question string | target string | label int64 | answer string |
|---|---|---|---|
G1.3 A six-digit number 1234xy is divisible by both 8 and 9 . Given that $x+y=c$, find the value of $c$. | 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/63 - 32) multiplied by √(y/63 - not sure if that's immediately helpful.
Alternatively, since the product (x -. Let me set a = x - 2016 and b = y - 2016. Then, ab = 2016². So, a and b are positive numbers (assuming x and y are positive, which they probably are since we're dealin... | 1 | 32 |
10. It is required to fence a rectangular area adjacent to a wall. The fence should have a length of \(60 \mathrm{~m}\). What should be the length and width of this area so that its area is the largest? | First, I need to visualize the problem. There's a rectangular area adjacent to a wall, so one side of the rectangle doesn't require fencing.
Let’s denote the length parallel to the wall as \( L \) and the width perpendicular[
L = 60 - 2W
\]
Substituting 4W = 0
\]
Solving for \( W \):
\[
4W = 60 \\
W = 15 \text{ mete... | 0 | 450 |
31. Given that $\alpha$ is an acute angle satisfying
$$
\sqrt{369-360 \cos \alpha}+\sqrt{544-480 \sin \alpha}-25=0
$$
find the value of $40 \tan \alpha$. | Alright, so I have this problem here where I need to find the value of 40 tan α, given that α is an acute angle and it satisfies the equation:
√(369 - 360 cos α) + √(544 - 480 sin α) - 25 = 0
Hmm, okay. Let me try to break this down step by step.
First, I notice that both square roots have similar structures. They a... | 1 | 30 |
## Task 7 - V00607 = V00503
To an imagined number, 16 is added, then multiplied by 7, followed by subtracting 8, and finally dividing by 9. The result is 22 remainder 4.
What is the imagined number? | 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]
Multiplying both sides by 9 gives:
\[
7(x + 16) - 8 = 202
\]
Expanding and simplifying the equation:
\[
7x + 112 - 8 = 202
\]
\[
7x + 104 = 202
\... | 1 | 14 |
IMO 1974 Problem A1 Three players play the following game. There are three cards each with a different positive integer. In each round the cards are randomly dealt to the players and each receives the number of counters on his card. After two or more rounds, one player has received 20, another 10 and the third 9 counte... | Alright, so I’ve got this problem from IMO 1974, Problem A1. It involves three players who each start with three cards, each having a different positive integer. The game they play involves dealing the cards randomly each round, and each player receives counters equal to the number on their card. After two or more roun... | 0 | 9 |
9.1. Solve the equation:
$$
\left(x^{2}-20\right)^{2}+\left(x^{2}-19\right)^{2}=2019
$$ | Okay, so I've got this equation to solve: \((x^2 - 20)^2 + (x^2 - 19)^2 = 2019\). Hmm, let me try to figure out how to approach this. It looks a bit complicated, but maybe if I simplify it somehow, it will become easier.
First off, both terms on the left side are squared, so it might help to expand them. But before I ... | 0 | \\sqrt{\frac{39+\sqrt{4037}}{2}} |
19. (CAN 5) The triangular array $\left(a_{n, k}\right)$ of numbers is given by $a_{n, 1}=1 / n$, for $n=1,2, \ldots, a_{n, k+1}=a_{n-1, k}-a_{n, k}$, for $1 \leq k \leq n-1$. Find the harmonic mean of the 1985th row. | Okay, so I have this problem here about a triangular array of numbers, and I need to find the harmonic mean of the 1985th row. Hmm, let's try to break this down step by step.
First,'re given a triangular array \( (a_{n, k}) \) where - a_{n, k} \).
And the question is asking for the harmonic mean of the 1985th row. Ok... | 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) forn.
First, let me understand what exactly is being asked. I need to need a different approach. Let me consider n as a number consisting of all So, n must end with a digit that when multiplied b... | 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 this3/2)s.
I need to find the circumradius of triangle MAB... | 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 \^3 + 3x^2 \cdot \frac{1}{ximplifying each term:
- \( x^3 \) stays as it is.
- \( 3x^2 \cdot \frac{1}{x^3} + 3y = y^3 \). Then rearranged to \( x^3 + \frac{1}{x^3} =). That seems consistent.
Is there another way to verify this result? Maybe by finding the value of \( 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 a^2 = 0 \) is 0.28. Hmm, the sum of the squares of the roots is \( (sum)^2 - 2(product) \). From earlier, the sum of the roots is \( 3a \) and the product is \( a^2 \). Therefore, the sum ... | 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 |
$A_1, A_2, ..., A_n$ are the subsets of $|S|=2019$ such that union of any three of them gives $S$ but if we combine two of subsets it doesn't give us $S$. Find the maximum value of $n$. | Okay, so I have this problem where I need to find the maximum number of subsets, n, of a set S with |S| = 2019. The conditions are that the union of any three subsets gives me the entire set S, but if I take just two subsets and union them, they don't cover all of S. Hmm, interesting.
Let me try to wrap my head around... | 0 | 64 |
1. Find the sum of all fractions in lowest terms with value greater than 10 but smaller than 100 and with denominator equal to 3 .
求所有大於 10 且小於 100 , 且以 3 為分母的最簡分數之和。 | 嗯，好的，我现在要解决的问题是：求所有大于10且小于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，题目计算这些数的和。
首先，我想找到所有满足条件的a的这些数中被3整除的数有多少个，然后用总数减去被3整除的数量，就能得到不被3整除的数量。
首先，找到31到299之间被3整除的最小。300是3的倍数，但299比300少1，所以最大的被3整除的数是297=3×99。
那么，从33到2977。
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的和，然后... | 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 length. The prism is inscribed in a sphere, so all its vertices lie on the surface of the sphere.
The problem mentions segment CD is the diameter of the sphere. S... | 1 | 36 |
Example 3 As shown in Figure 4, in a $4 \times 4$ grid square,
construct a grid point $\triangle A B C$
$(A B=\sqrt{5}, B C=$
$\sqrt{13}, C A=\sqrt{10}$ ). How many
grid point triangles congruent to
$\triangle A B C$ (including
$\triangle A B C)$ can be
constructed in Figure 4? | Okay, so I've got this problem here about constructing grid point triangles that are congruent to triangle ABC. Let me try to understand what it's asking.
First, the triangle ABC has sides AB = sqrt(5), BC = sqrt(13), and CA = sqrt(10). The grid is 4x4, so I guess that means it's a grid of 4x4 squares, making 5x5 poi... | 0 | 32 |
20. (6 points) If the ten-digit number $\mathrm{a} 2016 \mathrm{~b} 2017$ is divisible by 33, then, how many such ten-digit numbers are there? | Okay, so I have this problem here: I need to find how many ten-digit numbers of the form a2016b2017 are 3.
So, let's compute the sum:
a + 2 + + 1 + 7
Let me compute this step by step:
a + 2 = a + 2
+ 0 = a + 2
+ 1 = a + 3
+ 6 = a 3. Let's note that down:
Condition 1: a + b + 19 ≡ 0 mod 3
Simplify this:
a + b ... | 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 0)/2 = (2n+1)/2, which is not an integer unless n is a half-integer, which it isn't becau... | 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 area of t... | 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 + 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, I need to find values of c that make abc a valid three-digit number. Since a = c + 3, c must be less tha... | 1 | 964 |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 23