question stringlengths 15 6.18k | target stringlengths 264 28.9k | 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 x + y in terms of c, where c = x + y. Since8, as 10 + 8 = 18, which is divisible by 9.
Finally, I'll verify that with c = 8, the last three digits 4xy form a number divisible by 8. The possible values for xy are 16, 24, 32, 40, 48, 56, 64, 72, and 80. Among these, only 4... | 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 not sure if that's immediately helpful.
Alternatively, since the product (x -. Let me set a = x - 2016 and b =6². So, a and b are positive numbers (assuming x and y are positive, which they probably are since we're dealing with squa... | 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]
Multiplying both sides by 9 gives:
\[
7(x + 16) - 8 = 202
\]
Expanding and simplifying the equation:
\[
7x + 112 - 8 = 202
\]
\[
7x + 104 = 202
\]
Subtracting 104 from both sides:
\[
7x = 98
\]
Fina... | 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, harmonic mean of the 1985th row. Okay, so before anything else, I need to recall what a harmonic mean is. The harmonic mean \( H \) of a ... | 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 of triangle MAB, and its area is given as 36π. I need to find the perimeter of triangle ABC.
Hmm, let's break this Let's denote the length of each side as 's'. So, the perimeter will just be 3s once I find 's'.
Now, the midpoint6, so r = 6. Theref... | 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 \( x > 0 \2 = 25 \), I need to find the value of \( x.
- \( 3x^2 \cdot \frac{1}{x to \( x^3 + \frac{1}{x^3} =). That seems consistent.
Is there another way to verify this result? Maybe by finding the value of \( x \) explicitly and then computing \( x^33 + \frac{1}{x^3} \), anothe... | 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 should be \( (3a)^2 - ). Taking square roots, \( a = \sqrt{0.04}0.2 \) and \( a = -0.2 \) are -0.2 \).
Wait, but hold on. Let me double-che... | 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 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 to find the number of students who said "1" in both counts. So second round are equivalent to 1994 mod 4, but since k=... | 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且和。听起来有点复杂，但我慢慢来，一步一步分析。
首先，题目00少1，所以最大的被3整除的数是291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×项数 /2。
首项是31，末项是299，项数是269，所以总和是(3。
计算一下165×269：
首先，165=1485，
加起来：33000+95。
所以，从31到299的总和是44385。
接... | 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.
First, let me recall the formula for the volume of a prism: Volume = Base Area × Height. Since it's a regular triangular prism, the base... | 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 this2
+ 1 = a + 3
+ 6 = a a + b ≡ 2 mod 3.
Alright, that's our, let's recall that a and b are digits, meaning they can take integer values from 0 to 9, inclusive. How... | 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 an integer unless n... | 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 + y²).
B... | 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 to find values of c that make abc a valid three-digit number. Since a to 6:
- For c = 0:
a = 3, b = 2
abc = 320
def = 320 × 3 + 4 = 964
- For c = 1:
a = 4, b = 3
abc = 431
def = 431 × 3 + 4 = 1297
This is not a thr... | 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 need to find x² + y² + z². Hmm, okay.
+ |**c**|² + 2(-**a**·**² plus twice some combination of the dot products. Let me denote S = x S.
If I add all four equations together, I get:
4S +. So the total is 0.
Therefore, adding all four equations ... | 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 points K and L, respectively. I need to express the area of the parallelogram ABCD, which is denoted as 's', in terms of these areas. Maybe using similar triangles or some ratio of sides.
Since ABCD. The parametric equations are x = t, y = 0, where (2p)/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. Hmm, okay.
I think drawing a diagram might help. Since it's better to use vectors or coordinate geometry to find the coordinates of point C. Maybe sin(60°) = √3/2 ≈ 0 2 units, we re... | 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 |
Three, (This question 20 points) Given a triangle with one side length of 2, the median on this side is 1, and the sum of the other two sides is $1+\sqrt{3}$. Find the area of the triangle.
| Okay, so I have this triangle problem here, and I need to find its area. Let me try to figure this out step by step. The problem says that one side of the triangle has a length of 2, and the median on this side is 1. Also, the sum of the other two sides is \(1 + \sqrt{3}\). Hmm, okay, let me break this down.
First, le... | 1 | \frac{\sqrt{3}}{2} |
1. Sei $A B C$ ein spitzwinkliges Dreieck mit $A B \neq B C$ und Umkreis $k$. Seien $P$ und $Q$ die Schnittpunkte von $k$ mit der Winkelhalbierenden beziehungsweise der Aussenwinkelhalbierenden von $\angle C B A$. Sei $D$ der Schnittpunkt von $A C$ und $P Q$. Bestimme das Verhältnis $A D: D C$.
## 1st solution: | Okay, I have this geometry problem here about a triangle, its circumcircle, and some angle bisectors. Let me try to parse what the problem is saying and figure out a plan.
First, the setup: We have a triangle ABC, which is acute-angled, meaning all its angles are less than 90 degrees. Also, it's specified that AB is n... | 1 | 1 |
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