question stringlengths 15 6.18k | target stringlengths 264 39.8k | label int64 1 1 | answer stringlengths 1 136 |
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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, which are 4xy. For a number to be divisible by 8 So, 400 + 10x + y must be divisible 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 divisi... | 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/6/2016. Hmm, let me see. I remember that when single square root or perhaps square both sides of the equation to eliminate the radicals. But first, let me parse the given equation.
The equation is 1/x 2016². That seems useful.
So, (x -016)². Okay, so this gives me a relationsh... | 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 means:
\[
\frac{7(x + 16) - 8}{9} = 22 \frac{4}{9}
\]
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 simpli... | 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're given a triangular array \( (a_{n, k}) \) where 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 down. First, since ABC is equilateral, AB = BC = CA. L... | 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 25 \). I remember that squaring a binomial like \( (ab + b^2 \). So maybe I can expand t... | 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, so maybe I can use those.
... | 1 | -0.2 |
10. (3 points) 1994 students line up in a row, and they count off from the front to the back in a cycle of 1 to 3; then they count off from the back to the front in a cycle of 1 to 4. The number of people who report 1 in both rounds of counting is $\qquad$ people. | Alright, so I have this problem where 1994 students are lined up in a row. They first count off from the front to the back in cycles of 1 to 3. Then, they count off again from the figure out how many people will report the number 1 in both rounds of counting. Hmm, okay, let's break this down.
First, let me understand ... | 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9 -31 +1=269个数。
接下来，计算00少1，所以最大的被3整除的数是297=3×99。
那么，从33到297，这些数都是3的倍数，形成一个等差数列，公差是3，首项是33，末项是297。
计算这个等差数列的项数：项数=（ +1=88+1=89个数。
所以，在31到299之间，被3整除的数有89个，那么不被3整除的数就有269 -89=180个数。
现在，我需要计算这些a... | 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 on the surface of the sphere.
The problem mentions segment CD is the diameter of the midpoint of edge AA₁. We are given that CK = 2√6 and DK = 4. We need to find ... | 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 down step by step.
First, let me understand the structure of the number. It's a ten-digit number where the first digit is 'a', then followed. Got it.
Now, the num... | 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 states that each te... | 1 | \frac{1}{2}\left(3^{2n+1}+1\right) |
2. Positive integers $x, y$, for which $\gcd(x, y)=3$, are the coordinates of a vertex of a square with its center at the origin and an area of $20 \cdot \text{lcm}(x, y)$. Find the perimeter of the square. | Okay, so I have this problem here: there are positive integers x and y, with gcd(x, y) equal to 3. They are the coordinates of a vertex of a square centered at the origin, and the area of the square is 20 times the lcm of x and y. I need to find the perimeter of the square.
Hmm. Let me try to break this down step by s... | 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 abc. Given that a = b + 1 and b = c + 3.
Next, I'll represent the numbers abc and def numerically. The + f.
The equation provided is abc × 3 + 4 = defe + f
I also know that b = c + 2, so0e + f
333c + 964 = 100d + 10e + f
Now, ... | 1 | 964 |
7. In trapezoid $A B C D, A D$ is parallel to $B C$. If $A D=52, B C=65, A B=20$, and $C D=11$, find the area of the trapezoid. | Alright, so I have this trapezoid problem. Let me first visualize it. The trapezoid is ABCD, with AD parallel to BC. They gave me the lengths: AD is 52 units, BC is 65 units, AB is 20 units, and CD is 11 units. I need to find the area of this trapezoid.
Trapezoid area formula is pretty straightforward: (sum of the two... | 1 | 594 |
Let $\mathcal{P}$ be a parallelepiped with side lengths $x$, $y$, and $z$. Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$, $17$, $21$, and $23$. Compute $x^2+y^2+z^2$. | Okay, so I have this problem about a parallelepiped with side lengths x, y, and z. The four space diagonals are given as 1 need to find x² + y² + z². Hmm, okay.
First, I remember that in a parallelepiped, the space diagonals can be calculated using the formula similar to the space diagonal of a rectangular prism, but ... | 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 the areas 'p' and 'q' of triangles BKC and DCL.
First, let me vi... | 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 sides: AB is 1 unit, BC is 2 units, CD is √3 units. The angles at B and C are given: angle ABC. Hmm, okay.
I think drawing a diagram might help. Since it's step.
First, I'll start b... | 1 | \sqrt{7} |
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