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 |
|---|---|---|---|---|---|---|
$\frac{2}{10}+\frac{4}{100}+\frac{6}{1000}=$ | .246 | 1. **Convert fractions to a common denominator**: We start by converting each fraction to have a common denominator, which is 1000 in this case. This is done by multiplying the numerator and denominator of each fraction by the appropriate factor:
\[
\frac{2}{10} = \frac{2 \times 100}{10 \times 100} = \frac{200}{1... | 0.5 | 2,852.375 | 2,972.625 | 2,732.125 |
A bowling ball must have a diameter of 9 inches. Calculate both the surface area and the volume of the bowling ball before the finger holes are drilled. Express your answers as common fractions in terms of \(\pi\). | \frac{729\pi}{6} | 0 | 1,714.5 | -1 | 1,714.5 | |
Suppose six points are taken inside or on a rectangle with dimensions $1 \times 2$. Let $b$ be the smallest possible number with the property that it is always possible to select one pair of points from these six such that the distance between them is equal to or less than $b$. Calculate the value of $b$. | \frac{\sqrt{5}}{2} | 0 | 8,192 | -1 | 8,192 | |
Kristen has to clear snow from a driveway that is 30 feet long and 3 feet wide. If the snow is initially 8 inches deep, and compacting the snow reduces its volume by 10%, how much snow (in cubic feet) must Kristen move? | 54 | 0.5625 | 1,816.1875 | 1,056.111111 | 2,793.428571 | |
Jennifer wants to enclose her rectangular vegetable garden using 160 feet of fencing. She has decided that one side of the garden should be exactly 30 feet long. What is the maximum area that she can enclose, assuming the sides of the rectangle are natural numbers? | 1500 | 0.875 | 4,833.6875 | 4,353.928571 | 8,192 | |
How many ways are there to put 6 balls into 4 boxes if the balls are indistinguishable but the boxes are distinguishable, with the condition that no box remains empty? | 22 | 0 | 6,093.3125 | -1 | 6,093.3125 | |
If $9^{x + 2} = 240 + 9^x$, then the value of $x$ is: | 0.5 | 1. **Rewrite the given equation using properties of exponents:**
\[
9^{x+2} = 240 + 9^x
\]
We know that $9^{x+2} = 9^x \cdot 9^2$. Since $9^2 = 81$, we can substitute:
\[
81 \cdot 9^x = 240 + 9^x
\]
2. **Isolate terms involving $9^x$:**
To simplify, we can move all terms involving $9^x$ to one ... | 1 | 1,932.25 | 1,932.25 | -1 |
After shifting the graph of the function $y=\sin^2x-\cos^2x$ to the right by $m$ units, the resulting graph is symmetric to the graph of $y=k\sin x\cos x$ ($k>0$) with respect to the point $\left( \frac{\pi}{3}, 0 \right)$. Find the minimum positive value of $k+m$. | 2+ \frac{5\pi}{12} | 0.1875 | 7,995.875 | 7,146 | 8,192 | |
The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$.
$\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0 \\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}$ | 8 | 1. **Identify the Incorrect Sum**: First, we add the given numbers without changing any digits:
- $742586 + 829430 = 1572016$
- The provided sum is $1212016$.
2. **Analyze the Incorrectness**: The provided sum $1212016$ differs significantly from the actual sum $1572016$. We need to change one digit $d$ to anoth... | 0 | 8,192 | -1 | 8,192 |
Let the function \( g : \mathbb{R} \to \mathbb{R} \) satisfy the equation
\[ g(x) + 2g(2 - x) = 4x^3 - x^2 \]
for all \( x \). Find \( g(5) \). | -\frac{709}{3} | 0.5 | 7,229.0625 | 6,266.125 | 8,192 | |
How many positive odd integers greater than 1 and less than $200$ are square-free? | 79 | 0.0625 | 7,969.4375 | 7,197 | 8,020.933333 | |
Find all values of \( n \in \mathbf{N} \) for which there exist a number \( m \in \mathbf{N} \), a triangle \( ABC \) with sides \( AB = 33 \), \( AC = 21 \), \( BC = n \), and points \( D \), \( E \) on sides \( AB \), \( AC \) respectively, satisfying the conditions \( AD = DE = EC = m \). | 30 | 0 | 8,192 | -1 | 8,192 | |
A large rectangle has side lengths of $(x+7)$ and $(x+5)$. In the large rectangle, there is a rectangular hole with side lengths of $(2x-3)$ and $(x-2)$. What is the area of the large rectangle (not including the area of the hole)? Express your answer as a polynomial in $x$. | -x^2+19x+29 | 1 | 3,942.8125 | 3,942.8125 | -1 | |
In isosceles $\triangle ABC$, $|AB|=|AC|$, vertex $A$ is the intersection point of line $l: x-y+1=0$ with the y-axis, and $l$ bisects $\angle A$. If $B(1,3)$, find:
(I) The equation of line $BC$;
(II) The area of $\triangle ABC$. | \frac {3}{2} | 0.875 | 4,295.4375 | 3,738.785714 | 8,192 | |
A cylinder has a radius of 5 cm and a height of 12 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? | 2\sqrt{61} | 0.6875 | 4,448.3125 | 2,985.363636 | 7,666.8 | |
The function $f: \mathbb{R}\rightarrow \mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\forall$ $x\in \mathbb{R}$ and $f(x)=x(x-1)$ for $\forall$ $x\in (0,1]$ . Find the greatest real number $m$ , for which the inequality $f(x)\geq -\frac{8}{9}$ is true for $\forall$ $x\in (-\infty , m]$ .
| 7/3 | 0 | 8,001.9375 | -1 | 8,001.9375 | |
The vector $\vec{a} =(-1,3)$, $\vec{b} =(3,-4)$, then the projection of vector $\vec{a}$ in the direction of vector $\vec{b}$ is ______. | -3 | 0.5 | 2,429.875 | 1,946.25 | 2,913.5 | |
Fill the first eight positive integers in a $2 \times 4$ table, one number per cell, such that each row's four numbers increase from left to right, and each column's two numbers increase from bottom to top. How many different ways can this be done? | 14 | 0.1875 | 6,634.9375 | 3,752.666667 | 7,300.076923 | |
On Monday at work, David produces $w$ widgets per hour, and works for $t$ hours. Exhausted by this work, on Tuesday, he decides to work for $2$ fewer hours, but manages to produce $4$ additional widgets per hour. If $w = 2t$, how many more widgets did David produce on Monday than on Tuesday? | 8 | 1 | 1,647.625 | 1,647.625 | -1 | |
Add $956_{12} + 273_{12}$. Express your answer in base $12$, using $A$ for $10$ and $B$ for $11$ if necessary. | 1009_{12} | 0.625 | 5,077.375 | 3,755.1 | 7,281.166667 | |
Determine all integers $ k\ge 2$ such that for all pairs $ (m$, $ n)$ of different positive integers not greater than $ k$, the number $ n^{n\minus{}1}\minus{}m^{m\minus{}1}$ is not divisible by $ k$. | 2 \text{ and } 3 | Let us analyze the problem, which requires us to determine all integers \( k \ge 2 \) such that for all pairs \( (m, n) \) of different positive integers not greater than \( k \), the expression \( n^{n-1} - m^{m-1} \) is not divisible by \( k \).
### Step 1: Understand the condition
The condition states:
- For \( n,... | 0 | 8,135.3125 | -1 | 8,135.3125 |
$P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches. Then $PB$, in inches, equals: | $3\sqrt{2}$ | 1. **Identify the relationships given by the problem:**
- $PA = 3$ inches, $PD = 4$ inches, $PC = 5$ inches, and we need to find $PB = x$ inches.
2. **Use the perpendicular distances from $P$ to the sides of the rectangle:**
- Let $a, b, c, d$ be the perpendicular distances from $P$ to sides $AB, BC, CD, DA$ res... | 0 | 2,734.375 | -1 | 2,734.375 |
Determine the number of non-degenerate rectangles whose edges lie completely on the grid lines of the following figure. | 297 | First, let us count the total number of rectangles in the grid without the hole in the middle. There are $\binom{7}{2}=21$ ways to choose the two vertical boundaries of the rectangle, and there are 21 ways to choose the two horizontal boundaries of the rectangles. This makes $21^{2}=441$ rectangles. However, we must ex... | 0 | 7,430.125 | -1 | 7,430.125 |
Four prime numbers are randomly selected without replacement from the first twelve prime numbers. What is the probability that the sum of the four selected numbers is even? | \frac{2}{3} | 0.8125 | 4,702.25 | 4,387 | 6,068.333333 | |
Let \( T = 3 \times ((1 + i)^{15} - (1 - i)^{15}) \), where \( i = \sqrt{-1} \). Calculate \( |T| \). | 768 | 0.6875 | 5,979.75 | 4,974.181818 | 8,192 | |
The difference between two positive integers is 12 and their product is 45. What is the sum of the integers? | 18 | 1 | 1,793.75 | 1,793.75 | -1 | |
For an arithmetic sequence $a_1, a_2, a_3, \dots,$ let
\[ S_n = a_1 + a_2 + a_3 + \dots + a_n, \]
and let
\[ T_n = S_1 + S_2 + S_3 + \dots + S_n. \]
Given the value of $S_{2023}$, determine the smallest integer $n$ for which you can uniquely determine the value of $T_n$. | 3034 | 0.5625 | 6,914.5 | 6,054 | 8,020.857143 | |
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed. | 71 | Obviously $n\le 90$. We see that the problem's condition is equivalent to: 96 is the smallest number that can be formed which is 1 mod 5, and 92, 93, 94 can be formed (95 can always be formed). Now divide this up into cases. If $n\equiv 0\pmod{5}$, then 91 can be formed by using $n+1$ and some 5's, so there are no solu... | 0 | 8,192 | -1 | 8,192 |
Find the distance between the foci of the hyperbola $x^2 - 6x - 4y^2 - 8y = 27.$ | 4 \sqrt{10} | 1 | 2,787.375 | 2,787.375 | -1 | |
Find the natural number \( N \) such that it is divisible by 5 and 49, and it has exactly 10 divisors, including 1 and \( N \). | 12005 | 0.875 | 5,674.1875 | 5,314.5 | 8,192 | |
Calculate the value of $\cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7} = \_\_\_\_\_\_$. | -\frac{1}{8} | 0.4375 | 7,502.25 | 6,615.428571 | 8,192 | |
In parallelogram $ABCD$, if $\overrightarrow{AE}=2\overrightarrow{ED}$, $\overrightarrow{BF}=\overrightarrow{FC}$, and $\overrightarrow{AC}=λ\overrightarrow{AE}+\overrightarrow{AF}$, then $\lambda =$____. | \frac{3}{4} | 0.9375 | 3,602.125 | 3,296.133333 | 8,192 | |
John is 24 years younger than his dad. The sum of their ages is 68 years. How many years old is John? | 22 | 1 | 495.5 | 495.5 | -1 | |
The constant term in the expansion of the binomial $\left(\frac{1}{\sqrt{x}} - x^2\right)^{10}$ is ______. | 45 | 1 | 2,992.625 | 2,992.625 | -1 | |
When a single number is added to each member of the sequence 20, 50, 100, the sequence becomes expressible as $x, a x, a^{2} x$. Find $a$. | \frac{5}{3} | $\frac{5}{3}$. | 1 | 3,035.875 | 3,035.875 | -1 |
If \( b \) and \( n \) are positive integers with \( b, n \leq 18 \), what is the greatest number of positive factors \( b^n \) can have? | 703 | 0.1875 | 8,101.6875 | 7,710.333333 | 8,192 | |
At Pine Ridge Elementary School, one third of the students ride the school bus home. One fifth of the students are picked up by car. One eighth of the students go home on their skateboards. Another one tenth of the students share rides with classmates. The rest of the students walk home. What fractional part of the stu... | \frac{29}{120} | 1 | 747.8125 | 747.8125 | -1 | |
Last month, Xiao Ming's household expenses were 500 yuan for food, 200 yuan for education, and 300 yuan for other expenses. This month, the costs of these three categories increased by 6%, 20%, and 10%, respectively. What is the percentage increase in Xiao Ming's household expenses for this month compared to last month... | 10\% | 0.5625 | 678.375 | 680.444444 | 675.714286 | |
The graph below shows the number of home runs in April for the top hitters in the league. What is the mean (average) number of home runs hit by these players?
[asy]
draw((0,0)--(0,7)--(24,7)--(24,0)--cycle);
label("KEY:",(3,5));
fill((3,2.5)..(3.5,2)..(3,1.5)..(2.5,2)..cycle);
label("- one(1) baseball player",(14,2));... | 7 | 0.0625 | 5,177.75 | 2,156 | 5,379.2 | |
For any subset \( S \subseteq \{1, 2, \ldots, 15\} \), a number \( n \) is called an "anchor" for \( S \) if \( n \) and \( n+|S| \) are both members of \( S \), where \( |S| \) denotes the number of members of \( S \). Find the average number of anchors over all possible subsets \( S \subseteq \{1, 2, \ldots, 15\} \). | 13/8 | 0 | 8,192 | -1 | 8,192 | |
Evaluate $$\sin \left(1998^{\circ}+237^{\circ}\right) \sin \left(1998^{\circ}-1653^{\circ}\right)$$ | -\frac{1}{4} | We have \(\sin \left(1998^{\circ}+237^{\circ}\right) \sin \left(1998^{\circ}-1653^{\circ}\right)=\sin \left(2235^{\circ}\right) \sin \left(345^{\circ}\right)=\sin \left(75^{\circ}\right) \sin \left(-15^{\circ}\right)=-\sin \left(75^{\circ}\right) \sin \left(15^{\circ}\right)=-\sin \left(15^{\circ}\right) \cos \left(15^... | 0.875 | 4,216.1875 | 4,316.357143 | 3,515 |
The graph of the quadratic $y = ax^2 + bx + c$ has these properties: (1) The maximum value of $y = ax^2 + bx + c$ is 4, which occurs at $x = 2$. (2) The graph passes through the point $(0,-16)$. If the graph also passes through the point $(5,n)$, what is the value of $n$? | -41 | 1 | 2,078.5 | 2,078.5 | -1 | |
Chewbacca has 25 pieces of orange gum and 35 pieces of apple gum. Some of the pieces are in complete packs, while others are loose. Each complete pack has exactly $y$ pieces of gum. If Chewbacca loses two packs of orange gum, then the ratio of the number of pieces of orange gum he has to the number of pieces of apple g... | \frac{15}{4} | 0.125 | 8,035.75 | 6,942 | 8,192 | |
If $x-y>x$ and $x+y<y$, then | $x<0,y<0$ | We are given two inequalities:
1. \(x - y > x\)
2. \(x + y < y\)
#### Analyzing the first inequality:
Starting from the first inequality:
\[ x - y > x \]
Subtract \(x\) from both sides:
\[ -y > 0 \]
This implies:
\[ y < 0 \]
Thus, \(y\) is negative.
#### Analyzing the second inequality:
Starting from the second inequ... | 0 | 4,775.625 | -1 | 4,775.625 |
A rugby team scored 24 points, 17 points, and 25 points in the seventh, eighth, and ninth games of their season. Their mean points-per-game was higher after 9 games than it was after their first 6 games. What is the smallest number of points that they could score in their 10th game for their mean number of points-per-g... | 24 | 0.3125 | 5,246.1875 | 4,696.2 | 5,496.181818 | |
The slope angle of the tangent line to the curve $y= \sqrt {x}$ at $x= \frac {1}{4}$ is ______. | \frac {\pi}{4} | 1 | 2,150.1875 | 2,150.1875 | -1 | |
Suppose that there are real numbers $a, b, c \geq 1$ and that there are positive reals $x, y, z$ such that $$\begin{aligned} a^{x}+b^{y}+c^{z} & =4 \\ x a^{x}+y b^{y}+z c^{z} & =6 \\ x^{2} a^{x}+y^{2} b^{y}+z^{2} c^{z} & =9 \end{aligned}$$ What is the maximum possible value of $c$ ? | \sqrt[3]{4} | The Cauchy-Schwarz inequality states that given 2 sequences of $n$ real numbers $x_{1}, x_{2}, \ldots, x_{n}$ and $y_{1}, y_{2}, \ldots, y_{n}$, then $\left(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}\right)\left(y_{1}^{2}+y_{2}^{2}+\ldots+y_{n}^{2}\right) \geq\left(x_{1} y_{1}+x_{2} y_{2}+\ldots+x_{n} y_{n}\right)^{2}$ with ... | 0.125 | 7,930.8125 | 6,102.5 | 8,192 |
Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r^{\text{th}}$ root of $N$. What is $f(2) + f(3) + f(4) + f(5)+ f(6)$? | 8 | 1. **Define the number and function**: Let $N$ be the number $7777\ldots777$ with $313$ digits, where each digit is $7$. Define $f(r)$ as the leading digit of the $r$-th root of $N$.
2. **General property of leading digits under root transformation**: For any positive integer $k$ and real number $n > 10^k$, we have:
... | 0.3125 | 7,752.75 | 7,268.4 | 7,972.909091 |
If the sum of the coefficients of each term in the expansion of $(x- \frac {4}{x})^{n}$ is $81$, then the constant term in the expansion is ________. | 96 | 1 | 2,184.8125 | 2,184.8125 | -1 | |
Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be?
$\textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ \text{none of these}$
| 5 | 0 | 5,793 | -1 | 5,793 | |
How many different four-letter arrangements can be formed using the six letters $A, B, C, D, E$ and $F$, if the first letter must be $C$, one of the other letters must be $B$, and no letter can be used more than once in the arrangement? | 36 | 0.9375 | 4,149.5625 | 3,880.066667 | 8,192 | |
What is the smallest number of 3-cell L-shaped tiles that can be placed in an 8x8 square such that no more of these tiles can be placed in the square? | 11 | 0 | 7,960.4375 | -1 | 7,960.4375 | |
How many positive integer divisors of $1800^{1800}$ are divisible by exactly 180 positive integers? | 18 | 0 | 5,792.1875 | -1 | 5,792.1875 | |
In rectangle \(ABCD\), \(AB = 2\) and \(AD = 1\). Point \(P\) is a moving point on side \(DC\) (including \(D\) and \(C\)), and point \(Q\) is a moving point on the extension of side \(CB\) (including point \(B\)), such that \(|\overrightarrow{DP}| = |\overrightarrow{BQ}|\). Find the minimum value of the dot product \(... | 3/4 | 0.5625 | 4,162.3125 | 3,067.222222 | 5,570.285714 | |
In isosceles right-angled triangle $ABC$ , $CA = CB = 1$ . $P$ is an arbitrary point on the sides of $ABC$ . Find the maximum of $PA \cdot PB \cdot PC$ . | \frac{\sqrt{2}}{4} | 0 | 8,092.75 | -1 | 8,092.75 | |
A flag is made of three horizontal strips of fabric, each of a solid color, either red, white, blue or green. If no two adjacent strips can be the same color, how many distinct flags are possible? These two flags are different.
[asy]draw((0,0)--(0,6),linewidth(1));
filldraw((0,3)--(4,3)--(4,4)--(0,4)--(0,3)--cycle,whi... | 36 | 1 | 3,645.0625 | 3,645.0625 | -1 | |
Given two lines $l_1: y = 2x$, $l_2: y = -2x$, and a line $l$ passing through point $M(-2, 0)$ intersects $l_1$ and $l_2$ at points $A$ and $B$, respectively, where point $A$ is in the third quadrant, point $B$ is in the second quadrant, and point $N(1, 0)$;
(1) If the area of $\triangle NAB$ is 16, find the equation... | -\frac {1}{5} | 0.375 | 7,215.625 | 5,588.333333 | 8,192 | |
Evaluate $\log_2\frac{1}{16}$. | -4 | 1 | 2,171.5625 | 2,171.5625 | -1 | |
Given $\tan (\alpha +\beta )=7$ and $\tan (\alpha -\beta )=1$, find the value of $\tan 2\alpha$. | -\dfrac{4}{3} | 0.875 | 4,165.375 | 3,590.142857 | 8,192 | |
Given the set of 10 integers {1, 2, 3, ..., 9, 10}, choose any 3 distinct numbers to be the coefficients of the quadratic function f(x) = ax^2 + bx + c. Determine the number of ways to choose the coefficients such that f(1)/3 is an integer. | 252 | 0.4375 | 7,508.9375 | 6,630.714286 | 8,192 | |
In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the interior angles $A$, $B$, and $C$, respectively. If $a\cos \left(B-C\right)+a\cos A=2\sqrt{3}c\sin B\cos A$ and $b^{2}+c^{2}-a^{2}=2$, then the area of $\triangle ABC$ is ____. | \frac{\sqrt{3}}{2} | 0 | 7,652.75 | -1 | 7,652.75 | |
For any four-digit number $m$, if the digits of $m$ are all non-zero and distinct, and the sum of the units digit and the thousands digit is equal to the sum of the tens digit and the hundreds digit, then this number is called a "mirror number". If we swap the units digit and the thousands digit of a "mirror number" to... | \frac{{11}}{8} | 0 | 8,192 | -1 | 8,192 | |
In space, the following four propositions are given:
(1) Through a point, there is exactly one plane perpendicular to a given line;
(2) If the distances from two points outside a plane to the plane are equal, then the line passing through these two points must be parallel to the plane;
(3) The projections of two ... | (1)(4) | 0 | 4,682.9375 | -1 | 4,682.9375 | |
A number $N$ has three digits when expressed in base $7$. When $N$ is expressed in base $9$ the digits are reversed. Then the middle digit is: | 0 | 1. **Expressing $N$ in different bases**: Let $N$ be represented as $\overline{abc}_7$ in base $7$ and as $\overline{cba}_9$ in base $9$. This means:
- In base $7$: $N = 49a + 7b + c$
- In base $9$: $N = 81c + 9b + a$
2. **Setting up the equation**: Since both expressions represent the same number $N$, we equate... | 1 | 3,905.3125 | 3,905.3125 | -1 |
Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals.
[i] | f(x) = 2x |
To find all functions \( f: \mathbb{R}^{+} \to \mathbb{R}^{+} \) satisfying the given functional equation:
\[
f(x + f(y)) = f(x + y) + f(y)
\]
for all positive real numbers \( x \) and \( y \), we will proceed as follows.
### Step 1: Exploring the Functional Equation
Let's introduce \( f \) such that it satisfies ... | 0 | 7,733.9375 | -1 | 7,733.9375 |
In $\triangle ABC$, $A, B, C$ are the three interior angles, and $a, b, c$ are the sides opposite to angles $A, B, C$ respectively. It is given that $2 \sqrt{2}\left(\sin^2 A - \sin^2 C\right) = (a - b) \sin B$, and the radius of the circumcircle of $\triangle ABC$ is $\sqrt{2}$.
(1) Find angle $C$;
(2) Find the maximu... | \frac{3\sqrt{3}}{2} | 0 | 6,543.25 | -1 | 6,543.25 | |
Given a sequence of length 15 composed of zeros and ones, find the number of sequences where all zeros are consecutive, all ones are consecutive, or both. | 270 | 0 | 7,913 | -1 | 7,913 | |
There are 3 math clubs in the school district, with 5, 7, and 8 students respectively. Each club has two co-presidents. If I randomly select a club, and then randomly select three members of that club to give a copy of $\emph{Introduction to} \allowbreak\ \emph{Counting and} \allowbreak\ \emph{Probability}$, what is th... | \dfrac{11}{60} | 0.9375 | 3,300.1875 | 2,974.066667 | 8,192 | |
For how many triples $(x, y, z)$ of integers between -10 and 10 inclusive do there exist reals $a, b, c$ that satisfy $$\begin{gathered} a b=x \\ a c=y \\ b c=z ? \end{gathered}$$ | 4061 | If none are of $x, y, z$ are zero, then there are $4 \cdot 10^{3}=4000$ ways, since $x y z$ must be positive. Indeed, $(a b c)^{2}=x y z$. So an even number of them are negative, and the ways to choose an even number of 3 variables to be negative is 4 ways. If one of $x, y, z$ is 0 , then one of $a, b, c$ is zero at le... | 0 | 8,192 | -1 | 8,192 |
Let \( P \) be a point inside regular pentagon \( ABCDE \) such that \( \angle PAB = 48^\circ \) and \( \angle PDC = 42^\circ \). Find \( \angle BPC \), in degrees. | 84 | 0 | 8,192 | -1 | 8,192 | |
The equation $y=-4.9t^2+3.5t+5$ describes the height (in meters) of a ball thrown upward at $3.5$ meters per second from $5$ meters above the ground, where $t$ is the time in seconds. In how many seconds will the ball hit the ground? Express your answer as a common fraction. | \frac{10}{7} | 1 | 2,976.8125 | 2,976.8125 | -1 | |
Given that $\lg 2 = 0.3010$, determine the number of digits in the integer $2^{2015}$. | 607 | 0.6875 | 5,810.5625 | 4,728.090909 | 8,192 | |
Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch reads 12:57 and 36 seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads 10:00 PM? | 10:25 PM | 1. **Identify the rate of time loss:** Cassandra's watch loses time such that in 1 hour of actual time, her watch shows only 57 minutes and 36 seconds. We convert 36 seconds to minutes:
\[
36 \text{ seconds} = \frac{36}{60} \text{ minutes} = 0.6 \text{ minutes}
\]
Therefore, in 1 hour of actual time, her w... | 0 | 6,550.375 | -1 | 6,550.375 |
Given two arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ with respective sums of the first $n$ terms $S_n$ and $T_n$, if $\frac{S_n}{T_n} = \frac{2n-3}{4n-3}$ holds for any natural number $n$, find the value of $\frac{a_9}{b_5+b_7} + \frac{a_3}{b_8+b_4}$. | \frac{19}{41} | 0.75 | 5,136.8125 | 5,242.333333 | 4,820.25 | |
Starting with an equilateral triangle as shown in diagram a, each side of the triangle is divided into three equal parts, and at the middle segment, new equilateral triangles are constructed outward, as shown in diagram b, forming a "snowflake hexagon." Next, each of the 12 sides of the "snowflake hexagon" is divided i... | 40/27 | 0.4375 | 7,144.1875 | 6,226.714286 | 7,857.777778 | |
Let all possible $2023$ -degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$ ,
where $P(0)+P(1)=0$ , and the polynomial has 2023 real roots $r_1, r_2,\cdots r_{2023}$ [not necessarily distinct] so that $0\leq r_1,r_2,\cdots r_{2023}\leq1$ . What is the maximum value of $r_1... | 2^{-2023} | 0 | 7,998.5 | -1 | 7,998.5 | |
Let \( z = \frac{1+\mathrm{i}}{\sqrt{2}} \). Then the value of \( \left(\sum_{k=1}^{12} z^{k^{2}}\right)\left(\sum_{k=1}^{12} \frac{1}{z^{k^{2}}}\right) \) is ( ). | 36 | 0.375 | 7,004.125 | 5,705.666667 | 7,783.2 | |
Let $A_1A_2A_3\ldots A_{12}$ be a dodecagon ($12$-gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. Al... | 19 | We can solve the problem by removing $1$ frog, and calculate the expected time for the remaining $2$ frogs. In the original problem, when the movement stops, $2$ of the $3$ frogs meet. Because the $3$ frogs cannot meet at one vertex, the probability that those two specific frogs meet is $\frac13$. If the expected time ... | 0 | 8,192 | -1 | 8,192 |
Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra $P_{i}$ is defined recursively as follows: $P_{0}$ is a regular tetrahedron whose volume is 1. To obtain $P_{i + 1}$, replace the midpoint triangle of every face of $P_{i}$ by an outward-pointing regular t... | 101 | On the first construction, $P_1$, four new tetrahedra will be constructed with side lengths $\frac 12$ of the original one. Since the ratio of the volume of similar polygons is the cube of the ratio of their corresponding lengths, it follows that each of these new tetrahedra will have volume $\left(\frac 12\right)^3 = ... | 0.0625 | 7,867.375 | 7,142 | 7,915.733333 |
In a plane, there are 10 lines, among which 4 lines are parallel to each other. Then, these 10 lines can divide the plane into at most how many parts? | 50 | 0.125 | 8,126.875 | 7,671 | 8,192 | |
Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only $80\%$ of the problems she solved alone, but overall $88\%$ of her answers were correct. Zoe had c... | 93 | 1. **Define Variables:**
Let $t$ be the total number of problems in the homework assignment. Let $x$ be the number of problems that Chloe and Zoe solved correctly together.
2. **Calculate Chloe's Correct Answers:**
Chloe solved half of the problems alone and got $80\%$ of them correct. Therefore, the number of p... | 0.875 | 3,941.625 | 3,765 | 5,178 |
Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of these two circles? | \frac{9\pi - 18}{2} | 0.125 | 6,003.5625 | 5,533 | 6,070.785714 | |
Given a parabola \( y^2 = 6x \) with two variable points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), where \( x_1 \neq x_2 \) and \( x_1 + x_2 = 4 \). The perpendicular bisector of segment \( AB \) intersects the x-axis at point \( C \). Find the maximum area of triangle \( \triangle ABC \). | \frac{14}{3}\sqrt{7} | 0 | 8,042.375 | -1 | 8,042.375 | |
At 8:00 AM, Xiao Cheng and Xiao Chen set off from locations A and B respectively, heading towards each other. They meet on the way at 9:40 AM. Xiao Cheng says: "If I had walked 10 km more per hour, we would have met 10 minutes earlier." Xiao Chen says: "If I had set off half an hour earlier, we would have met 20 minute... | 150 | 0.1875 | 6,610.1875 | 3,350 | 7,362.538462 | |
Let $[x]$ represent the greatest integer less than or equal to the real number $x$. How many positive integers $n \leq 1000$ satisfy the condition that $\left[\frac{998}{n}\right]+\left[\frac{999}{n}\right]+\left[\frac{1000}{n}\right]$ is not divisible by 3? | 22 | 0 | 8,192 | -1 | 8,192 | |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{1}{2}t\\ y=\frac{\sqrt{3}}{2}t\end{array}\right.$ (where $t$ is a parameter). Taking $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equatio... | \frac{\sqrt{7}}{4} | 0 | 6,243.1875 | -1 | 6,243.1875 | |
From the set $\left\{ \frac{1}{3}, \frac{1}{2}, 2, 3 \right\}$, select a number and denote it as $a$. From the set $\{-2, -1, 1, 2\}$, select another number and denote it as $b$. Then, the probability that the graph of the function $y=a^{x}+b$ passes through the third quadrant is ______. | \frac{3}{8} | 0.125 | 7,896.875 | 6,150.5 | 8,146.357143 | |
Which digit will appear in the 534th place after the decimal point in the decimal representation of $\frac{5}{13}$? | 5 | 1 | 2,132.125 | 2,132.125 | -1 | |
The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107)$. The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum of the absolute values of all possible slopes for $\overline{AB}$ is $m/n$, ... | 131 | 0 | 8,192 | -1 | 8,192 | |
What is the coefficient of $a^3b^3$ in $(a+b)^6\left(c + \dfrac{1}{c}\right)^8$? | 1400 | 0.5 | 4,711.625 | 3,539.875 | 5,883.375 | |
The first few rows of a new sequence are given as follows:
- Row 1: $3$
- Row 2: $6, 6, 6, 6$
- Row 3: $9, 9, 9, 9, 9, 9$
- Row 4: $12, 12, 12, 12, 12, 12, 12, 12$
What is the value of the $40^{\mathrm{th}}$ number if this arrangement were continued? | 18 | 0.5 | 5,646.75 | 5,597.125 | 5,696.375 | |
The gravitational force that Earth exerts on an object is inversely proportional to the square of the distance between the center of the Earth and the object. When Bill is on the surface of Earth, 4,000 miles from the center, the gravitational force is 600 Newtons. What is the gravitational force (in Newtons) that the ... | \frac{1}{6} | 0.8125 | 2,731.1875 | 2,751.692308 | 2,642.333333 | |
This puzzle features a unique kind of problem where only one digit is known. It appears to have a single solution and, surprisingly, filling in the missing digits is not very difficult. Given that a divisor multiplied by 7 results in a three-digit number, we conclude that the first digit of the divisor is 1. Additional... | 124 | 0 | 8,192 | -1 | 8,192 | |
Evaluate the integral $$\int_{ -2 }^{ 2 }$$($$\sqrt {16-x^{2}}$$+sinx)dx=\_\_\_\_\_\_ | 4\sqrt{3} + \frac{8\pi}{3} | 0.3125 | 5,208.5 | 2,866.2 | 6,273.181818 | |
If $\frac{4}{3} (r + s + t) = 12$, what is the average of $r$, $s$, and $t$? | 3 | 1 | 1,193.875 | 1,193.875 | -1 | |
The number of students studying in the 5th-6th grades of the school is expressed as a three-digit number. From the digits of this number (without repetitions), 6 different two-digit numbers can be formed, the sum of which is twice the number of students in the 5th-6th grades. How many students are in these grades? | 198 | 0.875 | 3,839.1875 | 3,240.642857 | 8,029 | |
Aaron has 144 identical cubes, each with edge length 1 cm. He uses all of the cubes to construct a solid rectangular prism, which he places on a flat table. If the perimeter of the base of the prism is 20 cm, what is the sum of all possible heights of the prism? | 31 | Suppose that the base of the prism is $b \mathrm{~cm}$ by $w \mathrm{~cm}$ and the height of the prism is $h \mathrm{~cm}$. Since Aaron has 144 cubes with edge length 1 cm, then the volume of the prism is $144 \mathrm{~cm}^{3}$, and so $bwh = 144$. Since the perimeter of the base is 20 cm, then $2b + 2w = 20$ or $b + w... | 0.625 | 4,407.625 | 3,244.9 | 6,345.5 |
Given the odd function $f(x)$ that is increasing on the interval $[3,7]$ and has a minimum value of $5$, determine the behavior of $f(x)$ on the interval $[-7,-3]$. | -5 | 0 | 6,324.125 | -1 | 6,324.125 | |
Compute the number of integers \(n \in\{1,2, \ldots, 300\}\) such that \(n\) is the product of two distinct primes, and is also the length of the longest leg of some nondegenerate right triangle with integer side lengths. | 13 | Let \(n=p \cdot q\) for primes \(p<q\). If \(n\) is the second largest side of a right triangle there exist integers \(c, a\) such that \(a<p q\) and \((p q)^{2}=c^{2}-a^{2}=(c-a)(c+a)\). Since \(c-a<c+a\) there are three cases for the values of \(c-a, c+a\), and in each case we determine when \(a<p q\). (a) \(c-a=1\) ... | 0 | 8,192 | -1 | 8,192 |
$\left(\frac{(x+1)^{2}(x^{2}-x+1)^{2}}{(x^{3}+1)^{2}}\right)^{2}\cdot\left(\frac{(x-1)^{2}(x^{2}+x+1)^{2}}{(x^{3}-1)^{2}}\right)^{2}$ equals: | 1 | 1. **Simplify the given expression**:
Start by simplifying the expression inside the parentheses:
\[
\left(\frac{(x+1)^2(x^2-x+1)^2}{(x^3+1)^2}\right)^2 \cdot \left(\frac{(x-1)^2(x^2+x+1)^2}{(x^3-1)^2}\right)^2
\]
We can simplify each fraction by pulling out the squares:
\[
\left(\frac{(x+1)(x^2-x... | 1 | 3,136.25 | 3,136.25 | -1 |
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