 | 8,090.9375 |
Let $r,$ $s,$ and $t$ be the roots of the equation $x^3 - 20x^2 + 18x - 7 = 0.$ Find the value of $(1+r)(1+s)(1+t).$ | 46 | 1 | 2,842.3125 | 2,842.3125 | -1 | |
Let $x, y$, and $z$ be distinct real numbers that sum to 0. Find the maximum possible value of $$\frac{x y+y z+z x}{x^{2}+y^{2}+z^{2}}$$ | -1/2 | Note that $0=(x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 z x$. Rearranging, we get that $x y+y z+z x=-\frac{1}{2}\left(x^{2}+y^{2}+z^{2}\right)$, so that in fact the quantity is always equal to $-1 / 2$. | 0 | 6,416.0625 | -1 | 6,416.0625 |
For which integers $n \in\{1,2, \ldots, 15\}$ is $n^{n}+1$ a prime number? | 1, 2, 4 | $n=1$ works. If $n$ has an odd prime factor, you can factor, and this is simulated also by $n=8$: $$a^{2 k+1}+1=(a+1)\left(\sum_{i=0}^{2 k}(-a)^{i}\right)$$ with both parts larger than one when $a>1$ and $k>0$. So it remains to check 2 and 4, which work. Thus the answers are $1,2,4$. | 0 | 7,635.8125 | -1 | 7,635.8125 |
There are 8 white balls and 2 red balls in a bag. Each time a ball is randomly drawn and then a white ball is put back. What is the probability that all red balls are drawn exactly at the 4th draw? | 0.0434 | 0 | 7,148.3125 | -1 | 7,148.3125 | |
Convert the point $\left( 2 \sqrt{3}, \frac{2 \pi}{3} \right)$ in polar coordinates to rectangular coordinates. | (-\sqrt{3}, 3) | 1 | 1,641.8125 | 1,641.8125 | -1 | |
In parallelogram \(ABCD\), \(OE = EF = FD\). The area of the parallelogram is 240 square centimeters. The area of the shaded region is _______ square centimeters. | 20 | 0.0625 | 7,764.3125 | 8,192 | 7,735.8 | |
A fair coin is tossed six times and the sequence of heads and tails is recorded. What is the probability that the sequence contains exactly two heads? Express your answer as a common fraction. | \frac{15}{64} | 1 | 1,785.1875 | 1,785.1875 | -1 | |
Given that $\cos \alpha =-\dfrac{3}{4}, \sin \beta =\dfrac{2}{3}$, with $\alpha$ in the third quadrant and $\beta \in (\dfrac{\pi }{2}, \pi )$.
(I) Find the value of $\sin 2\alpha$;
(II) Find the value of $\cos (2\alpha + \beta )$. | -\dfrac{\sqrt{5} + 6\sqrt{7}}{24} | 0 | 3,700.25 | -1 | 3,700.25 | |
In the three-dimensional Cartesian coordinate system, the equation of the plane passing through $P(x_{0}, y_{0}, z_{0})$ with normal vector $\overrightarrow{m}=(a, b, c)$ is $a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0$, and the equation of the line passing through $P(x_{0}, y_{0}, z_{0})$ with direction vector $\overrightarrow... | \frac{\sqrt{55}}{11} | 0 | 6,615.5 | -1 | 6,615.5 | |
The integer points $(x, y)$ in the first quadrant satisfy $x + y > 8$ and $x \leq y \leq 8$. How many such integer points $(x, y)$ are there? | 20 | 0.75 | 6,056.5 | 5,344.666667 | 8,192 | |
If $ rac{1}{2n} + rac{1}{4n} = rac{3}{12}$, what is the value of $n$? | 3 | Since $ rac{1}{2n} + rac{1}{4n} = rac{2}{4n} + rac{1}{4n} = rac{3}{4n}$, then the given equation becomes $ rac{3}{4n} = rac{3}{12}$ or $4n = 12$. Thus, $n = 3$. | 1 | 1,540.625 | 1,540.625 | -1 |
How many ways can the integers from -7 to 7 be arranged in a sequence such that the absolute values of the numbers in the sequence are nonincreasing? | 128 | 0.125 | 7,990.875 | 6,583 | 8,192 | |
Let $\Delta ABC$ be an acute-angled triangle and let $H$ be its orthocentre. Let $G_1, G_2$ and $G_3$ be the centroids of the triangles $\Delta HBC , \Delta HCA$ and $\Delta HAB$ respectively. If the area of $\Delta G_1G_2G_3$ is $7$ units, what is the area of $\Delta ABC $ ? | 63 | 0.9375 | 5,269.6875 | 5,074.866667 | 8,192 | |
Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people so that exact... | 216 | We only need to figure out the number of ways to order the string $BBBCCCFFF$, where exactly one $B$ is in the first three positions, one $C$ is in the $4^{th}$ to $6^{th}$ positions, and one $F$ is in the last three positions. There are $3^3=27$ ways to place the first $3$ meals. Then for the other two people, there a... | 0 | 8,192 | -1 | 8,192 |
The volume of a cylinder circumscribed around a sphere with radius $r$ is $V_{1}$, and the volume of a cone circumscribed around the same sphere is $V_{2}$. What is the minimum value of the ratio $V_{2} / V_{1}$? | 4/3 | 0.25 | 7,709.9375 | 6,593.75 | 8,082 | |
Take a standard set of 28 dominoes and put back double 3, double 4, double 5, and double 6, as they will not be needed. Arrange the remaining dominoes to form 3 square frames, as shown in the image, so that the sum of the points along each side is equal. In the given example, these sums are equal to 15. If this is one ... | 15 | 0.0625 | 8,084.6875 | 8,192 | 8,077.533333 | |
Given the function $f(x)=(2-a)(x-1)-2\ln x$ $(a\in \mathbb{R})$.
(Ⅰ) If the tangent line at the point $(1,g(1))$ on the curve $g(x)=f(x)+x$ passes through the point $(0,2)$, find the decreasing interval of the function $g(x)$;
(Ⅱ) If the function $y=f(x)$ has no zeros in the interval $\left(0,\frac{1}{2}\right)$, fin... | 2-4\ln 2 | 0.1875 | 7,939.9375 | 7,015.333333 | 8,153.307692 | |
A rectangular park is to be fenced on three sides using a 150-meter concrete wall as the fourth side. Fence posts are to be placed every 15 meters along the fence, including at the points where the fence meets the concrete wall. Calculate the minimal number of posts required to fence an area of 45 m by 90 m. | 13 | 0.0625 | 8,013.5 | 7,379 | 8,055.8 | |
In a \(7 \times 7\) table, some cells are black while the remaining ones are white. In each white cell, the total number of black cells located with it in the same row or column is written; nothing is written in the black cells. What is the maximum possible sum of the numbers in the entire table? | 168 | 0 | 8,192 | -1 | 8,192 | |
Let $\triangle PQR$ have side lengths $PQ=13$, $PR=15$, and $QR=14$. Inside $\angle QPR$ are two circles: one is tangent to rays $\overline{PQ}$, $\overline{PR}$, and segment $\overline{QR}$; the other is tangent to the extensions of $\overline{PQ}$ and $\overline{PR}$ beyond $Q$ and $R$, and also tangent to $\overline... | 5\sqrt{13} | 0 | 7,117.25 | -1 | 7,117.25 | |
1. Calculate $\log _{2.5}6.25+ \lg 0.01+ \ln \sqrt {e}-2\,^{1+\log _{2}3}$
2. Given $\tan \alpha=-3$, and $\alpha$ is an angle in the second quadrant, find $\sin \alpha$ and $\cos \alpha$. | - \frac { \sqrt {10}}{10} | 0 | 2,972.375 | -1 | 2,972.375 | |
Alice celebrated her birthday on Friday, March 15 in the year 2012. Determine the next year when her birthday will next fall on a Monday.
A) 2021
B) 2022
C) 2023
D) 2024
E) 2025 | 2025 | 0 | 5,341.875 | -1 | 5,341.875 | |
Let $M = 123456789101112\dots4950$ be the $95$-digit number formed by writing integers from $1$ to $50$ in order, one after the other. What is the remainder when $M$ is divided by $45$? | 15 | 0.625 | 5,761 | 4,302.4 | 8,192 | |
What is the nearest integer to \((3+\sqrt{2})^6\)? | 7414 | 0 | 7,783.375 | -1 | 7,783.375 | |
Determine the longest side of the polygon formed by the system:
$$
\begin{cases}
x + y \leq 4 \\
x + 2y \geq 4 \\
x \geq 0 \\
y \geq 0
\end{cases}
$$ | 2\sqrt{5} | 0 | 4,646.8125 | -1 | 4,646.8125 | |
From a standard deck of 52 cards, what is the probability that the top four cards include only cards from the black suits ($\spadesuit$ and $\clubsuit$)? | \frac{276}{4998} | 0 | 6,844.375 | -1 | 6,844.375 | |
A point $Q$ is randomly chosen inside an equilateral triangle $DEF$. What is the probability that the area of $\triangle DEQ$ is greater than both $\triangle DFQ$ and $\triangle EFQ$? | \frac{1}{3} | 0.0625 | 8,191.8125 | 8,192 | 8,191.8 | |
Find the sum of squares of all distinct complex numbers $x$ satisfying the equation $0=4 x^{10}-7 x^{9}+5 x^{8}-8 x^{7}+12 x^{6}-12 x^{5}+12 x^{4}-8 x^{3}+5 x^{2}-7 x+4$ | -\frac{7}{16} | For convenience denote the polynomial by $P(x)$. Notice $4+8=7+5=12$ and that the consecutive terms $12 x^{6}-12 x^{5}+12 x^{4}$ are the leading terms of $12 \Phi_{14}(x)$, which is suggestive. Indeed, consider $\omega$ a primitive 14 -th root of unity; since $\omega^{7}=-1$, we have $4 \omega^{10}=-4 \omega^{3},-7 \om... | 0 | 8,068.25 | -1 | 8,068.25 |
Calculate the following powers to 4 decimal places:
a) \(1.02^{30}\)
b) \(0.996^{13}\) | 0.9492 | 0.25 | 7,981.1875 | 7,348.75 | 8,192 | |
Factor $w^4-16$ as far as possible, where the factors are monic polynomials with real coefficients. | (w-2)(w+2)(w^2+4) | 0.9375 | 1,956.3125 | 1,655.933333 | 6,462 | |
Four football teams participate in a round-robin tournament, where each team plays a match against every other team. In each match, the winning team earns 3 points, the losing team earns 0 points, and in the case of a draw, both teams earn 1 point each. After all matches are completed, it is known that the total points... | 120 | 0.125 | 7,867.625 | 5,597 | 8,192 | |
How many solid $1 imes 1 imes 1$ cubes are required to make a solid $2 imes 2 imes 2$ cube? | 8 | The volume of a $1 imes 1 imes 1$ cube is 1 . The volume of a $2 imes 2 imes 2$ cube is 8 . Thus, 8 of the smaller cubes are needed to make the larger cube. | 1 | 808.0625 | 808.0625 | -1 |
A student has 2 identical photo albums and 3 identical stamp albums. The student wants to give away 4 albums, one to each of four friends. How many different ways can the student give away the albums? | 10 | 0.375 | 6,783.1875 | 6,007.166667 | 7,248.8 | |
In the diagram, each of the two circles has center \(O\). Also, \(O P: P Q = 1:2\). If the radius of the larger circle is 9, what is the area of the shaded region? | 72 \pi | 0.8125 | 2,994.6875 | 2,252.461538 | 6,211 | |
Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth? | 37 | 1. Let $p$ represent the number of pennies Leah has, and $n$ represent the number of nickels Leah has. We know from the problem statement that the total number of coins Leah has is 13. Therefore, we can write the equation:
\[
n + p = 13
\]
2. The problem also states that if Leah had one more nickel, she would... | 1 | 1,705.9375 | 1,705.9375 | -1 |
For an integer $n$, let $f_{9}(n)$ denote the number of positive integers $d \leq 9$ dividing $n$. Suppose that $m$ is a positive integer and $b_{1}, b_{2}, \ldots, b_{m}$ are real numbers such that $f_{9}(n)=\sum_{j=1}^{m} b_{j} f_{9}(n-j)$ for all $n>m$. Find the smallest possible value of $m$. | 28 | Let $M=9$. Consider the generating function $$F(x)=\sum_{n \geq 1} f_{M}(n) x^{n}=\sum_{d=1}^{M} \sum_{k \geq 1} x^{d k}=\sum_{d=1}^{M} \frac{x^{d}}{1-x^{d}}$$ Observe that $f_{M}(n)=f_{M}\left(n+M\right.$ !) for all $n \geq 1$ (in fact, all $n \leq 0$ as well). Thus $f_{M}(n)$ satisfies a degree $m$ linear recurrence ... | 0 | 8,176.75 | -1 | 8,176.75 |
A total area of \( 2500 \, \mathrm{m}^2 \) will be used to build identical houses. The construction cost for a house with an area \( a \, \mathrm{m}^2 \) is the sum of the material cost \( 100 p_{1} a^{\frac{3}{2}} \) yuan, labor cost \( 100 p_{2} a \) yuan, and other costs \( 100 p_{3} a^{\frac{1}{2}} \) yuan, where \... | 156 | 0.0625 | 8,146.875 | 7,470 | 8,192 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.