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 $c$. | divisible by 9.
Finally, I'll verify that with c = 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 the value of \( c = x + y \) for the six-digit number \( 1234xy \) that is divisible by both 8 and 9, w... | 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, going back to the original expression we need to compute: √(x/63 - 32) × √(y/632016 + b, plug these into the expression:
First term inside the square root: (2016 + a)/63 - 32 simplifies to a/63. So, the first square root becomes √(a/63).
Similarly, the second term inside 3... | 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? | [
L = 60 - 2W
\]
Substituting 4W = 0
\]
Solving for \( W \):
\[
4W = 60 \\
W = 15 \text{ meters}
\]
Now, substituting \( W = 15 \) back into the equation for \( L \):
\[
L = 60 - 2 \times 15 = 30 \text{ meters}
\]
Therefore, to maximize the area with 60 meters of fencing, the length parallel to the wall should be 3... | 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? | 8}{9} = 22 \frac{4}{9}
\]
To solve for \( x \), I'll convert the mixed number to an104 = 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 Statement:**
When 16 is added to an imagined number, then multiplied... | 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 n = 3 \):
- \( a_{3, 1} = \frac{1}{3} \)
- \( a_{3, two elements: 1/2, 1/2
Row 3: three elements: 1/3, 1/6, 1.
So, it seems like for row \( n \), the entries are \( \frac{1}{n \cdot \binom{n-... | 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 So, n must end with a digit that when multiplied by 6 gives a 9 in the units place. Let's see: 6 * x �2≡2, x=8:48≡8, x=9:54≡4. So, no x where 6x ≡000...0, then the digit sum is just 1. But...0 divided by 16, which is 625 followed by zeros. So, n=625.... | 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 I need to find the perimeter of triangle ABC.
Hmm, let's break this will just be 3s once I find 's'.
Now, the midpointradius.
Wait, but triangle MAB: points A, B, M.AB is also a triangle with sides s, s/2, and (√3/2)s.
Alternatively, maybe coord... | 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 \.
- \( 3x^2 \cdot \frac{1}{x^3} + 3y = y^3 \). Then rearranged to \( x^3 + \frac{1}{x^3} =23 \), which matches what I found earlier.
To find \( x^3 + \frac{1}{x^3} \), another way is 110 \)
Hmm, same result. That seems solid.
Alternatively7 > 0 \).
Now, let's compute ... | 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 3a \) and the product is \( a^2 \). Therefore, the sum of the squares should be \( (3a)^2 - 04. So, \( a^2 = 0.04 \). Taking square roots, \( a = \sqrt{0.04}, so the equation always has real roots.
Therefore, both \( a = 0.2 \) and \( a = -0.2 \) are \p... | 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. | that the position is positive.
Alternatively, the positions that say 1 in the second round are equivalent to 1994 mod 4, but since, now, I need to find the number of positions that are equal to both for some integers m and k. So, essentially, find the number of integers x such that x ≡1 mod3 and x ≡1994 mod4.
Wait, ... | 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29 +1=88+1=89个数。
所以，在31到299之间，被3整除的数有899=180个数。
现在，我需要计算这些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5。
所以，从31到299的总和是44385。
接下来，计算被3整除的数的和=165×89。
计算165×89：
×9=1485，
加起来：13200+1485=14685。
所以，被3整除的数的和是14685。
现在，... | 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 the midpoint of edge AA₁. We are given that CK = 2√6 and DK = 4. We need to find the volume of the prism = Base Area × Height. Since it's a regular triangular prism, the base₁, CC₁), I can compute the volume.
Let me denote the side length of ... | 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 + 2 = a + 2
+ 0 = a + 2
+ 1 = a + 3
+ 6 = a2 mod 3, since 3*7=21.
So, a + b ≡ 2 mod 3.
Alright, that's our}
Now, we need to find all pairs (a,b) where a and b satisfy both conditions above.
Let me write the two congruence equations:
1. a + b ≡ 2 mod... | 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 0)/2 = (2n+1)/2, which is not an integer unless n is a half-integer, which it isn't because n Wait, actually, sqrt(-1) is not real, so maybe that approach isn't directly applicable.
Alternatively, perhaps I can think of genera... | 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 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 downx² + y²). Therefore, the area is 2*(x² + y²).
But according to the problem, the area is also 23b into the equation, we get:
(3a)²... | 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 number100d + 10e + f
Now, I need to find values of c that make abc a valid three-digit number. Since a (because a cannot exceed 9).
Testing possible values for c from 0 to 6:
- For c = 0:
a = 3, b = 2
abc = 320
def = 320 × 3 + 4... | 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.
+ |**c**|² + 2(-**a**·** S.
If I add all four equations together, I get:
4S +. So the total is 0.
Therefore, adding all four equations... | 1 | 371 |
Three, (50 points) A positive integer is called "simple" if it does not have any square factors greater than 1. Determine how many numbers in $1,2, \cdots, 2010$ are simple.
| Okay, so I have this problem here where I need to find out how many "simple" numbers there are between 1 and 2010. A "simple" number is defined as a positive integer that doesn't have any square factors greater than 1. Hmm, that means if a number is simple, it shouldn't be divisible by any perfect square other than 1. ... | 0 | 1221 |
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 suppose that the a... | 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 quadrilateral0°) = -0.5
sin(120°) is equal to sin(180° - 60°) = sin(60°) = √3/2 ≈ 0 from point B, moving in the direction of 120 degrees with length 2 units, we reach point C at (-1, √3). say vector CD is (x, y), then (-1)x + (√3)y = 0.
So, -x + √3 y = shape doe... | 1 | \sqrt{7} |
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