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 |
|---|---|---|---|---|---|---|
Define the sequences $(a_n),(b_n)$ by
\begin{align*}
& a_n, b_n > 0, \forall n\in\mathbb{N_+} \\
& a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}} \\
& b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}}
\end{align*}
1) If $a_{100}b_{100} = a_{101}b_{101}$, find the value of $a_1-b_1$;
2) If $a_{100} = b_{99}... | 199 |
Define the sequences \( (a_n) \) and \( (b_n) \) by
\[
\begin{align*}
& a_n, b_n > 0, \forall n \in \mathbb{N_+}, \\
& a_{n+1} = a_n - \frac{1}{1 + \sum_{i=1}^n \frac{1}{a_i}}, \\
& b_{n+1} = b_n + \frac{1}{1 + \sum_{i=1}^n \frac{1}{b_i}}.
\end{align*}
\]
1. If \( a_{100} b_{100} = a_{101} b_{101} \), find the value... | 0 | 8,156.0625 | -1 | 8,156.0625 |
A line passing through the point P(3/2, 1/2) intersects the ellipse x^2/6 + y^2/2 = 1 at points A and B, satisfying PA + PB = 0. If M is any point on the line AB and O is the origin, find the minimum value of |OM|. | \sqrt{2} | 0.9375 | 5,057.0625 | 4,848.066667 | 8,192 | |
In a certain football invitational tournament, 16 cities participate, with each city sending two teams, Team A and Team B. According to the competition rules, after several days of matches, it was found that aside from Team A from city $A$, the number of matches already played by each of the other teams was different. ... | 15 | 0.25 | 7,856.5 | 7,358 | 8,022.666667 | |
At the end of the school year, teachers of the third grade met with the parents of some of their students; exactly 31 people were present at this meeting. The Latin teacher was asked questions by 16 parents, the French teacher by 17 parents, the English teacher by 18 parents, and so on up to the Math teacher, who was a... | 23 | 0 | 7,250.5 | -1 | 7,250.5 | |
From among $2^{1/2}, 3^{1/3}, 8^{1/8}, 9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are | $3^{1/3},\ 2^{1/2}$ | To find the greatest and the next to the greatest values among $2^{1/2}, 3^{1/3}, 8^{1/8}, 9^{1/9}$, we can compare these values by considering their logarithmic forms or by direct calculation.
1. **Convert to comparable forms:**
We can compare $a^{1/b}$ by considering $b \cdot \log(a)$, since $a^{1/b} = e^{\log(a^... | 0 | 6,642.75 | -1 | 6,642.75 |
The side edge of a regular tetrahedron \( S-ABC \) is 2, and the base is an equilateral triangle with side length 1. A section passing through \( AB \) divides the volume of the tetrahedron into two equal parts. Find the cosine of the dihedral angle between this section and the base. | \frac{2}{\sqrt{15}} | 0 | 8,106.5 | -1 | 8,106.5 | |
If
\[\sin x + \cos x + \tan x + \cot x + \sec x + \csc x = 7,\]then find $\sin 2x.$ | 22 - 8 \sqrt{7} | 0.75 | 6,752.625 | 6,272.833333 | 8,192 | |
Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together ... | 334 | From the given information, we can see that Abe can paint $\frac{1}{15}$ of the room in an hour, Bea can paint $\frac{1}{15}\times\frac{3}{2} = \frac{1}{10}$ of the room in an hour, and Coe can paint the room in $\frac{1}{15}\times 2 = \frac{2}{15}$ of the room in an hour. After $90$ minutes, Abe has painted $\frac{1}{... | 0.9375 | 5,055 | 5,065.8 | 4,893 |
What is the value of $102^{4} - 4 \cdot 102^{3} + 6 \cdot 102^2 - 4 \cdot 102 + 1$? | 100406401 | 0 | 4,432.8125 | -1 | 4,432.8125 | |
Solve the equation \[\frac{x^2 + 3x + 4}{x + 5} = x + 6.\] | -\frac{13}{4} | 1 | 2,419.625 | 2,419.625 | -1 | |
During breaks, schoolchildren played table tennis. Any two schoolchildren played no more than one game against each other. At the end of the week, it turned out that Petya played half, Kolya - a third, and Vasya - one fifth of the total number of games played during the week. What could be the total number of games pla... | 30 | 0.0625 | 8,102.1875 | 6,755 | 8,192 | |
The perimeter of the triangles that make up rectangle \(ABCD\) is 180 cm. \(BK = KC = AE = ED\), \(AK = KD = 17 \) cm. Find the perimeter of a rectangle, one of whose sides is twice as long as \(AB\), and the other side is equal to \(BC\). | 112 | 0 | 8,113.5625 | -1 | 8,113.5625 | |
There is a parking lot with $10$ empty spaces. Three different cars, A, B, and C, are going to park in such a way that each car has empty spaces on both sides, and car A must be parked between cars B and C. How many different parking arrangements are there? | 40 | 0 | 8,042.75 | -1 | 8,042.75 | |
The line joining $(4,3)$ and $(7,1)$ divides the square shown into two parts. What fraction of the area of the square is above this line? Assume the square has vertices at $(4,0)$, $(7,0)$, $(7,3)$, and $(4,3)$. | \frac{5}{6} | 0 | 5,500.875 | -1 | 5,500.875 | |
Find $h(x)$, with terms in order of decreasing degree, if \[9x^3-3x+1+h(x)=3x^2-5x+3.\] | -9x^3+3x^2-2x+2 | 0.9375 | 2,013.125 | 1,601.2 | 8,192 | |
Find the greatest value of $t$ such that \[\frac{t^2 - t -56}{t-8} = \frac{3}{t+5}.\] | -4 | 1 | 2,236 | 2,236 | -1 | |
How many four-digit numbers starting with the digit $2$ and having exactly three identical digits are there? | 27 | 0.0625 | 7,312.4375 | 7,628 | 7,291.4 | |
What is the value of $b$ if $-x^2+bx-5<0$ only when $x\in (-\infty, 1)\cup(5,\infty)$? | 6 | 1 | 2,133.625 | 2,133.625 | -1 | |
The product of the digits of 1423 is 24. Find how many distinct four-digit positive integers have a product of their digits equal to 18. | 36 | 0.125 | 8,150.875 | 7,863 | 8,192 | |
Given two circles that intersect at two points $(2,3)$ and $(m,2)$, and both circle centers lie on the line $x+y+n=0$. Find the value of $m+n$. | -2 | 0 | 5,821.8125 | -1 | 5,821.8125 | |
Given the function $f(x)= \dfrac {2-\cos \left( \dfrac {\pi}{4}(1-x)\right)+\sin \left( \dfrac {\pi}{4}(1-x)\right)}{x^{2}+4x+5}(-4\leqslant x\leqslant 0)$, find the maximum value of $f(x)$. | 2+ \sqrt {2} | 0 | 7,761.5625 | -1 | 7,761.5625 | |
Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive intege... | 25 | Let Jane's age $n$ years from now be $10a+b$, and let Dick's age be $10b+a$. If $10b+a>10a+b$, then $b>a$. The possible pairs of $a,b$ are:
$(1,2), (1,3), (2,3), (1,4), (2,4), (3,4), \dots , (8,9)$
That makes 36. But $10a+b>25$, so we subtract all the extraneous pairs: $(1,2), (1,3), (2,3), (1,4), (2,4), (1,5), (2,5),... | 0.0625 | 8,192 | 8,192 | 8,192 |
How many of the 200 students surveyed said that their favourite food was sandwiches, given the circle graph results? | 20 | Since the angle in the sector representing cookies is $90^{\circ}$, then this sector represents $\frac{1}{4}$ of the total circle. Therefore, 25% of the students chose cookies as their favourite food. Thus, the percentage of students who chose sandwiches was $100\%-30\%-25\%-35\%=10\%$. Since there are 200 students in ... | 0 | 6,554.9375 | -1 | 6,554.9375 |
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[23]{3}$, and $a_n=a_{n-1}a_{n-2}^3$ for $n\geq 2$. Determine the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer. | 22 | 0.0625 | 8,053.375 | 6,709 | 8,143 | |
Determine the value of \( n \) such that \( 2^7 \cdot 3^4 \cdot n = 10! \). | 350 | 1 | 3,068.4375 | 3,068.4375 | -1 | |
With square tiles of a side length of an exact number of units, a room with a surface area of 18,144 square units has been tiled in the following manner: on the first day one tile was placed, the second day two tiles, the third day three tiles, and so on. How many tiles were necessary? | 2016 | 0.1875 | 7,547 | 5,530.666667 | 8,012.307692 | |
Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 being rational and the remaining 50 being irrational. Then, in each cell of the table, he recorded the sum of the numbers written next to its row an... | 1250 | 0.0625 | 7,943 | 4,208 | 8,192 | |
Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern? | 32/17 | 1. **Identify the original pattern dimensions and tile counts**: The original pattern consists of a square with 8 black tiles and 17 white tiles. The pattern is arranged in a way that suggests the square is 5x5 tiles in size (since $5^2 = 25$ and $8 + 17 = 25$).
2. **Understand the extension**: The problem states that... | 0.5625 | 4,453.125 | 3,506.555556 | 5,670.142857 |
Suppose a sequence $\{a\_n\}$ satisfies $\frac{1}{a\_{n+1}} - \frac{1}{a\_n} = d (n \in \mathbb{N}^*, d$ is a constant), then the sequence $\{a\_n\}$ is called a "harmonic sequence". It is known that the sequence $\{\frac{1}{x\_n}\}$ is a "harmonic sequence", and $x\_1 + x\_2 + ... + x\_{20} = 200$, find the maximum va... | 100 | 0 | 8,192 | -1 | 8,192 | |
Given that point \( P \) lies on the hyperbola \(\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1\), and the distance from \( P \) to the right directrix of the hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of the hyperbola, find the x-coordinate of point \( P \). | -\frac{64}{5} | 0.0625 | 8,108.0625 | 6,849 | 8,192 | |
If $100^a = 7$ and $100^b = 11,$ then find $20^{(1 - a - b)/(2(1 - b))}.$ | \frac{100}{77} | 0 | 8,192 | -1 | 8,192 | |
Given the function $f(x)$, for any $x \in \mathbb{R}$, it satisfies $f(x+6) + f(x) = 0$, and the graph of $y=f(x-1)$ is symmetric about the point $(1,0)$. If $f(2) = 4$, find the value of $f(2014)$. | -4 | 0.625 | 5,705 | 4,632.2 | 7,493 | |
A four-dimensional rectangular hyper-box has side lengths $W$, $X$, $Y$, and $Z$. It has "faces" (three-dimensional volumes) whose measures are $60$, $80$, $120$, $60$, $80$, $120$ cubic units. What is $W$ + $X$ + $Y$ + $Z$?
**A)** 200
**B)** 250
**C)** 300
**D)** 318.5
**E)** 400 | 318.5 | 0 | 8,192 | -1 | 8,192 | |
In 2005, the ages of a brother and sister were 16 and 10 years old, respectively. In which year was the brother's age twice that of the sister's? | 2001 | 0.6875 | 858.8125 | 934.545455 | 692.2 | |
Given the function $f(x)=(x-a)^{2}+(2\ln x-2a)^{2}$, where $x > 0, a \in \mathbb{R}$, find the value of the real number $a$ such that there exists $x_{0}$ such that $f(x_{0}) \leqslant \frac{4}{5}$. | \frac{1}{5} | 0.0625 | 8,192 | 8,192 | 8,192 | |
Given the function $g(x)=x-1$, and the function $f(x)$ satisfies $f(x+1)=-2f(x)-1$. When $x \in (0,1]$, $f(x)=x^{2}-x$. For any $x_1 \in (1,2]$ and $x_2 \in R$, determine the minimum value of $(x_1-x_2)^2+(f(x_1)-g(x_2))^2$. | \frac{49}{128} | 0.0625 | 7,662.1875 | 5,507 | 7,805.866667 | |
Given $x > 0$, $y > 0$, and $2x+8y-xy=0$, find the minimum value of $x+y$. | 18 | 0.9375 | 3,975.25 | 3,694.133333 | 8,192 | |
Let $f(x)$ be a function defined on $\mathbb{R}$ with a period of 2. On the interval $[-1,1)$, $f(x)$ is given by
$$
f(x) = \begin{cases}
x+a & \text{for } -1 \leq x < 0,\\
\left| \frac{2}{5} - x \right| & \text{for } 0 \leq x < 1,
\end{cases}
$$
where $a \in \mathbb{R}$. If $f\left(-\frac{5}{2}\right) = f\left(\frac{9... | -\frac{2}{5} | 0.75 | 4,257.875 | 3,731.333333 | 5,837.5 | |
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Given that $$\frac {sin2B}{ \sqrt {3}cos(B+C)-cosCsinB}= \frac {2b}{c}$$.
(I) Find the measure of angle A.
(II) If $$a= \sqrt {3}$$, find the maximum area of triangle ABC. | \frac { \sqrt {3}}{4} | 0 | 6,921 | -1 | 6,921 | |
What is $4+10\div2-2\cdot3$? | 3 | 1 | 1,663.6875 | 1,663.6875 | -1 | |
Find all pairs of integers $ (x,y)$, such that
\[ x^2 \minus{} 2009y \plus{} 2y^2 \equal{} 0
\] | (0,0)； (-588,784)； (588,784) |
To solve the equation \(x^2 - 2009y + 2y^2 = 0\) for integer pairs \((x, y)\), we begin by rearranging the equation as follows:
\[
x^2 = 2009y - 2y^2.
\]
The right-hand side must be a perfect square for some integer \(x\). Therefore, consider the expression:
\[
x^2 = 2y^2 - 2009y.
\]
To factor or simplify, we comp... | 0 | 8,192 | -1 | 8,192 |
Joel selected an acute angle $x$ (strictly between 0 and 90 degrees) and wrote the values of $\sin x$, $\cos x$, and $\tan x$ on three different cards. Then he gave those cards to three students, Malvina, Paulina, and Georgina, one card to each, and asked them to figure out which trigonometric function (sin, cos, or t... | \frac{1 + \sqrt{5}}{2} | 0 | 8,192 | -1 | 8,192 | |
Add $452_8$ and $167_8$ in base $8$, then subtract $53_8$ from the result. | 570_8 | 0 | 5,276.8125 | -1 | 5,276.8125 | |
What is the quantity equivalent to '2% of 1'? | \frac{2}{100} | The quantity $2 \%$ is equivalent to the fraction $\frac{2}{100}$, so '2% of 1' is equal to $\frac{2}{100}$. | 0 | 249.25 | -1 | 249.25 |
Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^3$ for all $x \ge 1.$ If $g(16) = 16,$ then compute $[g(4)]^3.$ | 16 | 0.375 | 6,606.75 | 3,964.666667 | 8,192 | |
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 499 | 0 | 8,192 | -1 | 8,192 | |
Griffin and Hailey run for $45$ minutes on a circular track. Griffin runs counterclockwise at $260 m/min$ and uses the outer lane with a radius of $50$ meters. Hailey runs clockwise at $310 m/min$ and uses the inner lane with a radius of $45$ meters, starting on the same radial line as Griffin. Determine how many times... | 86 | 0.5625 | 7,117.9375 | 6,429 | 8,003.714286 | |
Arithmetic sequences $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1<a_2 \le b_2$ and $a_n b_n = 2010$ for some $n$. What is the largest possible value of $n$? | 8 |
1. **Identify the form of the sequences**: Given that $\left(a_n\right)$ and $\left(b_n\right)$ are arithmetic sequences with integer terms and $a_1 = b_1 = 1$, we can express the $n$-th terms of these sequences as:
\[
a_n = 1 + (n-1)x \quad \text{and} \quad b_n = 1 + (n-1)y
\]
where $x$ and $y$ are the co... | 0.4375 | 7,518 | 6,651.428571 | 8,192 |
A gasoline tank is $\frac78$ full. After $12$ gallons have been used, it is half full. How many gallons does this tank hold when it is full? | 32 | 1 | 1,771.75 | 1,771.75 | -1 | |
Given $a\ln a=be^{b}$, where $b > 0$, find the maximum value of $\frac{b}{{{a^2}}}$ | \frac{1}{2e} | 0.625 | 5,601.625 | 4,047.4 | 8,192 | |
Given that $sin(x- \frac {π}{4})= \frac {2}{3}$, find the value of $sin2x$. | \frac{1}{9} | 1 | 3,661.8125 | 3,661.8125 | -1 | |
Let $\min \{a, b\}$ denote the smaller value between $a$ and $b$. When the positive numbers $x$ and $y$ vary, $t = \min \left\{ x, \frac{y}{x^{2}+y^{2}} \right\}$ also varies. What is the maximum value of $t$? | 1/2 | 0 | 7,297.9375 | -1 | 7,297.9375 | |
A rectangle was cut into three rectangles, two of which have dimensions 9 m x 12 m and 10 m x 15 m. What is the maximum possible area of the original rectangle? Express your answer in square meters. | 330 | 0 | 8,192 | -1 | 8,192 | |
A primary school conducted a height survey. For students with heights not exceeding 130 cm, there are 99 students with an average height of 122 cm. For students with heights not less than 160 cm, there are 72 students with an average height of 163 cm. The average height of students with heights exceeding 130 cm is 155 ... | 621 | 0 | 5,121.25 | -1 | 5,121.25 | |
Given the limit of the ratio of an infinite decreasing geometric series \(\{a_{n}\}\) satisfies \(\lim _{n \rightarrow \infty} \frac{a_{1} + a_{4} + a_{7} + \cdots + a_{3n-2}}{a_{1} + a_{2} + \cdots + a_{n}} = \frac{3}{4}\), find the common ratio of the series. | \frac{\sqrt{21} - 3}{6} | 0 | 3,923.6875 | -1 | 3,923.6875 | |
A circle with center O is tangent to the coordinate axes and to the hypotenuse of a $45^\circ$-$45^\circ$-$90^\circ$ triangle ABC, where AB = 2. Determine the exact radius of the circle. | 2 + \sqrt{2} | 0 | 4,719.125 | -1 | 4,719.125 | |
$ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$ , different from $C$ . What is the length of the segment $IF$ ? | 10 | 0.25 | 7,827.75 | 6,901.5 | 8,136.5 | |
In triangle $XYZ$, $XY = 540$ and $YZ = 360$. Points $N$ and $O$ are located on $\overline{XY}$ and $\overline{XZ}$ respectively, such that $XN = NY$, and $\overline{ZO}$ is the angle bisector of angle $Z$. Let $Q$ be the point of intersection of $\overline{YN}$ and $\overline{ZO}$, and let $R$ be the point on line $YN... | 216 | 0 | 8,192 | -1 | 8,192 | |
Given quadrilateral $ABCD,$ side $\overline{AB}$ is extended past $B$ to $A'$ so that $A'B = AB.$ Points $B',$ $C',$ and $D'$ are similarly constructed.
[asy]
unitsize(1 cm);
pair[] A, B, C, D;
A[0] = (0,0);
B[0] = (2,0);
C[0] = (1.5,2);
D[0] = (0.2,1.5);
A[1] = 2*B[0] - A[0];
B[1] = 2*C[0] - B[0];
C[1] = 2*D[0] - ... | \left( \frac{1}{15}, \frac{2}{15}, \frac{4}{15}, \frac{8}{15} \right) | 0.4375 | 6,757.6875 | 4,913.571429 | 8,192 | |
A piece of string fits exactly once around the perimeter of a rectangle with a length of 16 and a width of 10. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string? | 215 | 0.9375 | 4,584.125 | 4,343.6 | 8,192 | |
Let $a$, $b$, and $c$ be the roots of $x^3 - 20x^2 + 18x - 7 = 0$. Compute \[(a+b)^2 + (b+c)^2 + (c+a)^2.\] | 764 | 1 | 4,062.625 | 4,062.625 | -1 | |
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common en... | 1 | 1. **Understanding the Problem**: We need to find the shortest path on the surface of a regular tetrahedron from the midpoint of one edge to the midpoint of an opposite edge. The tetrahedron has edges of length 1.
2. **Unfolding the Tetrahedron**: To visualize the problem, we can unfold the tetrahedron into a flat pla... | 0 | 8,192 | -1 | 8,192 |
In the geometric sequence {a_n}, a_6 and a_{10} are the two roots of the equation x^2+6x+2=0. Determine the value of a_8. | -\sqrt{2} | 0.4375 | 7,006.75 | 6,142.285714 | 7,679.111111 | |
An inverted frustum with a bottom diameter of 12 and height of 18, filled with water, is emptied into another cylindrical container with a bottom diameter of 24. Assuming the cylindrical container is sufficiently tall, what will be the height of the water level in the cylindrical container? | 1.5 | 0.0625 | 7,038.9375 | 7,051 | 7,038.133333 | |
Given the lateral area of a cylinder with a square cross-section is $4\pi$, calculate the volume of the cylinder. | 2\pi | 0.3125 | 5,516.875 | 4,313.8 | 6,063.727273 | |
$A$,$B$,$C$,$D$,$E$,$F$ are 6 students standing in a row to participate in a literary performance. If $A$ does not stand at either end, and $B$ and $C$ must be adjacent, then the total number of different arrangements is ____. | 144 | 0.0625 | 7,996 | 8,192 | 7,982.933333 | |
A cube has six faces. Each face has some dots on it. The numbers of dots on the six faces are 2, 3, 4, 5, 6, and 7. Harry removes one of the dots at random, with each dot equally likely to be removed. When the cube is rolled, each face is equally likely to be the top face. What is the probability that the top face has ... | \frac{13}{27} | When a dot is removed from a face with an even number of dots, that face then has an odd number of dots. When a dot is removed from a face with an odd number of dots, that face then has an even number of dots. Initially, there are 3 faces with an even number of dots and 3 faces with an odd number of dots. If a dot is r... | 0.25 | 7,316.5625 | 5,989 | 7,759.083333 |
Find the sum of the absolute values of the roots of $x^4-4x^3-4x^2+16x-8=0$. | 2+2\sqrt{2}+2\sqrt{3} | 0.0625 | 8,011.25 | 8,192 | 7,999.2 | |
There are two rows of seats, with 11 seats in the front row and 12 seats in the back row. Now, we need to arrange for two people, A and B, to sit down. It is stipulated that the middle 3 seats of the front row cannot be occupied, and A and B cannot sit next to each other. How many different arrangements are there? | 346 | 0.0625 | 7,776.0625 | 4,051 | 8,024.4 | |
A telephone station serves 400 subscribers. For each subscriber, the probability of calling the station within an hour is 0.01. Find the probabilities of the following events: "within an hour, 5 subscribers will call the station"; "within an hour, no more than 4 subscribers will call the station"; "within an hour, at l... | 0.7619 | 0.125 | 7,721.3125 | 4,910 | 8,122.928571 | |
Given that in triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\angle BAC = 60^{\circ}$, $D$ is a point on side $BC$ such that $AD = \sqrt{7}$, and $BD:DC = 2c:b$, then the minimum value of the area of $\triangle ABC$ is ____. | 2\sqrt{3} | 0.375 | 7,610.125 | 7,040 | 7,952.2 | |
What is the greatest integer less than 150 for which the greatest common divisor of that integer and 18 is 6? | 138 | 1 | 4,362.9375 | 4,362.9375 | -1 | |
The number of elderly employees in a sample of 32 young employees from a workplace with a total of 430 employees, 160 of whom are young and the number of middle-aged employees is twice the number of elderly employees, can be found by determining the ratio of young employees in the population and the sample. | 18 | 0.125 | 6,121.125 | 6,434.5 | 6,076.357143 | |
In the final round of a giraffe beauty contest, two giraffes named Tall and Spotted have made it to this stage. There are 105 voters divided into 5 districts, each district divided into 7 sections, with each section having 3 voters. Voters select the winner in their section by majority vote; in a district, the giraffe ... | 24 | 0.0625 | 7,886.5 | 7,829 | 7,890.333333 | |
In right triangle $ABC$ with $\angle B = 90^\circ$, we have $AB = 8$ and $AC = 6$. Find $\cos C$. | \frac{4}{5} | 0.0625 | 7,600.1875 | 5,580 | 7,734.866667 | |
Find all values of $x$ that satisfy the equation $|x-3|=2x+4$. Express your answers in simplest fractional form. | -\frac13 | 1 | 2,363.3125 | 2,363.3125 | -1 | |
Natural numbers \( x, y, z \) are such that \( \operatorname{GCD}(\operatorname{LCM}(x, y), z) \cdot \operatorname{LCM}(\operatorname{GCD}(x, y), z) = 1400 \).
What is the maximum value that \( \operatorname{GCD}(\operatorname{LCM}(x, y), z) \) can take? | 10 | 0.0625 | 8,113.4375 | 6,935 | 8,192 | |
Given \( z \in \mathbf{C} \) and \( z^{7} = 1 \) (where \( z \neq 1 \)), find the value of \( \cos \alpha + \cos 2 \alpha + \cos 4 \alpha \), where \(\alpha\) is the argument of \(z\). | -\frac{1}{2} | 0.3125 | 7,586.75 | 6,255.2 | 8,192 | |
Given that \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\) and \(\sin \beta = 2 \cos (\alpha + \beta) \cdot \sin \alpha \left(\alpha + \beta \neq \frac{\pi}{2}\right)\), find the maximum value of \(\tan \beta\). | \frac{\sqrt{3}}{3} | 0 | 6,451.5 | -1 | 6,451.5 | |
In the diagram, three circles each with a radius of 5 units intersect at exactly one common point, which is the origin. Calculate the total area in square units of the shaded region formed within the triangular intersection of the three circles. Express your answer in terms of $\pi$.
[asy]
import olympiad; import geome... | \frac{150\pi - 75\sqrt{3}}{12} | 0 | 7,655.5 | -1 | 7,655.5 | |
A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction. | \frac{9}{2} | 0.6875 | 6,133.6875 | 5,198.090909 | 8,192 | |
There are $5$ people participating in a lottery, each drawing a ticket from a box containing $5$ tickets ($3$ of which are winning tickets) without replacement until all $3$ winning tickets have been drawn, ending the activity. The probability that the activity ends exactly after the $4$th person draws is $\_\_\_\_\_\_... | \frac{3}{10} | 0.3125 | 7,406.4375 | 5,678.2 | 8,192 | |
Given that one air conditioner sells for a 10% profit and the other for a 10% loss, and the two air conditioners have the same selling price, determine the percentage change in the shopping mall's overall revenue. | 1\% | 0.4375 | 5,548.25 | 4,309.428571 | 6,511.777778 | |
Given the numbers 2 and 8, find the product of three numbers that form a geometric sequence with these two numbers. | 64 | 0.0625 | 7,781.5625 | 8,192 | 7,754.2 | |
If the system of inequalities $\left\{\begin{array}{l}9x - a \geqslant 0, \\ 8x - b < 0\end{array}\right.$ has integer solutions only for $1, 2, 3$, how many ordered pairs of integers $(a, b)$ satisfy the system? | 72 | 0.25 | 7,388.875 | 7,252 | 7,434.5 | |
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that\[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\]What is $k?$ | 137 | 1. **Express the problem in binary**: We start by expressing the numbers in binary:
\[
\frac{2^{289}+1}{2^{17}+1} = \frac{1\#0\#0\#0\#0\#0\#0\#0\#0\#0\#0\#0\#0\#0\#0\#0\#0\#1_2}{1\#1_2}
\]
where $\#$ represents $16$ consecutive $0$s.
2. **Expand the denominator**: We consider the expansion of $(2^{17} + 1)... | 0 | 7,483.5 | -1 | 7,483.5 |
Two people agreed to meet at a specific location between 12 PM and 1 PM. The condition is that the first person to arrive will wait for the second person for 15 minutes and then leave. What is the probability that these two people will meet if each of them chooses their moment of arrival at the agreed location randomly... | 7/16 | 0.6875 | 5,671.3125 | 5,081 | 6,970 | |
Given $f(n) = n^2 \cos(n\pi)$ and $a_n = f(n) + f(n+1)$, find the sum of $a_1 + a_2 + a_3 + \cdots + a_{100}$. | -100 | 0.5625 | 6,369.875 | 4,952.666667 | 8,192 | |
Find the sum of all real roots of the equation \(3 \tan ^{2} x + 8 \tan x + 3 = 0\) in the range \(0 < x < 2\pi\). | 5\pi | 0.25 | 7,434.0625 | 6,297.25 | 7,813 | |
What is the smallest four-digit number that is divisible by $33$? | 1023 | 1 | 2,000.5625 | 2,000.5625 | -1 | |
If the real numbers \( x \) and \( y \) satisfy \( 3x + 2y - 1 \geqslant 0 \), then the minimum value of \( u = x^2 + y^2 + 6x - 2y \) is _______ | -66/13 | 0 | 6,500 | -1 | 6,500 | |
Compute: $9 \cdot \frac{1}{13} \cdot 26.$ | 18 | 1 | 1,164 | 1,164 | -1 | |
The integers $G$ and $H$ are chosen such that
\[\frac{G}{x+5}+\frac{H}{x^2-4x}=\frac{x^2-2x+10}{x^3+x^2-20x}\]for all real values of $x$ except $-5$, $0$, and $4$. Find $H/G$. | 2 | 0.875 | 2,898.625 | 2,142.428571 | 8,192 | |
What is the smallest positive odd integer having the same number of positive divisors as 360? | 31185 | 0 | 8,159.1875 | -1 | 8,159.1875 | |
One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that $133^5+110^5+84^5+27^5=n^{5}$. Find the value of $n$.
| 144 | 0.375 | 7,773.125 | 7,075 | 8,192 | |
Given that $\alpha$ is an angle in the third quadrant, $f\left( \alpha \right)=\dfrac{\sin (\alpha -\dfrac{\pi }{2})\cos (\dfrac{3\pi }{2}+\alpha )\tan (\pi -\alpha )}{\tan (-\alpha -\pi )\sin (-\alpha -\pi )}$.
(1) Simplify $f\left( \alpha \right)$
(2) If $\cos (\alpha -\dfrac{3\pi }{2})=\dfrac{1}{5}$, find the valu... | -\dfrac{2\sqrt{6}}{5} | 0 | 4,634.5 | -1 | 4,634.5 | |
Find the value of $a_0 + a_1 + a_2 + \cdots + a_6$ given that $(2-x)^7 = a_0 + a_1(1+x)^2 + \cdots + a_7(1+x)^7$. | 129 | 0.5625 | 6,519.5 | 5,218.666667 | 8,192 | |
The ratio of the dividend to the divisor is 9:2, and the ratio of the divisor to the quotient is ____. | \frac{2}{9} | 0.0625 | 526 | 643 | 518.2 | |
5. Let $S$ denote the set of all positive integers whose prime factors are elements of $\{2,3,5,7,11\}$ . (We include 1 in the set $S$ .) If $$ \sum_{q \in S} \frac{\varphi(q)}{q^{2}} $$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$ , find $a+b$ . (Here $\varphi$ denot... | 1537 | 0.4375 | 6,597 | 5,071.857143 | 7,783.222222 | |
Pegs are put in a board $1$ unit apart both horizontally and vertically. A rubber band is stretched over $4$ pegs as shown in the figure, forming a quadrilateral. Its area in square units is
[asy]
int i,j;
for(i=0; i<5; i=i+1) {
for(j=0; j<4; j=j+1) {
dot((i,j));
}}
draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7... | 6 |
#### Solution 1: Using Pick's Theorem
1. **Identify the number of interior and boundary points:**
- Interior points: $5$
- Boundary points: $4$
2. **Apply Pick's Theorem:**
- Pick's Theorem states that the area $A$ of a simple lattice polygon is given by:
\[
A = I + \frac{B}{2} - 1
\]
wher... | 0.9375 | 5,062.6875 | 4,854.066667 | 8,192 |
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