problem stringlengths 10 5.15k | answer dict |
|---|---|
How many positive integer solutions does the equation have $$ \left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1? $$ ( $\lfloor x \rfloor$ denotes the integer part of $x$ , for example $\lfloor 2\rfloor = 2$ , $\lfloor \pi\rfloor = 3$ , $\lfloor \sqrt2 \rfloor =1$ ) | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose a sequence of positive numbers $\left\{a_{n}\right\}$ satisfies: $a_{0} = 1, a_{n} = a_{n+1} + a_{n+2},$ for $n = 0, 1, 2, \ldots$. Find $a_{1}$. | {
"answer": "\\frac{\\sqrt{5} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a, b, c \in (0,1]\) and \(\lambda\) be a real number such that \(\frac{\sqrt{3}}{\sqrt{a+b+c}} \geq 1+\lambda(1-a)(1-b)(1-c)\) is always satisfied. Find the maximum value of \(\lambda\). | {
"answer": "64/27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Give an example of a number $x$ for which the equation $\sin 2017 x - \operatorname{tg} 2016 x = \cos 2015 x$ holds. Justify your answer. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three workers are digging a pit. They work in shifts, with each worker working as long as it takes for the other two to dig half of the pit. Working in this manner, they dug the pit. How many times faster would the three workers dig the same pit if they worked simultaneously? | {
"answer": "2.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The odd function $y=f(x)$ has a domain of $\mathbb{R}$, and when $x \geq 0$, $f(x) = 2x - x^2$. If the range of the function $y=f(x)$, where $x \in [a, b]$, is $[\frac{1}{b}, \frac{1}{a}]$, then the minimum value of $b$ is ______. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the origin of the rectangular coordinate system xOy as the pole and the positive semi-axis of the x-axis as the polar axis, establish a polar coordinate system with the same unit length. The parametric equation of the line l is $$\begin{cases} \overset{x=2+t}{y=1+t}\end{cases}$$ (t is the parameter), and the polar coordinate equation of the circle C is $$ρ=4 \sqrt {2}sin(θ+ \frac {π}{4})$$.
(1) Find the ordinary equation of line l and the rectangular coordinate equation of circle C.
(2) Suppose the curve C intersects with line L at points A and B. If point P has rectangular coordinates (2, 1), find the value of ||PA|-|PB||. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many values of $x$, $-30<x<120$, satisfy $\cos^2 x + 3\sin^2 x = 1$? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the company, there are elves, fairies, and dwarves. Each elf is friends with all fairies except for three of them, and each fairy is friends with twice as many elves. Each elf is friends with exactly three dwarves, and each fairy is friends with all the dwarves. Each dwarf is friends with exactly half of the total number of elves and fairies. How many dwarves are there in the company? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For lines $l_{1}$: $(3+a)x+4y=5-3a$ and $l_{2}$: $2x+(5+a)y=8$ to be parallel, $a=$ ______. | {
"answer": "-7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The region consisting of all points in three-dimensional space within 4 units of line segment $\overline{CD}$, plus a cone with the same height as $\overline{CD}$ and a base radius of 4 units, has a total volume of $448\pi$. Find the length of $\textit{CD}$. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S = \{1, 2, \ldots, 2005\}$. If any set of $n$ pairwise coprime numbers from $S$ always contains at least one prime number, find the smallest value of $n$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On Monday, Knight Milivoj traveled 25 miles and spent the night in Zubín. The next day, Tuesday, he reached Veselín. On the way back, he traveled 6 miles more on Thursday than on Monday and spent the night in Kostín. On Friday, he traveled the remaining 11 miles to Rytířov. Determine the distance between Zubín and Veselín. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Buses travel along a country road, at equal intervals in both directions and at equal speeds. A cyclist, traveling at $16 \, \mathrm{km/h}$, begins counting buses from the moment when two buses meet beside him. He counts the 31st bus approaching from the front and the 15th bus from behind, both meeting the cyclist again.
How fast were the buses traveling? | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We are laying railway tracks that are $15 \text{ meters }$ long at a temperature of $t = -8^{\circ} \text{C}$. What gap should we leave between each rail if the maximum expected temperature is $t = 60^{\circ} \text{C}$? The coefficient of expansion for the rail is $\lambda = 0.000012$. | {
"answer": "12.24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(a, b, c\) are positive integers and the quadratic equation \(a x^{2}+b x+c=0\) has two real roots whose absolute values are both less than \(\frac{1}{3}\), find the minimum value of \(a + b + c\). | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( p = \frac{21^{3}-11^{3}}{21^{2}+21 \times 11+11^{2}} \), find \( p \).
If \( p \) men can do a job in 6 days and 4 men can do the same job in \( q \) days, find \( q \).
If the \( q \)-th day of March in a year is Wednesday and the \( r \)-th day of March in the same year is Friday, where \( 18 < r < 26 \), find \( r \).
If \( a * b = ab + 1 \), and \( s = (3 * 4)^{*} \), find \( s \). | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The class monitor wants to buy soda in batches for all 50 students and teachers in the class for the sports day. According to the store's policy, every 5 empty bottles can be exchanged for one soda bottle, so there is no need to buy 50 bottles of soda. Then, the minimum number of soda bottles that need to be bought to ensure everyone gets one bottle of soda is . | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and }x^3+bx+a \text{ have at least a common real zero .}\}$ How many elements are there in $S$ ? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two irreducible fractions have their denominators equal to 600 and 700. Find the minimum value for the denominator of the sum of the fractions. | {
"answer": "168",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four different natural numbers, of which one is 1, have the following properties: the sum of any two of them is a multiple of 2, the sum of any three of them is a multiple of 3, and the sum of all four numbers is a multiple of 4. What is the minimum possible sum of these four numbers? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, where $\angle X = 90^\circ$, the hypotenuse $YZ = 13$, and $\tan Z = 3\cos Y$. What is the length of side $XY$? | {
"answer": "\\frac{2\\sqrt{338}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the sides opposite to angles \(A, B,\) and \(C\) are denoted by \(a, b,\) and \(c\) respectively. Given that \(c = 10\) and \(\frac{\cos A}{\cos B} = \frac{b}{a} = \frac{4}{3}\). Point \(P\) is a moving point on the incircle of triangle \(ABC\), and \(d\) is the sum of the squares of the distances from \(P\) to vertices \(A, B,\) and \(C\). Find \(d_{\min} + d_{\max}\). | {
"answer": "160",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose \( a \) is an integer. A sequence \( x_1, x_2, x_3, x_4, \ldots \) is constructed with:
- \( x_1 = a \),
- \( x_{2k} = 2x_{2k-1} \) for every integer \( k \geq 1 \),
- \( x_{2k+1} = x_{2k} - 1 \) for every integer \( k \geq 1 \).
For example, if \( a = 2 \), then:
\[ x_1 = 2, \quad x_2 = 2x_1 = 4, \quad x_3 = x_2 - 1 = 3, \quad x_4 = 2x_3 = 6, \quad x_5 = x_4 - 1 = 5, \]
and so on.
The integer \( N = 578 \) can appear in this sequence after the 10th term (e.g., \( x_{12} = 578 \) when \( a = 10 \)), but the integer 579 does not appear in the sequence after the 10th term for any value of \( a \).
What is the smallest integer \( N > 1395 \) that could appear in the sequence after the 10th term for some value of \( a \)? | {
"answer": "1409",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \(a, b, c, d\) belong to the interval \([-6.5 ; 6.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | {
"answer": "182",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Workshop A and Workshop B together have 360 workers. The number of workers in Workshop A is three times that of Workshop B. How many workers are there in each workshop? | {
"answer": "270",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four chess players - Ivanov, Petrov, Vasiliev, and Kuznetsov - played a round-robin tournament (each played one game against each of the others). A victory awards 1 point, a draw awards 0.5 points to each player. It was found that the player in first place scored 3 points, and the player in last place scored 0.5 points. How many possible distributions of points are there among the named chess players, if some of them could have scored the same number of points? (For example, the cases where Ivanov has 3 points and Petrov has 0.5 points, and where Petrov has 3 points and Ivanov has 0.5 points, are considered different!) | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the quadratic equation \( (5a + 2b)x^2 + ax + b = 0 \) has a unique solution for \( x \), find the value of \( x \). | {
"answer": "\\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using three rectangular pieces of paper (A, C, D) and one square piece of paper (B), an area of 480 square centimeters can be assembled into a large rectangle. It is known that the areas of B, C, and D are all 3 times the area of A. Find the total perimeter of the four pieces of paper A, B, C, and D in centimeters. | {
"answer": "184",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABC\) be a triangle with circumradius \(R = 17\) and inradius \(r = 7\). Find the maximum possible value of \(\sin \frac{A}{2}\). | {
"answer": "\\frac{17 + \\sqrt{51}}{34}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function \( f(x) \) satisfies the equation \( 2 f(x) + x^{2} f\left(\frac{1}{x}\right) = \frac{3 x^{3} - x^{2} + 4 x + 3}{x + 1} \) and \( g(x) = \frac{5}{x + 1} \), determine the minimum value of \( f(x) + g(x) \). | {
"answer": "\\frac{15}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On September 10, 2005, the following numbers were drawn in the five-number lottery: 4, 16, 22, 48, 88. All five numbers are even, exactly four of them are divisible by 4, three by 8, and two by 16. In how many ways can five different numbers with these properties be selected from the integers ranging from 1 to 90? | {
"answer": "15180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four princesses thought of two-digit numbers, and Ivan thought of a four-digit number. After they wrote their numbers in a row in some order, the result was 132040530321. Find Ivan's number. | {
"answer": "5303",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular 2017-gon, all diagonals are drawn. Petya randomly selects some number $\mathrm{N}$ of diagonals. What is the smallest $N$ such that among the selected diagonals there are guaranteed to be two diagonals of the same length? | {
"answer": "1008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Farmer Yang has a \(2015 \times 2015\) square grid of corn plants. One day, the plant in the very center of the grid becomes diseased. Every day, every plant adjacent to a diseased plant becomes diseased. After how many days will all of Yang's corn plants be diseased? | {
"answer": "2014",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular hexagon \( K L M N O P \) is inscribed in an equilateral triangle \( A B C \) such that the points \( K, M, O \) lie at the midpoints of the sides \( A B, B C, \) and \( A C \), respectively. Calculate the area of the hexagon \( K L M N O P \) given that the area of triangle \( A B C \) is \( 60 \text{ cm}^2 \). | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any real number \( x \), let \([x]\) denote the greatest integer less than or equal to \( x \). When \( 0 \leqslant x \leqslant 100 \), how many different integers are in the range of the function \( f(x) = [2x] + [3x] + [4x] + [5x] \)? | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A piece of alloy weighing 6 kg contains copper. Another piece of alloy weighing 8 kg contains copper in a different percentage than the first piece. A certain part was separated from the first piece, and a part twice as heavy was separated from the second piece. Each of the separated parts was then alloyed with the remainder of the other piece, resulting in two new alloys with the same percentage of copper. What is the mass of each part separated from the original alloy pieces? | {
"answer": "2.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From point \( A \), two rays are drawn intersecting a given circle: one at points \( B \) and \( C \), and the other at points \( D \) and \( E \). It is known that \( AB = 7 \), \( BC = 7 \), and \( AD = 10 \). Determine \( DE \). | {
"answer": "0.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, given the set of points $I=\{(x, y) \mid x$ and $y$ are integers, and $0 \leq x \leq 5,0 \leq y \leq 5\}$, find the number of distinct squares that can be formed with vertices from the set $I$. | {
"answer": "105",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An electronic clock always displays the date as an eight-digit number. For example, January 1, 2011, is displayed as 20110101. What is the last day of 2011 that can be evenly divided by 101? The date is displayed as $\overline{2011 \mathrm{ABCD}}$. What is $\overline{\mathrm{ABCD}}$? | {
"answer": "1221",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A company has calculated that investing x million yuan in project A will yield an economic benefit y that satisfies the relationship: $y=f(x)=-\frac{1}{4}x^{2}+2x+12$. Similarly, the economic benefit y from investing in project B satisfies the relationship: $y=h(x)=-\frac{1}{3}x^{2}+4x+1$.
(1) If the company has 10 million yuan available for investment, how should the funds be allocated to maximize the total investment return?
(2) If the marginal effect function for investment is defined as $F(x)=f(x+1)-f(x)$, and investment is not recommended when the marginal value is less than 0, how should the investment be allocated? | {
"answer": "6.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Natural numbers \( x_{1}, x_{2}, \ldots, x_{13} \) are such that \( \frac{1}{x_{1}} + \frac{1}{x_{2}} + \ldots + \frac{1}{x_{13}} = 2 \). What is the minimum value of the sum of these numbers? | {
"answer": "85",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many unordered pairs of coprime numbers are there among the integers 2, 3, ..., 30? Recall that two integers are called coprime if they do not have any common natural divisors other than one. | {
"answer": "248",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCDEFGH$ be a regular octagon, and let $I, J, K$ be the midpoints of sides $AB, DE, GH$ respectively. If the area of $\triangle IJK$ is $144$, what is the area of octagon $ABCDEFGH$? | {
"answer": "1152",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle \( \triangle ABC \) with circumcenter \( O \) and orthocenter \( H \), and \( O \neq H \). Let \( D \) and \( E \) be the midpoints of sides \( BC \) and \( CA \) respectively. Let \( D' \) and \( E' \) be the reflections of \( D \) and \( E \) with respect to \( H \). If lines \( AD' \) and \( BE' \) intersect at point \( K \), find the value of \( \frac{|KO|}{|KH|} \). | {
"answer": "3/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 11 children sitting in a circle playing a game. They are numbered clockwise from 1 to 11. The game starts with child number 1, and each child has to say a two-digit number. The number they say cannot have a digit sum of 6 or 9, and no child can repeat a number that has already been said. The game continues until someone cannot say a new number, and the person who cannot say a new number loses the game. Who will be the last person in the game? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A farmer had an enclosure with a fence 50 rods long, which could only hold 100 sheep. Suppose the farmer wanted to expand the enclosure so that it could hold twice as many sheep.
How many additional rods will the farmer need? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $3, n$, and $n+1$ cents, $115$ cents is the greatest postage that cannot be formed. | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere with a radius of \(\sqrt{3}\) has a cylindrical hole drilled through it; the axis of the cylinder passes through the center of the sphere, and the diameter of the base of the cylinder is equal to the radius of the sphere. Find the volume of the remaining part of the sphere. | {
"answer": "\\frac{9 \\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of triangle $ABC$ is $2 \sqrt{3}$, side $BC$ is equal to $1$, and $\angle BCA = 60^{\circ}$. Point $D$ on side $AB$ is $3$ units away from point $B$, and $M$ is the intersection point of $CD$ with the median $BE$. Find the ratio $BM: ME$. | {
"answer": "3 : 5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four spheres, each with a radius of 1, are placed on a horizontal table with each sphere tangential to its neighboring spheres (the centers of the spheres form a square). There is a cube whose bottom face is in contact with the table, and each vertex of the top face of the cube just touches one of the four spheres. Determine the side length of the cube. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At the end of $1997$, the desert area in a certain region was $9\times 10^{5}hm^{2}$ (note: $hm^{2}$ is the unit of area, representing hectares). Geologists started continuous observations from $1998$ to understand the changes in the desert area of this region. The observation results at the end of each year are recorded in the table below:
| Year | Increase in desert area compared to the original area (end of year) |
|------|--------------------------------------------------------------------|
| 1998 | 2000 |
| 1999 | 4000 |
| 2000 | 6001 |
| 2001 | 7999 |
| 2002 | 10001 |
Based on the information provided in the table, estimate the following:
$(1)$ If no measures are taken, approximately how much will the desert area of this region become by the end of $2020$ in $hm^{2}$?
$(2)$ If measures such as afforestation are taken starting from the beginning of $2003$, with an area of $8000hm^{2}$ of desert being transformed each year, but the desert area continues to increase at the original rate, in which year-end will the desert area of this region be less than $8\times 10^{5}hm^{2}$ for the first time? | {
"answer": "2021",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On each of the one hundred cards, a different non-zero number is written such that each number equals the square of the sum of all the others.
What are these numbers? | {
"answer": "\\frac{1}{99^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The *cross* of a convex $n$ -gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$ .
Suppose $S$ is a dodecagon ( $12$ -gon) inscribed in a unit circle. Find the greatest possible cross of $S$ . | {
"answer": "\\frac{2\\sqrt{66}}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ms. Linda teaches mathematics to 22 students. Before she graded Eric's test, the average score for the class was 84. After grading Eric's test, the class average rose to 85. Determine Eric's score on the test. | {
"answer": "106",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For real numbers \(x, y, z\), the matrix
\[
\begin{pmatrix}
x & y & z \\
y & z & x \\
z & x & y
\end{pmatrix}
\]
is not invertible. Find all possible values of
\[
\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y}.
\] | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a natural number $n \ge 3$ , we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the $n$ -gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of $n$ ?
| {
"answer": "41",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square has sides of length 3 units. A second square is formed having sides that are $120\%$ longer than the sides of the first square. This process is continued sequentially to create a total of five squares. What will be the percent increase in the perimeter from the first square to the fifth square? Express your answer to the nearest tenth. | {
"answer": "107.4\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two individuals, A and B, start traveling towards each other from points A and B, respectively, at the same time. They meet at point C, after which A continues to point B and B rests for 14 minutes before continuing to point A. Both A and B, upon reaching points B and A, immediately return and meet again at point C. Given that A walks 60 meters per minute and B walks 80 meters per minute, how far apart are points A and B? | {
"answer": "1680",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The height \( PO \) of the regular quadrilateral pyramid \( PABC D \) is 4, and the side of the base \( ABCD \) is 6. Points \( M \) and \( K \) are the midpoints of segments \( BC \) and \( CD \). Find the radius of the sphere inscribed in the pyramid \( PMKC \). | {
"answer": "\\frac{12}{13+\\sqrt{41}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
[asy]size(8cm);
real w = 2.718; // width of block
real W = 13.37; // width of the floor
real h = 1.414; // height of block
real H = 7; // height of block + string
real t = 60; // measure of theta
pair apex = (w/2, H); // point where the strings meet
path block = (0,0)--(w,0)--(w,h)--(0,h)--cycle; // construct the block
draw(shift(-W/2,0)*block); // draws white block
path arrow = (w,h/2)--(w+W/8,h/2); // path of the arrow
draw(shift(-W/2,0)*arrow, EndArrow); // draw the arrow
picture pendulum; // making a pendulum...
draw(pendulum, block); // block
fill(pendulum, block, grey); // shades block
draw(pendulum, (w/2,h)--apex); // adds in string
add(pendulum); // adds in block + string
add(rotate(t, apex) * pendulum); // adds in rotated block + string
dot(" $\theta$ ", apex, dir(-90+t/2)*3.14); // marks the apex and labels it with theta
draw((apex-(w,0))--(apex+(w,0))); // ceiling
draw((-W/2-w/2,0)--(w+W/2,0)); // floor[/asy]
A block of mass $m=\text{4.2 kg}$ slides through a frictionless table with speed $v$ and collides with a block of identical mass $m$ , initially at rest, that hangs on a pendulum as shown above. The collision is perfectly elastic and the pendulum block swings up to an angle $\theta=12^\circ$ , as labeled in the diagram. It takes a time $ t = \text {1.0 s} $ for the block to swing up to this peak. Find $10v$ , in $\text{m/s}$ and round to the nearest integer. Do not approximate $ \theta \approx 0 $ ; however, assume $\theta$ is small enough as to use the small-angle approximation for the period of the pendulum.
*(Ahaan Rungta, 6 points)* | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular tank with a horizontal cross-sectional area of \(S = 6 \ \text{m}^2\) is filled with water up to a height of \(H = 5 \ \text{m}\). Determine the time it takes for all the water to flow out of the tank through a small hole at the bottom with an area of \(s = 0.01 \ \text{m}^2\), assuming that the outflow speed of the water is \(0.6 \sqrt{2gh}\), where \(h\) is the height of the water level above the hole and \(g\) is the acceleration due to gravity. | {
"answer": "1010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of $\frac12\cdot\frac41\cdot\frac18\cdot\frac{16}{1} \dotsm \frac{1}{2048}\cdot\frac{4096}{1}$, and multiply the result by $\frac34$. | {
"answer": "1536",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle $PQR$, medians are drawn from $P$ and $Q$ to divide segments $\overline{QR}$ and $\overline{PR}$ in half, respectively. If the length of the median from $P$ to the midpoint of $QR$ is $8$ units, and the median from $Q$ to the midpoint of $PR$ is $4\sqrt{5}$ units, find the length of segment $\overline{PQ}$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the fractional equation in terms of $x$, $\frac{x-2}{x-3}=\frac{n+1}{3-x}$ has a positive root, then $n=\_\_\_\_\_\_.$ | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a \(4 \times 4\) square. Its cells are called neighboring if they share a common side. All cells are painted in two colors: red and blue. It turns out that each red cell has more red neighbors than blue ones, and each blue cell has an equal number of red and blue neighbors. It is known that cells of both colors are present. How many red cells are in the square? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of the expression for \( a, b > 0 \):
$$
\frac{|4a - 10b| + |2(a - b\sqrt{3}) - 5(a\sqrt{3} + b)|}{\sqrt{a^2 + b^2}}
$$ | {
"answer": "2 \\sqrt{87}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function \( y = \tan(2015x) - \tan(2016x) + \tan(2017x) \) has how many zeros in the interval \([0, \pi]\)? | {
"answer": "2017",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given in $\bigtriangleup ABC$, $AB = 75$, and $AC = 120$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. Find the length of $BC$. | {
"answer": "117",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Congcong performs a math magic trick by writing the numbers $1, 2, 3, 4, 5, 6, 7$ on the blackboard and lets others select 5 of these numbers. The product of these 5 numbers is then calculated and told to Congcong, who guesses the chosen numbers. If when it is Benben's turn to select, Congcong cannot even determine whether the sum of the 5 chosen numbers is odd or even, what is the product of the 5 numbers chosen by Benben? | {
"answer": "420",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a kingdom of animals, tigers always tell the truth, foxes always lie, and monkeys sometimes tell the truth and sometimes lie. There are 100 animals of each kind, divided into 100 groups, with each group containing exactly 2 animals of one kind and 1 animal of another kind. After grouping, Kung Fu Panda asked each animal in each group, "Is there a tiger in your group?" and 138 animals responded "yes." He then asked, "Is there a fox in your group?" and 188 animals responded "yes." How many monkeys told the truth both times? | {
"answer": "76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can two distinct squares be chosen from an $8 \times 8$ chessboard such that the midpoint of the line segment connecting their centers is also the center of a square on the board? | {
"answer": "480",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A box contains $3$ pennies, $5$ nickels, $7$ dimes, and $4$ quarters. Eight coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $1.00$ (100 cents)?
A) $\frac{325}{75582}$
B) $0$
C) $\frac{5000}{75582}$
D) $\frac{2345}{75582}$
E) $\frac{6000}{75582}$ | {
"answer": "\\frac{2345}{75582}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a football championship with 16 teams, each team played with every other team exactly once. A win was awarded 3 points, a draw 1 point, and a loss 0 points. A team is considered successful if it scored at least half of the maximum possible points. What is the maximum number of successful teams that could have participated in the tournament? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ be a point on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $F_{1}$ and $F_{2}$ be the two foci of the ellipse, and $e$ be the eccentricity of the ellipse. Given $\angle P F_{1} F_{2}=\alpha$ and $\angle P F_{2} F_{1}=\beta$, express $\tan \frac{\alpha}{2} \cdot \tan \frac{\beta}{2}$ in terms of $e$. | {
"answer": "\\frac{1 - e}{1 + e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Dima took the fractional-linear function \(\frac{a x + 2b}{c x + 2d}\), where \(a, b, c, d\) are positive numbers, and summed it with the remaining 23 functions obtained from it by permuting the numbers \(a, b, c, d\). Find the root of the sum of all these functions, independent of the numbers \(a, b, c, d\). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Every day, from Monday to Friday, an old man went to the blue sea and cast his net into the sea. Each day, he caught no more fish than the previous day. In total, he caught exactly 100 fish over the five days. What is the minimum total number of fish he could have caught on Monday, Wednesday, and Friday? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The largest divisor of a natural number \( N \), smaller than \( N \), was added to \( N \), producing a power of ten. Find all such \( N \). | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $B= \frac {\pi}{3}$ and $(a-b+c)(a+b-c)= \frac {3}{7}bc$.
(Ⅰ) Find the value of $\cos C$;
(Ⅱ) If $a=5$, find the area of $\triangle ABC$. | {
"answer": "10 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( M = \{1, 2, 3, \ldots, 1995\} \). Subset \( A \) of \( M \) satisfies the condition: If \( x \in A \), then \( 15x \notin A \). What is the maximum number of elements in \( A \)? | {
"answer": "1870",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We repeatedly toss a coin until we get either three consecutive heads ($HHH$) or the sequence $HTH$ (where $H$ represents heads and $T$ represents tails). What is the probability that $HHH$ occurs before $HTH$? | {
"answer": "2/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
100 participants came to the international StarCraft championship. The game is played in a knockout format, meaning two players participate in each match, the loser is eliminated from the championship, and the winner remains. Find the maximum possible number of participants who won exactly two matches. | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the parallelogram \(ABCD\), the longer side \(AD\) is 5. The angle bisectors of angles \(A\) and \(B\) intersect at point \(M\).
Find the area of the parallelogram, given that \(BM = 2\) and \(\cos \angle BAM = \frac{4}{5}\). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya wrote consecutive natural numbers \(N\), \(N+1\), \(N+2\), and \(N+3\) in rectangular boxes. Below each rectangle, he wrote the sum of the digits of the corresponding number in a circle.
The sum of the numbers in the first and second circles equals 200, and the sum of the numbers in the third and fourth circles equals 105. What is the sum of the numbers in the second and third circles? | {
"answer": "103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A \( 5 \mathrm{~cm} \) by \( 5 \mathrm{~cm} \) pegboard and a \( 10 \mathrm{~cm} \) by \( 10 \mathrm{~cm} \) pegboard each have holes at the intersection of invisible horizontal and vertical lines that occur in \( 1 \mathrm{~cm} \) intervals from each edge. Pegs are placed into the holes on the two main diagonals of both pegboards. The \( 5 \mathrm{~cm} \) by \( 5 \mathrm{~cm} \) pegboard is shown; it has 16 holes. The 8 shaded holes have pegs, and the 8 unshaded holes do not. How many empty holes does the \( 10 \mathrm{~cm} \) by \( 10 \mathrm{~cm} \) pegboard have?
| {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fill the numbers $1, 2, \cdots, 36$ into a $6 \times 6$ grid with each cell containing one number, such that each row is in ascending order from left to right. What is the minimum possible sum of the six numbers in the third column? | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many $n=2,3,4,\ldots,99,100$ is the base-$n$ number $215216_n$ a multiple of $5$? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular tetrahedron $S-ABC$ with a base that is an equilateral triangle of side length 1 and side edges of length 2. If a plane passing through line $AB$ divides the tetrahedron's volume into two equal parts, the cosine of the dihedral angle between the plane and the base is: | {
"answer": "$\\frac{2 \\sqrt{15}}{15}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular hexagon \(ABCDEF\), points \(M\) and \(K\) are taken on the diagonals \(AC\) and \(CE\) respectively, such that \(AM : AC = CK : CE = n\). Points \(B, M,\) and \(K\) are collinear. Find \(n\). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In right triangle \( ABC \), a point \( D \) is on hypotenuse \( AC \) such that \( BD \perp AC \). Let \(\omega\) be a circle with center \( O \), passing through \( C \) and \( D \) and tangent to line \( AB \) at a point other than \( B \). Point \( X \) is chosen on \( BC \) such that \( AX \perp BO \). If \( AB = 2 \) and \( BC = 5 \), then \( BX \) can be expressed as \(\frac{a}{b}\) for relatively prime positive integers \( a \) and \( b \). Compute \( 100a + b \). | {
"answer": "8041",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( M = \{1, 2, \cdots, 2005\} \), and \( A \) be a subset of \( M \). If for any \( a_i, a_j \in A \) with \( a_i \neq a_j \), an isosceles triangle can be uniquely determined with \( a_i \) and \( a_j \) as side lengths, find the maximum value of \( |A| \). | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many integers between $3250$ and $3500$ have four distinct digits arranged in increasing order? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On each side of a right-angled triangle, a semicircle is drawn with that side as a diameter. The areas of the three semicircles are \( x^{2} \), \( 3x \), and 180, where \( x^{2} \) and \( 3x \) are both less than 180. What is the area of the smallest semicircle? | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Oleg drew an empty 50×50 table and wrote a number above each column and next to each row.
It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 being irrational. Then, in each cell of the table, he wrote the sum of the numbers written next to its row and its column (a "sum table"). What is the maximum number of sums in this table that could be rational numbers? | {
"answer": "1250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Lyla and Isabelle run on a circular track both starting at point \( P \). Lyla runs at a constant speed in the clockwise direction. Isabelle also runs in the clockwise direction at a constant speed 25% faster than Lyla. Lyla starts running first and Isabelle starts running when Lyla has completed one third of one lap. When Isabelle passes Lyla for the fifth time, how many times has Lyla returned to point \( P \)? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A dark room contains 120 red socks, 100 green socks, 70 blue socks, 50 yellow socks, and 30 black socks. A person randomly selects socks from the room without the ability to see their colors. What is the smallest number of socks that must be selected to guarantee that the selection contains at least 15 pairs? | {
"answer": "146",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A snail crawls from one tree to another. In half a day, it covered \( l_{1}=5 \) meters. Then, it got tired of this and turned back, crawling \( l_{2}=4 \) meters. It got tired and fell asleep. The next day, the same process repeats. The distance between the trees is \( s=30 \) meters. On which day of its journey will the snail reach the tree? (10 points) | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( a \) and \( b \) are real numbers, and the following system of inequalities in terms of \( x \):
\[
\left\{\begin{array}{l}
20x + a > 0, \\
15x - b \leq 0
\end{array}\right.
\]
has integer solutions of only 2, 3, and 4, find the maximum value of \( ab \). | {
"answer": "-1200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer \( n \) such that \( n(n+1)(n+2) \) is divisible by 247. | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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