question stringlengths 15 6.18k | target stringlengths 359 55.9k | 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 a six-digit number where x and y are unknown digits.
To determine if the number is divisible by 8, I need to look at the last three digits, which are 4xy. For a number to be divisible by 8, the number formed by its last three digits must be divisible by 8. So, 400 + 10x ... | 1 | 8 |
8. Find $\sqrt{\frac{x}{63}-32} \times \sqrt{\frac{y}{63}-32}$, given that $\frac{1}{x}+\frac{1}{y}=\frac{1}{2016}$.
ANSWER: 32. | Okay, so I need to find the value of √(x/63 - 32) multiplied by √(y/63 - 32), given that 1/x + 1/y = 1/2016. Hmm, let me see. I remember that when dealing with square roots multiplied together, it might be helpful to combine them under a single square root or perhaps square both sides of the equation to eliminate the r... | 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 16 is added to \( x \), the expression becomes \( x + 16 \).
Next, multiplying this sum by 7 gives \( 7(x + 16) \).
Subtracting 8 from this product results in \( 7(x + 16) - 8 \).
Finally, dividing by 9 leads to the expression \( \frac{7(x + 16) - 8}{9} \).... | 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 first try to visualize it. Since it's equilateral, all sides are equal, and all angles are 60 degrees. The midpoint of BC is M, so that splits BC into two equal parts. The problem mentions the circumcircle of triangle MAB, and its area is given as 3... | 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 \( \left(x + \frac{1}{x}\right)^2 = 25 \). I remember that squaring a binomial like \( (... | 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 back to the front in cycles of 1 to 4. I need to figure out how many people will report the number 1 in both rounds of counting. Hmm, okay, ... | 1 | 166 |
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，题目里说以3为分母的最简分数，也就是说分数的形式是a/3，其中a是一个整数。最简分数的话，也就是说a和3互质，即a不能被3整除，对吧？比如，4/3，5/3这样的分数都是最简分数，而6/3其实可以约分成2，这样就不是最简分数了。
接下来，题目要求这些分数的值要大于10且小于100。也就是说，10 < a/3 < 100。那我可以先把这个不等式转化一下，找到a的范围。
把不等式两边都乘以3，得到：30 < a < 300。所以，a必须是31到299之间的整数，对吧？但是同时，a还不能被3整除，因... | 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 base is an equilateral triangle, and the lateral edges are all equal in length. The prism is inscribed in a sphere, so all its vertices lie on the surface of t... | 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 by 2, 0, 1, 6, then an... | 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 (a + b)^k, you can ... | 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 + 2, I can express a in terms of c as a = c + 3.
Next, I'll represent the numbers abc and def numerically. The number abc can be written as 100a + 10b + c, and def as 100d + 10e + f.
The equat... | 1 | 964 |
7. In trapezoid $A B C D, A D$ is parallel to $B C$. If $A D=52, B C=65, A B=20$, and $C D=11$, find the area of the trapezoid. | Alright, so I have this trapezoid problem. Let me first visualize it. The trapezoid is ABCD, with AD parallel to BC. They gave me the lengths: AD is 52 units, BC is 65 units, AB is 20 units, and CD is 11 units. I need to find the area of this trapezoid.
Trapezoid area formula is pretty straightforward: (sum of the two... | 1 | 594 |
Let $\mathcal{P}$ be a parallelepiped with side lengths $x$, $y$, and $z$. Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$, $17$, $21$, and $23$. Compute $x^2+y^2+z^2$. | Okay, so I have this problem about a parallelepiped with side lengths x, y, and z. The four space diagonals are given as 15, 17, 21, and 23. I need to find x² + y² + z². Hmm, okay.
First, I remember that in a parallelepiped, the space diagonals can be calculated using the formula similar to the space diagonal of a rec... | 1 | 371 |
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