question stringlengths 15 6.18k | target stringlengths 264 34.7k | 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 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/6 32), given that 1/x + 1/y = 1/2016. Hmm, let me see. I remember that when6². So, a and b are positive numbers (assuming x and y are positive, which they probably are since we're dealing with square roots).
Now, going back to the original expression we need to compute: √(x/63 ... | 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]
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 simplifying the equation:
\[
7x + 112 - 8 = 202
\]
\[
7x + ... | 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 * 3), 1/(4 * 3), 1/4ldots, n \).
Wait, let's test 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,0)
Distance is sqrt[ (0 - (-s/2))² + (0 - 0)² ] = sqr... | 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 = 23 \). That looks correct.
2. Then, I considered1}{x} = 5 \). Let me set \( t = 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 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, a^2 \). Therefore, discrimi... | 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 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 counting1, the second is 2, third is 3, fourth is 4, fifth is 1, etc. So, the position from the back is the number they say. So, the first ... | 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，题目所有满足31 ≤ a ≤ 299，且a不被3整除的数，然后把这些a代入a/3，再求这些分数的和。
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整除的所有数的和。
从31到299的总和可以用等差数列求和公式计算：和=（首项 +末项）×项数 /2。
首项是31，末项是299，项数是269，所以总和是(31+299)×269 /2 = ... | 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 sphere. So, points C and D are at opposite ends of the sphere, meaning the distance between them is twice the radius. Also, point K is the midpoint of edge AA₁... | 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 - b ≡ 1 mod 11
Alright, so now we have two conditions:
1. a + b ≡ 2 mod 3
2. a - b ≡ 1 mod 11
Now, let's recall that a and b are digits, meaning they can take i... | 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 over the coefficien... | 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 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 = 3
So, in this case, a =1, b=3
So, we have two possible pai... | 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 + 10e + f
303c + 30b + 904 = 100d + 10 to find values of c that make abc a valid three-digit number. Since a = c + 3, c must be less than or equal to 6 (because a cannot exceed 9).
Testing possible values for c from 0 to 6:
- Fo... | 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 x5, 17, 21, and 23. I need to find x² + y² + z². Hmm, okay.
2. D2² = S + 2(ab - ac - bc) = 17² = 289
3. D3² = S + 2(-ab + ac - bc) = 21² = 441
4. D4² = S + 2(-ab - ac + bc) = 23² = 9
Now, if I label these equations as (1), (2), (3), (4), respec... | 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 a c
Simplify each bracket:
First bracket: [c - (2q)/a - c] = - ... | 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 is 120 degrees, and angle BCD is 90 degrees. Hmm, okay.
I think drawing a diagram might help. Since it's say vector CD is (x, y), then (-1)x + (√3)y = 0.
So, -x + √3 y =�3, the magn... | 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 \( n \) such that both the digit sum of \( n \) and the digit sum of \( n + 1 \) are divisible by 5. Hmm, digit sums... okay, I remember that the digit sum is just the sum of all the digits in a number. For example, the digit sum of 1303 is... | 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 0 of (arcsin x / x) raised to the power of (2/(x + 5)). Hmm, that looks a bit complicated, but maybe I can break it down step by step.
First, I remember that when dealing with limits involving exponents, especially when the base approaches 1 and th... | 1 | 1 |
End of preview. Expand in Data Studio
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