problem stringlengths 10 5.15k | answer dict |
|---|---|
Find the largest positive number \( c \) such that for every natural number \( n \), the inequality \( \{n \sqrt{2}\} \geqslant \frac{c}{n} \) holds, where \( \{n \sqrt{2}\} = n \sqrt{2} - \lfloor n \sqrt{2} \rfloor \) and \( \lfloor x \rfloor \) denotes the integer part of \( x \). Determine the natural number \( n \) for which \( \{n \sqrt{2}\} = \frac{c}{n} \).
(This problem appeared in the 30th International Mathematical Olympiad, 1989.) | {
"answer": "\\frac{1}{2\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her? | {
"answer": "262144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There exists a positive number $m$ such that the positive roots of the equation $\sqrt{3} \sin x - \cos x = m$ form an arithmetic sequence in ascending order. If the point $A(1, m)$ lies on the line $ax + by - 2 = 0 (a > 0, b > 0)$, find the minimum value of $\frac{1}{a} + \frac{2}{b}$. | {
"answer": "\\frac{9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The square quilt block shown is made from sixteen unit squares, where eight of these squares have been divided in half diagonally to form triangles. Each triangle is shaded. What fraction of the square quilt is shaded? Express your answer as a common fraction. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Observe the following three rows of numbers and complete the subsequent questions:
①-2, 4, -8, 16, ...
②1, -2, 4, -8, ...
③0, -3, 3, -9, ...
(1) Consider the pattern in row ① and write the expression for the $n^{th}$ number.
(2) Denote the $m^{th}$ number in row ② as $a$ and the $m^{th}$ number in row ③ as $b$. Write the relationship between $a$ and $b$.
(3) Let $x$, $y$, and $z$ represent the $2019^{th}$ number in rows ①, ②, and ③, respectively. Calculate the value of $x + y + z$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the city of Autolândia, car license plates are numbered with three-digit numbers ranging from 000 to 999. The mayor, Pietro, has decided to implement a car rotation system to reduce pollution, with specific rules for each day of the week regarding which cars can be driven:
- Monday: only cars with odd-numbered plates;
- Tuesday: only cars with plates where the sum of the three digits is greater than or equal to 11;
- Wednesday: only cars with plates that are multiples of 3;
- Thursday: only cars with plates where the sum of the three digits is less than or equal to 14;
- Friday: only cars with plates containing at least two identical digits;
- Saturday: only cars with plates strictly less than 500;
- Sunday: only cars with plates where all three digits are less than or equal to 5.
a) On which days can the car with plate 729 be driven?
b) Maria, the mayor's wife, wants a car that can be driven every day except Sunday. Which plate should she have?
c) Mayor Pietro needs a plate that allows him to drive every day. Which plate should he have?
d) Why can all inhabitants of Autolândia drive at least once a week? | {
"answer": "255",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of the region described by $x \ge 0,$ $y \ge 0,$ and
\[50 \{x\} \ge \lfloor x \rfloor - \lfloor y \rfloor.\] | {
"answer": "25.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There exist $s$ unique nonnegative integers $m_1 > m_2 > \cdots > m_s$ and $s$ unique integers $b_k$ ($1\le k\le s$) with each $b_k$ either $1$ or $-1$ such that \[b_13^{m_1} + b_23^{m_2} + \cdots + b_s3^{m_s} = 1729.\]
Find $m_1 + m_2 + \cdots + m_s$. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of
\[x^3 + 12x + \frac{81}{x^4}\]
for $x > 0$. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance from point $\left(1,0\right)$ to the line $3x+4y-2+\lambda \left(2x+y+2\right)=0$, where $\lambda \in R$, needs to be determined. | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Joel now selects an acute angle $x$ (between 0 and 90 degrees) and writes $\sin x$, $\cos x$, and $\tan x$ on three different cards. Each student, Malvina, Paulina, and Georgina, receives one card, but Joel chooses a value $\sin x$ that is a commonly known value, $\sin x = \frac{1}{2}$. The three students know the angle is acute and share the values on their cards without knowing which function produced which. Only Paulina is able to surely identify the function and the specific angle for the value on her card. Determine all possible angles $x$ and find the value that Paulina could identify on her card. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the repetend in the decimal representation of $\frac{5}{17}$. | {
"answer": "294117647058823529",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the $6$ sides and the $9$ diagonals of a regular hexagon are randomly and independently colored red, blue, or green with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the hexagon such that all of its sides have the same color?
A) $\frac{3}{4}$
B) $\frac{880}{1000}$
C) $\frac{872}{1000}$
D) $\frac{850}{1000}$ | {
"answer": "\\frac{872}{1000}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangular pyramid $A B C D$ with a base $A B C$, the lateral edges are pairwise perpendicular, $D A=D B=5$, and $D C=1$. From a point on the base, a light ray is emitted. After reflecting exactly once from each of the lateral faces (without reflecting from the edges), the ray hits a point on the base of the pyramid. What is the minimum distance the ray could have traveled? | {
"answer": "\\frac{10\\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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. | {
"answer": "80660",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x \) and \( y \) be real numbers with \( y > x > 0 \), satisfying
\[ \frac{x}{y} + \frac{y}{x} = 8. \]
Find the value of \[ \frac{x + y}{x - y}. \] | {
"answer": "\\sqrt{\\frac{5}{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A new definition: $\left[a,b,c\right]$ represents the "graph number" of a quadratic function $y=ax^{2}+bx+c$ (where $a\neq 0$, and $a$, $b$, $c$ are real numbers). For example, the "graph number" of $y=-x^{2}+2x+3$ is $\left[-1,2,3\right]$.
$(1)$ The "graph number" of the quadratic function $y=\frac{1}{3}x^{2}-x-1$ is ______.
$(2)$ If the "graph number" of a quadratic function is $\left[m,m+1,m+1\right]$, and the graph intersects the $x$-axis at only one point, find the value of $m$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A flea is jumping on the vertices of square \(ABCD\), starting from vertex \(A\). With each jump, it moves to an adjacent vertex with a probability of \(\frac{1}{2}\). The flea stops when it reaches the last vertex it has not yet visited. Determine the probability that each vertex will be the last one visited. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The angle between vector $\overrightarrow{a}=(\sqrt{3},\;1)$ and vector $\overrightarrow{b}=(\sqrt{3},\;-1)$ is _______. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the triangles with natural number side lengths, a perimeter not exceeding 100, and the difference between the longest and shortest sides not greater than 2, there are a total of different triangles that are not congruent to each other. | {
"answer": "190",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of the base of a brick are 28 cm and 9 cm, and its height is 6 cm. A snail crawls rectilinearly along the faces of the brick from one vertex of the lower base to the opposite vertex of the upper base. The horizontal and vertical components of its speed $v_{x}$ and $v_{y}$ are related by the equation $v_{x}^{2}+4 v_{y}^{2}=1$ (for example, on the upper face, $v_{y}=0$ cm/min, hence $v_{x}=v=1$ cm/min). What is the minimum time the snail can spend on its journey? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A box of chocolates in the shape of a cuboid was full of chocolates arranged in rows and columns. Míša ate some of them, and the remaining chocolates were rearranged to fill three entire rows completely, except for one space. Míša ate the remaining chocolates from another incomplete row. Then he rearranged the remaining chocolates and filled five columns completely, except for one space. He again ate the chocolates from the incomplete column. In the end, one-third of the original number of chocolates remained in the box. Determine:
a) How many chocolates were there in the entire box originally?
b) How many chocolates did Míša eat before the first rearrangement? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two dice are made so that the chances of getting an even sum are twice that of getting an odd sum. What is the probability of getting an odd sum in a single roll of these two dice?
(a) \(\frac{1}{9}\)
(b) \(\frac{2}{9}\)
(c) \(\frac{4}{9}\)
(d) \(\frac{5}{9}\) | {
"answer": "$\\frac{4}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The edge of cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is equal to 1. Construct a cross-section of the cube by a plane that has the maximum perimeter. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[ e^{2 \pi i/17} + e^{4 \pi i/17} + e^{6 \pi i/17} + \dots + e^{32 \pi i/17}. \] | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An eight-sided die numbered from 1 to 8 is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$? | {
"answer": "192",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The square of a three-digit number ends with three identical digits different from zero. Write the smallest such three-digit number. | {
"answer": "462",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Board with dimesions $2018 \times 2018$ is divided in unit cells $1 \times 1$ . In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain black chips. If $W$ is number of remaining white chips, and $B$ number of remaining black chips on board and $A=min\{W,B\}$ , determine maximum of $A$ | {
"answer": "1018081",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $A$ and $C$ coincide, forming the pentagon $ABEFD$. What is the length of segment $EF$? Express your answer in simplest radical form. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = x^2 + ax + b$ ($a, b \in \mathbb{R}$).
(Ⅰ) Given $x \in [0, 1]$,
(i) If $a = b = 1$, find the range of the function $f(x)$;
(ii) If the range of the function $f(x)$ is $[0, 1]$, find the values of $a$ and $b$;
(Ⅱ) When $|x| \geq 2$, it always holds that $f(x) \geq 0$, and the maximum value of $f(x)$ in the interval $(2, 3]$ is 1, find the maximum and minimum values of $a^2 + b^2$. | {
"answer": "74",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a labor and technical competition among five students: A, B, C, D, and E, the rankings from first to fifth place were determined. When A and B asked about their results, the respondent told A, "Unfortunately, both you and B did not win the championship"; and told B, "You certainly are not the worst." Based on these responses, how many different possible ranking arrangements are there for the five students? (Fill in the number) | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)$ that satisfies $f\left(\frac{\pi}{2} - x\right) + f(x) = 0$ and $f(\pi + x) = f(-x)$, calculate the value of $f\left(\frac{79\pi}{24}\right)$. | {
"answer": "\\frac{\\sqrt{2} - \\sqrt{6}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Bethany has 11 pound coins and some 20 pence coins and some 50 pence coins in her purse. The mean value of the coins is 52 pence. Which could not be the number of coins in the purse? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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? | {
"answer": "499",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse M: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ (a>0, b>0) with two vertices A(-a, 0) and B(a, 0). Point P is a point on the ellipse distinct from A and B. The slopes of lines PA and PB are k₁ and k₂, respectively, and $$k_{1}k_{2}=- \frac {1}{2}$$.
(1) Find the eccentricity of the ellipse C.
(2) If b=1, a line l intersects the x-axis at D(-1, 0) and intersects the ellipse at points M and N. Find the maximum area of △OMN. | {
"answer": "\\frac { \\sqrt {2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Dave walks to school and averages 85 steps per minute, with each step being 80 cm long. It now takes him 15 minutes to get to school. Jack, walking the same route to school, takes steps that are 72 cm long and averages 104 steps per minute. Find the time it takes Jack to reach school. | {
"answer": "13.62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the number of multiples of 4 that are between 100 and 500. | {
"answer": "99",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The points $(2, 5), (10, 9)$, and $(6, m)$, where $m$ is an integer, are vertices of a triangle. What is the sum of the values of $m$ for which the area of the triangle is a minimum? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a,\) \(b,\) and \(c\) be positive real numbers such that \(abc = 27.\) Find the minimum value of
\[
a^2 + 6ab + 9b^2 + 4c^2.
\] | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Katrine has a bag containing 4 buttons with distinct letters M, P, F, G on them (one letter per button). She picks buttons randomly, one at a time, without replacement, until she picks the button with letter G. What is the probability that she has at least three picks and her third pick is the button with letter M?
| {
"answer": "1/12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. | {
"answer": "\\frac{4(3+2\\sqrt{2})}{1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the sum: 1 - 2 + 3 - 4 + $\cdots$ + 100 - 101 | {
"answer": "-151",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function \( g \), defined on the set of ordered pairs of positive integers, satisfies the following properties:
\[
\begin{align*}
g(x, x) &= x, \\
g(x, y) &= g(y, x), \quad \text{and} \\
(x + 2y)g(x, y) &= yg(x, x + 2y).
\end{align*}
\]
Calculate \( g(18, 66) \). | {
"answer": "198",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A boy is riding a scooter from one bus stop to another and looking in the mirror to see if a bus appears behind him. As soon as the boy notices the bus, he can change the direction of his movement. What is the maximum distance between the bus stops so that the boy is guaranteed not to miss the bus, given that he rides at a speed three times less than the speed of the bus and can see the bus at a distance of no more than 2 km? | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya has:
a) 2 different volumes from the collected works of A.S. Pushkin, each volume is 30 cm high;
b) a set of works by E.V. Tarle in 4 volumes, each volume is 25 cm high;
c) a book of lyrical poems with a height of 40 cm, published by Vasya himself.
Vasya wants to arrange these books on a shelf so that his own work is in the center, and the books located at the same distance from it on both the left and the right have equal heights. In how many ways can this be done?
a) $3 \cdot 2! \cdot 4!$;
b) $2! \cdot 3!$;
c) $\frac{51}{3! \cdot 2!}$;
d) none of the above answers are correct. | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a number is a multiple of 4 or contains the digit 4, we say this number is a "4-inclusive number", such as 20, 34. Arrange all "4-inclusive numbers" in the range \[0, 100\] in ascending order to form a sequence. What is the sum of all items in this sequence? | {
"answer": "1883",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of pairs of integers $x, y$ with different parities such that $\frac{1}{x}+\frac{1}{y} = \frac{1}{2520}$ . | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $0 \leq x \leq 2$, find the maximum and minimum values of the function $y = 4^{x- \frac {1}{2}} - 3 \times 2^{x} + 5$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability that Class A will be assigned exactly 2 of the 8 awards, with each of the 4 classes (A, B, C, and D) receiving at least 1 award is $\qquad$ . | {
"answer": "\\frac{2}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A 10-cm-by-10-cm square is partitioned such that points $A$ and $B$ are on two opposite sides of the square at one-third and two-thirds the length of the sides, respectively. What is the area of the new shaded region formed by connecting points $A$, $B$, and their reflections across the square's diagonal?
[asy]
draw((0,0)--(15,0));
draw((15,0)--(15,15));
draw((15,15)--(0,15));
draw((0,15)--(0,0));
draw((0,5)--(15,10));
draw((15,5)--(0,10));
fill((7.5,2.5)--(7.5,12.5)--(5,7.5)--(10,7.5)--cycle,gray);
label("A",(0,5),W);
label("B",(15,10),E);
[/asy] | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are enough cuboids with side lengths of 2, 3, and 5. They are neatly arranged in the same direction to completely fill a cube with a side length of 90. The number of cuboids a diagonal of the cube passes through is | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers between 10 and 13000, when read from left to right, are formed by consecutive digits in ascending order? For example, 456 is one of these numbers, but 7890 is not. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ticket price for a cinema is: 6 yuan per individual ticket, 40 yuan for a group ticket for every 10 people, and students enjoy a 10% discount. A school with 1258 students plans to watch a movie (teachers get in for free). The school should pay the cinema at least ____ yuan. | {
"answer": "4536",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(g(x)\) be the function defined on \(-2 \leq x \leq 2\) by the formula $$g(x) = 2 - \sqrt{4 - x^2}.$$ This function represents the upper half of a circle with radius 2 centered at \((0, 2)\). If a graph of \(x = g(y)\) is overlaid on the graph of \(y = g(x)\), then one fully enclosed region is formed by the two graphs. What is the area of that region, rounded to the nearest hundredth? | {
"answer": "1.14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many natural numbers greater than 10 but less than 100 are relatively prime to 21? | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a table, there are five clocks with hands. It is allowed to move any number of them forward. For each clock, the time by which it is moved will be referred to as the translation time. It is required to set all clocks such that they show the same time. What is the minimum total translation time needed to guarantee this? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer with exactly $12$ positive factors? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a basket, there are 41 apples: 10 green, 13 yellow, and 18 red. Alyona is sequentially taking out one apple at a time from the basket. If at any point the number of green apples she has taken out is less than the number of yellow apples, and the number of yellow apples is less than the number of red apples, then she will stop taking out more apples.
(a) What is the maximum number of yellow apples Alyona can take out from the basket?
(b) What is the maximum total number of apples Alyona can take out from the basket? | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the boat is leaking water at the rate of 15 gallons per minute, the maximum time to reach the shore is 50/15 minutes. If Amy rows towards the shore at a speed of 2 mph, then every 30 minutes she increases her speed by 1 mph, find the maximum rate at which Boris can bail water in gallons per minute so that they reach the shore safely without exceeding a maximum capacity of 50 gallons. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A semicircle with a radius of 1 is drawn inside a semicircle with a radius of 2. A circle is drawn such that it touches both semicircles and their common diameter. What is the radius of this circle? | {
"answer": "\\frac{8}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) where \( a > b > 0 \), with the left focus at \( F \). A tangent to the ellipse is drawn at a point \( A \) on the ellipse and intersects the \( y \)-axis at point \( Q \). Let \( O \) be the origin of the coordinate system. If \( \angle QFO = 45^\circ \) and \( \angle QFA = 30^\circ \), find the eccentricity of the ellipse. | {
"answer": "\\frac{\\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that three cultural courses (Chinese, Mathematics, and Foreign Language) and three other arts courses are randomly scheduled in six periods, find the probability that no two adjacent cultural courses are separated by more than one arts course. | {
"answer": "\\dfrac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x_1<x_2< \ldots <x_{2024}$ be positive integers and let $p_i=\prod_{k=1}^{i}(x_k-\frac{1}{x_k})$ for $i=1,2, \ldots, 2024$ . What is the maximal number of positive integers among the $p_i$ ? | {
"answer": "1012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x, y, z \in \mathbf{R} \) such that \( x^2 + y^2 + xy = 1 \), \( y^2 + z^2 + yz = 2 \), \( x^2 + z^2 + xz = 3 \), find \( x + y + z \). | {
"answer": "\\sqrt{3 + \\sqrt{6}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of six-digit palindromes. | {
"answer": "9000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$, with $b=6$, $c=10$, and $\cos C=-\frac{2}{3}$.
$(1)$ Find $\cos B$;
$(2)$ Find the height on side $AB$. | {
"answer": "\\frac{20 - 4\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are both unit vectors, if $|\overrightarrow{a}-2\overrightarrow{b}|=\sqrt{3}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ____. | {
"answer": "\\frac{1}{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\frac{x^2}{4} + \frac{y^2}{1} = 1$ with one of its foci at $F = (\sqrt{3}, 0)$, find a point $P = (p, 0)$ where $p > 0$ such that for any chord $\overline{AB}$ passing through $F$, the angles $\angle APF$ and $\angle BPF$ are equal. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x, y, z, w \) be different positive real numbers such that \( x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{w}=w+\frac{1}{x}=t \). Find \( t \). | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence of length 15 consisting of the letters $A$ and $B$ satisfies the following conditions:
For any two consecutive letters, $AA$ appears 5 times, $AB$, $BA$, and $BB$ each appear 3 times. How many such sequences are there?
For example, in $AA B B A A A A B A A B B B B$, $AA$ appears 5 times, $AB$ appears 3 times, $BA$ appears 2 times, and $BB$ appears 4 times, which does not satisfy the above conditions. | {
"answer": "560",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ and $c$ be positive real numbers such that $abc = 8.$ Find the minimum value of
\[(3a + b)(a + 3c)(2bc + 4).\] | {
"answer": "384",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Carl has 24 fence posts and places one on each of the four corners, with 3 yards between neighboring posts, where the number of posts on the longer side is three times the number of posts on the shorter side, determine the area, in square yards, of Carl's lawn. | {
"answer": "243",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance between locations A and B is 291 kilometers. Persons A and B depart simultaneously from location A and travel to location B at a constant speed, while person C departs from location B and heads towards location A at a constant speed. When person B has traveled \( p \) kilometers and meets person C, person A has traveled \( q \) kilometers. After some more time, when person A meets person C, person B has traveled \( r \) kilometers in total. Given that \( p \), \( q \), and \( r \) are prime numbers, find the sum of \( p \), \( q \), and \( r \). | {
"answer": "221",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Numbers between $1$ and $4050$ that are integer multiples of $5$ or $7$ but not $35$ can be counted. | {
"answer": "1273",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each positive integer \( x \), let \( f(x) \) denote the greatest power of 3 that divides \( x \). For example, \( f(9) = 9 \) and \( f(18) = 9 \). For each positive integer \( n \), let \( T_n = \sum_{k=1}^{3^n} f(3k) \). Find the greatest integer \( n \) less than 1000 such that \( T_n \) is a perfect square. | {
"answer": "960",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of the binomial coefficients in the expansion of $(5x- \frac{1}{\sqrt{x}})^n$ is 64, determine the constant term in its expansion. | {
"answer": "375",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the two points \(A(4,1)\) and \(B(2,5)\). For each point \(C\) with positive integer coordinates, we define \(d_C\) to be the shortest distance needed to travel from \(A\) to \(C\) to \(B\) moving only horizontally and/or vertically. The positive integer \(N\) has the property that there are exactly 2023 points \(C(x, y)\) with \(x > 0\) and \(y > 0\) and \(d_C = N\). What is the value of \(N\)? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( m = n^{4} + x \), where \( n \) is a natural number and \( x \) is a two-digit positive integer, what value of \( x \) will make \( m \) a composite number? | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ and $c$ be nonnegative numbers such that $a^2 + b^2 + c^2 = 1.$ Find the maximum value of
\[3ab \sqrt{2} + 6bc.\] | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles triangle \(ABC\) (\(AB = BC\)), the angle bisectors \(AM\) and \(BK\) intersect at point \(O\). The areas of triangles \(BOM\) and \(COM\) are 25 and 30, respectively. Find the area of triangle \(ABC\). | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of ways to arrange the letters of the word PERSEVERANCE. | {
"answer": "9,979,200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the convex quadrilateral \(ABCD\): \(AB = AC = AD = BD\) and \(\angle BAC = \angle CBD\). Find \(\angle ACD\). | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the 2015 natural numbers between 1 and 2015, what is the maximum number of numbers that can be found such that their product multiplied by 240 is a perfect square? | {
"answer": "134",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair six-sided die with faces numbered 1, 2, 3, 4, 5, and 6 is rolled twice. Let $a$ and $b$ denote the outcomes of the first and second rolls, respectively.
(1) Find the probability that the line $ax + by + 5 = 0$ is tangent to the circle $x^2 + y^2 = 1$.
(2) Find the probability that the segments with lengths $a$, $b$, and $5$ form an isosceles triangle. | {
"answer": "\\frac{7}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A king traversed a $9 \times 9$ chessboard, visiting each square exactly once. The king's route is not a closed loop and may intersect itself. What is the maximum possible length of such a route if the length of a move diagonally is $\sqrt{2}$ and the length of a move vertically or horizontally is 1? | {
"answer": "16 + 64 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In convex quadrilateral $ABCD$ , $\angle ADC = 90^\circ + \angle BAC$ . Given that $AB = BC = 17$ , and $CD = 16$ , what is the maximum possible area of the quadrilateral?
*Proposed by Thomas Lam* | {
"answer": "529/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arrange 1, 2, 3, a, b, c in a row such that letter 'a' is not at either end and among the three numbers, exactly two are adjacent. The probability is $\_\_\_\_\_\_$. | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$ .
Find $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$ . | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $17^{-1} \equiv 31 \pmod{53}$, find $36^{-1} \pmod{53}$ as a residue modulo 53 (i.e., a value between 0 and 52 inclusive). | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ferry "Yi Rong" travels at a speed of 40 kilometers per hour. On odd days, it travels downstream from point $A$ to point $B$, while on even days, it travels upstream from point $B$ to point $A$ (with the water current speed being 24 kilometers per hour). On one odd day, when the ferry reached the midpoint $C$, it lost power and drifted downstream to point $B$. The captain found that the total time taken that day was $\frac{43}{18}$ times the usual time for an odd day.
On another even day, the ferry again lost power as it reached the midpoint $C$. While drifting, the repair crew spent 1 hour repairing the ferry, after which it resumed its journey to point $A$ at twice its original speed. The captain observed that the total time taken that day was exactly the same as the usual time for an even day. What is the distance between points $A$ and $B$ in kilometers? | {
"answer": "192",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let tetrahedron $ABCD$ have $AD=BC=30$, $AC=BD=40$, and $AB=CD=50$. For any point $X$ in space, suppose $g(X)=AX+BX+CX+DX$. Determine the least possible value of $g(X)$, expressed as $p\sqrt{q}$ where $p$ and $q$ are positive integers with $q$ not divisible by the square of any prime. Report the sum $p+q$. | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with an area of 40 is inscribed in a semicircle. If another square is inscribed in a full circle with the same radius, what is the area of this square? | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the 3rd grade, the boys wear blue swim caps, and the girls wear red swim caps. The male sports commissioner says, "I see 1 more blue swim cap than 4 times the number of red swim caps." The female sports commissioner says, "I see 24 more blue swim caps than red swim caps." Based on the sports commissioners' statements, calculate the total number of students in the 3rd grade. | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x \) satisfies \(\log _{5 x} 2 x = \log _{625 x} 8 x\), find the value of \(\log _{2} x\). | {
"answer": "\\frac{\\ln 5}{2 \\ln 2 - 3 \\ln 5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eight students from a university are preparing to carpool for a trip. There are two students from each grade level—freshmen, sophomores, juniors, and seniors—divided into two cars, Car A and Car B, with each car seating four students. The seating arrangement of the four students in the same car is not considered. However, the twin sisters, who are freshmen, must ride in the same car. The number of ways for Car A to have exactly two students from the same grade is _______. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If five pairwise coprime distinct integers \( a_{1}, a_{2}, \cdots, a_{5} \) are randomly selected from \( 1, 2, \cdots, n \) and there is always at least one prime number among them, find the maximum value of \( n \). | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Every day at noon, a scheduled ship departs from Moscow to Astrakhan and from Astrakhan to Moscow. The ship traveling from Moscow takes exactly four days to reach Astrakhan, then stays there for two days, and at noon two days after its arrival in Astrakhan, it departs back to Moscow. The ship traveling from Astrakhan takes exactly five days to reach Moscow and, after resting for two days in Moscow, departs back to Astrakhan. How many ships must be operational on the Moscow - Astrakhan - Moscow route under the described travel conditions? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing it on top of the existing stack (if there are already bags there). If given a choice to carry a bag from the truck or from the gate, the worker randomly chooses each option with a probability of 0.5. Eventually, all the bags end up in the shed.
a) (7th grade level, 1 point). What is the probability that the bags end up in the shed in the reverse order compared to how they were placed in the truck?
b) (7th grade level, 1 point). What is the probability that the bag that was second from the bottom in the truck ends up as the bottom bag in the shed? | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression:
\[4(1+4(1+4(1+4(1+4(1+4(1+4(1+4(1))))))))\] | {
"answer": "87380",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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