question stringlengths 15 6.18k | target stringlengths 225 23.8k | label int64 1 1 | answer stringlengths 1 136 |
|---|---|---|---|
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 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 48 is divisible by 8.
Thus, the value of c is 8.
</think>
To determine... | 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 simplifies to a/63. So, the first square root becomes √(a/63).
Similarly, the second term inside2, so this simplifies to 32 + b/63 - 32, which is b/63. Therefore, the second square root is √(b/63).
So, the entire expression becomes √(a/63) × √(b/63). Multiplying these tog... | 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 similar44 - B² => y = (544... | 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 116) - 8 = 202
\]
Expanding and simplifying the equation:
\[
7x + 112 - 8 = 202
\]
\[
7x + 104 = 202
\]
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 ... | 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,n(n-1)}, \frac{1}{n(n-1)}, \ldots, \frac{1}{n(n-1)}, \frac{1}{n} \).
Wait, but hold on, for row \( n \), the entries are symmetric, as we... | 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 the624? 16*624=10000 - 16*1=16, so 16*624=10000-16=9984. S(n)=6+2+4=12, S(16n)=9+9+8+4=30. Ratio=12/30=0.4. Worse.
What if n=6251? 16*6251=100016. S(n)=6+2+5+1=14, S(16n)=1+0+0+0+1+6=8. Ratio=14/8=1.75. Still worse than 13.
How about n=6252? 16*6252=10... | 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, AM = (√3/2)s.
Let me check if this triangle is a right-angled triangle. If it is, then area is 1/2 * base * height.
Let me see: s² = (s/2)² + ( (√3/2)s )² ?
Compute RHS: (s²/4) + (3s²/4) = s², which equals LHS. So, yes, triangle MAB is right-ang... | 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 \( x}\right)^2 = \frac{25 + 10\sqrt{21} + 21}{4} = \frac{46 + 10\sqrt{21}}{4} = \frac{23 + 5\sqrt{21}}{2} \)
Now, compute \( x^3 \):
\( x^3 = x \cdot x^2 = \frac{5 + \sqrt{21}}{2} \cdot \frac{23 + 5\\sqrt{21} \)
Compute each term:... | 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 mathematical terms, the smallest real number is the most negative one, so \( -0.2 \) is smaller than \( 0.2 \). So, the answer is \( a = -0.2 \).
But the problem is, sometimes in problems like this, they might expect the answer in fractional form instea... | 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, now, I need to find the number of positions that are equal to both 1 + 3m and 1994 - 4k for some integers m and k. So, essentially, find the number of integers x such that x ≡1 mod3 and x ≡1994 mod4.
Wait, but 1994 is congruent to 2 mod4, right? So, x ≡1 m... | 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=1485，
加起来：33000+9900=42900，再加上1485，得42900+1485=44385。
所以，从31到299的总和是44385。
接下来，计算被3整除的数的和，也就是从33到297的和，同样用等差数列求和公式：
首项是33，末项是297，项数是89，和=(33+297)×89 /2=330×89 /2=165×89。
计算165×89：
165×80=13200，
165×9=1485，
加起来：13200+1485=14685。
所以，被3整除的数的和是14685。
现在，不被3整除的数的和就是44385 ... | 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.
But we also have from CK: a² + h²/4 = 24, and from DK: sqrt(a²/3 + h²/4) = R
Wait, but we also have that the center is at (a/2, (√3/6)a, h/2). Since point D is on the sphere, its distance from the center should also be R.
Compute the dista... | 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 divisible by 33. Hmm, okay. Let's try to break this2 mod 3
Simplify:
2b + 1 ≡ 2 mod 3
Subtract 1 from both sides:
2b ≡ 1 mod 3
Now, we need to solve for b in this congruence.
We can multiply both sides by the modular inverse of 2 mod 3. The inverse of... | 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 + the coefficients of the terms where the exponent of x is an integer. Hmm,n+1} + 1]/2.
The sum when m is odd: [3^{2n+1} - 1]/2.
Wait, that's conflicting with the earlier result.
Wait, no, if I consider f(1) = sum_{m} C(2n+1, m) * 2^m = 3^{2n+1}
f(-2) = sum_{m} ... | 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 yy, x) or (y, -x), depending on the direction.
Wait, actually, if you have a square with a vertex at (x, y), the next vertex can be obtained by rotating (x, y) by 90 degrees around the origin. The rotation of (x, y) by 90 degrees is (-y, x). So, the ... | 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).
- k t1 t2.
Wait,... | 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 = 320 × 3 + 4 = 964
- For c = 1:
a = 4, b = 3
abc = 431
def = 4 Not a three-digit number.
- For c = 3:
a = 6, b = 5
abc = 653
def = 653 × 3 + 4 = 1959 + 4 = 1963
Not a three-digit number.
- For c = 4:
a = 7, b =... | 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 3: 441 - 371 = 70 = 2(-ab + ac - bc), so -ab + ac - bc = 35.
From equation 4: 529 - I can try to solve for ab, ac, bc.
Let me write these equations:
Let me denote:
Equation (1): ab + ac + bc = -73
Equation (2): ab - ac - bc = -41
Equation (3)... | 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 and height 2. So, area p = (1/2)*2*2=2.
Similarly, triangle DCL: Points D(0,2), C(2,2), L(0,4). It's a right triangle with base 2 and height 2. Area q = ( AB.
Wait, actually, if the line through C is drawn such that it intersects AB and AD beyond B and D, ... | 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, √3), which is in the second quadrant relative to the origin, moving to (1/2, (3√3)/2) would place D in the first quadrant. This might cause the quadrilateral ABCD to be convex.
Option 2: D is at (-5/2, √ in the third quadrant. This might make the q... | 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 number with \(k = 4\) trailing 9s but with a smaller digit before. For example, \(n = 49999\). Or, is there a smaller number with \(k = 4\) but with a smaller digit before? Wait, if the digit before is smaller, say 0, but then the number wo... | 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 (2/(x + 5)). Hmm, that looks a bit complicated,^2/6], the entire expression becomes:
[1 + x^2/6]^{(2/5)(1 - x/5 + x^2/25 - ...)}
Now, raising this to the power of (2/5)(1 - x/5 + ...), we can use the approximation (1 + a)^b ≈ 1 + a*b +, and b = (2... | 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 |
Three. (20 points) Given that the graph of the linear function $y=a x+b$ passes through the points $A(\sqrt{3}, \sqrt{3}+2)$, $B(-1, \sqrt{3})$, and $C(c$, $2-c)$. Find the value of $a^{2}+b^{2}+c^{2}-a b-b c-c a$.
| Alright, so I have this problem about a linear function y = ax + b2), B(-1, √3), and C(c, 2 - c). And I need to find the value of a² + b² + 2b² + 2c² - 2ab - 2bc - 2ca]
= (a² + b² + c² - ab - bc - ca)
So, yes, that formula is correct. Therefore, another way to compute the expression is to compute ½[(a - b)² + (b - c)... | 1 | 4+\sqrt{3} |
1. 12 large monkeys stand evenly on a large circle with a circumference of 480 meters. Between each pair of adjacent large monkeys, 3 small monkeys are added, requiring the distance between adjacent monkeys on the circle to be equal. This distance is $\qquad$ meters. | 36 = 48 monkeys altogether.
To find the new equal distance between each adjacent monkey:
Divide the total circumference by the number of gaps between monkeys, which is 48.
Therefore, the distance is 480 meters divided by 48 equals 10 meters.
</think>
**Solution:**
1. **Determine the Initial Distance Between Large M... | 1 | 10 |
Task B-3.4. A group of children found a wooden board in the shape of a quadrilateral and decided to use it for the game "pick-a-point". Since the target was of an unusual shape, they had to adapt the game rules. They asked for advice from Mark's older brother, a good mathematician. He measured, calculated, and wrote so... | Alright, so I've got this problem about a quadrilateral board where some kids are playing a game called "pick-a-point." They found a wooden board that's a quadrilateral, and they want to play a game where you get points based on where you pick a point on the board. The mathematician brother of√5)/(10π) - 1.
Alternativ... | 1 | \frac{7}{10\pi}(5\sqrt{3}-\sqrt{5})-1 |
10. Let $a$, $b$, $c$ represent three different non-zero digits, forming the largest and smallest three-digit numbers, respectively. Denote the difference between these two three-digit numbers as $m$, and the sum of the digits of $m$ exactly divides $m$. Then the sum of all different values of $m$ is $\qquad$ | Okay, so I need to solve this problem where I have three different non-zero 921, n = 129, p = 921 - 129 = 792. Sum of digits: 7 + 9 + 2 = 18, and 792 ÷ 18 = 44. So that works too.
So all four p's are valid and can be achieved with appropriate digits a, b, c.
Therefore, the valid p's are 198, 396, 594, and 792. Now, t... | 1 | 1980 |
## Problem Statement
Calculate the lengths of the arcs of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=2 \cos ^{3} t \\
y=2 \sin ^{3} t
\end{array}\right. \\
& 0 \leq t \leq \frac{\pi}{4}
\end{aligned}
$$ | Okay, so I have this problem here where I need to find the lengths of = 0.
Next, compute cos(0) which is 1. So, the second term is (-3/2)*1 = -3/2. But since we have [(-3/2) cos(...) - (-3/2) cos(...)], it becomes [0] - [(-3/2)(1)] = 0 - (-3/2) = 3/2.
Therefore, the arc length is 3/2 units. Let me just double-check m... | 1 | \frac{3}{2} |
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