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 |
|---|---|---|---|---|---|---|
In a scalene triangle with integer side lengths $a, b, c$, the following relation holds. What is the smallest height of the triangle?
$$
\frac{a^{2}}{c}-(a-c)^{2}=\frac{b^{2}}{c}-(b-c)^{2}
$$ | 2.4 | 0 | 7,194.125 | -1 | 7,194.125 | |
In triangle $XYZ$, $XY=25$ and $XZ=14$. The angle bisector of $\angle X$ intersects $YZ$ at point $E$, and point $N$ is the midpoint of $XE$. Let $Q$ be the point of the intersection of $XZ$ and $YN$. The ratio of $ZQ$ to $QX$ can be expressed in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive ... | 39 | 0 | 5,827.6875 | -1 | 5,827.6875 | |
What is the smallest possible area, in square units, of a right triangle with two sides measuring $7$ units and $8$ units? | \frac{7\sqrt{15}}{2} | 0 | 4,949.375 | -1 | 4,949.375 | |
Given $m= \int_{0}^{\frac{\pi}{2}} 9\cos xdx$, then the constant term in the expansion of $\left( \frac{1}{\sqrt{x}}-x\right)^{m}$ is ______. | -84 | 1 | 2,964.25 | 2,964.25 | -1 | |
In $\triangle ABC, AB = 10, BC = 9, CA = 8$ and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. Find the length of $PC$. | 16 | 0.875 | 4,612.8125 | 4,101.5 | 8,192 | |
For which value of \( x \) is \( x^3 < x^2 \)? | \frac{3}{4} | If \( x = 1 \), then \( x^2 = 1 \) and \( x^3 = 1 \) and so \( x^3 = x^2 \). If \( x > 1 \), then \( x^3 \) equals \( x \) times \( x^2 \); since \( x > 1 \), then \( x \) times \( x^2 \) is greater than \( x^2 \) and so \( x^3 > x^2 \). Therefore, if \( x \) is positive with \( x^3 < x^2 \), we must have \( 0 < x < 1 ... | 0 | 4,485.5625 | -1 | 4,485.5625 |
What is the volume of a cube whose surface area is twice that of a cube with volume 1? | 2\sqrt{2} | 1. **Understanding the problem**: We need to find the volume of a cube whose surface area is twice that of another cube with a volume of 1.
2. **Calculating the surface area of the first cube**:
- The volume of the first cube is given as 1 cubic unit.
- The formula for the volume of a cube is $V = s^3$, where $s... | 1 | 1,865.4375 | 1,865.4375 | -1 |
The sides of an isosceles triangle are $\cos x,$ $\cos x,$ and $\cos 7x,$ and its vertex angle is $2x.$ (All angle measurements are in degrees.) Enter all possible values of $x,$ separated by commas. | 10^\circ, 50^\circ, 54^\circ | 0 | 7,973.0625 | -1 | 7,973.0625 | |
Consider the sequence: $x_1=19,x_2=95,x_{n+2}=\text{lcm} (x_{n+1},x_n)+x_n$, for $n>1$, where $\text{lcm} (a,b)$ means the least common multiple of $a$ and $b$. Find the greatest common divisor of $x_{1995}$ and $x_{1996}$. | 19 |
Consider the sequence given by \( x_1 = 19, x_2 = 95, x_{n+2} = \text{lcm}(x_{n+1}, x_n) + x_n \) for \( n > 1 \). We are tasked with finding the greatest common divisor (GCD) of \( x_{1995} \) and \( x_{1996} \).
First, observe that the relationship between consecutive terms in the sequence needs to be investigated ... | 0.5625 | 6,454.4375 | 5,103 | 8,192 |
We have 21 pieces of type $\Gamma$ (each formed by three small squares). We are allowed to place them on an $8 \times 8$ chessboard (without overlapping, so that each piece covers exactly three squares). An arrangement is said to be maximal if no additional piece can be added while following this rule. What is the smal... | 16 | 0.0625 | 8,014.1875 | 7,581 | 8,043.066667 | |
Given the function $y=ax^2+bx+c$, where $a, b, c \in \{0, 1, 2, 3, 4\}$, calculate the total number of different quadratic functions. | 100 | 0.9375 | 2,905.6875 | 2,997.533333 | 1,528 | |
Find the positive real number $x$ such that $\lfloor x \rfloor \cdot x = 54$. Express $x$ as a decimal. | 7.714285714285714 | 0 | 5,022.6875 | -1 | 5,022.6875 | |
In bag A, there are 3 white balls and 2 red balls, while in bag B, there are 2 white balls and 4 red balls. If a bag is randomly chosen first, and then 2 balls are randomly drawn from that bag, the probability that the second ball drawn is white given that the first ball drawn is red is ______. | \frac{17}{32} | 0.1875 | 6,860.875 | 5,472 | 7,181.384615 | |
A rectangular room has a perimeter of 42 feet and an area of 108 square feet. What is the length, in feet, of the shorter side? | 9 | 1 | 1,881.375 | 1,881.375 | -1 | |
Suppose we need to divide 12 dogs into three groups, where one group contains 4 dogs, another contains 6 dogs, and the last contains 2 dogs. How many ways can we form the groups so that Rover is in the 4-dog group and Spot is in the 6-dog group? | 2520 | 0.4375 | 7,159 | 6,016.571429 | 8,047.555556 | |
The value of $10^{\log_{10}7}$ is: | 7 | 1. **Understanding the Expression**: The expression given is $10^{\log_{10}7}$. This can be interpreted using the definition of logarithms. Recall that if $b^y = x$, then $\log_b x = y$.
2. **Applying the Definition of Logarithms**: Here, $\log_{10}7 = x$ implies that $10^x = 7$. This is because the logarithm $\log_{... | 1 | 2,161.0625 | 2,161.0625 | -1 |
Given the hyperbola $\frac {x^{2}}{16}- \frac {y^{2}}{9}=1$, and a chord AB with a length of 6 connected to the left focus F₁, calculate the perimeter of △ABF₂ (F₂ being the right focus). | 28 | 0.25 | 7,900.875 | 8,042 | 7,853.833333 | |
Five cards with different numbers are given: $-5$, $-4$, $0$, $+4$, $+6$. Two cards are drawn from them. The smallest quotient obtained by dividing the numbers on these two cards is ____. | -\dfrac{3}{2} | 0.375 | 4,166.5 | 5,666 | 3,266.8 | |
Each cell of a $100 \times 100$ board is painted in either blue or white. We call a cell balanced if it has an equal number of blue and white neighboring cells. What is the maximum number of balanced cells that can be found on the board? (Cells are considered neighbors if they share a side.) | 9608 | 0 | 8,192 | -1 | 8,192 | |
Given the hyperbola $x^{2}- \frac{y^{2}}{24}=1$, let the focal points be F<sub>1</sub> and F<sub>2</sub>, respectively. If P is a point on the left branch of the hyperbola such that $|PF_{1}|=\frac{3}{5}|F_{1}F_{2}|$, find the area of triangle $\triangle PF_{1}F_{2}$. | 24 | 0.875 | 4,252.5625 | 3,689.785714 | 8,192 | |
Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC = 8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI = EJ = EK = 2$. A solid $S$ is obtained by drilling a tunnel... | 417 | 0 | 8,192 | -1 | 8,192 | |
The average age of Andras, Frances, and Gerta is 22 years. Given that Andras is 23 and Frances is 24, what is Gerta's age? | 19 | Since the average of the three ages is 22, the sum of the three ages is \( 3 \times 22 = 66 \). Since Andras' age is 23 and Frances' age is 24, then Gerta's age is \( 66 - 23 - 24 = 19 \). | 1 | 440.9375 | 440.9375 | -1 |
How many distinct arrangements of the letters in the word "balloon" are there, considering the repeated 'l' and 'o'? | 1260 | 0.375 | 1,918.9375 | 1,735.5 | 2,029 | |
A triangle has three different integer side lengths and a perimeter of 30 units. What is the maximum length of any one side? | 14 | 0.9375 | 4,583.1875 | 4,342.6 | 8,192 | |
Let $R = (8,6)$. The lines whose equations are $8y = 15x$ and $10y = 3x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\overline{PQ}$. The length of $PQ$ equals $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
| 67 | 0.875 | 3,632.8125 | 2,981.5 | 8,192 | |
Dean scored a total of 252 points in 28 basketball games. Ruth played 10 fewer games than Dean. Her scoring average was 0.5 points per game higher than Dean's scoring average. How many points, in total, did Ruth score? | 171 | 0.9375 | 1,815.8125 | 1,416.733333 | 7,802 | |
Sandy and Sam each selected a positive integer less than 250. Sandy's number is a multiple of 15, and Sam's number is a multiple of 20. What is the probability that they selected the same number? Express your answer as a common fraction. | \frac{1}{48} | 1 | 3,067.1875 | 3,067.1875 | -1 | |
In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $(x-1)^{2}+y^{2}=1$, and the parametric equation of curve $C_{2}$ is:
$$
\begin{cases}
x= \sqrt {2}\cos \theta \\
y=\sin \theta
\end{cases}
$$
($\theta$ is the parameter), with $O$ as the pole and the positive half-axis of $x$ as the polar axi... | \sqrt {3}- \frac {2 \sqrt {10}}{5} | 0 | 7,553.25 | -1 | 7,553.25 | |
The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$, is precisely: | 0, 12 | 1. **Testing $m = 0$:**
Substitute $m = 0$ into the expression $x^2 + 3xy + x + my - m$:
\[
x^2 + 3xy + x
\]
Factoring out $x$:
\[
x(x + 3y + 1)
\]
This expression has two factors, $(x)$ and $(x + 3y + 1)$, both linear in $x$ and $y$ with integer coefficients. Thus, $m = 0$ is a valid solut... | 0 | 8,192 | -1 | 8,192 |
It is known that P and Q are two points on the unit circle centered at the origin O, and they are located in the first and fourth quadrants, respectively. The x-coordinate of point P is $\frac{4}{5}$, and the x-coordinate of point Q is $\frac{5}{13}$. Then, $\cos \angle POQ = \_\_\_\_\_\_$. | \frac{56}{65} | 0 | 3,735.3125 | -1 | 3,735.3125 | |
Given a triangle \( \triangle ABC \) with \(\angle B = 90^\circ\). The incircle touches sides \(BC\), \(CA\), and \(AB\) at points \(D\), \(E\), and \(F\) respectively. Line \(AD\) intersects the incircle at another point \(P\), and \(PF \perp PC\). Find the ratio of the side lengths of \(\triangle ABC\). | 3:4:5 | 0 | 8,192 | -1 | 8,192 | |
Given that $\{a_{n}\}$ is an arithmetic sequence, with the sum of its first $n$ terms denoted as $S_{n}$, and $a_{4}=-3$, choose one of the following conditions as known: <br/>$(Ⅰ)$ The arithmetic sequence $\{a_{n}\}$'s general formula; <br/>$(Ⅱ)$ The minimum value of $S_{n}$ and the value of $n$ when $S_{n}$ reaches i... | -30 | 0.125 | 7,401.5625 | 6,293.5 | 7,559.857143 | |
Simplify first, then evaluate: $(1- \frac {2}{x+1})÷ \frac {x^{2}-x}{x^{2}-1}$, where $x=-2$. | \frac {3}{2} | 0.9375 | 2,307.875 | 2,312.066667 | 2,245 | |
With the rapid development of the "Internet + transportation" model, "shared bicycles" have appeared successively in many cities. In order to understand the satisfaction of users in a certain area with the services provided, a certain operating company randomly surveyed 10 users and obtained satisfaction ratings of 92,... | 50\% | 0.1875 | 7,853.25 | 8,052.333333 | 7,807.307692 | |
Let the positive numbers \( x \) and \( y \) satisfy \( x^{3} + y^{3} = x - y \). Find the maximum value of the real number \( \lambda \) such that \( x^{2} + \lambda y^{2} \leq 1 \) always holds. | 2 + 2\sqrt{2} | 0.375 | 7,351.375 | 5,950.333333 | 8,192 | |
Simplify $\sqrt{9800}$. | 70\sqrt{2} | 1 | 2,171.625 | 2,171.625 | -1 | |
A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and
\[a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\]for all $n \geq 3$. Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q$? | 8078 | 1. **Base Case Verification**:
- For $n=1$, $a_1 = 1 = \frac{3}{3}$, which matches the formula $\frac{3}{4n-1}$ when $n=1$.
- For $n=2$, $a_2 = \frac{3}{7}$, which matches the formula $\frac{3}{4n-1}$ when $n=2$.
2. **Inductive Hypothesis**:
- Assume that for some $m \geq 2$, $a_{m-1} = \frac{3}{4(m-1)-1}$ ... | 0.875 | 4,841.125 | 4,362.428571 | 8,192 |
Let $\mathrm {Q}$ be the product of the roots of $z^8+z^6+z^4+z^3+z+1=0$ that have a positive imaginary part, and suppose that $\mathrm {Q}=s(\cos{\phi^{\circ}}+i\sin{\phi^{\circ}})$, where $0<s$ and $0\leq \phi <360$. Find $\phi$. | 180 | 0 | 8,192 | -1 | 8,192 | |
Find all ordered triples of primes $(p, q, r)$ such that \[ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1. \] [i]Reid Barton[/i] | (2, 3, 5), (2, 5, 3), (3, 2, 5), (3, 5, 2), (5, 2, 3), (5, 3, 2) |
We are tasked with finding all ordered triples of primes \((p, q, r)\) such that
\[ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1. \]
Assume \( p = \min(p, q, r) \) and \( p \neq 2 \). Note the following conditions:
\[
\begin{align*}
\text{ord}_p(q) &\mid 2r \implies \text{ord}_p(q) = 2 \text{ or } 2r, \... | 0 | 8,192 | -1 | 8,192 |
Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and $P$ is a point on the ellipse, with $\overrightarrow{PF_{1}} \cdot (\overrightarrow{OF_{1}} + \overrightarrow{OP}) = 0$, if $|\overrightarrow{PF_{1}}| = \sqrt{2}|\overrightarrow{PF_{2}}|$, de... | \sqrt{6} - \sqrt{3} | 0.6875 | 6,941.3125 | 6,599.454545 | 7,693.4 | |
In a jumbo bag of bows, $\frac{1}{5}$ are red, $\frac{1}{2}$ are blue, $\frac{1}{10}$ are green and the remaining 30 are white. How many of the bows are green? | 15 | 1 | 1,720.8125 | 1,720.8125 | -1 | |
Compute $\frac{x}{w}$ if $w \neq 0$ and $\frac{x+6 y-3 z}{-3 x+4 w}=\frac{-2 y+z}{x-w}=\frac{2}{3}$. | \frac{2}{3} | We have $x+6 y-3 z=\frac{2}{3}(-3 x+4 w)$ and $-2 y+z=\frac{2}{3}(x-w)$, so $$\frac{x}{w}=\frac{(x+6 y-3 z)+3(-2 y+z)}{(-3 x+4 w)+3(x-w)}=\frac{\frac{2}{3}(-3 x+4 w)+3 \cdot \frac{2}{3}(x-w)}{(-3 x+4 w)+3(x-w)}=\frac{\frac{2}{3}[(-3 x+4 w)+3(x-w)]}{(-3 x+4 w)+3(x-w)}=\frac{2}{3}$$ | 0.875 | 4,834.75 | 4,560.571429 | 6,754 |
How many ways are there to put 4 balls in 3 boxes if the balls are not distinguishable but the boxes are? | 15 | 0.8125 | 4,268.6875 | 3,363.307692 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\sin \frac{B}{2}-\cos \frac{B}{2}= \frac{1}{5}$.
(I) Find the value of $\cos B$;
(II) If $b^{2}-a^{2}=ac$, find the value of $\frac{\sin C}{\sin A}$. | \frac{11}{25} | 0.875 | 3,964.5 | 3,757.428571 | 5,414 | |
Given that vertex E of right triangle ABE, where AE=BE, is in the interior of unit square ABCD, let R be the region consisting of all points inside ABCD and outside triangle ABE whose distance from AD is between 1/4 and 1/2. Calculate the area of R. | \frac{1}{8} | 0 | 7,465.375 | -1 | 7,465.375 | |
If the product of the first 2011 positive integers $1 \times 2 \times \ldots \times 2011$ is divisible by $2010^k$, then the maximum value of the positive integer $k$ is. | 30 | 0.75 | 4,330.8125 | 4,417.166667 | 4,071.75 | |
Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . Compute $m+n$ .
*Proposed by Nathan Xiong* | 65 | 0.0625 | 7,926 | 3,936 | 8,192 | |
Given the parametric equations of curve $C_1$ are $$\begin{cases} x=2\cos\theta \\ y=\sin\theta\end{cases}(\theta \text{ is the parameter}),$$ and the parametric equations of curve $C_2$ are $$\begin{cases} x=-3+t \\ y= \frac {3+3t}{4}\end{cases}(t \text{ is the parameter}).$$
(1) Convert the parametric equations of ... | \frac {12-2 \sqrt {13}}{5} | 0 | 6,865.8125 | -1 | 6,865.8125 | |
How many positive integers smaller than $1{,}000{,}000{,}000$ are powers of $2$, but are not powers of $16$? | 23 | 0.0625 | 6,268.625 | 6,356 | 6,262.8 | |
Given that the prime factorization of a positive integer \( A \) can be written as \( A = 2^{\alpha} \times 3^{\beta} \times 5^{\gamma} \), where \( \alpha, \beta, \gamma \) are natural numbers. If half of \( A \) is a perfect square, one-third of \( A \) is a perfect cube, and one-fifth of \( A \) is a fifth power of ... | 31 | 0.9375 | 3,961.1875 | 3,679.133333 | 8,192 | |
In a press conference before a championship game, ten players from four teams will be taking questions. The teams are as follows: three Celtics, three Lakers, two Warriors, and two Nuggets. If teammates insist on sitting together and one specific Warrior must sit at the end of the row on the left, how many ways can the... | 432 | 0.0625 | 7,538.875 | 6,968 | 7,576.933333 | |
Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\clubsuit$ and the second card dealt is a $\heartsuit$? | \frac{13}{204} | 0.5625 | 6,183.125 | 5,144.666667 | 7,518.285714 | |
Find the last two digits of the sum $$6! + 1 + 12! + 1 + 18! + 1 + \cdots + 96! + 1.$$ | 36 | 0.75 | 4,523.4375 | 3,824.916667 | 6,619 | |
In the ancient Chinese mathematical work "Nine Chapters on the Mathematical Art," there is a problem as follows: "There is a golden rod in China, five feet long. When one foot is cut from the base, it weighs four catties. When one foot is cut from the end, it weighs two catties. How much does each foot weigh in success... | 15 | 0 | 8,000.75 | -1 | 8,000.75 | |
A clock has a second, minute, and hour hand. A fly initially rides on the second hand of the clock starting at noon. Every time the hand the fly is currently riding crosses with another, the fly will then switch to riding the other hand. Once the clock strikes midnight, how many revolutions has the fly taken? $\emph{(O... | 245 | 0 | 7,883.9375 | -1 | 7,883.9375 | |
The value of $\frac{1}{16}a^0+\left (\frac{1}{16a} \right )^0- \left (64^{-\frac{1}{2}} \right )- (-32)^{-\frac{4}{5}}$ is: | 1 | We start by simplifying each term in the expression $\frac{1}{16}a^0+\left (\frac{1}{16a} \right )^0- \left (64^{-\frac{1}{2}} \right )- (-32)^{-\frac{4}{5}}$.
1. **Simplify $\frac{1}{16}a^0$:**
Since any non-zero number raised to the power of 0 is 1, we have $a^0 = 1$. Thus,
\[
\frac{1}{16}a^0 = \frac{1}{16}... | 0 | 2,685.375 | -1 | 2,685.375 |
Given two lines $l_1: y=a$ and $l_2: y= \frac {18}{2a+1}$ (where $a>0$), $l_1$ intersects the graph of the function $y=|\log_{4}x|$ from left to right at points A and B, and $l_2$ intersects the graph of the function $y=|\log_{4}x|$ from left to right at points C and D. Let the projection lengths of line segments AC an... | \frac {5}{2} | 0.5625 | 6,915.6875 | 5,923 | 8,192 | |
Which pair of numbers does NOT have a product equal to $36$? | {\frac{1}{2},-72} | To solve this problem, we need to calculate the product of each pair of numbers given in the options and check which pair does not have a product equal to $36$.
1. **Option A: $\{-4,-9\}$**
\[
(-4) \times (-9) = 36
\]
Here, the product is $36$.
2. **Option B: $\{-3,-12\}$**
\[
(-3) \times (-12) = 36... | 0 | 6,754.6875 | -1 | 6,754.6875 |
Given a revised graph for Lambda Corp., the number of employees at different tenure periods is represented with the following marks:
- Less than 1 year: 3 marks
- 1 to less than 2 years: 6 marks
- 2 to less than 3 years: 5 marks
- 3 to less than 4 years: 4 marks
- 4 to less than 5 years: 2 marks
- 5 to less than 6 year... | 24.14\% | 0.5625 | 5,406.5625 | 5,252.555556 | 5,604.571429 | |
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$ | \frac{7}{2} | 0.875 | 794.875 | 822.071429 | 604.5 | |
Let $a_1, a_2, a_3,\dots$ be an increasing arithmetic sequence of integers. If $a_4a_5 = 13$, what is $a_3a_6$? | -275 | 1 | 3,383.375 | 3,383.375 | -1 | |
The equation of one of the axes of symmetry for the graph of the function $f(x)=\sin \left(x- \frac {\pi}{4}\right)$ $(x\in\mathbb{R})$ can be found. | -\frac{\pi}{4} | 0 | 4,144 | -1 | 4,144 | |
Consider a sequence $x_1,x_2,\cdots x_{12}$ of real numbers such that $x_1=1$ and for $n=1,2,\dots,10$ let \[ x_{n+2}=\frac{(x_{n+1}+1)(x_{n+1}-1)}{x_n}. \] Suppose $x_n>0$ for $n=1,2,\dots,11$ and $x_{12}=0$ . Then the value of $x_2$ can be written as $\frac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers ... | 622 | 0 | 8,192 | -1 | 8,192 | |
Sara makes a staircase out of toothpicks as shown:
[asy] size(150); defaultpen(linewidth(0.8)); path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45); for(int i=0;i<=2;i=i+1) { for(int j=0;j<=3-i;j=j+1) { filldraw(shift((i,j))*h,black); filldraw(shift((j,i))*v,black); } } [/asy]
This is a 3-step stairca... | 12 |
To solve this problem, we need to determine the number of steps, $n$, in a staircase that uses 180 toothpicks. We will derive a formula for the number of toothpicks used in a staircase with $n$ steps and then solve for $n$.
#### Step 1: Establish the pattern
Each step in the staircase consists of a horizontal and a v... | 0.4375 | 6,493.125 | 4,308.857143 | 8,192 |
What is the tens digit of $7^{2011}$? | 4 | 1. **Identify the goal:** We need to find the tens digit of $7^{2011}$. This is equivalent to finding $7^{2011} \mod 100$ because the tens digit of a number is the second last digit from the right.
2. **Use modular arithmetic to simplify the problem:** We start by calculating the powers of $7$ modulo $100$ to identify... | 1 | 3,572.1875 | 3,572.1875 | -1 |
In a right-angled triangle, the lengths of the two legs are 12 and 5, respectively. Find the length of the hypotenuse and the height from the right angle to the hypotenuse. | \frac{60}{13} | 1 | 2,058.3125 | 2,058.3125 | -1 | |
A positive integer is called a perfect power if it can be written in the form \(a^b\), where \(a\) and \(b\) are positive integers with \(b \geq 2\). The increasing sequence \(2, 3, 5, 6, 7, 10, \ldots\) consists of all positive integers which are not perfect powers. Calculate the sum of the squares of the digits of th... | 21 | 0 | 8,054.25 | -1 | 8,054.25 | |
The graph of the function in the form \( y=\frac{b}{|x|-a} \) (where \( a, b > 0 \)) resembles the Chinese character "唄". It is referred to as the "唄 function", and the point symmetric to its intersection with the y-axis about the origin is called the "目 point". A circle with its center at the 明 point that intersects t... | 3\pi | 0.25 | 7,443.5 | 6,781.25 | 7,664.25 | |
In the right parallelopiped $ABCDA^{\prime}B^{\prime}C^{\prime}D^{\prime}$ , with $AB=12\sqrt{3}$ cm and $AA^{\prime}=18$ cm, we consider the points $P\in AA^{\prime}$ and $N\in A^{\prime}B^{\prime}$ such that $A^{\prime}N=3B^{\prime}N$ . Determine the length of the line segment $AP$ such that for any posit... | 27/2 | 0.5625 | 5,556.9375 | 3,920.666667 | 7,660.714286 | |
What is the greatest product obtainable from two integers whose sum is 246? | 15129 | 1 | 2,271.5625 | 2,271.5625 | -1 | |
In the cube ABCD-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>, the angle formed by the skew lines A<sub>1</sub>B and AC is \_\_\_\_\_\_°; the angle formed by the line A<sub>1</sub>B and the plane A<sub>1</sub>B<sub>1</sub>CD is \_\_\_\_\_\_\_\_\_°. | 30 | 0.1875 | 4,572.4375 | 6,095.666667 | 4,220.923077 | |
Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$. | 36^\circ |
Given five points \( A_1, A_2, A_3, A_4, \) and \( A_5 \) in the plane such that no three are collinear, we are tasked with determining the maximum possible minimum value for the angles \( \angle A_i A_j A_k \), where \( i, j, k \) are distinct integers between \(1\) and \(5\).
### Key Observations:
1. In a convex p... | 0.25 | 8,064.6875 | 7,780.25 | 8,159.5 |
In quadrilateral $EFGH$, $EF = 6$, $FG = 18$, $GH = 6$, and $HE = x$ where $x$ is an integer. Calculate the value of $x$. | 12 | 0.0625 | 7,240.25 | 8,192 | 7,176.8 | |
In trapezoid \(ABCD\), \(AD\) is parallel to \(BC\). \(\angle A = \angle D = 45^\circ\), while \(\angle B = \angle C = 135^\circ\). If \(AB = 6\) and the area of \(ABCD\) is 30, find \(BC\). | 2\sqrt{2} | 0.75 | 5,113.5625 | 5,295.166667 | 4,568.75 | |
Find the smallest positive integer $ K$ such that every $ K$-element subset of $ \{1,2,...,50 \}$ contains two distinct elements $ a,b$ such that $ a\plus{}b$ divides $ ab$. | 26 |
To find the smallest positive integer \( K \) such that every \( K \)-element subset of \( \{1, 2, \ldots, 50\} \) contains two distinct elements \( a \) and \( b \) such that \( a + b \) divides \( ab \), we need to analyze the properties of the set and the divisibility condition.
Consider the set \( \{1, 2, \ldots,... | 0 | 8,192 | -1 | 8,192 |
Let $k$ and $n$ be positive integers and let $$ S=\left\{\left(a_{1}, \ldots, a_{k}\right) \in \mathbb{Z}^{k} \mid 0 \leq a_{k} \leq \cdots \leq a_{1} \leq n, a_{1}+\cdots+a_{k}=k\right\} $$ Determine, with proof, the value of $$ \sum_{\left(a_{1}, \ldots, a_{k}\right) \in S}\binom{n}{a_{1}}\binom{a_{1}}{a_{2}} \cdots\... | \[
\binom{k+n-1}{k} = \binom{k+n-1}{n-1}
\] | Answer: $\binom{k+n-1}{k}=\binom{k+n-1}{n-1}$ Solution 1: Let $$ T=\left\{\left(b_{1}, \ldots, b_{n}\right) \mid 0 \leq b_{1}, \ldots, b_{n} \leq k, b_{1}+\cdots+b_{n}=k\right\} $$ The sum in question counts $|T|$, by letting $a_{i}$ be the number of $b_{j}$ that are at least $i$. By stars and bars, $|T|=\binom{k+n-1}{... | 0 | 8,124.125 | -1 | 8,124.125 |
In $\triangle ABC$, $P$ is a point on the side $BC$ such that $\overrightarrow{BP} = \frac{1}{2}\overrightarrow{PC}$. Points $M$ and $N$ lie on the line passing through $P$ such that $\overrightarrow{AM} = \lambda \overrightarrow{AB}$ and $\overrightarrow{AN} = \mu \overrightarrow{AC}$ where $\lambda, \mu > 0$. Find th... | \frac{8}{3} | 0.4375 | 7,039.25 | 5,557.142857 | 8,192 | |
In triangle $ABC$, $AB = 10$, $BC = 14$, and $CA = 16$. Let $D$ be a point in the interior of $\overline{BC}$. Let points $I_B$ and $I_C$ denote the incenters of triangles $ABD$ and $ACD$, respectively. The circumcircles of triangles $BI_BD$ and $CI_CD$ meet at distinct points $P$ and $D$. The maximum possible area of ... | 150 | 0 | 8,192 | -1 | 8,192 | |
A square is inscribed in the ellipse whose equation is $x^2 + 3y^2 = 3$. One vertex of the square is at $(0, 1)$, and one diagonal of the square lies along the y-axis. Determine the square of the length of each side of the square. | \frac{5}{3} - 2\sqrt{\frac{2}{3}} | 0 | 7,748.5 | -1 | 7,748.5 | |
The first three terms of a geometric progression are $\sqrt 3$, $\sqrt[3]3$, and $\sqrt[6]3$. What is the fourth term? | 1 | 1. **Identify the pattern in the terms**: The given terms of the geometric progression are $\sqrt{3}$, $\sqrt[3]{3}$, and $\sqrt[6]{3}$. We can rewrite these terms using exponents:
- $\sqrt{3} = 3^{1/2} = 3^{3/6}$
- $\sqrt[3]{3} = 3^{1/3} = 3^{2/6}$
- $\sqrt[6]{3} = 3^{1/6} = 3^{1/6}$
2. **Recognize the seque... | 1 | 3,137.625 | 3,137.625 | -1 |
What is the area of the shaded region in the figure below? Round your answer to the nearest square centimeter.
[asy]
draw((0,0) -- (3,0) -- (3,3) -- (0,3)--cycle) ; draw((3,0)-- (12,0) -- (12,9) -- (3, 9)--cycle);
label ( "3 cm", (0,1.5), W); label ( "3 cm", (1.5 ,0), S);
label ( "9 cm", (3+9/2 ,0), S);label ( "9 cm"... | 30 | 1 | 3,796.1875 | 3,796.1875 | -1 | |
Call an positive integer almost-square if it can be written as $a \cdot b$, where $a$ and $b$ are integers and $a \leq b \leq \frac{4}{3} a$. How many almost-square positive integers are less than or equal to 1000000 ? Your score will be equal to $25-65 \frac{|A-C|}{\min (A, C)}$. | 130348 | To get a good estimate for the number of almost-square integers, note that any number of the form $a \cdot b$, with $b \leq \frac{4}{3} a$, will be by definition almost-square. Let's assume that it's relatively unlikely that a number is almost-square in more than one way. Then the number of almostsquare numbers less th... | 0 | 8,192 | -1 | 8,192 |
A certain real estate property is holding a lottery for homebuyers, with the following rules: For homeowners who purchase the property, they can randomly draw 2 balls from box $A$, which contains 2 red balls and 2 white balls, and 2 balls from box $B$, which contains 3 red balls and 2 white balls. If all 4 balls drawn ... | 3,675 | 0 | 7,675.125 | -1 | 7,675.125 | |
If the line $L$ in the $xy$-plane has half the slope and twice the $y$-intercept of the line $y = \frac{2}{3} x + 4$, then an equation for $L$ is: | $y = \frac{1}{3} x + 8$ | 1. **Identify the slope and $y$-intercept of the given line:**
The given line equation is $y = \frac{2}{3}x + 4$. Here, the slope ($m$) is $\frac{2}{3}$ and the $y$-intercept ($c$) is $4$.
2. **Determine the slope and $y$-intercept of the new line $L$:**
According to the problem, the new line $L$ has half th... | 0 | 1,277.625 | -1 | 1,277.625 |
Let $G$ be a simple graph with 100 vertices such that for each vertice $u$, there exists a vertice $v \in N \left ( u \right )$ and $ N \left ( u \right ) \cap N \left ( v \right ) = \o $. Try to find the maximal possible number of edges in $G$. The $ N \left ( . \right )$ refers to the neighborhood. | 3822 |
Let \( G \) be a simple graph with 100 vertices such that for each vertex \( u \), there exists a vertex \( v \in N(u) \) and \( N(u) \cap N(v) = \emptyset \). We aim to find the maximal possible number of edges in \( G \).
We claim that the maximal number of edges is \( \boxed{3822} \).
To prove this, we consider t... | 0 | 6,953.4375 | -1 | 6,953.4375 |
In right triangle $\triangle ABC$ with $\angle BAC = 90^\circ$, medians $\overline{AD}$ and $\overline{BE}$ are given such that $AD = 18$ and $BE = 24$. If $\overline{AD}$ is the altitude from $A$, find the area of $\triangle ABC$. | 432 | 0 | 5,997.375 | -1 | 5,997.375 | |
There are four groups of numbers with their respective averages specified as follows:
1. The average of all multiples of 11 from 1 to 100810.
2. The average of all multiples of 13 from 1 to 100810.
3. The average of all multiples of 17 from 1 to 100810.
4. The average of all multiples of 19 from 1 to 100810.
Among the... | 50413.5 | 0.4375 | 6,380.875 | 5,002 | 7,453.333333 | |
If lines $l_{1}$: $ax+2y+6=0$ and $l_{2}$: $x+(a-1)y+3=0$ are parallel, find the value of $a$. | -1 | 0.1875 | 5,890.5625 | 5,557.666667 | 5,967.384615 | |
The line \(y = -\frac{2}{3}x + 10\) crosses the \(x\)-axis at point \(P\) and the \(y\)-axis at point \(Q\). Point \(T(r, s)\) is on the segment \(PQ\). If the area of \(\triangle POQ\) is four times the area of \(\triangle TOP\), find the value of \(r + s\). | 13.75 | 0 | 4,502.375 | -1 | 4,502.375 | |
In the Cartesian coordinate system, it is known that the terminal side of an angle $\alpha$ with the origin as the vertex and the non-negative half-axis of the $x$-axis as the initial side passes through the point $(-3,-4)$.
$(1)$ Find the value of $\frac{sin\alpha}{tan\alpha}$;
$(2)$ Find the value of $\frac{sin(\... | \frac{3}{5} | 0.875 | 4,551.3125 | 4,290.714286 | 6,375.5 | |
Let \( x_{i} \in \mathbf{R} \), \( x_{i} \geqslant 0 \) for \( i=1,2,3,4,5 \), and \( \sum_{i=1}^{5} x_{i} = 1 \). Find the minimum value of \( \max \left\{x_{1}+x_{2}, x_{2}+x_{3}, x_{3}+x_{4}, x_{4} + x_{5}\right\} \). | \frac{1}{3} | 0.3125 | 7,449.3125 | 5,815.4 | 8,192 | |
$H O W, B O W$, and $D A H$ are equilateral triangles in a plane such that $W O=7$ and $A H=2$. Given that $D, A, B$ are collinear in that order, find the length of $B A$. | 11 | Note that $H \neq B$ since otherwise $D A B$ is an equilateral triangle. Let $M$ be the midpoint of $D A$, so $H B=7 \sqrt{3}$ and $H M=\sqrt{3}$, and $\angle H M B=90^{\circ}$. By the Pythagorean theorem, $$ B M=\sqrt{(7 \sqrt{3})^{2}-(\sqrt{3})^{2}}=12 $$ Then $B A=B M-A M=11$. | 0 | 8,192 | -1 | 8,192 |
A four-digit number $\overline{abcd}$ has the properties that $a + b + c + d = 26$, the tens digit of $b \cdot d$ equals $a + c$, and $bd - c^2$ is a multiple of 2. Find this four-digit number (provide justification). | 1979 | 0 | 7,887.1875 | -1 | 7,887.1875 | |
The sequence $\{a_n\}$ satisfies $a_{n+1}+(-1)^n a_n = 2n-1$. Find the sum of the first $80$ terms of $\{a_n\}$. | 3240 | 0.0625 | 7,578.375 | 5,270 | 7,732.266667 | |
In the Cartesian coordinate system, the coordinates of the two foci of an ellipse are $F_{1}(-2\sqrt{2},0)$ and $F_{2}(2\sqrt{2},0)$. The minimum distance from a point on ellipse $C$ to the right focus is $3-2\sqrt{2}$.
$(1)$ Find the equation of ellipse $C$;
$(2)$ Suppose a line with a slope of $-2$ intersects curv... | \dfrac{3}{2} | 0.5 | 7,493.0625 | 7,013.625 | 7,972.5 | |
Find the positive real number $x$ for which
\[5 \sqrt{1 + x} + 5 \sqrt{1 - x} = 7 \sqrt{2}.\] | \frac{7}{25} | 1 | 2,502.5625 | 2,502.5625 | -1 | |
A cuboctahedron is a polyhedron whose faces are squares and equilateral triangles such that two squares and two triangles alternate around each vertex. What is the volume of a cuboctahedron of side length 1? | 5 \sqrt{2} / 3 | We can construct a cube such that the vertices of the cuboctahedron are the midpoints of the edges of the cube. Let $s$ be the side length of this cube. Now, the cuboctahedron is obtained from the cube by cutting a tetrahedron from each corner. Each such tetrahedron has a base in the form of an isosceles right triangle... | 0 | 8,144.3125 | -1 | 8,144.3125 |
The three numbers $5, a, b$ have an average (mean) of 33. What is the average of $a$ and $b$? | 47 | Since $5, a, b$ have an average of 33, then $\frac{5+a+b}{3}=33$. Multiplying by 3, we obtain $5+a+b=3 \times 33=99$, which means that $a+b=94$. The average of $a$ and $b$ is thus equal to $\frac{a+b}{2}=\frac{94}{2}=47$. | 1 | 1,020.625 | 1,020.625 | -1 |
Given $\cos\alpha + \cos\beta = \frac{1}{2}$, $\sin\alpha + \sin\beta = \frac{\sqrt{3}}{2}$, then $\cos(\alpha - \beta) =$ \_\_\_\_\_\_. | -\frac{1}{2} | 1 | 3,681.5625 | 3,681.5625 | -1 | |
Given three points \(A, B, C\) forming a triangle with angles \(30^{\circ}\), \(45^{\circ}\), and \(105^{\circ}\). Two of these points are chosen, and the perpendicular bisector of the segment connecting them is drawn. The third point is then reflected across this perpendicular bisector to obtain a fourth point \(D\). ... | 12 | 0 | 7,650.0625 | -1 | 7,650.0625 |
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