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 |
|---|---|---|---|---|---|---|
Jane starts to make a list, in increasing order, of the positive integers that have a first digit of 2. She writes $2, 20, 21, 22, \ldots$ and wants to find the three-digit number formed by the 998th, 999th, and 1000th digits. Help her determine what those digits are. | 216 | 0.25 | 7,832.875 | 6,755.5 | 8,192 | |
Let $A = \{1, 2, 3, 4, 5, 6, 7\}$, and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$.
| 399 | 0 | 7,747.5625 | -1 | 7,747.5625 | |
Yannick has a bicycle lock with a 4-digit passcode whose digits are between 0 and 9 inclusive. (Leading zeroes are allowed.) The dials on the lock is currently set at 0000. To unlock the lock, every second he picks a contiguous set of dials, and increases or decreases all of them by one, until the dials are set to the ... | (12,2) | To simplify the solution, we instead consider the equivalent problem of reducing a passcode to 0000 using the given move. Given a passcode $a_{1} a_{2} a_{3} a_{4}$, define a differential of the passcode to be a quintuple ( $d_{1}, d_{2}, d_{3}, d_{4}, d_{5}$ ) such that $d_{i} \equiv a_{i}-a_{i-1}(\bmod 10)$ for $i=1,... | 0 | 8,192 | -1 | 8,192 |
Let $A_{1}, A_{2}, \ldots, A_{m}$ be finite sets of size 2012 and let $B_{1}, B_{2}, \ldots, B_{m}$ be finite sets of size 2013 such that $A_{i} \cap B_{j}=\emptyset$ if and only if $i=j$. Find the maximum value of $m$. | \binom{4025}{2012} | In general, we will show that if each of the sets $A_{i}$ contain $a$ elements and if each of the sets $B_{j}$ contain $b$ elements, then the maximum value for $m$ is $\binom{a+b}{a}$. Let $U$ denote the union of all the sets $A_{i}$ and $B_{j}$ and let $|U|=n$. Consider the $n$ ! orderings of the elements of $U$. Note... | 0 | 7,864.25 | -1 | 7,864.25 |
If \( a, b, c, d \) are four distinct positive integers such that \( a \times b \times c \times d = 2277 \), what is the maximum value of \( a + b + c + d \)? | 84 | 0.125 | 8,037.125 | 8,192 | 8,015 | |
Let \( x \) and \( y \) be real numbers, \( y > x > 0 \), such that
\[ \frac{x}{y} + \frac{y}{x} = 4. \]
Find the value of
\[ \frac{x + y}{x - y}. \] | \sqrt{3} | 0 | 4,536.25 | -1 | 4,536.25 | |
From 8 female students and 4 male students, 3 students are to be selected to participate in a TV program. Determine the number of different selection methods when the selection is stratified by gender. | 112 | 0.8125 | 1,946.9375 | 1,662.769231 | 3,178.333333 | |
In the expansion of $(x^{2}+1)^{2}(x-1)^{6}$, the coefficient of $x^{5}$ is ____. | -52 | 0.6875 | 7,407.625 | 7,051.090909 | 8,192 | |
Julio cuts off the four corners, or vertices, of a regular tetrahedron. How many vertices does the remaining shape have? | 12 | 0.75 | 5,591.0625 | 4,943.083333 | 7,535 | |
Consider a unit cube in a coordinate system with vertices $A(0,0,0)$, $A'(1,1,1)$, and other vertices placed accordingly. A regular octahedron has vertices placed at fractions $\frac{1}{3}$ and $\frac{2}{3}$ along the segments connecting $A$ with $A'$'s adjacent vertices and vice versa. Determine the side length of thi... | \frac{\sqrt{2}}{3} | 0 | 8,192 | -1 | 8,192 | |
Of the 50 students in the drama club, 36 take mathematics, 27 take physics and 20 students take both mathematics and physics. How many drama club students take neither mathematics nor physics? | 7 | 1 | 1,189.5 | 1,189.5 | -1 | |
For each \(i \in\{1, \ldots, 10\}, a_{i}\) is chosen independently and uniformly at random from \([0, i^{2}]\). Let \(P\) be the probability that \(a_{1}<a_{2}<\cdots<a_{10}\). Estimate \(P\). | 0.003679 | The probability that \(a_{2}>a_{1}\) is \(7/8\). The probability that \(a_{3}>a_{2}\) is \(7/9\). The probability that \(a_{4}>a_{3}\) is \(23/32\). The probability that \(a_{5}>a_{4}\) is \(17/25\). The probability that \(a_{6}>a_{5}\) is \(47/72\). The probability that \(a_{7}>a_{6}\) is \(31/49\). The probability th... | 0 | 8,192 | -1 | 8,192 |
A banquet has invited 44 guests. There are 15 identical square tables, each of which can seat 1 person per side. By appropriately combining the square tables (to form rectangular or square tables), ensure that all guests are seated with no empty seats. What is the minimum number of tables in the final arrangement? | 11 | 0.375 | 7,288.625 | 5,783 | 8,192 | |
(1) Given $\tan(\alpha+\beta)= \frac{2}{5}$ and $\tan\left(\beta- \frac{\pi}{4}\right)= \frac{1}{4}$, find the value of $\frac{\cos\alpha+\sin\alpha}{\cos\alpha-\sin\alpha}$;
(2) Given $\alpha$ and $\beta$ are acute angles, and $\cos(\alpha+\beta)= \frac{\sqrt{5}}{5}$, $\sin(\alpha-\beta)= \frac{\sqrt{10}}{10}$, find... | \frac{\pi}{4} | 0.8125 | 5,719.375 | 5,148.769231 | 8,192 | |
Given a sequence $\{a_{n}\}$ such that $a_{4}+a_{7}=2$, $a_{5}a_{6}=-8$. If $\{a_{n}\}$ is an arithmetic progression, then $a_{1}a_{10}=$____; if $\{a_{n}\}$ is a geometric progression, then $a_{1}+a_{10}=$____. | -7 | 0.625 | 5,555.625 | 4,655.7 | 7,055.5 | |
A fair six-sided die is rolled twice. Let $a$ and $b$ be the numbers obtained from the first and second roll respectively. Determine the probability that three line segments of lengths $a$, $b$, and $5$ can form an isosceles triangle. | \frac{7}{18} | 0.3125 | 7,749 | 7,684.6 | 7,778.272727 | |
Hagrid has 100 animals. Among these animals, each is either striped or spotted but not both, each has either wings or horns but not both, there are 28 striped animals with wings, there are 62 spotted animals, and there are 36 animals with horns. How many of Hagrid's spotted animals have horns? | 26 | Each of the animals is either striped or spotted, but not both. Since there are 100 animals and 62 are spotted, then there are $100 - 62 = 38$ striped animals. Each striped animal must have wings or a horn, but not both. Since there are 28 striped animals with wings, then there are $38 - 28 = 10$ striped animals with h... | 0.625 | 4,956.6875 | 3,648.6 | 7,136.833333 |
The price of an item is an integer number of yuan. With 100 yuan, you can buy up to 3 items. Person A and Person B each have a certain number of 100-yuan bills. The amount of money Person A has can buy at most 7 items, and the amount of money Person B has can buy at most 14 items. Together, they can buy 1 more item tha... | 27 | 0 | 7,736.5625 | -1 | 7,736.5625 | |
In the diagram, $D$ is on side $A C$ of $\triangle A B C$ so that $B D$ is perpendicular to $A C$. Also, $\angle B A C=60^{\circ}$ and $\angle B C A=45^{\circ}$. If the area of $\triangle A B C$ is $72+72 \sqrt{3}$, what is the length of $B D$? | 12 \sqrt[4]{3} | 0 | 7,933.75 | -1 | 7,933.75 | |
The table below shows the Gross Domestic Product (GDP) of China from 2012 to 2018 in trillion US dollars:
| Year | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 |
|------|------|------|------|------|------|------|------|
| Year Code $x$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| GDP $y$ (trillion US dollars) | 8.5 | 9.6 | 10.4 ... | 2028 | 0.25 | 6,320.9375 | 6,249 | 6,344.916667 | |
If $f(x)=\frac{16}{5+3x}$, what is the value of $\left[f^{-1}(2)\right]^{-2}$? | 1 | 1 | 2,228.6875 | 2,228.6875 | -1 | |
Riquinho distributed 1000,00 reais among his friends Antônio, Bernardo, and Carlos in the following manner: he gave, successively, 1 real to Antônio, 2 reais to Bernardo, 3 reais to Carlos, 4 reais to Antônio, 5 reais to Bernardo, etc. How much money did Bernardo receive? | 345 | 0.1875 | 7,161.9375 | 6,256.666667 | 7,370.846154 | |
How many positive factors of 72 are perfect cubes? | 2 | 1 | 2,417.375 | 2,417.375 | -1 | |
Given that the asymptote equation of the hyperbola $y^{2}+\frac{x^2}{m}=1$ is $y=\pm \frac{\sqrt{3}}{3}x$, find the value of $m$. | -3 | 0.6875 | 3,521.0625 | 4,062.818182 | 2,329.2 | |
Calculate the area, in square units, of the triangle formed by the $x$ and $y$ intercepts of the curve $y = (x-3)^2 (x+2) (x-1)$. | 45 | 0.125 | 5,208.5 | 5,679.5 | 5,141.214286 | |
Given an equilateral triangle $DEF$ with $DE = DF = EF = 8$ units and a circle with radius $4$ units tangent to line $DE$ at $E$ and line $DF$ at $F$, calculate the area of the circle passing through vertices $D$, $E$, and $F$. | \frac{64\pi}{3} | 0.3125 | 6,377.4375 | 4,729.2 | 7,126.636364 | |
Given $\sqrt{2.1}=1.449$ and $\sqrt{21}=4.573$, find the value of $\sqrt{21000}$. | 144.9 | 0.25 | 7,095.375 | 5,670 | 7,570.5 | |
Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents send him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns $16$ dollars per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper rout... | 25 | 1. **Identify the total amount of money Chandler needs:** Chandler wants to buy a mountain bike that costs $500.
2. **Calculate the total amount of birthday money Chandler received:**
- From his grandparents: $50$
- From his aunt: $35$
- From his cousin: $15$
- Total birthday money: $50 + 35 + 15 = 100$
3... | 1 | 1,273.5625 | 1,273.5625 | -1 |
Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $e = \dfrac{\sqrt{6}}{3}$, and the distance between the left focus and one endpoint of the minor axis is $\sqrt{3}$.
$(I)$ Find the standard equation of the ellipse;
$(II)$ Given the fixed point $E(-1, 0)$, if the line $y = kx... | \dfrac{7}{6} | 0.875 | 5,428.9375 | 5,116.5 | 7,616 | |
Find the largest positive integer $k$ such that $\phi ( \sigma ( 2^k)) = 2^k$ . ( $\phi(n)$ denotes the number of positive integers that are smaller than $n$ and relatively prime to $n$ , and $\sigma(n)$ denotes the sum of divisors of $n$ ). As a hint, you are given that $641|2^{32}+1$ . | 31 | 0.3125 | 7,928.875 | 7,411.2 | 8,164.181818 | |
Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x - 19,$ the remainder is 99, and when $P(x)$ is divided by $x - 99,$ the remainder is 19. What is the remainder when $P(x)$ is divided by $(x - 19)(x - 99)$? | -x + 118 | 1 | 1,607.25 | 1,607.25 | -1 | |
The quadratic $x^2 - 40x + 121$ can be written in the form $(x+b)^2+c$, where $b$ and $c$ are constants. What is $b+c$? | -299 | 0.9375 | 2,225.0625 | 2,237.6 | 2,037 | |
Let \( m \) and \( n \) (with \( m > n \)) be positive integers such that \( 70^2 \) divides \( 2023^m - 2023^n \). What is the smallest value of \( m+n \)? | 24 | 0.0625 | 8,128.3125 | 7,173 | 8,192 | |
The divisors of a natural number \( n \) (including \( n \) and 1) which has more than three divisors, are written in ascending order: \( 1 = d_{1} < d_{2} < \ldots < d_{k} = n \). The differences \( u_{1} = d_{2} - d_{1}, u_{2} = d_{3} - d_{2}, \ldots, u_{k-1} = d_{k} - d_{k-1} \) are such that \( u_{2} - u_{1} = u_{3... | 10 | 0 | 8,192 | -1 | 8,192 | |
By multiplying a natural number by the number that is one greater than it, the product takes the form $ABCD$, where $A, B, C, D$ are different digits. Starting with the number that is 3 less, the product takes the form $CABD$. Starting with the number that is 30 less, the product takes the form $BCAD$. Determine these ... | 8372 | 0 | 8,044.9375 | -1 | 8,044.9375 | |
Given a function $f(x) = m\ln{x} + nx$ whose tangent at point $(1, f(1))$ is parallel to the line $x + y - 2 = 0$, and $f(1) = -2$, where $m, n \in \mathbb{R}$,
(Ⅰ) Find the values of $m$ and $n$, and determine the intervals of monotonicity for the function $f(x)$;
(Ⅱ) Let $g(x)= \frac{1}{t}(-x^{2} + 2x)$, for a positi... | \frac{e(e - 2)}{e - 1} | 0.3125 | 7,890.5 | 7,321.8 | 8,149 | |
A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?
[asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(1,1/2,1/4); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white,... | \frac{7}{30} | 1. **Calculate the Volume:**
Each unit cube has a volume of $1$ cubic unit. Since there are seven unit cubes joined together, the total volume \( V \) is:
\[
V = 7 \times 1 = 7 \text{ cubic units}
\]
2. **Calculate the Surface Area:**
To find the surface area, we need to consider the exposed faces of ea... | 0.3125 | 6,856.5 | 4,900.6 | 7,745.545455 |
Given that tetrahedron $ABCD$ is inscribed in sphere $O$, and $AD$ is the diameter of sphere $O$. If triangles $\triangle ABC$ and $\triangle BCD$ are equilateral triangles with side length 1, what is the volume of tetrahedron $ABCD$? | $\frac{\sqrt{2}}{12}$ | 0 | 8,192 | -1 | 8,192 | |
Given two non-zero planar vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy the condition: for any $λ∈R$, $| \overrightarrow{a}-λ \overrightarrow{b}|≥slant | \overrightarrow{a}- \frac {1}{2} \overrightarrow{b}|$, then:
$(①)$ If $| \overrightarrow{b}|=4$, then $\overrightarrow{a}· \overrightarrow{b}=$ ______... | \sqrt {3} | 0 | 5,649.1875 | -1 | 5,649.1875 | |
What is the smallest possible area, in square units, of a right triangle with side lengths $7$ units and $10$ units? | 35 | 0 | 4,427.1875 | -1 | 4,427.1875 | |
A shooter's probability of hitting the 10, 9, and 8 rings in a single shot are respectively 0.2, 0.3, and 0.1. Express the probability that the shooter scores no more than 8 in a single shot as a decimal. | 0.5 | 0.4375 | 4,558.0625 | 3,129.142857 | 5,669.444444 | |
Given a triangle \(ABC\) with the midpoints of sides \(BC\), \(AC\), and \(AB\) denoted as \(D\), \(E\), and \(F\) respectively, it is known that the medians \(AD\) and \(BE\) are perpendicular to each other, with lengths \(\overline{AD} = 18\) and \(\overline{BE} = 13.5\). Calculate the length of the third median \(CF... | 22.5 | 0.1875 | 6,101.125 | 5,487.666667 | 6,242.692308 | |
Given complex numbers \(z_1 = -3 + 4i\) and \(z_2 = 2 - i\), find an equation in the form \(az + b\overline{z} = c\) where \(a\), \(b\), and \(c\) are some complex numbers that represents the line joining \(z_1\) and \(z_2\), and determine the product \(ab\). | 50 | 0 | 6,845.6875 | -1 | 6,845.6875 | |
Gretchen has ten socks, two of each color: red, blue, green, yellow, and purple. She randomly draws five socks. What is the probability that she has exactly two pairs of socks with the same color? | \frac{5}{42} | 0 | 6,129.3125 | -1 | 6,129.3125 | |
$|{\sqrt{2}-\sqrt{3}|-tan60°}+\frac{1}{\sqrt{2}}$. | -\frac{\sqrt{2}}{2} | 0 | 2,122 | -1 | 2,122 | |
A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width? | 2\sqrt{2}+2 |
Let's denote the side length of the square as $s$ and the width of the brush as $w$. We are given that half the area of the square is painted, so the painted area is $\frac{s^2}{2}$.
#### Step 1: Analyze the geometry
The brush sweeps along both diagonals of the square. Each diagonal divides the square into two equal ... | 0 | 7,147.0625 | -1 | 7,147.0625 |
In a "clearance game," the rules stipulate that in round \( n \), a dice is to be rolled \( n \) times. If the sum of the points of these \( n \) rolls is greater than \( 2^{n} \), the player clears the round.
(1) What is the maximum number of rounds a player can clear in this game?
(2) What is the probability that t... | \frac{100}{243} | 0.5625 | 5,407.9375 | 4,375.666667 | 6,735.142857 | |
Given in parallelogram $ABCD$, point $E$ is the midpoint of side $BC$. A point $F$ is arbitrarily chosen on side $AB$. The probability that the area ratio of $\triangle ADF$ to $\triangle BFE$ is not less than $1$ is ______. | \frac{2}{3} | 0.9375 | 4,954.25 | 4,738.4 | 8,192 | |
Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
| 097 | 0 | 8,052.625 | -1 | 8,052.625 | |
Five cards are lying on a table as shown.
\[\begin{matrix} & \qquad & \boxed{\tt{P}} & \qquad & \boxed{\tt{Q}} \\ \\ \boxed{\tt{3}} & \qquad & \boxed{\tt{4}} & \qquad & \boxed{\tt{6}} \end{matrix}\]
Each card has a letter on one side and a whole number on the other side. Jane said, "If a vowel is on one side of any c... | 3 | To solve this problem, we need to test Jane's statement: "If a vowel is on one side of any card, then an even number is on the other side." We need to find a counterexample to disprove this statement.
1. **Understanding the statement**: The statement can be interpreted using logical implication. If there is a vowel on... | 0.625 | 7,052.0625 | 6,368.1 | 8,192 |
Given \( f(x) = x^{5} + a_{1} x^{4} + a_{2} x^{3} + a_{3} x^{2} + a_{4} x + a_{5} \), and \( f(m) = 2017m \) for \( m = 1, 2, 3, 4 \), find \( f(10) - f(-5) \). | 75615 | 0.625 | 5,636.75 | 4,784.1 | 7,057.833333 | |
Square $EFGH$ has sides of length 4. Segments $EK$ and $EL$ divide the square's area into two equal parts. Calculate the length of segment $EK$. | 4\sqrt{2} | 0.0625 | 7,900.9375 | 3,535 | 8,192 | |
Arrange the sequence $\{2n+1\} (n\in\mathbb{N}^{*})$ sequentially in brackets with one number in the first bracket, two numbers in the second bracket, three numbers in the third bracket, four numbers in the fourth bracket, one number in the fifth bracket, and so on in a cycle, then calculate the sum of the numbers in t... | 2392 | 0 | 8,076.25 | -1 | 8,076.25 | |
There exist positive integers $A,B$ and $C$, with no common factor greater than $1$, such that
\[A \log_{200} 5 + B \log_{200} 2 = C.\]What is $A + B + C$? | 6 | 1. **Express the equation using logarithm properties**:
Given the equation:
\[
A \log_{200} 5 + B \log_{200} 2 = C
\]
We can use the change of base formula for logarithms, $\log_b a = \frac{\log_k a}{\log_k b}$, to rewrite the logarithms in terms of base 10 (or any common base, but base 10 is standard)... | 1 | 2,131.75 | 2,131.75 | -1 |
Angry reviews about the work of an online store are left by $80\%$ of dissatisfied customers (those who were poorly served in the store). Of the satisfied customers, only $15\%$ leave a positive review. A certain online store earned 60 angry and 20 positive reviews. Using this statistic, estimate the probability that t... | 0.64 | 0.4375 | 4,978.0625 | 3,914.857143 | 5,805 | |
Two passenger trains, A and B, are 150 meters and 200 meters long, respectively. They are traveling in opposite directions on parallel tracks. A passenger on train A measures that train B passes by his window in 10 seconds. How long does a passenger on train B see train A pass by his window in seconds? | 7.5 | 0.8125 | 2,501.5 | 1,963.538462 | 4,832.666667 | |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $| \overrightarrow {a} |=5$, $| \overrightarrow {b} |=3$, and $( \overrightarrow {a} - \overrightarrow {b} )(2 \overrightarrow {a} + 3 \overrightarrow {b} )=13$.
1. Find the cosine value of the angle between $\overrightarrow {a}$ and $\overrigh... | \sqrt{21} | 1 | 2,707 | 2,707 | -1 | |
Given a sequence $\{a\_n\}$ that satisfies $a\_1=1$ and $a\_n= \frac{2S\_n^2}{2S\_n-1}$ for $n\geqslant 2$, where $S\_n$ is the sum of the first $n$ terms of the sequence, find the value of $S\_{2016}$. | \frac{1}{4031} | 0.75 | 4,617.375 | 3,858 | 6,895.5 | |
A line passes through the distinct vectors $\mathbf{u}$ and $\mathbf{v}.$ For some value of $k$, the vector
\[k \mathbf{u} + \frac{5}{8} \mathbf{v}\] must also lie on the line. Find $k$. | \frac{3}{8} | 1 | 2,494.8125 | 2,494.8125 | -1 | |
I have a drawer with 6 forks, 6 spoons, and 6 knives in it. If I reach in and randomly remove three pieces of silverware, what is the probability that I get one fork, one spoon, and one knife? | \frac{9}{34} | 1 | 3,638.375 | 3,638.375 | -1 | |
The volume of the top portion of the water tower is equal to the volume of a sphere, which can be calculated using the formula $V = \frac{4}{3}\pi r^3$. The top portion of the real tower has a volume of 50,000 liters, so we can solve for the radius:
$\frac{4}{3}\pi r^3 = 50,000$
Simplifying, we get $r^3 = \frac{50,00... | 0.95 | 0.0625 | 1,445.5 | 1,103 | 1,468.333333 | |
A sphere has a volume of \( 288\pi \) cubic inches. Determine the surface area of the sphere. Also, if the sphere were to be perfectly cut in half, what would be the circumference of the flat circular surface of one of the halves? Express your answers in terms of \( \pi \). | 12\pi | 0.9375 | 1,223.625 | 1,259.2 | 690 | |
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(xy) = f(x)f(y) + f(f(x + y))$$
holds for all $x, y \in \mathbb{R}$. | f(x) = 0 \text{ and } f(x) = x - 1. |
We are tasked with finding all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfying the equation:
\[
f(xy) = f(x)f(y) + f(f(x+y))
\]
for all \( x, y \in \mathbb{R} \).
### Step 1: Consideration of Simple Cases
First, let us consider the case where \( x = 0 \):
\[
f(0) = f(0)f(y) + f(f(y))
\]
for all \... | 0 | 8,192 | -1 | 8,192 |
Let \( a < b < c < d < e \) be real numbers. Among the 10 sums of the pairs of these numbers, the least three are 32, 36, and 37, while the largest two are 48 and 51. Find all possible values of \( e \). | 27.5 | 0 | 7,926 | -1 | 7,926 | |
12 real numbers x and y satisfy \( 1 + \cos^2(2x + 3y - 1) = \frac{x^2 + y^2 + 2(x+1)(1-y)}{x-y+1} \). Find the minimum value of xy. | \frac{1}{25} | 0.5625 | 6,880.375 | 6,293.444444 | 7,635 | |
There are $N$ points marked on a plane. Any three of them form a triangle whose angles are expressible in degrees as natural numbers. What is the maximum possible $N$ for which this is possible? | 180 | 0.0625 | 8,182.75 | 8,044 | 8,192 | |
At the rally commemorating the 60th anniversary of the Chinese people's victory in the War of Resistance against Japan, two schools each send 3 representatives to speak in turns, criticizing the heinous crimes committed by the Japanese aggressors and praising the heroic deeds of the Chinese people in their struggle aga... | 72 | 0.3125 | 5,993.875 | 5,152.2 | 6,376.454545 | |
Add 78.1563 to 24.3981 and round to the nearest hundredth. | 102.55 | 1 | 356.4375 | 356.4375 | -1 | |
A retailer purchases a gadget at $50$ minus $10%$. He aims to sell the gadget at a gain of $25%$ on his cost price after offering a $15%$ discount on the marked price. Determine the gadget's marked price in dollars. | 66.18 | 0.875 | 3,979.5625 | 3,703.357143 | 5,913 | |
I have two 10-sided dice where each die has 3 gold sides, 4 silver sides, 2 diamond sides, and 1 rainbow side. If I roll both dice, what is the probability that they come up showing the same color or pattern? | \frac{3}{10} | 0.3125 | 6,367.8125 | 3,775 | 7,546.363636 | |
Triangle $PQR$ has side lengths $PQ=160, QR=300$, and $PR=240$. Lines $m_P, m_Q$, and $m_R$ are drawn parallel to $\overline{QR}, \overline{RP}$, and $\overline{PQ}$, respectively, such that the intersections of $m_P, m_Q$, and $m_R$ with the interior of $\triangle PQR$ are segments of lengths $75, 60$, and $20$, respe... | 155 | 0 | 8,192 | -1 | 8,192 | |
Given the function $f(x) = x^3 - 3x - 1$, if for any $x_1$, $x_2$ in the interval $[-3,2]$, it holds that $|f(x_1) - f(x_2)| \leq t$, then the minimum value of the real number $t$ is ______. | 20 | 1 | 2,653.6875 | 2,653.6875 | -1 | |
What is the least four-digit positive integer, with all different digits, that is divisible by each of its digits? | 1236 | 0 | 8,192 | -1 | 8,192 | |
If a four-digit natural number $\overline{abcd}$ has digits that are all different and not equal to $0$, and satisfies $\overline{ab}-\overline{bc}=\overline{cd}$, then this four-digit number is called a "decreasing number". For example, the four-digit number $4129$, since $41-12=29$, is a "decreasing number"; another ... | 8165 | 0.3125 | 7,315.125 | 6,323.6 | 7,765.818182 | |
Positive integers $a$ and $b$ satisfy the condition \[\log_2(\log_{2^a}(\log_{2^b}(2^{1000}))) = 0.\] Find the sum of all possible values of $a+b$. | 881 | To simplify, we write this logarithmic expression as an exponential one. Just looking at the first log, it has a base of 2 and an argument of the expression in parenthesis. Therefore, we can make 2 the base, 0 the exponent, and the argument the result. That means $\log_{2^a}(\log_{2^b}(2^{1000}))=1$ (because $2^0=1$). ... | 0.9375 | 3,900 | 4,044.6 | 1,731 |
Add $5.467$ and $3.92$ as a decimal. | 9.387 | 1 | 2,116.6875 | 2,116.6875 | -1 | |
In $\triangle XYZ$, angle XZY is a right angle. There are three squares constructed such that each side adjacent to angle XZY has a square on it. The sum of the areas of these three squares is 512 square centimeters. Also, XZ is 20% longer than ZY. What's the area of the largest square? | 256 | 1 | 3,520.875 | 3,520.875 | -1 | |
There are $10$ seats in each of $10$ rows of a theatre and all the seats are numbered. What is the probablity that two friends buying tickets independently will occupy adjacent seats? | \dfrac{1}{55} | 0.125 | 6,726.625 | 6,759 | 6,722 | |
In an isosceles triangle, the center of the inscribed circle divides the altitude in the ratio $17: 15$. The base is 60. Find the radius of this circle. | 7.5 | 0.1875 | 6,097.1875 | 5,498.666667 | 6,235.307692 | |
Point $P$ is inside triangle $\triangle ABC$. Line $AC$ intersects line $BP$ at $Q$, and line $AB$ intersects line $CP$ at $R$. Given that $AR = RB = CP$, and $CQ = PQ$, find $\angle BRC$. | 120 | 0 | 8,192 | -1 | 8,192 | |
For how many $n$ in $\{1, 2, 3, ..., 100 \}$ is the tens digit of $n^2$ odd? | 20 | 1. **Express $n$ in terms of its digits**: Let $n = 10a + b$, where $a$ is the tens digit and $b$ is the units digit of $n$. Then $n^2 = (10a + b)^2$.
2. **Expand $n^2$**:
\[
n^2 = (10a + b)^2 = 100a^2 + 20ab + b^2
\]
- The term $100a^2$ contributes only to the hundreds place and higher, so it does not af... | 0.25 | 7,951.625 | 7,230.5 | 8,192 |
A line segment begins at $(1, 3)$. It is 5 units long and ends at the point $(4, x)$ and $x > 0$. What is the value of $x$? | 7 | 1 | 1,397 | 1,397 | -1 | |
Let \(a\) be a positive real number. Find the value of \(a\) such that the definite integral
\[
\int_{a}^{a^2} \frac{\mathrm{d} x}{x+\sqrt{x}}
\]
achieves its smallest possible value. | 3 - 2\sqrt{2} | 0.75 | 6,017.75 | 5,620.333333 | 7,210 | |
Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2.$ Given that the length of the chord... | 405 | Call our desired length $x$. Note for any $X$ on $\overline{AB}$ and $Y$ on $\overline{O_1O_2}$ such that $\overline{XY}\perp\overline{AB}$ that the function $f$ such that $f(\overline{O_1Y})=\overline{XY}$ is linear. Since $(0,4)$ and $(14,10)$, we can quickly interpolate that $f(10)=\overline{O_3T}=\frac{58}{7}$. The... | 0.125 | 7,919.4375 | 7,496.5 | 7,979.857143 |
How many ways are there to list the numbers 1 to 10 in some order such that every number is either greater or smaller than all the numbers before it? | 512 | 0.1875 | 7,625.25 | 5,909 | 8,021.307692 | |
The prime factorization of 2160 is $2^4 \times 3^3 \times 5$. How many of its positive integer factors are perfect squares? | 6 | 1 | 2,009.1875 | 2,009.1875 | -1 | |
To rebuild homes after an earthquake for disaster relief, in order to repair a road damaged during the earthquake, if Team A alone takes 3 months to complete the work, costing $12,000 per month; if Team B alone takes 6 months to complete the work, costing $5,000 per month. How many months will it take for Teams A and B... | 34,000 | 0.3125 | 726.3125 | 739 | 720.545455 | |
Group the set of positive odd numbers {1, 3, 5, ...} from smallest to largest, where the $n$-th group contains $2n-1$ odd numbers. That is, the first group, the second group, the third group... consist of the sets {1}, {3, 5, 7}, {9, 11, 13, 15, 17}, ..., respectively. In which group does 2007 belong? | 32 | 0.375 | 7,380 | 6,791 | 7,733.4 | |
An up-right path from $(a, b) \in \mathbb{R}^{2}$ to $(c, d) \in \mathbb{R}^{2}$ is a finite sequence $(x_{1}, y_{1}), \ldots,(x_{k}, y_{k})$ of points in $\mathbb{R}^{2}$ such that $(a, b)=(x_{1}, y_{1}),(c, d)=(x_{k}, y_{k})$, and for each $1 \leq i<k$ we have that either $(x_{i+1}, y_{i+1})=(x_{i}+1, y_{i})$ or $(x_... | 0.2937156494680644 | Note that any up-right path must pass through exactly one point of the form $(n,-n)$ (i.e. a point on the upper-left to lower-right diagonal), and the number of such paths is $\binom{800}{400-n}^{2}$ because there are $\binom{800}{400-n}$ up-right paths from $(-400,-400)$ to $(n,-n)$ and another $\binom{800}{400-n}$ fr... | 0 | 8,192 | -1 | 8,192 |
What is the area of the region defined by the equation $x^2+y^2 + 2x - 4y - 8 = 3y - 6x + 9$? | \frac{153\pi}{4} | 0 | 3,106.0625 | -1 | 3,106.0625 | |
What is the value of $(x - y)(x + y)$ if $x = 10$ and $y = 15$? | -125 | 1 | 1,608.3125 | 1,608.3125 | -1 | |
It is known that 999973 has exactly three distinct prime factors. Find the sum of these prime factors. | 171 | 0.875 | 4,755.75 | 4,264.857143 | 8,192 | |
Let $a_1 , a_2 , \dots$ be a sequence for which $a_1=2$ , $a_2=3$, and $a_n=\frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$. What is $a_{2006}$? | 3 | 1 | 2,578.375 | 2,578.375 | -1 | |
Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves:
(a) He clears every piece of rubbish from a single pile.
(b) He clears one piece of rubbish from each pile.
However, every evening, a demon sneaks... | 199 | 0 | 8,046.6875 | -1 | 8,046.6875 | |
Let $\{a_{n}\}$ be an arithmetic sequence with the sum of the first $n$ terms denoted as $S_{n}$. Given $a_{1} \gt 0$, $a_{8}$ and $a_{9}$ are the two roots of the equation $x^{2}+x-2023=0$. Calculate the maximum value of $n$ that satisfies $S_{n} \gt 0$. | 15 | 0.5625 | 7,037.0625 | 6,138.777778 | 8,192 | |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $\sqrt{3}\sin C - c\cos A = c$.
$(1)$ Find the value of angle $A$.
$(2)$ If $b = 2c$, point $D$ is the midpoint of side $BC$, and $AD = \sqrt{7}$, find the area of triangle $... | 2\sqrt{3} | 0.0625 | 8,029.875 | 6,625 | 8,123.533333 | |
Caroline can make eleven lassis out of two mangoes. How many lassis can she make out of twelve mangoes? | 66 | 0.9375 | 1,656.0625 | 1,220.333333 | 8,192 | |
Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $f(n)$ points, where \[f(n) = \left\{
\begin{array}{cl} 6 & \text{ if }n\text{ is a multiple of 2 and 3}, \\
2 & \text{ if }n\text{ is only a multiple of 2}, \\
0 & \text{ if }n\text{ is not a multiple of 2}... | 32 | 0.8125 | 1,546.1875 | 1,563.461538 | 1,471.333333 | |
Given $A=3x^{2}-x+2y-4xy$ and $B=2x^{2}-3x-y+xy$.
$(1)$ Simplify $2A-3B$.
$(2)$ When $x+y=\frac{6}{7}$ and $xy=-1$, find the value of $2A-3B$.
$(3)$ If the value of $2A-3B$ is independent of the value of $y$, find the value of $2A-3B$. | \frac{49}{11} | 1 | 3,668.8125 | 3,668.8125 | -1 | |
Alice plays a game where she rolls a fair eight-sided die each morning. If Alice rolls a number divisible by 3, she wears red; otherwise, she wears blue. If she rolls a perfect square (1, 4), then she rolls again. In a leap year, what is the expected number of times Alice will roll her die? | 488 | 0 | 5,295.125 | -1 | 5,295.125 |
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