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 $c$. | divisible by 9.
Finally, I'll verify that with c = 8 The possible values for xy are 16, 24, 32, 40, 48, 56, 64, 72, and 80. Among these, only 48 is divisible by 8.
Thus, the value of c is 8.
</think>
To determine the value of \( c = x + y \) for the six-digit number \( 1234xy \) that is divisible by both 8 and 9, w... | 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/6, going back to the original expression we need to compute: √(x/63 - 32) × √(y/632016 + b, plug these into the expression:
First term inside the square root: (2016 + a)/63 - 32 simplifies to a/63. So, the first square root becomes √(a/63).
Similarly, the second term inside 3... | 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? | [
L = 60 - 2W
\]
Substituting 4W = 0
\]
Solving for \( W \):
\[
4W = 60 \\
W = 15 \text{ meters}
\]
Now, substituting \( W = 15 \) back into the equation for \( L \):
\[
L = 60 - 2 \times 15 = 30 \text{ meters}
\]
Therefore, to maximize the area with 60 meters of fencing, the length parallel to the wall should be 3... | 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? | 8}{9} = 22 \frac{4}{9}
\]
To solve for \( x \), I'll convert the mixed number to an104 = 202
\]
Subtracting 104 from both sides:
\[
7x = 98
\]
Finally, dividing by 7 yields:
\[
x = 14
\]
</think>
Sure, let's solve the problem step by step.
**Problem Statement:**
When 16 is added to an imagined number, then multiplied... | 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 n = 3 \):
- \( a_{3, 1} = \frac{1}{3} \)
- \( a_{3, two elements: 1/2, 1/2
Row 3: three elements: 1/3, 1/6, 1.
So, it seems like for row \( n \), the entries are \( \frac{1}{n \cdot \binom{n-... | 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 So, n must end with a digit that when multiplied by 6 gives a 9 in the units place. Let's see: 6 * x �2≡2, x=8:48≡8, x=9:54≡4. So, no x where 6x ≡000...0, then the digit sum is just 1. But...0 divided by 16, which is 625 followed by zeros. So, n=625.... | 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 I need to find the perimeter of triangle ABC.
Hmm, let's break this will just be 3s once I find 's'.
Now, the midpointradius.
Wait, but triangle MAB: points A, B, M.AB is also a triangle with sides s, s/2, and (√3/2)s.
Alternatively, maybe coord... | 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 \.
- \( 3x^2 \cdot \frac{1}{x^3} + 3y = y^3 \). Then rearranged to \( x^3 + \frac{1}{x^3} =23 \), which matches what I found earlier.
To find \( x^3 + \frac{1}{x^3} \), another way is 110 \)
Hmm, same result. That seems solid.
Alternatively7 > 0 \).
Now, let's compute ... | 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 3a \) and the product is \( a^2 \). Therefore, the sum of the squares should be \( (3a)^2 - 04. So, \( a^2 = 0.04 \). Taking square roots, \( a = \sqrt{0.04}, so the equation always has real roots.
Therefore, both \( a = 0.2 \) and \( a = -0.2 \) are \p... | 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. | that the position is positive.
Alternatively, the positions that say 1 in the second round are equivalent to 1994 mod 4, but since, now, I need to find the number of positions that are equal to both for some integers m and k. So, essentially, find the number of integers x such that x ≡1 mod3 and x ≡1994 mod4.
Wait, ... | 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29 +1=88+1=89个数。
所以，在31到299之间，被3整除的数有899=180个数。
现在，我需要计算这些a的和，然后除以3，得到分数的和。
首先，计算从31到299所有数的和，然后减去被3整除的所有数的和，剩下的就是不被3整除的所有数5。
所以，从31到299的总和是44385。
接下来，计算被3整除的数的和=165×89。
计算165×89：
×9=1485，
加起来：13200+1485=14685。
所以，被3整除的数的和是14685。
现在，... | 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 the midpoint of edge AA₁. We are given that CK = 2√6 and DK = 4. We need to find the volume of the prism = Base Area × Height. Since it's a regular triangular prism, the base₁, CC₁), I can compute the volume.
Let me denote the side length of ... | 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 + 2 = a + 2
+ 0 = a + 2
+ 1 = a + 3
+ 6 = a2 mod 3, since 3*7=21.
So, a + b ≡ 2 mod 3.
Alright, that's our}
Now, we need to find all pairs (a,b) where a and b satisfy both conditions above.
Let me write the two congruence equations:
1. a + b ≡ 2 mod... | 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 0)/2 = (2n+1)/2, which is not an integer unless n is a half-integer, which it isn't because n Wait, actually, sqrt(-1) is not real, so maybe that approach isn't directly applicable.
Alternatively, perhaps I can think of genera... | 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 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 downx² + y²). Therefore, the area is 2*(x² + y²).
But according to the problem, the area is also 23b into the equation, we get:
(3a)²... | 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 number100d + 10e + f
Now, I need to find values of c that make abc a valid three-digit number. Since a (because a cannot exceed 9).
Testing possible values for c from 0 to 6:
- For c = 0:
a = 3, b = 2
abc = 320
def = 320 × 3 + 4... | 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**·** S.
If I add all four equations together, I get:
4S +. So the total is 0.
Therefore, adding all four equations... | 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 quadrilateral0°) = -0.5
sin(120°) is equal to sin(180° - 60°) = sin(60°) = √3/2 ≈ 0 from point B, moving in the direction of 120 degrees with length 2 units, we reach point C at (-1, √3). say vector CD is (x, y), then (-1)x + (√3)y = 0.
So, -x + √3 y = shape doe... | 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 asking for both the maximum number of pages possible and the minimum number of leftover lead types.
Wait, hold on. If we have 2011 lead11 - 9 = 2002 lead types left. Each subs... | 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, maybe I should use the expansion of arcsin x to get a more accurate approximation near 0 ... | 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 looks messy. Maybe I should rationalize the denominator.
Multiply numerator and denominator by = [2√3 - 2] / (3 - 1) = [2√3 - 2] / 2 = √3Now, solve for c:
c = (3 - 2√3) / √3
Again, rationalize the denominator:
Multiply numerator and denominator be r... | 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 divide p.
Alright, so let me formalize this. Let me denote the + a. Therefore, p = m - n = (100a + 10b + c) - (100c + 99:
Sum of digits: 9 + 9 = 1. So 99 is invalid.
2. p = 198:
Sum of digits: 1 + 9 + 8 = . These are the four valid valu... | 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 parameter t ranges from 0 to π/4. Hmm, okay. I/dt)² = 36 cos⁴t sin²t.
Then, (dy/dt)² is (6 sin²t cos t)². Similarly, squaring 6 gives 36, squaring sin²t gives sin⁴t, and squaring cos t gives cos t|. Since t is between 0 and π/4, both cos t and sin t... | 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 sum is 101. Hmm, interesting. Let me try to break maximum \( n \) for which this condition holds. That means if \( n where no two add up to 101. I need to find the threshold where thi... | 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. Therefore, rₙ = k^(n-1) * r₁.
Given that, since r₁ is 1, r₂ = k, r₃ = k², r₄ = k³, etc. So₂) / sinα.
Wait, maybe not. Let's think again.
ₙ = rₙ / sinα.
Since each circle is tangent to the previous one, the distance between their centers should b... | 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 the30 = 10n - 10
\]
Subtracting \( 10n \) from both sides:
\[
n - 30 = -10
\]
Adding 30 to both sides:
\[
n = 20
\]
Therefore, there are 20 people in the classroom.
</think>
Let's solve the problem step-by-step.
**Given:**
1. The average age of everyone in the ... | 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 accuracy2 to 202,790 results in 202,832.
So, the product of 7 and 28,976 is 202,832.
Next, I need to add 84,567 to this product.
Adding 202,832 and 84,567 gives me 287,399.
Therefore, the final result ... | 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 \( p \) by each term in \( -p - 8 \):
\(cdot (-8) = -8p \)
Then, multiply \( -7 \) by each term in \( -p - 8 \):
\( -(1)(-56)}}{2(1)} \]
\[ p = \frac{-10 \pm \sqrt{100 + 224}}{2} \]
\[ p = \frac{-1 = -14 \). But wait, I need to check if these are valid ... | 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 4 to eliminate denominators and make it easier to handle.
Multiply each term by 4:
\[ (2n + 5)^2 + 27 = 4k^2 \]
So,
\[ (2n + (l - m) + (l + m) = 1 + 27 \)
\( 2l = 28 \)
\( l = 14 \)
Subtracting the two equations:
\( (l + ... | 1 | 4 |
I4.1 If $\frac{6 \sqrt{3}}{3 \sqrt{2}+2 \sqrt{3}}=3 \sqrt{\alpha}-6$, determine the value of $\alpha$. | = 18.
Calculating b²: (2√3)² = 4 * 3 = 12.
So, the denominator becomes√6 - 6.
Now, let's go back to the original equation:
3√6 - 6 = 3√α - 6.
α.
Now, both sides have a coefficient of 3. To isolate √α, I can divide both sides by 3:
(3√6)/3 = (3√α)/3.
Which simplifies to:
√6 = √α.
Now, to find α, I can square ... | 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,*A.
Therefore, the entire expression simplifies to:
200*A + Wait, 200A is 200 times A, which is a multiple of 100, but 200 is not A: 3, 6, 9.
Let me consider each case.
Case 1: A = 3
Then, N = (8 = 2
So, A=3, B=9, C=2
Now, let's check if ABC is 392.
So... | 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 filling the 5-liter jug completely. So, jug A (the 5. Step 6: (4, 3).
Wait, that's exactly what I needed! So, jug A now has exactly 4 liters. Therefore, the answer is yes, we can measure exactly 4 liters... | 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 $\qquad$ . | apex.
But wait, I don't know h yet. I do know the + 27 + h^2 = 36 + h^2 the problem says side edge is 4. Hmm, maybe it's a regular tetrahedron but with two different edge lengths? No, that's notWait, no, actually, the centroid of the base is at (3, (0 + 0 + 3*sqrt(3))/3, 0). That distance should be 4.
So, the distan... | 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:
\}^{n} a_i \right )
\]
But wait, in our \geq 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/ solutions for x must be exactly -44 or 44. Because if x is 44 or more, the right side's square root only allows up to 44. So x must be either -44 or 44.
Let me test x=44 first. Plugging into the -2. So, 6 ≠ -2, so x=44 is not a solut... | 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 labeled2 units long. I need to find the angle between two skew lines, DG 2)
- C₁: (2, 2, 2)
average of coordinates.
E_x = (0 + 2)/2 = 1
E_y = (2 + 0)/2 = 1
E_z = (2 + 0)/2 = 1
So,1)
Similarly, for EF, the direction vector is F - E.
EF vector: F - E ... | 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 it2^{x}\) multiplied by \(2^{x}\). ... | 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 $\qquad$. | a unique sum. However, in this case, the sums are not for all subsets, but only for subsets of size 1, 2, or , and triples because each is a unique number.
Therefore, the maximum number of different sums would be 6 (singles) + 12 (pairs) + 8 (triples) = 26.
But hold on number of different sums?
But wait, hold on. I... | 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 is multiplying to 1. So, if I keep grouping these terms in pairs just be 1.
Wait, but how many such pairs are there? Let1)), so the number of terms is 2m. Therefore, there are m pairs. Each pair multiplies to 1, so the entire product is 1^m = 1.
But wait... | 1 | 1 |
Solve the following equation:
$$
2^{x+3}+4^{x+1}=320 .
$$ | Okay, so I need to solve this equation: 2^{x+3 2. Perfect!
So, the quadratic factors as:
(y + 156 = 320. Yep, that works.
Hmm, just to make sure I didn't make a mistake earlier when substituting, let me go through the steps again.
Original equation: 2^{x+3}2 + 8y - 320 = 0 => y^2 + 2y - 80 = 0.
Factoring gives y =... | 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... | 60. So, the sum of these fractions is (47/6. Let me compute that.
1/3 is approximately 0.333, 1/4 is 0.25, and beans. Therefore, the remaining land, which is corn, is 1 - 47/60 = 13/60 of the total land.
And we know that this remaining 13/60 of the total land is 26 acres. So, 13/60 of x is 26. Therefore, x = 26 * ... | 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. Maybe I can squ... | 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 +(2a) = 98a \)
Right side: \( 4(a^2 + a \cdot a + a^2) = 1
So, discriminant \( D = -48b^2 + \( t = 4k \), where \( k \) is integer, then:
\[
s... | 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 terms from the first two terms. Let's see.
Looking at \(2^{113} \cdot 5^{106}\) and \(2^{110} \cdot 5^{105}\), number that ends with k nines. So, 1248 * 10^{105} is 1248 followed by11 followed by two nines.
Ah, so in this case, 1248 * 10^{105} is 1, the... | 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?
# | 'll use 2400. Converting the rates, I get 10/2400 - 3/2400, which equals 7/2400 per second.
Finally, to determine the time required to fill the bathtub, I'll take the reciprocal of the net rate: 2400/7 seconds. Converting that back to minutes gives approximately 4 minutes and 6 seconds.
</think>
To determine how long... | 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 lengths2)².
Let me compute each part:
First, n² + (.
Then, (n+2)² = n² + 4n n ≥ 1, the expression is positive when n - 3 > 0, i.e., n > 3.
So, the inequality holds when, the triangles are obtuse and right, respectively, and starting from n=4, they... | 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. | vertices. Specifically, there are:
- 6 faces, each can be rotated in 4 ways (0°, 90°, 18 two faces of the same color. Thus, no coloring can be fixed by a 180° rotation either. So, fixed colorings = 0 So, same reasoning applies: all three faces in each cycle must be the same color, which is impossible with all differe... | 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? | Combine like terms:
(2m - 0.25m) + (0.5w - 0.25w) + 33 from both sides:
1.75m + 0.25 test m = 4 first:
If m = 4,
w = 36 - 7*4 = 36 - 28 = 8
c = 6*4 - 24 = 24 - 24 = 0
So, c = 0. possible solution.
Now, let's test m = 5:
If m =5 = 12. Good.
So, this also works.
Now, which isn't possible.
So, only m=4 and m=5 ... | 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 position.
Therefore, the entire expression inside the absolute value is the sum of the signs of each \( a_i \), from \( i = 1 \) states that the \( a_i \)'s are distinct non-zero real numbers. to be positive and 1010 to be negative.
Hence, in that case, the s... | 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 value. Hmm, okay. Let me start by writing down the equation they gave factor sin β on the left side:
sin β [1 + 2 sin² α] = 2 cos α cos β sin α.
Now, I can write this as:
sin β = ... | 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 $\qquad$ . | , 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 that2, which occurs at x = 4 and x = 1/4. The minimum value of y is 0, which occurs at x = difference between the maximum and minimum values of the length of [a, b],=4, the domain is [1/4,... | 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... | ,0625.
Na konačnoj fazi, m svaku kombinaciju, što je 6 * 0,0625 = 0,375. Ovaj rezultat može biti prepišan u obliku 3/8.
Dakle, verovatnoća je 3/8.
</think>
Koristimo se binomnom distribucijom za izračun verovatnoće dogodka "dva pisma i dva glave" prijednom bacanju četiri simetrične kovanice.
1. **Broj načina (kombi... | 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 all those terms.
First, I remember that when dealing with recursive sequences and G, a₂₀₁₀). Let's denote d as the GCD of aₙ₊₁ and aₙ, so d divides both aₙ₊₁ and aₙ. From the recursive formula, aₙ.
Wait... | 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? | 1 to 9. That gives us 9 options for A.
Next,0 choices here.
So, to find the total number of 5-digit palindrome numbers, I just multiply the number of choices for each digit together. That would900. So, there are 900 different 5-digit palindrome numbers.
Wait, let me double-check that. If A is from 1-9, that's 9 opti... | 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 +48.9² = (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 fast. x +4y must equal 23 m², not 23² m². Wait, hold on, let me make sure.
Wait,, 22 solutions.
Total solutions: 5 + 22=27.
Wait, hold on, that seems straightfo... | 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 s = 2R \sin(18^\circ) \]
Wait angle in the right triangle. Yeah, that seems right.
So, the side length \( s \) is \( 2R \sin(18^\sqrt{5} - 1}{4} \) multiplied by 2. Hmm, let me recall the exact... | 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! | y + 2x) = 8x - 9y
\ can infer that \( 8x > 9y \). Therefore:
\[
8x - 9y > 0
\]
This means that:
\[
10x + 2y > 11y + 2x
\]
Thus, 10 pencils and 2 notebooks are more expensive than 11 notebooks and 2 pencils.
</think>
To determine which option is more expensive, let's analyze the given information step by step.
1. **De... | 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? | </think>
**Solution:**
1. **Number of Pioneers from Class 1a:**
- There are **3 Pioneers** from class 1a who receive certificates.
2. **Number of Pioneers from Class 1b:**
- It is stated that an **equal number** of Pioneers from class 1b also receive certificates.
- Therefore, there are also **3 Pioneers** ... | 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 rememberequiv 1 \mod 9 \).
- If \( x \equiv \( x \equiv 4 \mod 9 \... | 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: Hmm, again using the stars and bars theorem.
But wait, if Jia, which uses up another 3 quotas. So, we're left with 6 - 3 = 3 quotas to distribute without any restrictions, except t... | 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.
# | 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 sides by 37:
10X + Y = 1184 / 37
Let me calculate 1184 ÷ digits XY are 32. So, the original number is 432.
Let me check if that works.
... | 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 radius 2, and point P is on circle M, which is centered at (5 cosθ, 5 sinθ) with radius 1. So let α be half of angle EPF, then sin α = OE / OP = 2 / OP. Therefore, α = arcsin(2 / OP), so the total angle between the two tangents is φ = 2 arcs = 8 -... | 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 \( f(x \cdot \frac{\pi}{4} \). Since cosine is decreasing in the interval \( [0, \pi] \), to ensure \( \cos(\omega x) \) remains non-negative throughout \( \left[0, \frac{\pi}{4}\right] \), we need the maximum angle), which is less than 2.
This suggests that th... | 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:
$$
\int_{0}^{\frac{\pi}{4}} \frac{2 \cos x + denominator is $(2 \sin x - 3 \cos x)^3$, and the numerator is $2 \cos x + 3 \sin x$.}} \frac{1}{u^3} du
$$
Alright, integrating $\frac{1}{u^3}$ is straightforward. Remember that:
7}{18}
$$
Wait, but I think I might have messed up the... | 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 of2.
10^4 mod 13 = (10^37=3.
10^5 mod 13 = 3*13 is 6.
Now let's do the same for 1768=2.
10^11 mod 17 = 2*10 = 20 mod 17 = 3.
10^12 mod 17 = 3*10 =7. 17*7=119, so 120-119=1.
So 10^16 ≡ 1 mod 17. So t... | 1 | 48 |
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