question stringlengths 15 6.18k | target stringlengths 225 23.8k | 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 divisible by 9.
Finally, I'll verify that with c = 8, the last three digits 4xy form a number divisible by 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... | 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 simplifies to a/63. So, the first square root becomes √(a/63).
Similarly, the second term inside2, so this simplifies to 32 + b/63 - 32, which is b/63. Therefore, the second square root is √(b/63).
So, the entire expression becomes √(a/63) × √(b/63). Multiplying these tog... | 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 similar44 - B² => y = (544... | 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 116) - 8 = 202
\]
Expanding and simplifying the equation:
\[
7x + 112 - 8 = 202
\]
\[
7x + 104 = 202
\]
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 ... | 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,n(n-1)}, \frac{1}{n(n-1)}, \ldots, \frac{1}{n(n-1)}, \frac{1}{n} \).
Wait, but hold on, for row \( n \), the entries are symmetric, as we... | 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 the624? 16*624=10000 - 16*1=16, so 16*624=10000-16=9984. S(n)=6+2+4=12, S(16n)=9+9+8+4=30. Ratio=12/30=0.4. Worse.
What if n=6251? 16*6251=100016. S(n)=6+2+5+1=14, S(16n)=1+0+0+0+1+6=8. Ratio=14/8=1.75. Still worse than 13.
How about n=6252? 16*6252=10... | 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, AM = (√3/2)s.
Let me check if this triangle is a right-angled triangle. If it is, then area is 1/2 * base * height.
Let me see: s² = (s/2)² + ( (√3/2)s )² ?
Compute RHS: (s²/4) + (3s²/4) = s², which equals LHS. So, yes, triangle MAB is right-ang... | 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 \( x > 0 \2 = 25 \), I need to find the value of \( x}\right)^2 = \frac{25 + 10\sqrt{21} + 21}{4} = \frac{46 + 10\sqrt{21}}{4} = \frac{23 + 5\sqrt{21}}{2} \)
Now, compute \( x^3 \):
\( x^3 = x \cdot x^2 = \frac{5 + \sqrt{21}}{2} \cdot \frac{23 + 5\\sqrt{21} \)
Compute each term:... | 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 mathematical terms, the smallest real number is the most negative one, so \( -0.2 \) is smaller than \( 0.2 \). So, the answer is \( a = -0.2 \).
But the problem is, sometimes in problems like this, they might expect the answer in fractional form instea... | 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, now, I need to find the number of positions that are equal to both 1 + 3m and 1994 - 4k for some integers m and k. So, essentially, find the number of integers x such that x ≡1 mod3 and x ≡1994 mod4.
Wait, but 1994 is congruent to 2 mod4, right? So, x ≡1 m... | 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=1485，
加起来：33000+9900=42900，再加上1485，得42900+1485=44385。
所以，从31到299的总和是44385。
接下来，计算被3整除的数的和，也就是从33到297的和，同样用等差数列求和公式：
首项是33，末项是297，项数是89，和=(33+297)×89 /2=330×89 /2=165×89。
计算165×89：
165×80=13200，
165×9=1485，
加起来：13200+1485=14685。
所以，被3整除的数的和是14685。
现在，不被3整除的数的和就是44385 ... | 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.
But we also have from CK: a² + h²/4 = 24, and from DK: sqrt(a²/3 + h²/4) = R
Wait, but we also have that the center is at (a/2, (√3/6)a, h/2). Since point D is on the sphere, its distance from the center should also be R.
Compute the dista... | 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 divisible by 33. Hmm, okay. Let's try to break this2 mod 3
Simplify:
2b + 1 ≡ 2 mod 3
Subtract 1 from both sides:
2b ≡ 1 mod 3
Now, we need to solve for b in this congruence.
We can multiply both sides by the modular inverse of 2 mod 3. The inverse of... | 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 + the coefficients of the terms where the exponent of x is an integer. Hmm,n+1} + 1]/2.
The sum when m is odd: [3^{2n+1} - 1]/2.
Wait, that's conflicting with the earlier result.
Wait, no, if I consider f(1) = sum_{m} C(2n+1, m) * 2^m = 3^{2n+1}
f(-2) = sum_{m} ... | 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 yy, x) or (y, -x), depending on the direction.
Wait, actually, if you have a square with a vertex at (x, y), the next vertex can be obtained by rotating (x, y) by 90 degrees around the origin. The rotation of (x, y) by 90 degrees is (-y, x). So, the ... | 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).
- k t1 t2.
Wait,... | 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 = 320 × 3 + 4 = 964
- For c = 1:
a = 4, b = 3
abc = 431
def = 4 Not a three-digit number.
- For c = 3:
a = 6, b = 5
abc = 653
def = 653 × 3 + 4 = 1959 + 4 = 1963
Not a three-digit number.
- For c = 4:
a = 7, b =... | 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 3: 441 - 371 = 70 = 2(-ab + ac - bc), so -ab + ac - bc = 35.
From equation 4: 529 - I can try to solve for ab, ac, bc.
Let me write these equations:
Let me denote:
Equation (1): ab + ac + bc = -73
Equation (2): ab - ac - bc = -41
Equation (3)... | 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 and height 2. So, area p = (1/2)*2*2=2.
Similarly, triangle DCL: Points D(0,2), C(2,2), L(0,4). It's a right triangle with base 2 and height 2. Area q = ( AB.
Wait, actually, if the line through C is drawn such that it intersects AB and AD beyond B and D, ... | 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, √3), which is in the second quadrant relative to the origin, moving to (1/2, (3√3)/2) would place D in the first quadrant. This might cause the quadrilateral ABCD to be convex.
Option 2: D is at (-5/2, √ in the third quadrant. This might make the q... | 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 number with \(k = 4\) trailing 9s but with a smaller digit before. For example, \(n = 49999\). Or, is there a smaller number with \(k = 4\) but with a smaller digit before? Wait, if the digit before is smaller, say 0, but then the number wo... | 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 (2/(x + 5)). Hmm, that looks a bit complicated,^2/6], the entire expression becomes:
[1 + x^2/6]^{(2/5)(1 - x/5 + x^2/25 - ...)}
Now, raising this to the power of (2/5)(1 - x/5 + ...), we can use the approximation (1 + a)^b ≈ 1 + a*b +, and b = (2... | 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 + b2), B(-1, √3), and C(c, 2 - c). And I need to find the value of a² + b² + 2b² + 2c² - 2ab - 2bc - 2ca]
= (a² + b² + c² - ab - bc - ca)
So, yes, that formula is correct. Therefore, another way to compute the expression is to compute ½[(a - b)² + (b - c)... | 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. | 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, the distance is 480 meters divided by 48 equals 10 meters.
</think>
**Solution:**
1. **Determine the Initial Distance Between Large M... | 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√5)/(10π) - 1.
Alternativ... | 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 921, n = 129, p = 921 - 129 = 792. Sum of digits: 7 + 9 + 2 = 18, and 792 ÷ 18 = 44. So that works too.
So all four p's are valid and can be achieved with appropriate digits a, b, c.
Therefore, the valid p's are 198, 396, 594, and 792. Now, t... | 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 = 0.
Next, compute cos(0) which is 1. So, the second term is (-3/2)*1 = -3/2. But since we have [(-3/2) cos(...) - (-3/2) cos(...)], it becomes [0] - [(-3/2)(1)] = 0 - (-3/2) = 3/2.
Therefore, the arc length is 3/2 units. Let me just double-check m... | 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'sn+1)^3 - 1, which makes sense because there are ( points. But ... | 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 numbers from 1 to 50 (50 numbers) and then pick one more number. If we pick 101, 102, or another number from 51 to 100.
But if we pick another number from 51 to 100, say 51, then 51 is paired with 50. But 50 is already in our selection, so 50... | 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.1 + sinα)]² = 9
Thus, (sinα)/(1 + sinα) = 3 or -3.
Again, sinα can't be negative, so:
sinα = 3*(1 + sinα)
Wait, that gives sinα = 3 + 3 sinα
So, sinα - 3 sinα = 3 ⇒ -2 sinα = 3 ⇒ sinα = -3/2
Which is impossible because sinα cannot be less than ... | 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, gives me 202,790.
Finally, adding 42 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 is 287,399.
</think>
Certainly! Le... | 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 \() back into the original terms:
1. First term: \( -4 - 8 = -12 \)
2. Second term: \( 3 \cdot \sqrt{4} = 3 \cdot 2 = 6 \)
3. Third term: \( 4 - 7 = -3 \)
Now, check if these form a geometric progression.
In a GP, the ratio between consecutive terms sho... | 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 n > 0 \).
So, \( k \) is between \( n + 2 \) and \( n + 3 \). Since \( k \) must be an integer, the only possible integer value in that interval is \( n + 3 \) when the difference is such that \( k^2 \) is exactly \( n^2 + 5n + 13 \). Wait, but for \( n > 4 \), as ... | 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, theHmm, interesting. On both sides, we have a term minus 6. Let me write that down:
3√6 - 6 = 3√α - 6.
So, if I add 6 to both sides, I eliminate the -6 on each side:
3√6 = 3√α.
Now, both sides have a coefficient of 3. To isolate √α, I can divide both sides... | 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,2
Now, let's check if ABC is 392.
So, ABC = 392
Then, adding BC (92) twice:
392 + 92 + 92 = 392 + 184 = 576
But the problem392 = 876
Ah, yes! 876. So, that's correct.
Wait, so earlier I thought the result was 576, but that was wrong.
Wait, 92 + 392 is ... | 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 full because it already has 2 liters. So, I pour 1 liter from jug A to jug B. After this, jug A has 5 - 1 = 4 liters, and jug B is full with 3 liters. Step 6: (4, 3).
Wait, that's exactly what I needed! So, jug A now has exactly 4 liters. Therefore,... | 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 (2)
Subtract Equation (1) from Equation (2):
(6 - a)^2 + b^2 + c^2 - (a^2 + b^2 + c^2) = 0
Expanding (6 - a)^2: 36 - 12a + a^2
So, 36 - 12a + a^2 - a^2 = 0 => 36 - 12a = 0 => 12a = 36 => a = 3
So, a = 3.
Now, plug in point C: (3, 3*sqrt(3), 0):
(3 - 3)... | 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 \( |x| - 1 < 0 \). Specifically, for \( x \geq 0 \}{2} \).... | 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 \( indeed \( \frac{2}{2*2 - 1} = \frac{2}{3} \).
What if \( a_1 \) is not equal? Let's say \( a_1 = \frac{1}{4} \), \( a_2 = \frac{3}{4} \):
\( M = \frac{\frac{1}{4}}{2 - \frac{1}{4}} + \frac{\frac{3}{4}}{1 + \frac{1}{4}} = \frac{\frac{1}{4}}{\fr... | 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 - 1936) = 7 - 9 = -2, which is negative, but the left side is 6, so it's not equal. So x=44 isn't a solution.
Are there any other integers between -44 and 44 that I missed... | 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:
$$
\sqrt} - 21 x + 21} \geq 5 x^{2} - 26 x + 32
$$
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 domain of the variables invo... | 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 labeled60°, which is π/3 radians.
Wait, but in the problem statement, they say "the size of the angle formed by the skew lines". So, since it's a cube, perhaps the angle is standard, maybe 60°, 45°, 30°, 90°, etc.
So, 60° seems correct.
But to make sure... | 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 with the reflection of P over Q. So, reflection over Q would mean that if I have point P, reflecting it over Q give... | 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:
$$
x^{x}\left(31 - 2x - 2^{x}\right)
$$
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 it0.
Looking for a pair that adds up to 31. 11 and 20:... | 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. single numbers, all pair sums, and all triple sums are distinct. This is similar to a set where all subset sums of size 1, 2, and 3 are unique.
Such a set is called a sum-distinct set for subsets of size 1, 2, and 3. So, how large can such a set be?
... | 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^m = 1.
But wait, let me verify this with/5). Grouping the first two and the last two: [(3/2)(2/3)] * [(5/4)(4/5)] = 1 * 1 = 1. Correct.
Another test: m = 3: Terms are (1 + 1/2), (1 - 1/3), (1 + 1/4), (1 - 1/5), (1 + 1/6), (1 - 1/7). Group them into thr... | 1 | 1 |
Solve the following equation:
$$
2^{x+3}+4^{x+1}=320 .
$$ | Okay, so I need to solve this equation: 2^{x+3 + 4y^2 = 320 => 4y^2 + 8y - 320 = 0 => y^2 + 2y - 80 = 0.
Factoring gives (y + 10)(y - 8) = 0 => y = -10 or y = 8. Since y is positive, y = 8 => 2^x = 8 => x = 3.
Yep, that all checks out. So x = 3 is the solution.
I don't think I missed anything here. The substitution ... | 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,((2 + (-k))^n = (2 - k)^n\).
So, the equation becomes:
\(-k^n + (2 - k)^n + (2 - k)^n = 0\).
Simplify:
\(2(2 - k)^n - k^n = 0\).
... | 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. land.
And we know that this remaining 13/60 of (60/13). Again, 26 divided by 10. So, x is 120 acres.
Wait, but does this count as using algebra? I think it does because I set up an equation and solved for x, even though I used fractions. The user said... | 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 \( lo... | 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 \( bp^2 + 48n^2 = 2401
\]
Hmm, this looks like a Diophantine equation. So, we have:
\[
p^2 + 48n^2 = 2401
\]
We need to find integer solutions \( p \), \( n \) such that this holds.
Let me think about possibl... | 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 \(2^{63}} - 2^{22} \cdot 4^{44}^{3} \cdot 5^{1} - 1 = 40 - 1 = 39\).
4. So, first two terms became \(2^{110} \cdot 5^{105} \cdot 39 = 39 \cdot 10^{105}\) because \(2^{105} \cdot 5^{105} = 10^{105}\) and \(2^{105 + 5}... | 1 | 959 |
1. Among the triangles with side lengths being consecutive natural numbers and a perimeter not exceeding 100, the number of acute triangles is $\qquad$ . | Okay, so I need to find the number of acute triangles where the side lengths are consecutive natural numbers and the perimeter doesn't exceed 100. Hmm1: 31,32,33.
31² +32²=961+1024=1985.
33²=1089, so 1985>1089, also acute.
So, n=32 is acute.
Wait, what about n=1: sides 1,2,3.
But 1+2=3, which is not greater than 3... | 1 | 29 |
Exercise 11. A palette of 6 different colors is given. In how many ways can a cube be painted, using all 6 colors, and exactly one color per face? Two ways of coloring are considered identical if one can be obtained from the other by any rotation in space. | Alright, so I have this problem here where I need to figure out how many rotations (90° and 270°), so 3 × 2 = 6
- 180° face rotations: Each of the 3 axes contributes one rotation, so 3
- 180° edge rotations: 6 axes, each contributing one rotation, so 6
- 120° and 240° vertex rotations: For each of the 4 axes, two rotat... | 1 | 30 |
3. $a_{1}, a_{2}, a_{3}, \cdots, a_{2020}$ are distinct non-zero real numbers, so $\left|\frac{a_{1}}{\left|a_{1}\right|}+\frac{\left|a_{2}\right|}{a_{2}}+\cdots+\frac{a_{2019}}{\left|a_{2019}\right|}+\frac{\left|a_{2020}\right|}{a_{2020}}\right|$ has the minimum value of $\qquad$ | Okay, so I have this problem where there are 2020 distinct2} \), and so on. Wait, but these are both equal to \( \text{sign}(a_i) \), as I realized before.
So, regardless of whether \( i \) is odd or even, each term is \( \text{sign}(a_i) \). Therefore, the entire sum is \( \sum_{i=1}^{2020} \text{sign}(a_i) \).
Thus... | 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 again1.5 > 0
So, f'(1/4) > 0
At t = 3/4:
f'(3/4) numerator is 2 - 8*(9/16) = 2 - 4.5 = -2.5 < 0
So, f'(t) changes from positive to negative as t increases through 1/2, so t=1/2 is inde... | 1 | \frac{\sqrt{3}}{3} |
Let $S$ be a set of $2020$ distinct points in the plane. Let
\[M=\{P:P\text{ is the midpoint of }XY\text{ for some distinct points }X,Y\text{ in }S\}.\]
Find the least possible value of the number of points in $M$. | Okay, so I have this problem where I have a set S of 2020 distinct points in the plane. Then M is defined as the set of all midpoints P where P is the midpoint of XY for some distinct points X and Y in S. I need to find the least possible number of points in M, meaning I want to arrange the 2020 points in such a way th... | 1 | 4037 |
5. Define the length of the interval $\left[x_{1}, x_{2}\right]$ as $x_{2}-x_{1}$. If the domain of the function $y=\left|\log _{2} x\right|$ is $[a, b]$, and the range is $[0,2]$, then the difference between the maximum and minimum values of the length of the interval $[a, b]$ is $\qquad$ . | Alright, so I have this problem here about the function y equals the absolute value x=1 onwards. So, if I take an interval that includes x=1, the function will have a minimum at x=1.
To get the range [0,2], the interval must include both sides where the function reaches 2. But if I take an interval that just includes ... | 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... | (0,5)^2 * (0,5)^2 = 0,0625.
Na konačnoj fazi, množimo broj načina sa verovatnoćom za svaku kombinaciju, što je 6 * 0,0625 = 0,375. Ovaj rezultat može biti prepišan u obliku 3/8.
Dakle, verovatnoća je 3/8.
</think>
Koristimo se binomnom distribucijom za izračun verovatnoće dogodka "dva pisma i dva glave" prijednom b... | 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. So maybe the GCD of consecutive terms is always 1. That would mean that GCD(a₂₀₁₂, a₂₀₁₁) = 1.
But wait, let me check another term. Maybe a₆.
a₆ = a₅ + 3a₄ = 116 + 141 = 257.
GCD(116, 257).
257 ÷ 116 = 2 with a remainder of 257 - 232 = 25.
... | 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 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 remember what a geometric pr... | 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 \( n \) such4 and 2015. Also, this polynomial must have a real root, meaning there exists some real number \( \xi \) where \( Py^{2n} - a_{2n - 1}y^{2n - 1} + a_{2n - 2}y^{2n - 2} - \dots - a_1 y + a_0 = 0 \).
So, to have this equal to zero, the positive terms must... | 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 10 (for C).
Calculating that: 9 * 10 = 90, and 90 * 10 = 900. So, there are 900 different 5-digit palindrome numbers.
Wait, let me double-check that. If A is from 1-9, that's 9 options. B and C are from 0-9, so each has 10 options. Since the choi... | 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 +,392. I need to find the number of all positive integer pairs3, and for m=2 is 92. So, x +4y=23 and 92.
But, in the problem statement, N=23x +92y is a perfect square, but in our substitution, we considered k=23m, so N=23(x +4y)=k², so k=23m, which makes N=23² m².
Wait, b... | 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 3} is 2, and the median of {1, 2, 3, 4} is (2 + 3)/2 = 2.5.
Now, the problem is a... | 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) \]
Now, solve for \( x \):
\[
1000 - x = 2560 - 4x \\
3x = 1560 \\
x = 520
\]
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:** 640
- **Guagua's point... | 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 [ (sin2907° + sin2908° + ... + sin6507°)^(cos2880° + cos2881° + ... + cos6480°) ]
Hmm, okay. Let me try to ... | 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 insin \theta \). Setting this equal to 1, we get the equation:
\[ 16x^5 - 20x^3 + 5x = 1 \]
where \( x = \sin(18^\circ) \).
This simplifies to:
\[ 16x^5 - 20x^3 + 5x - 1 = 0 \]
Factoring this equation, we can find that \( x = \frac{\sqrt{5} - 1}{4} \) is ... | 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? | a to the number from class 1b: 3 + 3.
Therefore, a total of 6 Pioneers will be awarded.
</think>
**Solution:**
1. **Number of Pioneers from Class 1a:**
- There are **3 Pioneers** from class 1a who receive certificates.
2. **Number of Pioneers from Class 1b:**
- It is stated that an **equal number** of Pioneer... | 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_2^3 + \cdots + x_n^3 = 2002^{2002} \) has integer solutions. Hmm, interesting. Let me think through this step by step.
First, I remember.
But since we are trying to minimize \( n \), using four +1s gives us the minimal \( n = 4 \).
There... | 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 J, Y, B, D ≥ 1. That is equivalent to the number of solutions to j' + y' + b' + d' = 4, where j' = J -1, y' = Y -1, etc. Each j', y', b', d' ≥ 0. So, that's C(4 + 4 -1, 4 -1) = C(7,3) = 35, which matches our previous result.
Similarly, the number o... | 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 withX + 37Y = 1184.
Which simplified to 10X + Y = 32, meaning XY = 32, so the original number is 432.
Wait, but let me see if there are other possible values for X and Y. Since X and Y are digits (0-9), 10X + Y can only be numbers from 0 to 99. ... | 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 me5.656.
Wait, but our u ranges from 16 to 36. 4√2 is approximately 5.6516 - 12 + 32 / 16 = 4 + 2 = 6.
Compute f(36):
... | 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 \( f(x \frac{\sqrt{3}}{2} \)
I know that \( \sin(\theta) = \frac{\sqrt{3}}{2} \) when \( \theta = \frac{\pi}{3} + 2k\pi \) or \( \theta = \frac{2\pi}{3} + 2k\pi \) for any integer \(q \frac{\pi}{2} \).
Looking at the sine values, \( \sin(\frac{\pi}{3}) = \fra... | 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:
$$
\int_{0} -3$ to $u = -0.5 \sqrt{2}$ is:
$$
F(u) = -\frac{1}{2 u^2}
$$
So,
$$
F(-0.5 \sqrt{2}) - F(-3) = \left( -\frac{1}{2 (-0.5 \sqrt{2})^2} \right) - \left( -\frac{1}{2 (-3)^2} \right)
$$
Which is:
$$
\left( -\frac{1}{2 \cdot 0.5} \right) - \left( -\frac{1}{2 \cdot 9} \ri... | 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 expansion10^12 mod 17 = 3*10 =14 mod 17 = 11*10 = 110 mod 17. 17*6=102, so 110-102=8.
10^15 mod 17 = 8*10 = 80 mod 17. 17*4=68, 80-68=12.
10^16 mod 17 = 12*10 = 120 mod 17. 17*7=119, so 120-119=1.
So 10^16 ≡ 1 mod 17. So the order of 10 modulo 17 is 16,... | 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 inscribed2 = 4 * 6 * r^2 = 24 * r^2. Therefore, SA = sqrt(3) * 24 * r^2 = 24 * sqrt(3) * r^2.
Wait, but the question says to find the minimum total surface area. Hmm, is there a way to minimize this? Or is this the expression we need? Since the tetrahedron is reg... | 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 defines012 * 1005 - 1006] / 1006
Let me compute that.
First, factor out 1006 from the numerator:
But 2012 * 1005 can be written as 2 * 1006 * 1005 / 2?
Wait, maybe let's compute ... | 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 (2y) equals 1, then I need to find the minimum value of is less than 2. But earlier, I thought the minimum was at x=2, y=0, which gave |x| - |y| = 2.
So, that suggests that the minimum is actually sqrt(3), which is lower than 2.
But let me make sure that this is ind... | 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 integer5 * 3^2 * 7.
So, 2016= 32*9*7= 32*63.
So, 2016=32*63.
So, k² -3^n=32*63=32*63.
Hmm, perhaps I can write k²=3^n +32*63.
Wait, but that's similar to the earlier step.
Alternatively, perhaps I can write ... | 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 make sure, maybe I can plug in a value for x and see if it makes sense. Let's take x = 0. Then, y = arcsin(0) = 0. The derivative at x=0 would be 0 / sqrt(1 - 0) = 0, which seems correct because the function is flat at x=0.
What about x = 1/2? Then, x³ = 1... | 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_{ctan 2) \) and \( \sec(2 \arctan 2) \).
Let \( \theta = \arctan 2 \), so \( \tan \theta = 2 \). Then \( 2 \theta = 2 \arctan 2 \).
Using the double-angle formula for tan:
\( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} = \frac{4}{1 - 4} = \f... | 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^8 = 6x^8
x^{10}: 2x^{10} + 1x^{10} = 3x^{10}
x^{12}: 1x^{12}
So, overall:
1 + 3x^2 + 6x^4 + 7x^6 + 6x^8 + 3x^{10} + x^{12}
Therefore, (1 + x^2 + x^4)^3 = 1 + 3x^2 + 6x^4 + 7x^6 + 6x^8 + 3x^{10} + x^{12}
Therefore, [G(x)]^3 = x^{-6}*(1 + 3x^... | 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.5 - 5, (5√3)/2 - 0) (-7.5, (15√3)/2) = (0 + 7.5, 5√3 - (15√3)/2)
Let me compute the components:
x-component: 0 + 7.5 = 7.5
y-component: 5√3 - (15√3)/2. Let's convert 5√3 to halves: 10√3/2 - 15√3/2 = (-5√3)/2
So vector FB - 3 vector AB is (7.5, (... | 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 (3ᵏ) / x, because 4ᵏ = (3ᵏ) / (3/4)ᵏ.
Thus, 12ᵏ = 3ᵏ * (3ᵏ / x) = (3²ᵏ)/x = (9ᵏ)/x.
Similarly, 9ᵏ + 12ᵏ = 9ᵏ + 9ᵏ / x = 9ᵏ (1 + 1/x).
But 9ᵏ = (3²)ᵏ = 3²ᵏ, and x = (3/4)ᵏ.
So, 1 + 1/x = 1 + (4/3)ᵏ = 1 + (4/3)... | 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 we10 + B
So adding them:
100 +10A + B +100A +10 + B=100 +10 +10A +100A + B + B=110 +110A +2B=728
So 110A +2B=618
Divide by 2:55A + B=309
But if N=57, which is 10A + B=57, so A=5, B=7.
55A ... | 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:
² + 1)] + [-1 / sqrt(x² + 1) + sqrt(x² + 1)/x²] = [1 - 1]/sqrt(x² + 1) + sqrt(x² + 1)/x² = 0 + sqrt(x² + 1)/x²
Therefore, dy/dx = sqrt(x² + 1) / x²
Hmm, let me see if I can write this in a different form if possible. Maybe rationalize or simplify fur... | 1 | \frac{\sqrt{x^{2}+1}}{x^{2}} |
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