question stringlengths 15 6.18k | target stringlengths 264 34.7k | label int64 1 1 | answer stringlengths 1 136 |
|---|---|---|---|
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/6 32), given that 1/x + 1/y = 1/2016. Hmm, let me see. I remember that when6². So, a and b are positive numbers (assuming x and y are positive, which they probably are since we're dealing with square roots).
Now, going back to the original expression we need to compute: √(x/63 ... | 1 | 32 |
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]
To solve for \( x \), I'll convert the mixed number to an improper fraction:
\[
\frac{7(x + 16) - 8}{9} = \frac{202}{9}
\]
Multiplying both sides by 9 gives:
\[
7(x + 16) - 8 = 202
\]
Expanding and simplifying the equation:
\[
7x + 112 - 8 = 202
\]
\[
7x + ... | 1 | 14 |
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, let me parse the problem statement again to make sure I understand it correctly. We * 3), 1/(4 * 3), 1/4ldots, n \).
Wait, let's test th... | 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) for a positive integer n. Here, S(n) is the sum of the digits of n. I remember that in problems like this, digit sums can sometimes be tricky because of carries when multiplying, especially when de... | 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 this,0)
Distance is sqrt[ (0 - (-s/2))² + (0 - 0)² ] = sqr... | 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 \( x > 0 \) and \( \left(x + \frac{1}{x}\right)^2 = 25 \), I need to find the value of \( x^3 + \frac{1}{x^3} \). Hmm, let me think about how to approach this.
First off, the equation given is = 23 \). That looks correct.
2. Then, I considered1}{x} = 5 \). Let me set \( t = 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 the roots of the quadratic equation \( x^2 - 3a x + a^2 = 0 \) is 0.28. Hmm, okay. I think I remember some properties about quadratic equations and their roots, a^2 \). Therefore, discrimi... | 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 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 counting1, the second is 2, third is 3, fourth is 4, fifth is 1, etc. So, the position from the back is the number they say. So, the first ... | 1 | 166 |
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，题目所有满足31 ≤ a ≤ 299，且a不被3整除的数，然后把这些a代入a/3，再求这些分数的和。
不过，这样算起来可能会比较麻烦，我计算这些数的和。
首先，我想找到所有满足条件的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整除的所有数的和。
从31到299的总和可以用等差数列求和公式计算：和=（首项 +末项）×项数 /2。
首项是31，末项是299，项数是269，所以总和是(31+299)×269 /2 = ... | 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 the sphere. So, points C and D are at opposite ends of the sphere, meaning the distance between them is twice the radius. Also, point K is the midpoint of edge AA₁... | 1 | 36 |
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 divisible by 33. Hmm, okay. Let's try to break this - b ≡ 1 mod 11
Alright, so now we have two conditions:
1. a + b ≡ 2 mod 3
2. a - b ≡ 1 mod 11
Now, let's recall that a and b are digits, meaning they can take i... | 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 step by step.
First, I remember that when you expand a binomial like over the coefficien... | 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 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 = 3
So, in this case, a =1, b=3
So, we have two possible pai... | 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 + 10e + f
303c + 30b + 904 = 100d + 10 to find values of c that make abc a valid three-digit number. Since a = c + 3, c must be less than or equal to 6 (because a cannot exceed 9).
Testing possible values for c from 0 to 6:
- Fo... | 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 x5, 17, 21, and 23. I need to find x² + y² + z². Hmm, okay.
2. D2² = S + 2(ab - ac - bc) = 17² = 289
3. D3² = S + 2(-ab + ac - bc) = 21² = 441
4. D4² = S + 2(-ab - ac + bc) = 23² = 9
Now, if I label these equations as (1), (2), (3), (4), respec... | 1 | 371 |
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 a c
Simplify each bracket:
First bracket: [c - (2q)/a - c] = - ... | 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 quadrilateral ABCD. Let me try to visualize it first. The quadrilateral has four is 120 degrees, and angle BCD is 90 degrees. Hmm, okay.
I think drawing a diagram might help. Since it's say vector CD is (x, y), then (-1)x + (√3)y = 0.
So, -x + √3 y =�3, the magn... | 1 | \sqrt{7} |
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 |
## 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. | First, there are 12 large monkeys evenly spaced around a circle with a small monkeys.
Including the large monkeys, there are now 12 + 36 = 48 monkeys altogether.
To find the new equal distance between each adjacent monkey:
Divide the total circumference by the number of gaps between monkeys, which is 48.
Therefore, ... | 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 find the difference m - n, which is another number, let's call it p. The condition is that the sum of the digits of p exactly divides p. Finally, I need to find all different values of p and add them up a and c are digits such that a > c, we ca... | 1 | 1980 |
## 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 2 cos³t and y = 2 sin³t, and the parameter t ranges from 0 to π/4. Hmm, okay. I remember that to find the arc length of a parametric curve, there's a specific formula. Let me try to recall it.
I think the formula is an/dt = 2 * 3 cos²t (-sin t) whic... | 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.ferences) of the second and fourth circles. They tell me that the length of it. So, the first circle is the smallest, then the second is bigger/sinα:
(r₁ - r₂) / sinα = r₁ + r₂
Multiply both sides by sinα:
r₁ - r₂ = (r₁ + r₂) sinα
Bring all terms... | 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 the1} = 10 \quad \Rightarrow \quad S - 30 = 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 peo... | 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,000 and 56,000 gives me 196,000.
Adding 6,300 to 196,000 results in 202,300.
Adding 490 to 202,300 gives me 202,790.
Finally, adding 42 to 202,790 results in 202,832.
So, the product of 7 and ... | 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 \( 3 \cdot \sqrt{p} \), 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,frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Plugging in the values:
\[ p = \frac{-10 \pm \s... | 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 The conjugate of (a + b) is (a - b), right: 6√3 * (-2√3) = -12 * (√3 * √3) = -12 * 3 = -36.
So, the numerator becomes 18√6 -... | 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,100 and 999.
The problem shows an addition:
/ 3 = 292
And 200A divided by 3 is (200/3)*A, which is approximately 66.666*A, but again, not an integer unless a multiple of 3. So, 876 is divisible by 3, so 200A must also be divisible by 3 for (876 - 200A) to... | 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 remember something about these kinds of problems from before, maybe like the water jug puzzle. Let me try to figure it out step by step.
First, I'll and jug A is empty. Step 4: (0, 2).
Now, I fill jug A... | 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 (3, (3*sqrt(3))/3, 0) = (3, sqrt(3), 0). Wait, no, the centroid of an equilateral triangle is at ( (0 + 6 + 3)/3, (0 + 0 + 3*sqrt(3))/3, 0 ) = (3, sqrt(3), 0). So the apex is directly above the centroid, so its coordinates are (3, sqrt(3), h). Wait, that's ... | 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 \( a_1, a_2, \ldots, a_n \) such that their sum is 1. I need to find the minimum value of the expression:
\[
M = \sum_{i=1}^{n total sum is 1, right? So this simplifies each denominator to \( 1 + (1 - a_i) = 2 - a_i \} \frac{a_i}{2 - a_i} \) given... | 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: �4) - 4| - √(2017 - x²). Hmm, looks a bit complicated, but let's try to tackle it step by step.
First off, I notice that there are square roots on both integers, x must be ≤ -44 or ≥ 44.
Now, looking at the right side: |(x/4) - 4936 and 45² is 2025, so... | 1 | -44 |
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. three) are visible at the same time, I can add up the numbers on those faces to get a sum. The question is asking for the maximum number of, but in our problem, the numbers are on the faces, and when two faces are adjacent, their sum is considered. So... | 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 1.
Again, it's 1. Hmm, seems like each pair is multiplying to 1. So, if I keep grouping these terms in pairs, each pair will contribute a factor of 1, so the entire product will just be 1.
Wait, but how many such pairs are there? Let's see. The last term... | 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 this, it's often helpful to express everything in terms of the same base.^3 = 8 * 2^x = 8y,
and5 & 16, 8 & 10.
Considering that one number is positive and the other is negative to get a product of -80. Let's try 10 and -8: 10 * (-8) = -80, and 10 + ... | 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. Let me try to figure out how to solve it. The problem says that on out the total number of acres the farmer has.
Hmm, let's see..
So, if I add up all these fractions, they should add up to the total land minus the 26 acres. So, (1/3)x + (1/4)x + (1/5)x... | 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 step by step.
First off, I notice that both the numerator and the deno... | 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 |
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, let's break this down step by step.
First, consecutive natural numbers. that for n=3, the triangle is right-angled, not acute.
Similarly, for n=2, n² - 2n -3 = 4 -4 -... | 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. 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 symmetry. I remember something about group theory and Burn these rotati... | 1 | 30 |
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_- = 2020 \).
So, we can rephrase: we need to split 2020 into two integers \( n_+ \) and \( n_- \) such that \( |n_+ - n_-| \) is as small as possible.
Now, since 2020 is an even number, it can be split exactly into two equal halves, each of 1010. That is, \( ... | 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 valuea, b], and the range is [0, 2]. I need to find the difference between the maximum and minimum values of the length of the interval [a, b]. Hmm, okay, let me try to break this down,2]. So, the length of [a, b] would vary depending on wher... | 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... | Prvu koristim se osnovnim konceptom vjeroj6. 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 prepiša... | 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, 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 thinking there must the first few terms.
a₁ = 5, a₂ = 8. GCD(5, 8) = 1.
a₃ = a₂ + 3a₁ = 8 + 1... | 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 because it's in the middle and doesn't affect the leading digit. So, 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 total number of 5-digit palindrome n... | 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's break this down.
First, let me note that 23 and 9² m². Wait, hold on, let me make sure.
Wait,1: x +4y=23*1... | 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 - x \).
According to the problem, the points Guagua has left are four times the points Bobo has left. This can be expressed as:
\[ 1000 - x = 4(640 - x) \]
Now, solve for \( x \):
\[
1000 - x = 2560 - 4x \\
3x = 1560 \\
x = 520
\]
Therefore, they e... | 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. Hmm18^\circ \) (half of 36 degrees), a hypotenuse of \( R \), and the opposite side will be half of the base, which is \( \frac{s}{2} \).
Using trigonometry, specifically the sine function, I can re... | 1 | \frac{\sqrt{5}-1}{2}R |
## 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? | First, I note that there are 3 Pioneers from class 1 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 r... | 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 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. So the total number of distributions should be C(8-1, 2*1) = 35. So... | 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 is 0.75 of the original number. I need to figure out what the original number is. Hmm, let me think through this step by step.
First, let me understand what the problem is saying. We have a three
4*9.25Y = 37Y
4* 37 out of the left-hand si... | 1 | 432 |
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 to find the minimum value of the dot product of two vectors, PE and PF. Let me try to break it down step by step.
First, let me understand the given information. There's a circle O with equation x² + y let α be half of angle EPF, then sin α = OE ... | 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 the interval \( \left[0, \frac{\pi}{4}\right] \), and the maximum value on this interval is \( \sqrt{3} \). I need to find the value of \( \omega \).
Alright, let's break this down. First, I know that a function is montheta = \omega x = \frac{\pi}{2} \) ... | 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:
$$
\int_{0} 3 \sin x}{(2 \sin x - 3 \cos x)^{3}} d x
$$
Hmm, it looks a bit complicated, but maybe I can simplify it by substitution. Let me see... The integral becomes:
$$
\int \frac{du}{dx} \cdot \frac{1}{u^3} dx
$$
Which simplifies to:
$$
\int \frac{1}{u^3} du
$$
But wait, ... | 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/17, I need to find the orders of 10 modulo 13 and modulo 17 respectively.
Let's start with 13. I need to find the smallest k where 10^k ≡ 1 mod 13.
Compute powers of 10 modulo 13:
10^1 mod 13 = 10
... | 1 | 48 |
[Example 1.3.7] A sphere with radius $r$ is inscribed in a regular tetrahedron. Find the minimum total surface area of this regular tetrahedron. | Okay, so I have this problem: A sphere with radius r is inscribed in a regular tetrahedron. I need to find the minimum total surface area of this regular tetrahedron. Hmm, let's see. I, I need the total surface area of the regular tetrahedron. A regular tetrahedron has four equilateral triangular faces. The area of one... | 1 | 24\sqrt{3}r^{2} |
6) Let $\left\{a_{n}\right\}$ be a sequence with the sum of the first $n$ terms denoted as $S_{n}$. Let $T_{n}=\frac{S_{1}+S_{2}+\cdots+S_{n}}{n}$, which is called the "average" of the sequence $a_{1}, a_{2}, \cdots, a_{n}$. Given that the "average" of the sequence $a_{1}, a_{2}, \cdots, a_{1005}$ is 2012, what is the ... | Alright, so I have this problem here about sequences and their "averages." Let me try to understand it step by step.
First, the problem defines are the same as the original sequence. We need to find T₁₀₀₆ for this new sequence.
Let me break this down.
First,.
Wait, actually, for any k >=1, S'_k = S bit confusing.... | 1 | 2009 |
1. If $\log _{4}(x+2 y)+\log _{4}(x-2 y)=1$, then the minimum value of $|x|-|y|$ is $\qquad$ | Okay, so I have this problem here: If log base 4 of (x + 2y) plus log base 4 of (x - 2y) equals 1, then I need to find the minimum value of |x| - |y|. Hmm, okay, let me try to figure this out step by step.
First, I remember that when you have the sum + e^{-t}) = 1/2.
Let me solve for t:
Let me set e^t = u, so tanh t... | 1 | \sqrt{3} |
4. If $2016+3^{n}$ is a perfect square, then the positive integer $n=$ . $\qquad$ | Okay, so I have this problem: If 2016 plus 3 to the power of n is a perfect square, then find the positive integer n. Hmm, let's see how I can approach this.
First, a if 2016 + 3^n is a perfect square, then there must be some integer k such that 2016 + 3^n = k². My goal is to find the value of n that satisfies large, ... | 1 | 2 |
343. Find $y^{\prime}$, if $y=\arcsin x^{3}$. | Okay, so I need to find the derivative of y equals arcsin of x cubed. Hmm, I remember that when dealing with inverse trigonometric functions.
Simplifying the denominator, (x³)² is x�, the derivative dy/dx is 3x² divided by sqrt(1 - x⁶).
Wait, let me double-check if I did everything correctly. The chain rule applies... | 1 | \frac{3x^{2}}{\sqrt{1-x^{6}}} |
3 Find all real numbers $k$ such that the inequality
$$
a^{3}+b^{3}+c^{3}+d^{3}+1 \geqslant k(a+b+c+d)
$$
holds for all $a, b, c, d \in[-1,+\infty)$. (Xu Wanyi) | Okay, so I've got this problem here where I need to find all real numbers \( k \) such that the inequality
\[
a^{3} + b^{3} + c^{3} + d^{3} + 1 \geqslant k(a + b + c + d)
\]
holds for all \( a, b, c, d \in [-1, +\infty) \). Hmm, that looks a bit intimidating, but let me try to break it down step by step.
First, I noti... | 1 | \frac{3}{4} |
## Problem Statement
Calculate the definite integral:
$$
\int_{2 \operatorname{arctan} 2}^{2 \operatorname{arctan} 3} \frac{d x}{\cos x(1-\cos x)}
$$ | Alright, so I need to calculate this definite integral:
$$
\int_{} \frac{dx}{\cos x(1 - \cos x)}
$$
Hmm, okay. Let me start by understanding the integral and the limits. The limits are from \(2 \arctan 2\) to \( \theta = 2 \arctan 3 \).
We can use double-angle formula for tan:
\( \tan 2\theta = \frac{2 \tan \theta}... | 1 | \frac{1}{6}+\ln2-\ln3 |
(4) Let $x, y, z \in \mathbf{R}^{+}, x+y+z=1$, then the maximum value of the function $f(x, y, z)=x y^{2} z^{3}$ is $\qquad$ | Okay, so I have this problem here where I need to find the maximum value of the function f(x, y, z) = xy²z³, given that x, y, z are positive real numbers and x + y + z = 1. Hmm, let me think about how to approach this.
First, this seems like an optimization problem with a constraint. The function to maximize is f(x, y... | 1 | \frac{1}{432} |
Let \( x, y \) be positive real numbers such that \( x^{3} + y^{3} \leq x^{2} + y^{2} \). Find the greatest possible value of the product \( x y \). | Alright, so I've got this problem here: Let \( x, y \) be positive real numbers such that \( x^{3} + y^{3} \leq x^{2} + y^{2} \). I need to find the greatest possible value of the product \( xy \). Hmm, okay, let me try to figure this out step by step.
First, since both \( x \) and \( y \) are positive real numbers, a... | 1 | 1 |
4. Two students, A and B, play chess. Winning a game earns 2 points, drawing a game earns 1 point each, and losing a game earns 0 points. They play three consecutive games, and the one with more points wins. What is the probability that A wins? | Okay, so I need to figure out the probability that student A wins a chess points wins. Each game can result in a win, loss, or draw. The scoring is 2 points for a win, 1 point each for a draw, and 0 points for a loss.
First, I should model the, [G(x)]^3 = [x^{-2}(1 + x^2 + x^4)]^3 / 27 = x^{-6}(1 + x^2 + x^4)^3 / 27.
... | 1 | \frac{10}{27} |
Problem 4. Let $L$ be the intersection point of the diagonals $C E$ and $D F$ of a regular hexagon $A B C D E F$ with side length 5. Point $K$ is such that $\overrightarrow{L K}=\overrightarrow{F B}-3 \overrightarrow{A B}$. Determine whether point $K$ lies inside, on the boundary, or outside of $A B C D E F$, and also ... | Okay, so I have this problem about a regular hexagon named ABCDEF with side length 5. The problem mentions the intersection point L of the diagonals CE and DF. Then there's a point K defined by the vector equation: vector LK equals vector FB minus 3 times vector AB. I need to figure out whether K lies inside, on the bo... | 1 | \frac{5\sqrt{3}}{3} |
35. Let $f(n)$ be the integer nearest to $\sqrt{n}$. Find the value of
$$
\sum_{n=1}^{\infty} \frac{\left(\frac{3}{2}\right)^{f(n)}+\left(\frac{3}{2}\right)^{-f(n)}}{\left(\frac{3}{2}\right)^{n}}
$$ | Okay, so I have this problem where I need to find the value of an infinite sum. The function f(n) is defined as the integer closest to the square root of n. The sum is given by:
$$
\sum_{n=1}^{\infty} \frac{\left(\frac{3}{2}\right)^{f(n)} + \left(\frac{3}{2}\right)^{-f(n)}}{\left(\frac{3}{2}\right)^{n}}
$$
I need to ... | 1 | 5 |
$\begin{array}{l}\text { 1. Let } p, q \in \mathbf{R}_{+} \text {, and satisfy } \\ \log _{9} p=\log _{12} q=\log _{16}(p+q) \text {. } \\ \text { Then } \frac{q}{p}=\end{array}$ | Okay, so I have this problem where I need to find the ratio of qq with bases 9, 12, and 16 respectively are all equal. Hmm, let me try to break this down step by step.
First, the problem states that log base 9 of p is equal to log using the quadratic formula:
x = [-1 ± sqrt(1 + 412ᵏ, so:
q/p = (12ᵏ)/(9ᵏ) = (12/9)ᵏ =... | 1 | \frac{1+\sqrt{5}}{2} |
80. There is a two-digit number. When the digit 1 is added in front of it, a three-digit number is obtained; when the digit 1 is added behind it, another three-digit number is obtained. If the sum of these two three-digit numbers is 728, then the original two-digit number is $\qquad$. | Okay, so I've got this problem here about a two-digit number. Let me try to understand it step by step.
First, it says when we7=627
Yes, so N=57.
Wait, so the original number is 57.
Let me verify that.
Adding led to the equation 110A + 2B + 110=728. But perhaps that's harder because it's introducing more variabl... | 1 | 57 |
## Task Condition
Find the derivative.
$$
y=\ln \left(x+\sqrt{x^{2}+1}\right)-\frac{\sqrt{1+x^{2}}}{x}
$$ | Okay, so I need to find the derivative of the function y given by:
1}\right) - \frac{\sqrt{1 + x^2}}{x}
$$
Hmm, this looks a bit complicated, but I think I can handle it by breaking it down into two parts. Let me first write down the function again to make sure I have it correctly:
$$
y(1 + x²)
Then, compute d/dx ... | 1 | \frac{\sqrt{x^{2}+1}}{x^{2}} |
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