question string | target string | label int64 | answer string |
|---|---|---|---|
G1.3 A six-digit number 1234xy is divisible by both 8 and 9 . Given that $x+y=c$, find the value of $c$. | First, I'll analyze the number 1234xy, which is y must be divisible by 9.
Combining both conditions, I'll express x + y in terms of c, where c = x + y. Since 10 + c must be divisible by 9, c must be 8, as 10 + 8 = 18, which is divisible by 9.
Finally, I'll verify that with c = 8, the last three digits 4xy form a numb... | 1 | 8 |
8. Find $\sqrt{\frac{x}{63}-32} \times \sqrt{\frac{y}{63}-32}$, given that $\frac{1}{x}+\frac{1}{y}=\frac{1}{2016}$.
ANSWER: 32. | Okay, so I need to find the value of √(x/63 - 32) multiplied by √(y/63 - not sure if that's immediately helpful.
Alternatively, since the product (x -. Let me set a = x - 2016 and b = y - 2016. Then, ab = 2016². So, a and b are positive numbers (assuming x and y are positive, which they probably are since we're dealin... | 1 | 32 |
10. It is required to fence a rectangular area adjacent to a wall. The fence should have a length of \(60 \mathrm{~m}\). What should be the length and width of this area so that its area is the largest? | First, I need to visualize the problem. There's a rectangular area adjacent to a wall, so one side of the rectangle doesn't require fencing.
Let’s denote the length parallel to the wall as \( L \) and the width perpendicular[
L = 60 - 2W
\]
Substituting 4W = 0
\]
Solving for \( W \):
\[
4W = 60 \\
W = 15 \text{ mete... | 0 | 450 |
31. Given that $\alpha$ is an acute angle satisfying
$$
\sqrt{369-360 \cos \alpha}+\sqrt{544-480 \sin \alpha}-25=0
$$
find the value of $40 \tan \alpha$. | Alright, so I have this problem here where I need to find the value of 40 tan α, given that α is an acute angle and it satisfies the equation:
√(369 - 360 cos α) + √(544 - 480 sin α) - 25 = 0
Hmm, okay. Let me try to break this down step by step.
First, I notice that both square roots have similar structures. They a... | 1 | 30 |
## Task 7 - V00607 = V00503
To an imagined number, 16 is added, then multiplied by 7, followed by subtracting 8, and finally dividing by 9. The result is 22 remainder 4.
What is the imagined number? | First, I'll represent the imagined number as \( x \).
When 1 means:
\[
\frac{7(x + 16) - 8}{9} = 22 \frac{4}{9}
\]
To solve for \( x \), I'll convert the mixed number to an]
Multiplying both sides by 9 gives:
\[
7(x + 16) - 8 = 202
\]
Expanding and simplifying the equation:
\[
7x + 112 - 8 = 202
\]
\[
7x + 104 = 202
\... | 1 | 14 |
IMO 1974 Problem A1 Three players play the following game. There are three cards each with a different positive integer. In each round the cards are randomly dealt to the players and each receives the number of counters on his card. After two or more rounds, one player has received 20, another 10 and the third 9 counte... | Alright, so I’ve got this problem from IMO 1974, Problem A1. It involves three players who each start with three cards, each having a different positive integer. The game they play involves dealing the cards randomly each round, and each player receives counters equal to the number on their card. After two or more roun... | 0 | 9 |
9.1. Solve the equation:
$$
\left(x^{2}-20\right)^{2}+\left(x^{2}-19\right)^{2}=2019
$$ | Okay, so I've got this equation to solve: \((x^2 - 20)^2 + (x^2 - 19)^2 = 2019\). Hmm, let me try to figure out how to approach this. It looks a bit complicated, but maybe if I simplify it somehow, it will become easier.
First off, both terms on the left side are squared, so it might help to expand them. But before I ... | 0 | \\sqrt{\frac{39+\sqrt{4037}}{2}} |
19. (CAN 5) The triangular array $\left(a_{n, k}\right)$ of numbers is given by $a_{n, 1}=1 / n$, for $n=1,2, \ldots, a_{n, k+1}=a_{n-1, k}-a_{n, k}$, for $1 \leq k \leq n-1$. Find the harmonic mean of the 1985th row. | Okay, so I have this problem here about a triangular array of numbers, and I need to find the harmonic mean of the 1985th row. Hmm, let's try to break this down step by step.
First,'re given a triangular array \( (a_{n, k}) \) where - a_{n, k} \).
And the question is asking for the harmonic mean of the 1985th row. Ok... | 1 | \frac{1}{2^{1984}} |
. For a positive integer $n$, let $S(n)$ denote the sum of its digits. Find the largest possible value of the expression $\frac{S(n)}{S(16 n)}$.
## Answer: 13 | Alright, so I'm trying to solve this problem where I need to find the largest possible value of the expression S(n)/S(16n) forn.
First, let me understand what exactly is being asked. I need to need a different approach. Let me consider n as a number consisting of all So, n must end with a digit that when multiplied b... | 1 | 13 |
In equilateral triangle $ABC$, the midpoint of $\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has area $36\pi$, then find the perimeter of the triangle.
[i]Proposed by Isabella Grabski
[/i] | Okay, so I have this problem about an equilateral triangle ABC. Let me, so that splits BC into two equal parts. The problem mentions the circumcircle of triangle MAB, and its area is given as 36π. I need to find the perimeter of triangle ABC.
Hmm, let's break this3/2)s.
I need to find the circumradius of triangle MAB... | 1 | 36 |
18. If $x>0$ and $\left(x+\frac{1}{x}\right)^{2}=25$, find the value of $x^{3}+\frac{1}{x^{3}}$. | Okay, so I have this problem here: If \( x > 0 \^3 + 3x^2 \cdot \frac{1}{ximplifying each term:
- \( x^3 \) stays as it is.
- \( 3x^2 \cdot \frac{1}{x^3} + 3y = y^3 \). Then rearranged to \( x^3 + \frac{1}{x^3} =). That seems consistent.
Is there another way to verify this result? Maybe by finding the value of \( x \... | 1 | 110 |
2.1. Find the smallest value of $a$, for which the sum of the squares of the roots of the equation $x^{2}-3 a x+a^{2}=0$ is $0.28$. | Alright, let's see. I have this problem here where I need to find the smallest value of \( a \) such that the sum of the squares of a^2 = 0 \) is 0.28. Hmm, the sum of the squares of the roots is \( (sum)^2 - 2(product) \). From earlier, the sum of the roots is \( 3a \) and the product is \( a^2 \). Therefore, the sum ... | 1 | -0.2 |
10. (3 points) 1994 students line up in a row, and they count off from the front to the back in a cycle of 1 to 3; then they count off from the back to the front in a cycle of 1 to 4. The number of people who report 1 in both rounds of counting is $\qquad$ people. | Alright, so I have this problem where 1994 students are lined up in a row. They first count off from the front to the back in cycles of 1 to 3. Then, they count off again from the back to the front in cycles of 1 to 4. I need to figure out how many people will report the number 1 in both rounds of counting. Hmm, okay, ... | 1 | 166 |
$A_1, A_2, ..., A_n$ are the subsets of $|S|=2019$ such that union of any three of them gives $S$ but if we combine two of subsets it doesn't give us $S$. Find the maximum value of $n$. | Okay, so I have this problem where I need to find the maximum number of subsets, n, of a set S with |S| = 2019. The conditions are that the union of any three subsets gives me the entire set S, but if I take just two subsets and union them, they don't cover all of S. Hmm, interesting.
Let me try to wrap my head around... | 0 | 64 |
1. Find the sum of all fractions in lowest terms with value greater than 10 but smaller than 100 and with denominator equal to 3 .
求所有大於 10 且小於 100 , 且以 3 為分母的最簡分數之和。 | 嗯，好的，我现在要解决的问题是：求所有大于10且小于100，且以3为分母的最简分数之和。听起来有点复杂，但我慢慢来，一步一步分析。
首先，题目计算这些数的和。
首先，我想找到所有满足条件的a的这些数中被3整除的数有多少个，然后用总数减去被3整除的数量，就能得到不被3整除的数量。
首先，找到31到299之间被3整除的最小。300是3的倍数，但299比300少1，所以最大的被3整除的数是297=3×99。
那么，从33到2977。
计算这个等差数列的项数：项数=（ +1=88+1=89个数。
所以，在31到299之间，被3整除的数有89个，那么不被3整除的数就有269 -89=180个数。
现在，我需要计算这些a的和，然后... | 1 | 9900 |
Problem 6. (30 points) A regular triangular prism $A B C A_{1} B_{1} C_{1}$ with base $A B C$ and lateral edges $A A_{1}, B B_{1}, C C_{1}$ is inscribed in a sphere. Segment $C D$ is the diameter of this sphere, and point $K$ is the midpoint of edge $A A_{1}$. Find the volume of the prism if $C K=2 \sqrt{6}, D K=4$. | Okay, so I have this problem about a regular triangular prism inscribed in a sphere. Let me try to visualize it first. A regular triangular prism means that length. The prism is inscribed in a sphere, so all its vertices lie on the surface of the sphere.
The problem mentions segment CD is the diameter of the sphere. S... | 1 | 36 |
Example 3 As shown in Figure 4, in a $4 \times 4$ grid square,
construct a grid point $\triangle A B C$
$(A B=\sqrt{5}, B C=$
$\sqrt{13}, C A=\sqrt{10}$ ). How many
grid point triangles congruent to
$\triangle A B C$ (including
$\triangle A B C)$ can be
constructed in Figure 4? | Okay, so I've got this problem here about constructing grid point triangles that are congruent to triangle ABC. Let me try to understand what it's asking.
First, the triangle ABC has sides AB = sqrt(5), BC = sqrt(13), and CA = sqrt(10). The grid is 4x4, so I guess that means it's a grid of 4x4 squares, making 5x5 poi... | 0 | 32 |
20. (6 points) If the ten-digit number $\mathrm{a} 2016 \mathrm{~b} 2017$ is divisible by 33, then, how many such ten-digit numbers are there? | Okay, so I have this problem here: I need to find how many ten-digit numbers of the form a2016b2017 are 3.
So, let's compute the sum:
a + 2 + + 1 + 7
Let me compute this step by step:
a + 2 = a + 2
+ 0 = a + 2
+ 1 = a + 3
+ 6 = a 3. Let's note that down:
Condition 1: a + b + 19 ≡ 0 mod 3
Simplify this:
a + b ... | 1 | 3 |
5. In the expansion of $(\sqrt{x}+2)^{2 n+1}$, the sum of the coefficients of the terms where the exponent of $x$ is an integer is | Okay, so I have this problem: In the expansion of (√x + 2)^{2n+1}, I need to find the sum of the coefficients of the terms where the exponent of x is an integer. Hmm, that sounds a bit tricky, but let me see if I can figure it out 0)/2 = (2n+1)/2, which is not an integer unless n is a half-integer, which it isn't becau... | 1 | \frac{1}{2}\left(3^{2n+1}+1\right) |
2. Positive integers $x, y$, for which $\gcd(x, y)=3$, are the coordinates of a vertex of a square with its center at the origin and an area of $20 \cdot \text{lcm}(x, y)$. Find the perimeter of the square. | Okay, so I have this problem here: there are positive integers x and y, with gcd(x, y) equal to 3. They are the coordinates of a vertex of a square centered at the origin, and the area of the square is 20 times the lcm of x and y. I need to find the perimeter of the square.
Hmm. Let me try to break this down area of t... | 1 | 24\sqrt{5} |
36th Putnam 1975 Problem B3 Let n be a fixed positive integer. Let S be any finite collection of at least n positive reals (not necessarily all distinct). Let f(S) = (∑ a∈S a) n , and let g(S) = the sum of all n-fold products of the elements of S (in other words, the nth symmetric function). Find sup S g(S)/f(S). Solut... | Alright, so I've got this problem from the 36th Putnam exam, Problem B3. Let me try to wrap my head around it step by step.
First, the problem says: Let n be a fixed positive integer. We have a finite collection S of at least n positive real numbers. Then, they define two functions, f(S) and g(S).
- f(S) is the nth... | 1 | \frac{1}{n!} |
3. (5 points) If $\overline{\mathrm{abc}}$ and $\overline{\operatorname{def}}$ are two three-digit numbers, and $a=b+1, b=c+2, \overline{\mathrm{abc}} \times 3+4=\overline{\operatorname{def}}$, then $\overline{\mathrm{def}}$ $=$ $\qquad$ | First, I need to establish the relationships between the digits of the three-digit number + f.
The equation provided is abc × 3 + 4 = defe + f
I also know that b = c + 2, so0e + f
333c + 964 = 100d + 10e + f
Now, I need to find values of c that make abc a valid three-digit number. Since a = c + 3, c must be less tha... | 1 | 964 |
7. In trapezoid $A B C D, A D$ is parallel to $B C$. If $A D=52, B C=65, A B=20$, and $C D=11$, find the area of the trapezoid. | Alright, so I have this trapezoid problem. Let me first visualize it. The trapezoid is ABCD, with AD parallel to BC. They gave me the lengths: AD is 52 units, BC is 65 units, AB is 20 units, and CD is 11 units. I need to find the area of this trapezoid.
Trapezoid area formula is pretty straightforward: (sum of the two... | 1 | 594 |
Let $\mathcal{P}$ be a parallelepiped with side lengths $x$, $y$, and $z$. Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$, $17$, $21$, and $23$. Compute $x^2+y^2+z^2$. | Okay, so I have this problem about a parallelepiped with side lengths x, y, and z. The four space diagonals are given as 15, 17, 21, and 23. I need to find x² + y² + z². Hmm, okay.
+ |**c**|² + 2(-**a**·**529
So, if I write these out:
Equation 19
Now, if I label these equations as (1), (2), (3), (4), respectively.... | 1 | 371 |
Three, (50 points) A positive integer is called "simple" if it does not have any square factors greater than 1. Determine how many numbers in $1,2, \cdots, 2010$ are simple.
| Okay, so I have this problem here where I need to find out how many "simple" numbers there are between 1 and 2010. A "simple" number is defined as a positive integer that doesn't have any square factors greater than 1. Hmm, that means if a number is simple, it shouldn't be divisible by any perfect square other than 1. ... | 0 | 1221 |
1. A line is drawn through the vertex $C$ of parallelogram $A B C D$ which intersects the rays $A B$ and $A D$ at points $K$ and $L$ respectively. Express the area $s$ of parallelogram $A B C D$ in terms of the areas $p$ and $q$ of triangles $BKC$ and $D C L$. | Alright, so I have this problem about a parallelogram ABCD. A line is drawn through vertex C, and it intersects the rays AB and AD at points K and L, respectively. I need to express the area of the parallelogram ABCD, which is denoted as 's', in terms of the areas 'p' and 'q' of triangles BKC and DCL suppose that the a... | 1 | 2\sqrt{pq} |
Example 1 Let $a, b, c$ be the lengths of the three sides of a right-angled triangle, where $c$ is the length of the hypotenuse. Find the maximum value of $k$ such that $\frac{a^{3}+b^{3}+c^{3}}{a b c} \geqslant k$ holds. ${ }^{[2]}$ | Okay, so I have this problem here where I need to find the maximum value of \( k \) such that the inequality \( \frac{a^3 + b^3 + c^3}{a b c} \geq k \) holds for the sides of a right-angled triangle. Let me try to break this down step by step.
First, since it's a right-angled triangle, by the Pythagorean theorem, we k... | 1 | 2+\sqrt{2} |
2. As shown in Figure 4, in the convex quadrilateral $ABCD$, $AB=1$, $BC=2$, $CD=\sqrt{3}$, $\angle ABC=120^{\circ}$, $\angle BCD=90^{\circ}$.
Then $AD=$ $\qquad$ | Okay, so I need to find the length of AD in this convex quadrilateralLet me calculate these:
cos(120°) is equal to cos0°) = -0.5
sin(120°) is equal to sin(180° - 60°) = sin(60°) = √3/2 ≈ 0 B is at (-1, √3). But wait, since we placed √3). Hmm, that seems okay.
But wait, let me think 1 unit to the right along the x-ax... | 1 | \sqrt{7} |
3. (10 points) Each lead type is engraved with a digit. If printing twelve pages of a book, the lead types for page numbers used are the following 15: $1,2,3,4,5,6,7,8,9,1,0,1,1,1,2$.
Now, a new book is to be printed, and 2011 lead types for page numbers are taken from the warehouse, with some left over after typesetti... | Alright, let me try to figure this problem out. So, we have a number of pages this book can have, and the minimum number of leftover lead types.
So, essentially, we need to use as many as possible of the asking for both the maximum number of pages possible and the minimum number of leftover lead types.
Wait, hold on... | 0 | 706 |
B1. The digit sum of a number is obtained by adding the digits of the number. The digit sum of 1303, for example, is $1+3+0+3=7$.
Find the smallest positive integer $n$ for which the digit sum of $n$ and the digit sum of $n+1$ are both divisible by 5. | Okay, so I have this problem where I need to find the smallest positive integer \( n \) such that both the digit sum of \( n \) and the digit sum of \( n + 1 \) are divisible by 5. Hmm, digit sums... okay, I remember that the digit sum is just the sum of all the digits in a number. For example, the digit sum of 1303 is... | 1 | 49999 |
5. Find the largest positive integer $x$ such that $x$ is divisible by all the positive integers $\leq \sqrt[3]{x}$. | Okay, so I have this problem here: Find the largest positive integer \( x \) such that \( x \) is divisible by all the positive integers \( \leq \sqrt[3]{x} \). Hmm, that sounds interesting. Let me try to break it down step by step.
First, let me parse the problem. I need to find the largest \( x \) where \( x \) is d... | 0 | 420 |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(\frac{\arcsin x}{x}\right)^{\frac{2}{x+5}}$ | Okay, so I have this limit problem to solve: the limit as x approaches 0 of (arcsin x / x) raised to the power of (2/(x + 5)). Hmm, that looks a bit complicated, but maybe I can break it down step by step.
First, I remember that when dealing with limits involving exponents, especially when the base approaches 1 and th... | 1 | 1 |
Three, (This question 20 points) Given a triangle with one side length of 2, the median on this side is 1, and the sum of the other two sides is $1+\sqrt{3}$. Find the area of the triangle.
| Okay, so I have this triangle problem here, and I need to find its area. Let me try to figure this out step by step. The problem says that one side of the triangle has a length of 2, and the median on this side is 1. Also, the sum of the other two sides is \(1 + \sqrt{3}\). Hmm, okay, let me break this down.
First, le... | 1 | \frac{\sqrt{3}}{2} |
1. Sei $A B C$ ein spitzwinkliges Dreieck mit $A B \neq B C$ und Umkreis $k$. Seien $P$ und $Q$ die Schnittpunkte von $k$ mit der Winkelhalbierenden beziehungsweise der Aussenwinkelhalbierenden von $\angle C B A$. Sei $D$ der Schnittpunkt von $A C$ und $P Q$. Bestimme das Verhältnis $A D: D C$.
## 1st solution: | Okay, I have this geometry problem here about a triangle, its circumcircle, and some angle bisectors. Let me try to parse what the problem is saying and figure out a plan.
First, the setup: We have a triangle ABC, which is acute-angled, meaning all its angles are less than 90 degrees. Also, it's specified that AB is n... | 1 | 1 |
Three. (20 points) Given that the graph of the linear function $y=a x+b$ passes through the points $A(\sqrt{3}, \sqrt{3}+2)$, $B(-1, \sqrt{3})$, and $C(c$, $2-c)$. Find the value of $a^{2}+b^{2}+c^{2}-a b-b c-c a$.
| Alright, so I have this problem about a linear function y = ax + b that passes through three points: A(√3, √3 + 2), B(-1, √3), and C(c, 2 - c). And I need to find the value of a² + b² + c² - ab - bc - ca. Hmm, okay.
First, since it's a linear function, the equation has to satisfy all three points. So I can plug each p... | 1 | 4+\sqrt{3} |
1. 12 large monkeys stand evenly on a large circle with a circumference of 480 meters. Between each pair of adjacent large monkeys, 3 small monkeys are added, requiring the distance between adjacent monkeys on the circle to be equal. This distance is $\qquad$ meters. | each adjacent monkey:
Divide the total circumference by the number of gaps between monkeys, which is 48.
Therefore, the distance is 480 meters divided by 48 equals 10 meters.
</think>
**Solution:**
1. **Determine the Initial Distance Between Large Monkeys:**
- Total circumference of the circle: \(480\) meter... | 1 | 10 |
Task B-3.4. A group of children found a wooden board in the shape of a quadrilateral and decided to use it for the game "pick-a-point". Since the target was of an unusual shape, they had to adapt the game rules. They asked for advice from Mark's older brother, a good mathematician. He measured, calculated, and wrote so... | Alright, so I've got this problem about a quadrilateral board where some kids are playing a game called "pick-a-point." They found a wooden board that's a quadrilateral, and they want to play a game where you get points based on where you pick a point on the board. The mathematician brother of one of the kids, Mark, ca... | 1 | \frac{7}{10\pi}(5\sqrt{3}-\sqrt{5})-1 |
10. Let $a$, $b$, $c$ represent three different non-zero digits, forming the largest and smallest three-digit numbers, respectively. Denote the difference between these two three-digit numbers as $m$, and the sum of the digits of $m$ exactly divides $m$. Then the sum of all different values of $m$ is $\qquad$ | Okay, so I need to solve this problem where I have three different non-zero digits a, b, c. They form the largest and smallest three-digit numbers possible, which are denoted as m and n respectively. Then I need to find the difference m - n, which is another number, let's call it p. The condition is that the sum of the... | 1 | 1980 |
3. 3.1. Petya marked 5 points on a face of a cube, turned it and marked 6 points on an adjacent face, then turned it again and marked 7 points, and so on. He marked points on each face this way. What is the maximum number of points that can be on two opposite faces? | Okay, so I've got this problem here about Petya marking points on a cube's faces. Let me try to wrap my head around it step by step. Maybe if I break it down, it'll make more sense.
First, the cube has six faces, right? Each face is a square, and opposite faces are parallel and don't touch each other. So, if I imagine... | 0 | 18 |
3.1. How many numbers from 1 to 1000 can be represented as $[2 x]+[4 x]+[6 x]+[8 x]$ with positive $x$? | Okay, so I have this problem: I need to find how many numbers from 1 to 1000 can be represented as [2x] + [4x] + [6x] + [8x] where x is a positive number. The brackets [ ] probably mean the floor function, right? So [2x] is the greatest integer less than or equal to 2x.
First, I should try to understand what the expr... | 0 | 600 |
## Problem Statement
Calculate the lengths of the arcs of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=2 \cos ^{3} t \\
y=2 \sin ^{3} t
\end{array}\right. \\
& 0 \leq t \leq \frac{\pi}{4}
\end{aligned}
$$ | Okay, so I have this problem here where I need to find the lengths of the arcs of a curve given by parametric equations. The equations are x = 2 cos³t and y = 2 sin³t, and the parameter t ranges from 0 to π/4. Hmm, okay. IThen, (dy/dt)² is (6 sin²t cos t)². Similarly, squaring 6 gives 36, squaring sin²t gives sin⁴t, an... | 1 | \frac{3}{2} |
Let $n$ be a positive integer. Consider the set of $(n+1)^{3}-1$ points in three-dimensional space:
$$
S=\{(x, y, z): x, y, z \in\{0,1, \ldots, n\}, x+y+z>0\}
$$
Determine the minimum number of planes such that their union contains all points of the set $S$, but does not contain the point $(0,0,0)$. | Okay, so I have this problem here where I need to find the minimum number of planes such that their union contains all points of the set S, but doesn't include the point (0,0,0). Let me try to parse this step by step.
First, let's understand the set S. It's defined as all points (x, y, z) where x, y, z are integers fr... | 1 | 3n |
1. Let the natural number $n$ have the following property: from $1,2, \cdots, n$, any 51 different numbers are chosen. Among these 51 numbers, there must be two numbers whose sum is 101. What is the largest such $n$? | Okay, so I have this problem where I need to find the largest natural number \( n \) such that if I choose any 51 different numbers from the set \( \{1, 2, \ldots, n\} \), there must be at least two numbers among these 51 whose sum is 101. Hmm, interesting. Let me try to break this down.
First, I need to understand wh... | 1 | 100 |
[ $[\quad$ Similar figures $\quad]$
Several circles are inscribed in an angle, their radii increasing. Each subsequent circle touches the previous one. Find the sum of the lengths of the second and fourth circles, if the length of the third is $18 \pi$, and the area of the circle bounded by the first circle is $\pi$.
... | Okay, so I have this problem here about circles inscribed in an angle. The circles are increasing in size, and each subsequent circle touches the previous one. I need to find the sum of the lengths (which I think means the circumferences) of the second and fourth circles. They tell me that the length of Therefore, rₙ =... | 1 | 60\pi |
7. (10 points) The average age of everyone in the classroom is 11 years old. If we exclude a 30-year-old teacher, the average age of the rest is 10 years old. How many people are in the classroom? | First, I'll define \( n \) as the number of people in the equation:
\[
\frac{S - 30}{n - 0 = 10(n - 1)
\]
Substituting the first equation into the second, I get:
\[
11n - 30 = 10n - 10
\]
Subtracting \( 10n \) from both sides:
\[
n - 30 = -10
\]
Adding 30 to both sides:
\[
n = 20
\]
Therefore, there are 20 people i... | 1 | 20 |
## Task 5
Add to the product of 7 and 28976 the number 84567. | First, I need to calculate the product of 7 and 28,976.
I'll break down the multiplication step by step to ensure accuracy.
Starting with 7 multiplied by 20,000, which, I'll add all these intermediate results together: 140,0 gives me 202,790.
Finally, adding 42 to 202,790 results in 202,832.
So, the product of 7 an... | 1 | 287399 |
1. Find all values of $p$, for each of which the numbers $-p-8$, $3 \cdot \sqrt{p}$, and $p-7$ are the first, second, and third terms, respectively, of some geometric progression. | Okay, so I have this problem where I need to find all values of \( p \) such that the numbers \( -p - 8 \), \( 3 \cdot \sqrt{p} \), and \( p - 7 \) form the first, second, and third terms of a geometric progression, respectively. Hmm, let's break this down step by step.
First, I remember that in a geometric progressio... | 1 | 4 |
GS. 1 Given that $n$ is a positive integer. If $n^{2}+5 n+13$ is a perfect square, find the value of $n$. | Okay, so I have this problem: Given that \( n \) is a positive integer, and \( n^2 + 5n + 13 \) is a perfect square, I need to find the value of \( n \).
Hmm, perfect squares. That means \( n^2 + 5n + 13 \) should equal some integer squared, say \( k^2 \), where \( k \) is also a positive integer. So, I can write the ... | 1 | 4 |
I4.1 If $\frac{6 \sqrt{3}}{3 \sqrt{2}+2 \sqrt{3}}=3 \sqrt{\alpha}-6$, determine the value of $\alpha$. | Alright, let me try to solve this problem step by step. So, the equation given is:
(6√3) divided by (3√2 + 2√3) equals 3√α minus 6. We need to find α.
First, let me write it down clearly:
6√3 - 6.
Hmm, okay. So, the left side is a fractionating a²: (3√2)² = 9 * 2 = 18.
Calculating b²: (2√3)² = 4 * 3 = 12.
So, the... | 1 | 6 |
Let $A, B$ and $C$ be non-zero digits, so that $B C$ is a two-digit positive integer and $A B C$ is a three-digit positive integer made up of the digits $A, B$ and $C$. Suppose that
$$
\begin{array}{r}
B C \\
A B C \\
+\quad A B C \\
\hline 876
\end{array}
$$
What is the value of $A+B+C$ ? | Okay, let's try to figure out this problem step by step. So, we have three non-zero digits A, B, and C. BC is a100 and 999.
The problem shows an addition:
-------
876
```
So, essentially, we're adding the two-digit number BC twice to the three-digit number ABC, and the result is 876. Hmm, let me write this out mo... | 1 | 14 |
Task 1. We have two jugs, one holds exactly 3 liters, and the other holds exactly 5 liters. Can we measure exactly 4 liters using the two jugs? | Okay, so I have this problem where I need to measure exactly 4 liters using a 3-liter jug and a 5-liter jug. Hmm, I 4 liters in one of them. I guess I can fill, pour, is empty. That's step 1: (5, 0).
Now (5, 2).
Then, I pour from jug A into jug B until jug B is full. Jug B can take 1 more liter to be full because it ... | 1 | 4 |
(3) Given that the side length of the base of a regular tetrahedron is 6, and the side edge is 4, then the radius of the circumscribed sphere of this regular tetrahedron is $\qquad$ . | Okay, so I need to find the radius of the circumscribed sphere ( all four points, and finally compute the radius.
Let me try that. Letsqrt(3), h), where h is the height from the base to the apex.
But wait, I don't know h yet. I do know the - 0)^2 + (h - 0)^2 = 9 + 27 + h^2 = 36 + h^2 the problem says side edge is 4. ... | 1 | 4 |
10.1. Find the area of the figure defined on the coordinate plane by the inequalities $|x|-1 \leq y \leq \sqrt{1-x^{2}}$ | Alright, so I have this problem here where I need to find the area of a figure defined on the coordinate plane by the inequalities \( |x| - 1 \leq y \leq \sqrt{1 - x^2} \). Hmm, okay. Let me try to visualize and break this down step by step.
First, I know that \( |x| - 1 \) is a V-shaped graph that's shifted down by 1... | 1 | \frac{\pi}{2}+1 |
Example 7 Let $a_{i} \in \mathbf{R}^{+}, i=1,2, \cdots, n$, and $\sum_{i=1}^{n} a_{i}=1$, find
$$M=\sum_{i=1}^{n} \frac{a_{i}}{1+\sum_{j \neq i, j=1}^{n} a_{j}}$$
the minimum value. | Okay, so I have this problem here where I'm given positive real numbers \( Jensen's inequality.
First, let's compute the second derivative of \( f(x) \) to check for convexity.
First derivative:
\[
f'(2 - x + x}{(2 - x)^2} = \frac{2}{(2 - x)^2}
\]
Second derivative:
\ \geq n \cdot f\left( \frac{1}{n} \sum_{i=1}^{n... | 1 | \frac{n}{2n-1} |
2. Solve in the set of integers the equation: $\sqrt{x^{2}-1900}=\left|\frac{x}{4}-4\right|-\sqrt{2017-x^{2}}$.
( Mathematical Gazette ) | Alright, so I have this equation to solve in the set of integers: √(x² - 1900) = |(x/²). Hmm, looks a bit complicated, but let's try to tackle it integers, x must be ≤ -44 or ≥ 44.
Now, x must be between -44 and 44. Since we're dealing with integers, x can be from -44 to 44.
Wait, hold on, from the left side, x has t... | 1 | -44 |
19. Given that $A$ is a moving point on the parabola $y^{2}=2 x$, and the fixed point $B(2,0)$, a circle $\odot C$ is constructed with $A B$ as its diameter. If the chord length of the circle $\odot C$ intercepted by the line $l$ : $x+k y-\frac{3}{2}=0$ is a constant, find this chord length and the value of the real nu... | Alright, so I've got this geometry problem here. Let me try to understand it step by step.
First, the problem says: Given that A is a moving point on the parabola \( y^2 = 2x \), and the fixed point B(2, 0), a circle \( \odot C \) is constructed with AB as its diameter. If the chord length of the circle \( \odot C \) ... | 0 | \sqrt{3} |
5-3. Solve the inequality
$$
\sqrt{5 x-11}-\sqrt{5 x^{2}-21 x+21} \geqslant 5 x^{2}-26 x+32
$$
In your answer, specify the sum of all integer values of $x$ that satisfy the inequality. | Alright, so I've got this inequality to solve:
$$
\sqrt{5 x - 11} - \sqrt{5 x^{2} - 21 x + 21} \geq 5 x^{2} - 26 x + 32
$$
Hmm, okay. Let me break this down step by step.
First, I know that dealing with square roots can be tricky, especially when they're part of an inequality. So, I need to be careful about the dom... | 1 | 3 |
Example 2 Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 2, connect $D_{1} A$, $D_{1} B$, $E$ is the midpoint of $D_{1} B$, $F$ is the midpoint of $B C$, $G$ is the midpoint of $A D_{1}$, find the size of the angle formed by the skew lines $D G$ and $E F$. | Okay, so I have this geometry problem about a cube. The cube is labeled ABCD-A₁B₁C₁D₁, and each edge is 2 units long. I need to find the angle between two skew lines, DG and EF. Hmm, skew lines are lines that don't intersect and aren't parallel, right? So, they lie on different planes. To find the angle between them, I... | 1 | 60 |
Given a set of points in space, a [i]jump[/i] consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$. Find the smallest number $n$ such that for any set of $n$ lattice points in $10$-dimensional-space, it is possible to perform a finite number of jumps so that some two points ... | Okay, so I've come across this interesting problem about jumps in 10-dimensional space. Let me try to understand it step by step.
First, the problem says: Given a set of points in 10-dimensional space, a "jump" consists of taking two points, P and Q, and replacing P with the reflection of P over Q. So, reflection over... | 1 | 1025 |
2. Equation
$$
x^{2}-31 x+220=2^{x}\left(31-2 x-2^{x}\right)
$$
The sum of the squares of all real roots is $\qquad$ | Alright, so I've got this equation to solve:
$$
x^{2} - 31x + 220 = 2^{x}\left(31 - 2x - 2^{x}\right)
$$
And I need to find the sum of the squares of all real roots. Hmm, okay. Let me try to figure this out step by step.
First, let me rewrite the equation to see if I can simplify it or spot any patterns. So, the le... | 1 | 25 |
Cynthia and Lynnelle are collaborating on a problem set. Over a $24$-hour period, Cynthia and Lynnelle each independently pick a random, contiguous $6$-hour interval to work on the problem set. Compute the probability that Cynthia and Lynnelle work on the problem set during completely disjoint intervals of time. | Alright, so Cynthia and Lynnelle are working on a problem set over a 24-hour period. Each of them independently picks a random, contiguous 6-hour interval to work. We need to find the probability that their chosen intervals are completely disjoint. Hmm, okay, let me break this down step by step.
First, let's visualize... | 1 | \frac{4}{9} |
131 On each face of an opaque cube, a natural number is written. If several (one, two, or three) faces of the cube can be seen at the same time, then find the sum of the numbers on these faces. Using this method, the maximum number of different sums that can be obtained is $\qquad$. | Okay, so I have this problem about a cube with numbers on each face. to the faces such that the sums from singles, pairs, and triples are all these sums are unique.
I remember that in such problems, often assigning numbers in a way that each face has a unique power of 2 can be useful, because when you take subsets, th... | 1 | 26 |
# 7.1. (7 points)
Find the value of the expression
$$
\left(1+\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\left(1-\frac{1}{5}\right) \ldots\left(1+\frac{1}{2 m}\right)\left(1-\frac{1}{2 m+1}\right)
$$ | Okay, so I have this math problem here that I need to solve. It's an expression that involves a product of several terms, each of which is either n seems to be increasing by 1 each time. The product starts with (1 + 1/(2m)) and (1 - 1/(2m+1)). The question is asking me to find the value of this entire expression.
Hmm,... | 1 | 1 |
Solve the following equation:
$$
2^{x+3}+4^{x+1}=320 .
$$ | Okay, so I need to solve this equation: 2^{x+3} + 4^{x+1} = 320. Hmm can factor out the smaller exponent or make a substitution to simplify this. Let me see.
Let me denote y = 2^x. That might help becausey^2 + 2y - 80 = 0.
Alright0 = 0. Let me try to factor this quadratic. I'm looking 2. Perfect!
So, the quadratic f... | 1 | 3 |
$4 \cdot 247$ Find all positive integers $n$ such that the following equation has an integer solution.
$$
x^{n}+(2+x)^{n}+(2-x)^{n}=0 \text {. }
$$ | Alright, so I've got this equation to solve: \(x^{n} + (2 + x)^{n} + (2 - x)^{n} = 0\). I need to find all positive integers \(n\) for which there's an integer solution \(x\). Hmm, okay, let's break this down.
First, let me understand what the equation is saying. It's a sum of three terms: \(x^n\), \((2 + x)^n\), and ... | 1 | 1 |
61. Another task about a piece of land
- Here's another task,- said the Black Queen. Another farmer had a piece of land. On one third of his land, he grew pumpkins, on one fourth he planted peas, on one fifth he sowed beans, and the remaining twenty-six acres he allocated for corn.
How many acres of land did the farm... | Okay, so I've got this problem here about a farmer and his land. but the problem says not to use algebra, which is a bit concerning. Maybe0)x. Adding these together: 20/60 + 15/60 + 12/60 = 47/60. So, the sum of these fractions is (47/620. Therefore, x is 120 acres.
Wait, but the user mentioned not to use algebra, so... | 1 | 120 |
Example 6.1.1 Let $k$ be a positive real number. Find the minimum value of the following expression
$$k\left(x^{2}+y^{2}\right)+z^{2}$$
where $x, y, z>0$, and satisfy $x y+y z+z x=1$ | Okay, so I've got this problem where I need to find the minimum value of the expression \( k(x^2 + y^2) + z^2 \) with the constraint that \( xy + yz + zx = 1 \), where \( x, y, z > 0 \) and \( k \) is a positive real number. Hmm, let's see how to approach this.
First off, the expression I'm trying to minimize is \( k(... | 1 | \frac{-1+\sqrt{1+8k}}{2} |
Task B-3.8. For the angles $\alpha$ and $\beta$ of triangle $ABC$, the following holds:
$$
\begin{aligned}
& 3 \sin \alpha + 4 \cos \beta = 6 \\
& 4 \sin \beta + 3 \cos \alpha = 1
\end{aligned}
$$
Determine the measure of angle $\gamma$ of this triangle. | Okay, so I have this problem where I'm given two equations involving angles α and β of a triangle ABC. The goal is to find the measure of angle γ. Hmm, let me recall that in a triangle, the sum of angles is π radians, so γ = π - α - β. So if I can find α and β, I can easily find γ. Cool.
The equations given are:
3 si... | 1 | 30 |
1. Given that $a$ and $b$ are positive integers, and satisfy $\frac{a+b}{a^{2}+a b+b^{2}}=\frac{4}{49}$. Then the value of $a+b$ is $\qquad$ ـ. | Alright, so I've got this problem here where I need to find the value of \( a + b \) given that \( a \) and \( b \) are positive integers satisfying the equation:
\[
\frac{a + b}{a^2 + ab + b^2} = \frac{4}{49}
\]
Hmm, okay. Let me break this down(2a) = 98a \)
Right side: \( 4(a^2 + a \cdot a + a^2) = . Let me rearra... | 1 | 16 |
1. Find the sum of the digits of the number $A$, if $A=2^{63} \cdot 4^{25} \cdot 5^{106}-2^{22} \cdot 4^{44} \cdot 5^{105}-1$. Answer: 959. | Okay, so I have this problem where I need to find the sum of the digits of the number A. A is given as \(2^{63} \cdot 4^{25} \cdot 5^{106} - 2^{22} \cdot 4^{44} \cdot 5^{105} - 1\). Hmm, that looks a bit intimidating at first glance with all those exponents, but maybe I can simplify it somehow.
First, I notice that th... | 1 | 959 |
The bathtub fills with cold water in 6 minutes 40 seconds, and with hot water in 8 minutes. In addition, if the plug is removed from a full bathtub, the water will drain in 13 minutes 20 seconds. How long will it take to fill the bathtub completely, given that both taps are open, but the bathtub is not plugged?
# | First, I'll convert all the given times into seconds to make the calculations easier The combined filling rate is 1/400 + 1/480, which equals 1/240 per second.
However, there's a drain open as well, so I need to subtract the draining rate. The drain empties the bathtub at a rate of 1/800 per second. Therefore, the net... | 0 | 5 |
1. Among the triangles with side lengths being consecutive natural numbers and a perimeter not exceeding 100, the number of acute triangles is $\qquad$ . | Okay, so I need to find the number of acute triangles where the side lengths are consecutive natural numbers and the perimeter doesn't exceed 100. Hmm how many such triangles are acute with a perimeter ≤ 100.
Alright32. So n can range from 1 to 32.
But to exist. So n ranges from 2 to 32.
So now, for each n from 2 to... | 1 | 29 |
Exercise 11. A palette of 6 different colors is given. In how many ways can a cube be painted, using all 6 colors, and exactly one color per face? Two ways of coloring are considered identical if one can be obtained from the other by any rotation in space. | Alright, so I have this problem here where I need to figure out how many ways a cube can be painted using 6 different colors, one color per face. And importantly, two colorings are considered the same if one can be rotated to look like the other. Hmm, okay. So this is a problem about counting colorings up to rotational... | 1 | 30 |
7. Twelve people are carrying 12 loaves of bread. Each man carries 2 loaves, each woman carries half a loaf, and each child carries a quarter of a loaf, and all 12 people are involved in the carrying. How many men, how many women, and how many children were there? | Okay, let me try to figure this problem out. So, we have twelveCombine like terms:
(2m - 0.25m) + (0.5w - 0.25w) + 30.25w + 3 = 12.
Subtract 3 from both sides:
1.75m + 0.254.
So, the number of men must be at least 4.
Also see what possible values m can take.
From w = 36 - 5.
So, m can be 4 or 5.
Let's test m =... | 0 | 5 |
3. $a_{1}, a_{2}, a_{3}, \cdots, a_{2020}$ are distinct non-zero real numbers, so $\left|\frac{a_{1}}{\left|a_{1}\right|}+\frac{\left|a_{2}\right|}{a_{2}}+\cdots+\frac{a_{2019}}{\left|a_{2019}\right|}+\frac{\left|a_{2020}\right|}{a_{2020}}\right|$ has the minimum value of $\qquad$ | Okay, so I have this problem where there are 2020 distinct non-zero real numbers: \( a_1, a_2, a_3, \ldots, a_{2020} \). The question is about finding the minimum value of the absolute value of a sum involving each}} \). I need to figure out the minimum possible value of this expression.
position.
Therefore, the ent... | 1 | 0 |
2. Given $\alpha, \beta \in\left(0, \frac{\pi}{2}\right), \sin \beta=2 \cos (\alpha+\beta) \cdot \sin \alpha\left(\alpha+\beta \neq \frac{\pi}{2}\right)$, then the maximum value of $\tan \beta$ is | Okay, so I have this trigonometry problem here, and I need to find the maximum value of tan β given some conditions. Let me read it again carefully.
It says: Given α, β are in the interval (0, π/2), so both angles are between 0 and 90 degrees. The equation given is sin β = 2 cos(α + β) · sin α, and it also mentions th... | 1 | \frac{\sqrt{3}}{3} |
Let $S$ be a set of $2020$ distinct points in the plane. Let
\[M=\{P:P\text{ is the midpoint of }XY\text{ for some distinct points }X,Y\text{ in }S\}.\]
Find the least possible value of the number of points in $M$. | Okay, so I have this problem where I have a set S of 2020 distinct points in the plane. Then M is defined as the set of all midpoints P where P is the midpoint of XY for some distinct points X and Y in S. I need to find the least possible number of points in M, meaning I want to arrange the 2020 points in such a way th... | 1 | 4037 |
5. Define the length of the interval $\left[x_{1}, x_{2}\right]$ as $x_{2}-x_{1}$. If the domain of the function $y=\left|\log _{2} x\right|$ is $[a, b]$, and the range is $[0,2]$, then the difference between the maximum and minimum values of the length of the interval $[a, b]$ is $\qquad$ . | Alright, so I have this problem here about the function y equals the absolute value to find the difference between the maximum and minimum values of the length of the interval, the domain is all positive real numbers, but since we have the range restricted to [0, 2], we need to find the specific x-values that x ranges ... | 1 | 3 |
5. Bacamo istovremeno četiri simetrična novčića (za simetričan novčić vjerojatnosti da će pasti pismo ili glava su jednake). Kolika je vjerojatnost da su pri jednom bacanju ta četiri novčića pala dva pisma i dvije glave?
## Zadaci za 10 bodova:
Translates to:
5. We toss four symmetrical coins simultaneously (for a s... | j je jednak binomu od 4 po 2, što je 6. Ovo se može dobiti izračunavajući broj kombinacija.
Zatim, izračunamo verovatnoću za svaku kombinaciju, što je (0,5)^2 * (0,5)^2 = 0,0625.
Na konačnoj fazi, množimo broj načina sa verovatnoćom za svaku kombinaciju, što je 6 * 0,0625 = 0,375. Ovaj rezultat može biti prepišan u o... | 1 | \frac{3}{8} |
[b]Q11.[/b] Let be given a sequense $a_1=5, \; a_2=8$ and $a_{n+1}=a_n+3a_{n-1}, \qquad n=1,2,3,...$ Calculate the greatest common divisor of $a_{2011}$ and $a_{2012}$. | Alright, so I have this problem here where I'm given a sequence defined by a₁ = 5, a₂ = 8, and then each subsequent term is defined by aₙ₊₁ = aₙ + 3aₙ₋₁ for n = 1, 2, 3, and so on. The question is asking me to find the greatest common divisor (GCD) of a₂₀₁₁ and a₂₀₁₂. That seems pretty far out in the sequence, so I'm t... | 1 | 1 |
11.3. The numbers $a_{1}, a_{2}, a_{3}, a_{4}$ and $a_{5}$ form a geometric progression. Among them, there are both rational and irrational numbers. What is the maximum number of terms in this progression that can be rational numbers? | Alright, so I have this problem where there are five numbers in a geometric progression: a₁, a₂, a₃, a₄, and a₅. Among these, some are rational and some are irrational. The question is asking for the maximum number of terms that can be rational. Hmm, okay, let me think about this step by step.
First, I need to remembe... | 1 | 3 |
Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$.
- There is a real number $\xi$ with $P(\xi)=0$. | Okay, so I need to find the smallest positive integer \( n \) such that there's a polynomial \( P(x) \) of degree \( 2n \) with coefficients \( a_i \) all between 2014 and 2015. Also, this polynomial must have a real root, meaning there exists some real number \( \xi \) where \( P(\xi) = 0 \).
Hmm, let's break this do... | 1 | 2014 |
20. A palindrome number is the same either read from left to right or right to left, for example. 121 is a palindrome number. How many 5 -digit palindrome numbers are there together? | Okay, so I need to figure out how many 5-digit palindrome numbers there palindrome number. It has the form ABCBA, where A, B, and1 to 9. That gives us 9 options for A.
Next, there are 10 possible choices for B.
Similarly, C is the third digit, also ranging from 0 to 9. So, another 10 choices here.
So, to find the to... | 1 | 900 |
4. Let $N=23 x+92 y$ be a perfect square, and $N$ does not exceed 2,392. Then the number of all positive integer pairs $(x, y)$ that satisfy the above conditions is $\qquad$ pairs.
| Alright, so I have this problem here: N = 23x + 92y is a perfect square, and N doesn't exceed 2,392. I need to find the number of all positive integer pairs (x, y) that satisfy these conditions. Hmm, okay, let's48.9² = (48 + 0.9)² = 48² + 2*48*0.9 + 92 => y < 23. So, y can be any integer from 1 to 22.
But wait, not so... | 1 | 27 |
Let $ S \equal{} \{1,2,\ldots,2008\}$. For any nonempty subset $ A\in S$, define $ m(A)$ to be the median of $ A$ (when $ A$ has an even number of elements, $ m(A)$ is the average of the middle two elements). Determine the average of $ m(A)$, when $ A$ is taken over all nonempty subsets of $ S$. | Alright, so I have this problem where I need to find the average of the median of all nonempty subsets of the set S = {1, 2, ..., 2008}. That sounds a bit complex, but let me try to break it down step by step.
First, I remember that the median of a set is the middle value when the numbers are arranged in order. If the... | 1 | \frac{2009}{2} |
# Task 4. (12 points)
A numerical sequence is such that $x_{n}=\frac{n+2}{n x_{n-1}}$ for all $n \geq 2$. Find the product $x_{1} x_{2} x_{3} \ldots x_{2016} x_{2017}$, if $x_{1}=1$.
# | Alright, so I've got this problem here where a numerical sequence is defined by the recurrence relation \( x_{n} = \frac{n + 2}{n x_{n-1}} \) for all \( n \geq 2 \), and I know that \( x_{1} = 1 \). My task is to find the product \( x_{1} x_{2} x_{3} \ldots x_{2016} x_{2017} \). Hmm, okay, let me try to figure this out... | 1 | 673 |
87. Bobo has 640 points, Guagua has 1000 points. Now both of them exchange their points for the same watch. As a result, the points Guagua has left are 4 times the points Bobo has left. Therefore, exchanging for such a watch requires $\qquad$ points. | First, let's define the variables:
Let \( x \) represent the number of points Bobo uses to buy the watch.
Since both Bobo and Gu0 - x = 2560 - 4x \\
3x = 1560 \\
x = 520
\]
Therefore, they each need to exchange 520 points to buy the watch.
</think>
Let's solve the problem step-by-step.
**Given:**
- **Bobo's points... | 1 | 520 |
1. (5 points) Find the degree measure of the angle
$$
\delta=\arccos \left(\left(\sin 2907^{\circ}+\sin 2908^{\circ}+\cdots+\sin 6507^{\circ}\right)^{\cos } 2880^{\circ}+\cos 2881^{\circ}+\cdots+\cos 6480^{\circ}\right)
$$ | Okay, so I have this problem to solve, and it's about finding the degree measure of an angle δ. The angle is defined as the arccosine of a pretty complicated expression involving sums of sines and cosines over a large range of angles. Specifically, the expression is:
δ = arccos [ (sin2907° + sin2908° + ... + sin6507°)... | 1 | 63 |
10.367 Express the side of a regular decagon in terms of the radius $R$ of the circumscribed circle. | Okay, so I need to find the side length of a regular decagon in terms of the radius \( R \) of its circumscribed circle. Hmm the hypotenuse. So, in this case:
\[ \sin(18^\circ) = \frac{\frac{s}{2}}{R} \]
I can rearrange this equation to solve for \( s \):
\[ s = 2R \sin(18^\circ) \]
Wait, a hexagon's side length.
... | 1 | \frac{\sqrt{5}-1}{2}R |
## Task 3 - 300613
Lina buys 7 pencils and 8 notebooks and notices that 7 pencils are more expensive than 8 notebooks. Which is more expensive: 10 pencils and 2 notebooks or 11 notebooks and 2 pencils?
Justify your answer! | First, let's define the costs. Let the cost of one pencil be \(\]
Now, we need to compare the total costs of two options:
1 pencils, which costs \( 11y + 2x \).
To determine which option is more expensive, we'll subtract the second option from the first can infer that \( 8x > 9y \). Therefore:
\[
8x - 9y > 0
\]
This ... | 0 | 102211 |
## Task 4
From class 1a, 3 Pioneers receive certificates for good performance. An equal number of Pioneers from class 1b also receive certificates.
How many Pioneers will be awarded? | total number of Pioneers awarded, I add the number from class 1a to the number from class 1b: 3 + 3.
Therefore, a total of 6 Pioneers will be awarded.
</think>
**Solution:**
1. **Number of Pioneers from Class 1a:**
- There are **3 Pioneers** from class 1a who receive certificates.
2. **Number of Pioneers from C... | 1 | 6 |
Problem 9.5. Given a right triangle $A B C\left(\Varangle A C B=90^{\circ}\right)$, let $C H, H \in A B$, be the altitude to $A B$ and $P$ and $Q$ be the tangent points of the incircle of $\triangle A B C$ to $A C$ and $B C$, respectively. If $A Q \perp H P$ find the ratio $\frac{A H}{B H}$.
Stoyan Atanasov | Okay, so I have this geometry problem here, and I need to figure out the ratio of AH to BH in a right triangle ABC where angle C is 90 degrees. Let me try to break this down step by step.
First, let me visualize the triangle. Triangle ABC is right-angled at C, so sides AC and BC are the legs, and AB is the hypotenuse.... | 1 | \frac{1+\sqrt{5}}{2} |
Find the smallest positive integer $n$, such that $x_{1}{ }^{3}+x_{2}{ }^{3}+\cdots+x_{n}{ }^{3}=2002^{2002}$ has integer solutions. | Okay, so I have this problem where I need to find the smallest positive integer \( n \) such that the equation \( x_1^3 + x_2^3 + \cdots + x_n^3 = 2002^{2002} \) has integer solutions. Hmm, interesting. Let me think through this step by step.
First, I remember that every integer can be expressed as a sum of four squar... | 1 | 4 |
4. Distribute 8 model student quotas to 4 classes, Jia, Yi, Bing, and Ding, with each class getting at least 1 quota. The probability that Class Jia gets exactly 2 quotas is $\qquad$ . | Okay, so I need to figure out this probability problem. Let me read it again: I have 8 model student quotas to distribute among 4 classes: Jia, Yi, Bing, and Ding. Each class has to get at least 1 quota. I need to find the probability that Class Jia gets exactly 2 quotas.
Hmm, okay. So, this is a probability question ... | 1 | \frac{2}{7} |
[Decimal numeral system]
A three-digit number. A three-digit number starts with the digit 4. If this digit is moved to the end of the number, the resulting number is 0.75 of the original number. Find the original number.
# | Okay, so I have this problem here about a three-digit number that starts with 4. If I move that first digit to the end, the new number so now I have an equation that relates X and Y. My goal is to find X and Y such that this equation holds true, with X and Y being digits (i.e., integers from 0 to 9).
Let me write:
37... | 1 | 432 |
31. The number of poplars planted by the students is expressed as a three-digit number, and the number of lindens as a two-digit number. In total, they planted 144 trees. If the extreme digits in these two numbers are swapped and the resulting numbers are added, the sum is 603. How many poplars and how many lindens wer... | Alright, so I've got this problem here. It says that the number of poplars planted by the students is a three-digit number, and the number of lindens is a two-digit number. Together, they planted a total of 144 trees. Then, if we swap the extreme digits in these two numbers and add the resulting numbers, the sum is 603... | 0 | 115 |
9. (16 points) Given $\odot O: x^{2}+y^{2}=4$, circle $M$ :
$$
(x-5 \cos \theta)^{2}+(y-5 \sin \theta)^{2}=1(\theta \in \mathbf{R}) \text {, }
$$
Through any point $P$ on circle $M$, draw two tangents $P E$ and $P F$ to $\odot O$, with the points of tangency being $E$ and $F$. Try to find the minimum value of $\overri... | Alright, I have this problem here about circles and tangents, and I need θ or some other parameter, I can then compute t² cosφ and find its minimum.
First, let's find the length of the tangent from P to circle O. Since circle O is centered at (0,0) with radius 2, and point P is on circle M, which is centered at (5 cos... | 1 | 6 |
13. Given the function $f(x)=2 \sin \omega x(\omega>0)$ is monotonically increasing on $\left[0, \frac{\pi}{4}\right]$, and the maximum value on this interval is $\sqrt{3}$. Then $\omega=$ $\qquad$ . | Okay, so I have this problem where I'm given the function \( f(x) = 2 \sin(\omega x) \) with \( \omega > 0 \). It says that this function is monotonically increasing on] \), and the maximum value on this interval is \( \sqrt{3} \). I need to find the value of \( \omega \).
Alright \cdot \frac{\pi}{4} \). Since cosine ... | 1 | \frac{4}{3} |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{\frac{\pi}{4}} \frac{2 \cos x + 3 \sin x}{(2 \sin x - 3 \cos x)^{3}} d x
$$ | Okay, I have this integral to solve:
$$
\int_{0}^{\frac{\pi}{4}} \frac{2 \cos x + 3 \sin x}{(2 \sin x - 3 \cos x)^{3}} d x
$$
Hmm, it looks a bit complicated, but maybe I can simplify it by substitution. Let me see... The denominator is $(2 \sin x - 3 \cos x)^3$, and the numerator is $2 \cos x + 3 \sin x$. That's exa... | 1 | -\frac{17}{18} |
7. The ordinary fraction $\frac{1}{221}$ is represented as a periodic decimal fraction. Find the length of the period. (For example, the length of the period of the fraction $\frac{25687}{99900}=0.25712712712 \ldots=0.25$ (712) is 3.) | Okay, so I need to find the length of the period of the decimal expansion of 1/221. Hmm, I remember that the period of a fraction's decimal expansion depends on the denominator after factoring out all 2s and 5s. Since 221 is an odd number and doesn't end with 0, 2, or 5, it shouldn't have any factors of 2 or 5. So I gu... | 1 | 48 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.