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 an International track meet, 256 sprinters participate in a 100-meter dash competition. If the track has 8 lanes, and only the winner of each race advances to the next round while the others are eliminated, how many total races are needed to determine the champion sprinter? | 37 | 0.5 | 6,254.1875 | 4,628.875 | 7,879.5 | |
Let \( A \subseteq \{0, 1, 2, \cdots, 29\} \) such that for any integers \( k \) and any numbers \( a \) and \( b \) (possibly \( a = b \)), the expression \( a + b + 30k \) is not equal to the product of two consecutive integers. Determine the maximum possible number of elements in \( A \). | 10 | 0 | 8,192 | -1 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $\overrightarrow{m}=(\cos A,\cos B)$ and $\overrightarrow{n}=(b+2c,a)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.
(1) Find the measure of angle $A$.
(2) If $a=4 \sqrt {3}$ and $b+c=8$, find the length ... | 2 \sqrt {3} | 0 | 5,877.5625 | -1 | 5,877.5625 | |
Find the sum of the first eight prime numbers that have a units digit of 3. | 404 | 0 | 2,733.5625 | -1 | 2,733.5625 | |
A six-digit number 111aaa is the product of two consecutive positive integers \( b \) and \( b+1 \). Find the value of \( b \). | 333 | 0.4375 | 6,810.1875 | 5,033.571429 | 8,192 | |
Define the annoyingness of a permutation of the first \(n\) integers to be the minimum number of copies of the permutation that are needed to be placed next to each other so that the subsequence \(1,2, \ldots, n\) appears. For instance, the annoyingness of \(3,2,1\) is 3, and the annoyingness of \(1,3,4,2\) is 2. A ran... | \frac{2023}{2} | For a given permutation \(p_{1}, \ldots, p_{n}\), let \(f_{k}(p)\) be the smallest number of copies of \(p\) that need to be placed next to each other to have \(1, \ldots, k\) appear as a subsequence. We are interested in finding the expectation of \(f_{n}(p)\). Notice that if \(k\) appears before \(k+1\) in \(p\), the... | 0 | 8,192 | -1 | 8,192 |
Consider the operation "minus the reciprocal of," defined by $a \diamond b = a - \frac{1}{b}$. What is $((1 \diamond 2) \diamond 3) - (1 \diamond (2 \diamond 3))$? | -\frac{7}{30} | 1. **Calculate \(1 \diamond 2\):**
\[
1 \diamond 2 = 1 - \frac{1}{2} = \frac{2}{2} - \frac{1}{2} = \frac{1}{2}
\]
2. **Calculate \((1 \diamond 2) \diamond 3\):**
\[
\left(\frac{1}{2}\right) \diamond 3 = \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}
\]
3. **Calculate \(2 \diamon... | 1 | 2,648.6875 | 2,648.6875 | -1 |
The lengths of the edges of a regular tetrahedron \(ABCD\) are 1. \(G\) is the center of the base \(ABC\). Point \(M\) is on line segment \(DG\) such that \(\angle AMB = 90^\circ\). Find the length of \(DM\). | \frac{\sqrt{6}}{6} | 0 | 5,197.875 | -1 | 5,197.875 | |
Find the smallest positive integer $b$ for which $x^2 + bx + 1760$ factors into a product of two polynomials, each having integer coefficients. | 108 | 0 | 5,550.8125 | -1 | 5,550.8125 | |
Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15} \dotsm \frac{3n+6}{3n} \dotsm \frac{3003}{2997}.\] | 1001 | 0.375 | 4,394 | 4,169.666667 | 4,528.6 | |
Given that line $l$ is perpendicular to plane $\alpha$, and line $m$ is contained in plane $\beta$. Consider the following propositions:
(1) If $\alpha \parallel \beta$, then $l \perp m$.
(2) If $\alpha \perp \beta$, then $l \parallel m$.
(3) If $l \parallel m$, then $\alpha \perp \beta$.
(4) If $l \perp m$, then $... | (3) | 0 | 3,805.75 | -1 | 3,805.75 | |
Let $p, q, r$ be primes such that $2 p+3 q=6 r$. Find $p+q+r$. | 7 | First, it is known that $3 q=6 r-2 p=2(3 r-p)$, thus $q$ is even. The only even prime is 2 so $q=2$. Further, $2 p=6 r-3 q=3(2 r-q)$, which means that $p$ is a multiple of 3 and thus $p=3$. This means that $2 \cdot 3+3 \cdot 2=6 r \Longrightarrow r=2$. Therefore, $p+q+r=3+2+2=7$. | 1 | 2,964.0625 | 2,964.0625 | -1 |
Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars? | 80 | 1. **Identify the total sales**: The problem provides the sales for each month from January through April as $100$, $60$, $40$, and $120$ dollars respectively.
2. **Calculate the total sales**: Add the sales for each month to find the total sales over the four months.
\[
100 + 60 + 40 + 120 = 320 \text{ dollars}... | 0 | 3,800.8125 | -1 | 3,800.8125 |
Given the function $f(x) = |\log_2 x|$, let $m$ and $n$ be positive real numbers such that $m < n$ and $f(m)=f(n)$. If the maximum value of $f(x)$ on the interval $[m^2, n]$ is $2$, find the value of $n+m$. | \frac{5}{2} | 0.8125 | 4,643.25 | 3,852.846154 | 8,068.333333 | |
Compute the product of the roots of the equation \[3x^3 - x^2 - 20x + 27 = 0.\] | -9 | 1 | 3,814.8125 | 3,814.8125 | -1 | |
The on-time arrival rate of bus No. 101 in a certain city is 90%. Calculate the probability that the bus arrives on time exactly 4 times out of 5 rides for a person. | 0.32805 | 1 | 2,974.125 | 2,974.125 | -1 | |
There exist constants $b_1,$ $b_2,$ $b_3,$ $b_4$ such that
\[\sin^4 \theta = b_1 \sin \theta + b_2 \sin 2 \theta + b_3 \sin 3 \theta + b_4 \sin 4 \theta\]for all angles $\theta.$ Find $b_1^2 + b_2^2 + b_3^2 + b_4^2.$ | \frac{17}{64} | 0 | 8,192 | -1 | 8,192 | |
Majka examined multi-digit numbers in which odd and even digits alternate regularly. Those that start with an odd digit, she called "funny," and those that start with an even digit, she called "cheerful" (for example, the number 32387 is funny, the number 4529 is cheerful).
Majka created one three-digit funny number a... | 635040 | 0 | 8,125.75 | -1 | 8,125.75 | |
Three $1 \times 1 \times 1$ cubes are joined face to face in a single row and placed on a table. The cubes have a total of 11 exposed $1 \times 1$ faces. If sixty $1 \times 1 \times 1$ cubes are joined face to face in a single row and placed on a table, how many $1 \times 1$ faces are exposed? | 182 | 0.125 | 7,901.5 | 5,868 | 8,192 | |
Cory made a complete list of the prime numbers between 1 and 25. What is the sum of the smallest prime number and the largest prime number on his list? | 25 | 1 | 724.1875 | 724.1875 | -1 | |
Construct a cylindrical iron barrel with a volume of $V$. The lid of the barrel is made of aluminum alloy, and the price of aluminum alloy per unit area is three times that of iron. To minimize the cost of this container, the ratio of the bottom radius $r$ of the iron barrel to its height $h$ should be _______. | \frac{1}{4} | 0.125 | 6,189.5625 | 3,992.5 | 6,503.428571 | |
Given that the polynomial $x^2-kx+16$ has only positive integer roots, find the average of all distinct possibilities for $k$. | \frac{35}{3} | 0.9375 | 2,579.9375 | 2,205.8 | 8,192 | |
How many ordered quadruples \((a, b, c, d)\) of positive odd integers are there that satisfy the equation \(a + b + c + 2d = 15?\) | 34 | 0.3125 | 7,810.625 | 6,971.6 | 8,192 | |
Let $x,$ $y,$ $z$ be positive real number such that $xyz = \frac{2}{3}.$ Compute the minimum value of
\[x^2 + 6xy + 18y^2 + 12yz + 4z^2.\] | 18 | 0.25 | 7,974.375 | 7,321.5 | 8,192 | |
Three circles, each of radius $3$, are drawn with centers at $(14, 92)$, $(17, 76)$, and $(19, 84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the ab... | 24 | 0.3125 | 6,827.8125 | 3,826.6 | 8,192 | |
The distance from city $A$ to city $B$ is $999$ km. Along the highway leading from $A$ to $B$, there are kilometer markers indicating the distances from the marker to $A$ and $B$ as shown:
How many of these markers use only two different digits to indicate both distances? | 40 | 0 | 8,192 | -1 | 8,192 | |
Find the smallest natural number \( n \) such that the sum of the digits of each of the numbers \( n \) and \( n+1 \) is divisible by 17. | 8899 | 0 | 8,192 | -1 | 8,192 | |
Maria subtracts 2 from the number 15, triples her answer, and then adds 5. Liam triples the number 15, subtracts 2 from his answer, and then adds 5. Aisha subtracts 2 from the number 15, adds 5 to her number, and then triples the result. Find the final value for each of Maria, Liam, and Aisha. | 54 | 0.875 | 2,077.875 | 1,717 | 4,604 | |
Given the expression $(1296^{\log_6 4096})^{\frac{1}{4}}$, calculate its value. | 4096 | 0.8125 | 5,503.4375 | 4,883 | 8,192 | |
Given that the circumcenter of triangle $ABC$ is $O$, and $2 \overrightarrow{O A} + 3 \overrightarrow{O B} + 4 \overrightarrow{O C} = 0$, determine the value of $\cos \angle BAC$. | \frac{1}{4} | 0.375 | 7,289.375 | 5,785 | 8,192 | |
For polynomial $P(x)=1-\dfrac{1}{3}x+\dfrac{1}{6}x^{2}$, define $Q(x)=P(x)P(x^{3})P(x^{5})P(x^{7})P(x^{9})=\sum_{i=0}^{50} a_ix^{i}$. Then $\sum_{i=0}^{50} |a_i|=\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 275 | Multiply $P(x)P(x^3)$ and notice that the odd degree terms have a negative coefficient. Observing that this is probably true for all polynomials like this (including $P(x)P(x^3)P(x^5)P(x^7)P(x^9)$), we plug in $-1$ to get $\frac{243}{32} \implies \boxed{275}$. | 0.3125 | 6,750.75 | 4,351.2 | 7,841.454545 |
If I roll 5 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number? | \frac{485}{486} | 0.875 | 4,440.25 | 4,107.928571 | 6,766.5 | |
Compute $\sin 510^\circ$. | \frac{1}{2} | 1 | 2,080.125 | 2,080.125 | -1 | |
There are 42 stepping stones in a pond, arranged along a circle. You are standing on one of the stones. You would like to jump among the stones so that you move counterclockwise by either 1 stone or 7 stones at each jump. Moreover, you would like to do this in such a way that you visit each stone (except for the starti... | 63 | Number the stones $0,1, \ldots, 41$, treating the numbers as values modulo 42, and let $r_{n}$ be the length of your jump from stone $n$. If you jump from stone $n$ to $n+7$, then you cannot jump from stone $n+6$ to $n+7$ and so must jump from $n+6$ to $n+13$. That is, if $r_{n}=7$, then $r_{n+6}=7$ also. It follows th... | 0 | 8,192 | -1 | 8,192 |
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade? | 137 | 1. **Calculate the total weight of the lemonade**:
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water. The total weight of the lemonade is:
\[
100 \text{ grams (lemon juice)} + 100 \text{ grams (sugar)} + 400 \text{ grams (water)} = 600 \text{ grams}
\]
2. **Calculate the ... | 0.9375 | 1,894.8125 | 1,971.933333 | 738 |
Let $a_{1}, a_{2}, \ldots$ be an arithmetic sequence and $b_{1}, b_{2}, \ldots$ be a geometric sequence. Suppose that $a_{1} b_{1}=20$, $a_{2} b_{2}=19$, and $a_{3} b_{3}=14$. Find the greatest possible value of $a_{4} b_{4}$. | \frac{37}{4} | Solution 1. Let $\{a_{n}\}$ have common difference $d$ and $\{b_{n}\}$ have common ratio $r$; for brevity, let $a_{1}=a$ and $b_{1}=b$. Then we have the equations $a b=20,(a+d) b r=19$, and $(a+2 d) b r^{2}=14$, and we want to maximize $(a+3 d) b r^{3}$. The equation $(a+d) b r=19$ expands as $a b r+d b r=19$, or $20 r... | 0.625 | 6,671.375 | 5,979.3 | 7,824.833333 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and they satisfy the equation $\sin A + \sin B = [\cos A - \cos (π - B)] \sin C$.
1. Determine whether triangle $ABC$ is a right triangle and explain your reasoning.
2. If $a + b + c = 1 + \sqrt{2}$, find the maximum a... | \frac{1}{4} | 0.4375 | 7,155.25 | 5,822.285714 | 8,192 | |
Given that the heights of the first ten students formed a geometric sequence, with the fourth student 1.5 meters tall and the tenth student 1.62 meters tall, determine the height of the seventh student. | \sqrt{2.43} | 0 | 6,798.125 | -1 | 6,798.125 | |
Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\clubsuit$, the second card is a $\heartsuit$, and the third card is a king? | \frac{13}{2550} | 0 | 8,192 | -1 | 8,192 | |
If $f^{-1}(g(x))=x^4-1$ and $g$ has an inverse, find $g^{-1}(f(10))$. | \sqrt[4]{11} | 0.75 | 4,219.8125 | 2,895.75 | 8,192 | |
Given that the ratio of the length, width, and height of a rectangular prism is $4: 3: 2$, and that a plane cuts through the prism to form a hexagonal cross-section (as shown in the diagram), with the minimum perimeter of such hexagons being 36, find the surface area of the rectangular prism. | 208 | 0 | 8,192 | -1 | 8,192 | |
\(ABC\) is a triangle with \(AB = 15\), \(BC = 14\), and \(CA = 13\). The altitude from \(A\) to \(BC\) is extended to meet the circumcircle of \(ABC\) at \(D\). Find \(AD\). | \frac{63}{4} | 1 | 5,074.0625 | 5,074.0625 | -1 | |
One hundred people are in line to see a movie. Each person wants to sit in the front row, which contains one hundred seats, and each has a favorite seat, chosen randomly and independently. They enter the row one at a time from the far right. As they walk, if they reach their favorite seat, they sit, but to avoid steppi... | 10 | Let $S(i)$ be the favorite seat of the $i$ th person, counting from the right. Let $P(n)$ be the probability that at least $n$ people get to sit. At least $n$ people sit if and only if $S(1) \geq n, S(2) \geq n-1, \ldots, S(n) \geq 1$. This has probability: $$P(n)=\frac{100-(n-1)}{100} \cdot \frac{100-(n-2)}{100} \cdot... | 0 | 7,890.375 | -1 | 7,890.375 |
Given positive real numbers $x$, $y$, and $z$, find the minimum value of $\frac{x^2+2y^2+z^2}{xy+3yz}$. | \frac{2\sqrt{5}}{5} | 0 | 7,133.4375 | -1 | 7,133.4375 | |
The function $f(x)$ is defined by $f(x)=x^{2}-x$. What is the value of $f(4)$? | 12 | 1 | 1,450 | 1,450 | -1 | |
A spinner was created by drawing five radii from the center of a circle. The first four radii divide the circle into four equal wedges. The fifth radius divides one of the wedges into two parts, one having twice the area of the other. The five wedges are labeled with the wedge labeled by 2 having twice the area of the ... | 7/12 | 0.4375 | 6,898.0625 | 5,384.142857 | 8,075.555556 | |
What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?
| 180 | 0.6875 | 5,895.125 | 4,851.090909 | 8,192 | |
Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | 2028 | 0.1875 | 7,611 | 6,334.666667 | 7,905.538462 | |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 50\}$ have a perfect square factor other than one? | 19 | 0.25 | 7,866.25 | 6,889 | 8,192 | |
The number $27,000,001$ has exactly four prime factors. Find their sum. | 652 | First, we factor $$\begin{aligned} 27 x^{6}+1 & =\left(3 x^{2}\right)^{3}+1 \\ & =\left(3 x^{2}+1\right)\left(9 x^{4}-3 x^{2}+1\right) \\ & =\left(3 x^{2}+1\right)\left(\left(9 x^{4}+6 x^{2}+1\right)-9 x^{2}\right) \\ & =\left(3 x^{2}+1\right)\left(\left(3 x^{2}+1\right)^{2}-(3 x)^{2}\right) \\ & =\left(3 x^{2}+1\right... | 0.25 | 7,665.6875 | 7,470.75 | 7,730.666667 |
Given a decreasing arithmetic sequence $\{a_n\}$, if $a_1 + a_{100} = 0$, find the value of $n$ when the sum of the first $n$ terms, $S_n$, is maximized. | 50 | 0.75 | 5,243 | 4,260 | 8,192 | |
The slope of the tangent line to the curve $y=\frac{1}{3}{x^3}-\frac{2}{x}$ at $x=1$ is $\alpha$. Find $\frac{{sin\alpha cos2\alpha}}{{sin\alpha+cos\alpha}}$. | -\frac{3}{5} | 0 | 8,192 | -1 | 8,192 | |
Let $R^+$ be the set of positive real numbers. Determine all functions $f:R^+$ $\rightarrow$ $R^+$ such that for all positive real numbers $x$ and $y:$
$$f(x+f(xy))+y=f(x)f(y)+1$$
[i]Ukraine | f(x) = x + 1 |
Let \( R^+ \) be the set of positive real numbers. We need to determine all functions \( f: R^+ \rightarrow R^+ \) such that for all positive real numbers \( x \) and \( y \), the following equation holds:
\[
f(x + f(xy)) + y = f(x)f(y) + 1
\]
### Step-by-Step Solution:
1. **Assumption and Simplification:**
Let'... | 0 | 8,107.5625 | -1 | 8,107.5625 |
Let $ABC$ be a triangle in which $AB=AC$ . Suppose the orthocentre of the triangle lies on the incircle. Find the ratio $\frac{AB}{BC}$ . | 3/4 | 0.6875 | 6,000.875 | 5,004.909091 | 8,192 | |
Let $\triangle ABC$ be an isosceles triangle with $\angle A = 90^\circ.$ There exists a point $P$ inside $\triangle ABC$ such that $\angle PAB = \angle PBC = \angle PCA$ and $AP = 10.$ Find the area of $\triangle ABC.$
Diagram
[asy] /* Made by MRENTHUSIASM */ size(200); pair A, B, C, P; A = origin; B = (0,10*sqrt(5)); ... | 250 | Denote the area of $X$ by $[X].$ As in previous solutions, we see that $\angle APC = 90 ^\circ, \triangle BPC \sim \triangle APB$ with ratio $k = \sqrt{2}\implies$ \[\frac {PC}{PB} = \frac {PB}{PA} = k \implies PC = k^2 \cdot AP = 20 \implies [APC] = \frac {AP \cdot PC}{2} = 100.\] \[[BPC] = k^2 [APB] = 2 [APB].\] \[AB... | 0.5625 | 7,145.4375 | 6,331.444444 | 8,192 |
Given that point $P$ is the intersection point of the lines $l_{1}$: $mx-ny-5m+n=0$ and $l_{2}$: $nx+my-5m-n=0$ ($m$,$n\in R$, $m^{2}+n^{2}\neq 0$), and point $Q$ is a moving point on the circle $C$: $\left(x+1\right)^{2}+y^{2}=1$, calculate the maximum value of $|PQ|$. | 6 + 2\sqrt{2} | 0.0625 | 8,192 | 8,192 | 8,192 | |
It takes 60 grams of paint to paint a cube on all sides. How much paint is needed to paint a "snake" composed of 2016 such cubes? The beginning and end of the snake are shown in the illustration, while the rest of the cubes are represented by ellipsis. | 80660 | 0.375 | 6,136.3125 | 4,657.166667 | 7,023.8 | |
Consider the function $g(x) = \frac{x^2}{2} + 2x - 1$. Determine the sum of all distinct numbers $x$ such that $g(g(g(x))) = 1$. | -4 | 0.125 | 8,107.375 | 7,515 | 8,192 | |
In a polar coordinate system with the pole at point $O$, the curve $C\_1$: $ρ=6\sin θ$ intersects with the curve $C\_2$: $ρ\sin (θ+ \frac {π}{4})= \sqrt {2}$. Determine the maximum distance from a point on curve $C\_1$ to curve $C\_2$. | 3+\frac{\sqrt{2}}{2} | 0 | 7,169.6875 | -1 | 7,169.6875 | |
In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares. The measure of $\angle GDA$ is | 120^{\circ} | To find the measure of $\angle GDA$, we need to consider the geometric properties and the angles formed by the squares and the equilateral triangle.
1. **Identify the angles in the squares and the triangle:**
- Since $ABCD$ and $DEFG$ are squares, each angle in these squares is $90^\circ$.
- Since $CDE$ is an eq... | 0.25 | 6,791.125 | 5,458 | 7,235.5 |
If $\sqrt{25-\sqrt{n}}=3$, what is the value of $n$? | 256 | Since $\sqrt{25-\sqrt{n}}=3$, then $25-\sqrt{n}=9$. Thus, $\sqrt{n}=16$ and so $n=16^{2}=256$. | 1 | 1,225.6875 | 1,225.6875 | -1 |
If the graph of the function $f(x)=3\sin(2x+\varphi)$ is symmetric about the point $\left(\frac{\pi}{3},0\right)$ $(|\varphi| < \frac{\pi}{2})$, determine the equation of one of the axes of symmetry of the graph of $f(x)$. | \frac{\pi}{12} | 0.3125 | 7,789.75 | 7,318 | 8,004.181818 | |
Denis has cards with numbers from 1 to 50. How many ways are there to choose two cards such that the difference of the numbers on the cards is 11, and their product is divisible by 5?
The order of the selected cards does not matter: for example, selecting cards with numbers 5 and 16, as well as selecting cards with nu... | 15 | 0.5 | 6,577.9375 | 5,126.5 | 8,029.375 | |
Farmer Tim starts walking from the origin following the path \((t, \sin t)\) where \(t\) is the time in minutes. Five minutes later, Alex enters the forest and follows the path \((m, \cos t)\) where \(m\) is the time since Alex started walking. What is the greatest distance between Alex and Farmer Tim while they are wa... | \sqrt{29} | 0 | 6,598.8125 | -1 | 6,598.8125 | |
Given that a geometric sequence $\{a_n\}$ consists of positive terms, and $(a_3, \frac{1}{2}a_5,a_4)$ form an arithmetic sequence, find the value of $\frac{a_3+a_5}{a_4+a_6}$. | \frac{\sqrt{5}-1}{2} | 0 | 4,106.375 | -1 | 4,106.375 | |
In a right triangle with legs of 5 and 12, a segment is drawn connecting the shorter leg and the hypotenuse, touching the inscribed circle and parallel to the longer leg. Find its length. | 2.4 | 0 | 6,247.6875 | -1 | 6,247.6875 | |
Compute the definite integral:
$$
\int_{0}^{\sqrt{2}} \frac{x^{4} \cdot d x}{\left(4-x^{2}\right)^{3 / 2}}
$$ | 5 - \frac{3\pi}{2} | 0.25 | 7,393.1875 | 4,996.75 | 8,192 | |
Find the number of natural numbers \( k \) not exceeding 242400, such that \( k^2 + 2k \) is divisible by 303. | 3200 | 0.25 | 7,477.6875 | 7,447.5 | 7,487.75 | |
Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$ where $0<m<180$, will transform $\triangle ABC$ to $\triangle A'B'C'$. Find $m+x+y$. | 108 | After sketching, it is clear a $90^{\circ}$ rotation is done about $(x,y)$. Looking between $A$ and $A'$, $x+y=18$ and $x-y=24$. Solving gives $(x,y)\implies(21,-3)$. Thus $90+21-3=\boxed{108}$. ~mn28407 | 0.75 | 5,601.25 | 4,737.666667 | 8,192 |
Medians $BD$ and $CE$ of triangle $ABC$ are perpendicular, $BD=8$, and $CE=12$. The area of triangle $ABC$ is | 64 | Given that medians $BD$ and $CE$ of triangle $ABC$ are perpendicular, and their lengths are $BD = 8$ and $CE = 12$, we need to find the area of triangle $ABC$.
1. **Understanding the properties of medians**:
- The medians of a triangle intersect at the centroid, which divides each median into two segments, one of ... | 0.6875 | 5,749.8125 | 4,639.727273 | 8,192 |
Given positive real numbers \( a, b, c, d \) that satisfy the equalities
\[ a^{2}+d^{2}-ad = b^{2}+c^{2}+bc \quad \text{and} \quad a^{2}+b^{2} = c^{2}+d^{2}, \]
find all possible values of the expression \( \frac{ab+cd}{ad+bc} \). | \frac{\sqrt{3}}{2} | 0 | 8,192 | -1 | 8,192 | |
How many paths are there from $A$ to $B$, if every step must be up or to the right?[asy]size(4cm,4cm);int w=6;int h=5;int i;pen p=fontsize(9);for (i=0; i<h; ++i){draw((0,i) -- (w-1,i));}for (i=0; i<w; ++i){draw((i, 0)--(i,h-1));}label("$A$", (0,0), SW, p);label("$B$", (w-1,h-1), NE, p);[/asy] | 126 | 0.6875 | 3,072.6875 | 2,427.363636 | 4,492.4 | |
Given the hyperbola $\frac{x^{2}}{4}-y^{2}=1$ with its right focus $F$, and points $P_{1}$, $P_{2}$, …, $P_{n}$ on its right upper part where $2\leqslant x\leqslant 2 \sqrt {5}, y\geqslant 0$. The length of the line segment $|P_{k}F|$ is $a_{k}$, $(k=1,2,3,…,n)$. If the sequence $\{a_{n}\}$ is an arithmetic sequence wi... | 14 | 0.25 | 7,946.125 | 7,507.5 | 8,092.333333 | |
Given the polynomial $f(x) = 3x^6 + 5x^5 + 6x^4 + 20x^3 - 8x^2 + 35x + 12$ and $x = -2$, apply Horner's method to calculate the value of $v_4$. | -16 | 0.75 | 3,488.9375 | 3,278.25 | 4,121 | |
For a designer suit, Daniel must specify his waist size in centimeters. If there are $12$ inches in a foot and $30.5$ centimeters in a foot, then what size should Daniel specify, in centimeters, if his waist size in inches is $34$ inches? (You may use a calculator on this problem; answer to the nearest tenth.) | 86.4 | 1 | 2,377.9375 | 2,377.9375 | -1 | |
Let $ABC$ be an equilateral triangle of side length 6 inscribed in a circle $\omega$. Let $A_{1}, A_{2}$ be the points (distinct from $A$) where the lines through $A$ passing through the two trisection points of $BC$ meet $\omega$. Define $B_{1}, B_{2}, C_{1}, C_{2}$ similarly. Given that $A_{1}, A_{2}, B_{1}, B_{2}, C... | \frac{846\sqrt{3}}{49} | Let $A^{\prime}$ be the point on $BC$ such that $2BA^{\prime}=A^{\prime}C$. By law of cosines on triangle $AA^{\prime}B$, we find that $AA^{\prime}=2\sqrt{7}$. By power of a point, $A^{\prime}A_{1}=\frac{2 \cdot 4}{2\sqrt{7}}=\frac{4}{\sqrt{7}}$. Using side length ratios, $A_{1}A_{2}=2\frac{AA_{1}}{AA^{\prime}}=2\frac{... | 0 | 8,192 | -1 | 8,192 |
The New Year's gala has a total of 8 programs, 3 of which are non-singing programs. When arranging the program list, it is stipulated that the non-singing programs are not adjacent, and the first and last programs are singing programs. How many different ways are there to arrange the program list? | 720 | 0 | 7,947.125 | -1 | 7,947.125 | |
Given that three numbers are randomly selected from the set {1, 2, 3, 4, 5}, find the probability that the sum of the remaining two numbers is odd. | 0.6 | 0 | 5,414.1875 | -1 | 5,414.1875 | |
Given that both $α$ and $β$ are acute angles, and $\cos(α+β)= \frac{\sin α}{\sin β}$, find the maximum value of $\tan α$. | \frac{ \sqrt {2}}{4} | 0 | 7,144.75 | -1 | 7,144.75 | |
Find the sum of the squares of all real numbers satisfying the equation $x^{256} - 256^{32} = 0$. | 8 | 1. **Identify the form of the equation**: The given equation is $x^{256} - 256^{32} = 0$. We can rewrite this as $x^{256} = 256^{32}$.
2. **Simplify the equation**: Notice that $256 = 2^8$, so $256^{32} = (2^8)^{32} = 2^{256}$. Thus, the equation becomes $x^{256} = 2^{256}$.
3. **Roots of the equation**: The equation... | 1 | 2,840.75 | 2,840.75 | -1 |
Simplify first, then evaluate: $(a-2b)(a^2+2ab+4b^2)-a(a-5b)(a+3b)$, where $a$ and $b$ satisfy $a^2+b^2-2a+4b=-5$. | 120 | 0.5 | 4,325.875 | 3,559.375 | 5,092.375 | |
What real number is equal to the expression $2 + \frac{4}{1 + \frac{4}{2 + \frac{4}{1 + \cdots}}}$, where the $1$s and the $2$s alternate? | 4 | 0.75 | 4,913.9375 | 3,821.25 | 8,192 | |
Let $k$ be the answer to this problem. The probability that an integer chosen uniformly at random from $\{1,2, \ldots, k\}$ is a multiple of 11 can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$. | 1000 | We write $k=11 q+r$ for integers $q, r$ with $0 \leq r<11$. There are $q$ multiples of 11 from 1 to $k$, inclusive, so our probability is $\frac{a}{b}=\frac{q}{11 q+r}$. Let $d=\operatorname{gcd}(q, r)=\operatorname{gcd}(q, 11 q+r)$, so that the fraction $\frac{q / d}{(11 q+r) / d}$ is how we would write $\frac{q}{11 q... | 0 | 8,192 | -1 | 8,192 |
Out of 60 right-angled triangles with legs of 2 and 3, a rectangle was formed. What can be the maximum perimeter of this rectangle? | 184 | 0 | 8,192 | -1 | 8,192 | |
Let $\omega$ be a nonreal root of $x^3 = 1.$ Compute
\[(2 - \omega + 2\omega^2)^6 + (2 + \omega - 2\omega^2)^6.\] | 38908 | 0 | 8,192 | -1 | 8,192 | |
The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37$? | 217 | A brute-force solution to this question is fairly quick, but we'll try something slightly more clever: our numbers have the form ${\underline{(n+3)}}\,{\underline{(n+2)}}\,{\underline{( n+1)}}\,{\underline {(n)}}$$= 1000(n + 3) + 100(n + 2) + 10(n + 1) + n = 3210 + 1111n$, for $n \in \lbrace0, 1, 2, 3, 4, 5, 6\rbrace$.... | 0.75 | 4,282.3125 | 4,081.416667 | 4,885 |
Let $S$ be the sum of the interior angles of a polygon $P$ for which each interior angle is $7\frac{1}{2}$ times the exterior angle at the same vertex. Then | 2700^{\circ} | 1. **Understanding the relationship between interior and exterior angles**: Given that each interior angle $a_n$ is $7.5$ times its corresponding exterior angle $b_n$, we can write:
\[ a_n = 7.5 \times b_n \]
2. **Sum of exterior angles**: We know that the sum of the exterior angles of any polygon is always $360^\c... | 0.9375 | 2,586.0625 | 2,212.333333 | 8,192 |
A TV station is broadcasting 5 advertisements in a row, including 3 different commercial advertisements and 2 different public service advertisements. The last advertisement cannot be a commercial one, and the two public service advertisements cannot be broadcast consecutively. How many different broadcasting methods a... | 36 | 0.0625 | 8,179.75 | 7,996 | 8,192 | |
Let $ a,b,c,d>0$ for which the following conditions:: $a)$ $(a-c)(b-d)=-4$ $b)$ $\frac{a+c}{2}\geq\frac{a^{2}+b^{2}+c^{2}+d^{2}}{a+b+c+d}$ Find the minimum of expression $a+c$ | 4\sqrt{2} | 0.0625 | 8,166.4375 | 7,783 | 8,192 | |
$|3-\pi|=$ | \pi-3 | 1. **Identify the Expression**: We are given the expression $|3-\pi|$ and need to evaluate its absolute value.
2. **Understanding $\pi$**: The value of $\pi$ (pi) is approximately 3.14159, which is greater than 3.
3. **Calculate $3 - \pi$**: Since $\pi > 3$, the expression $3 - \pi$ results in a negative number:
\... | 0.8125 | 840.75 | 966.538462 | 295.666667 |
A bag contains 7 red chips and 4 green chips. One chip is drawn from the bag, then placed back into the bag, and a second chip is drawn. What is the probability that the two selected chips are of different colors? | \frac{56}{121} | 1 | 2,297.75 | 2,297.75 | -1 | |
If $f(x) = -7x^4 + 3x^3 + x - 5$, and $g(x)$ is a polynomial such that the degree of $f(x) + g(x)$ is 1, then what is the degree of $g(x)$? | 4 | 0.4375 | 5,468.5625 | 3,795 | 6,770.222222 | |
Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first gam... | \frac{1}{4} | To solve this problem, we need to consider all possible sequences of game outcomes where Team A wins the series and Team B wins the second game. We then determine the probability that Team B wins the first game under these conditions.
1. **Identify possible sequences:**
- Team A must win 3 games and Team B must win... | 0 | 8,060.1875 | -1 | 8,060.1875 |
From the set $\{1, 2, \cdots, 20\}$, choose 5 numbers such that the difference between any two numbers is at least 4. How many different ways can this be done? | 56 | 0.25 | 7,636 | 5,968 | 8,192 | |
Find the product of $10101_2$ and $101_2$. Express your answer in base $2$. | 1101001_2 | 0.3125 | 6,978.8125 | 4,309.8 | 8,192 | |
$2014$ points are placed on a circumference. On each of the segments with end points on two of the $2014$ points is written a non-negative real number. For any convex polygon with vertices on some of the $2014$ points, the sum of the numbers written on their sides is less or equal than $1$. Find the maximum possible va... | 507024.5 |
Given the problem, we are tasked with finding the maximum possible sum of numbers written on segments between 2014 points uniformly placed on a circumference, under the condition that for any convex polygon formed using these points as vertices, the sum of the numbers on its sides must not exceed 1.
Consider the foll... | 0 | 8,099.75 | -1 | 8,099.75 |
The function $f$ has the property that for each real number $x$ in its domain, $1/x$ is also in its domain and \[
f(x) + f\left(\frac{1}{x}\right) = x.
\]What is the largest set of real numbers that can be in the domain of $f$?
(a) ${\{x\mid x\ne0\}}$
(b) ${\{x\mid x<0\}}$
(c) ${\{x\mid x>0\}}$
(d) ${\{x\mid x\ne-1... | E | 0.6875 | 5,788.875 | 5,102.454545 | 7,299 | |
Let \( a_0 = -3 \), \( b_0 = 2 \), and for \( n \geq 0 \), let:
\[
\begin{align*}
a_{n+1} &= 2a_n + 2b_n + 2\sqrt{a_n^2 + b_n^2}, \\
b_{n+1} &= 2a_n + 2b_n - 2\sqrt{a_n^2 + b_n^2}.
\end{align*}
\]
Find \( \frac{1}{a_{2023}} + \frac{1}{b_{2023}} \). | \frac{1}{3} | 0 | 7,294.625 | -1 | 7,294.625 | |
Let $M$ be a set consisting of $n$ points in the plane, satisfying:
i) there exist $7$ points in $M$ which constitute the vertices of a convex heptagon;
ii) if for any $5$ points in $M$ which constitute the vertices of a convex pentagon, then there is a point in $M$ which lies in the interior of the pent... | 11 | 0.125 | 8,025.1875 | 7,759 | 8,063.214286 | |
What is the coefficient of $x^3y^5$ in the expansion of $\left(\frac{4}{3}x - \frac{2y}{5}\right)^8$? | -\frac{114688}{84375} | 0.1875 | 7,949.875 | 6,900.666667 | 8,192 |
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