question string | target string | label int64 | answer string |
|---|---|---|---|
2. In a class with 15 boys and 6 girls, we select one pair. What is the probability that the pair will include at least one girl? | First, I need to determine the total number of possible pairs that can be formed'll calculate the number of pairs that consist of only boys. There are 15 boys, so the number of all-boy pairs is C(15,) = 105.
To find the number of pairs that include at least one girl, I'll subtract the number of all-boy pairs from the ... | 1 | 0.5 |
25th Brazil 2003 Problem A1 Find the smallest positive prime that divides n 2 + 5n + 23 for some integer n. | Okay, so I have this problem where I need to find the smallest positive prime that divides \( n^2 + 5n + 23 \) for some integer \( n \). Hmm, let me try to think through this step by step.
First off, I know that primes are numbers greater than 1 that have no positive divisors other than 1 and themselves. The smallest ... | 1 | 17 |
Let $k$ be a positive real number. In the $X-Y$ coordinate plane, let $S$ be the set of all points of the form $(x,x^2+k)$ where $x\in\mathbb{R}$. Let $C$ be the set of all circles whose center lies in $S$, and which are tangent to $X$-axis. Find the minimum value of $k$ such that any two circles in $C$ have at least ... | Okay, so I need to find the minimum value of \( k \) such that any two circles in set \( C \) have at least one point of intersection. Hmm, let me start by understanding the problem.
First, \( S \) is the set of points \( (x, x^2 + k) \) where \( x \) is a real number. So, \( S \) is a parabola opening upwards with ve... | 1 | \frac{1}{4} |
7. Given $\alpha, \beta \geqslant 0, \alpha+\beta \leqslant 2 \pi$, then the minimum value of $\sin \alpha+2 \cos \beta$ is $\qquad$ | Okay, so I have this problem here: Given that α and β are both greater than or equal to zero, and their sum is less than or equal to 2π, I need to find the minimum value of sin α + 2 cos β. Hmm, okay, let's try to break this down step by step.
First, I should note the constraints: both α and β are non-negative, and th... | 1 | -\sqrt{5} |
8.2. On weekdays (from Monday to Friday), Petya worked out in the gym five times. It is known that in total he spent 135 minutes in the gym, and the time spent in the gym on any two different days differed by at least 7 minutes. What is the maximum duration that the shortest workout could have been? | Okay, so I have this problem where Petya went to the gym five at least 7 minutes. I need to find the maximum duration that the shortest workout could have been.
Hmm, let's see. So, I think this day. That usually means making the workout times as equal as possible, but with.
Let me denote the shortest workout time as ... | 1 | 13 |
5. Let $x y z+y+z=12$. Then the maximum value of $\log _{4} x+\log _{2} y+\log _{2} z$ is $\qquad$ . | Okay, so I have this problem here: Let \( xy z + y + z = 12 \). Then the maximum value of \( \log _ step by step.
First, I should probably simplify the expression we need to maximize1/2} y z \) subject to the constraint \( xyz + y + z = 12 \).
Okay, so I have an optimization problem substitution to reduce the number ... | 1 | 3 |
Example 3 Try to find all positive integers $k$, such that for any positive integers $a, b, c$ satisfying the inequality
$$
k(a b+b c+c a)>5\left(a^{2}+b^{2}+c^{2}\right)
$$
there must exist a triangle with side lengths $a$, $b$, and $c$.
(2002, Girls' Mathematical Olympiad) | Okay, so I've got this problem here: I need to find all positive integers k such that for any positive integers a, b, c, if the inequality k(ab + bc + ca) > 5(a² + b² + c²) holds, then a, b, c must form a triangle. Hmm, that sounds a bit tricky, but let me try to break it down step by step.
First off, I know that for ... | 1 | 6 |
1. [3 points] Find the number of eight-digit numbers, the product of the digits of each of which is equal to 3375. The answer should be presented as an integer. | Okay, so I have this problem: I need to find the number of eight-digit numbers where the product of the digits is equal to 3375. Hmm, let me think about how to approach this.
First, let's better to approach this as a problem of distributing the exponents of 3 and 5 among the eight digits.
So, each digit can be expres... | 1 | 1680 |
17. Economic forecast. Three chickens laid three eggs in three days. How many eggs will twelve chickens lay in twelve days? | .
Finally, to find out how many eggs twelve chickens will lay in twelve days, I multiply the daily rate by twelve days. So, 4 eggs/day * 12 days = 48 eggs.
</think>
Let's solve the problem step-by-step.
**Problem Statement:**
Three chickens laid three eggs in three days. How many eggs will twelve chickens lay in twe... | 1 | 48 |
Compute the $100^{\text{th}}$ smallest positive integer $n$ that satisfies the three congruences \[\begin{aligned} \left\lfloor \dfrac{n}{8} \right\rfloor &\equiv 3 \pmod{4}, \\ \left\lfloor \dfrac{n}{32} \right\rfloor &\equiv 2 \pmod{4}, \\ \left\lfloor \dfrac{n}{256} \right\rfloor &\equiv 1 \pmod{4}. \end{aligned}\] ... | Okay, so I have this problem where I need to find the 100th smallest positive integer n that satisfies three congruences involving floor functions. Let me try to parse this step by step.
First, the problem states:
Compute the 100th smallest positive integer n that satisfies the three congruences:
\[
\begin{aligned}
\... | 0 | 6491 |
A store offers packages of $12$ pens for $\$10$ and packages of $20$ pens for $\$15$. Using only these two types of packages of pens, find the greatest number of pens $\$173$ can buy at this store.
[i]Proposed by James Lin[/i] | Okay, so I need to figure out the greatest number of pens I can buy with $173 using only packages of 12 pens for $1.
Similarly, for a given y, x ≤ floor((173 - 15y)/10).
So, maybe I can iterate over possible more cost-effective. So, to maximize the number of pens, I should buy money for 12-pen packages.
Let me test ... | 1 | 224 |
## Task Condition
Find the derivative of the specified order.
$y=\frac{\ln x}{x^{3}}, y^{IV}=?$ | Okay, so I need to find the fourth derivative of the function y = (ln x) / x³. Hmm, that sounds a bit complicated, but x^(-4) + (1 - 3 ln x) * (-4x^(-5))
Simplify each term:
First term: (-3/x) * x^(-4) = -3x^(-5) = -3 / x⁵
Second term: (1 - 3 = u'v + uv'
So, y''' = (12/x) * x^(-5) + (12 ln x - 77 - 60 ln x) / x⁶
G... | 1 | \frac{-342+360\lnx}{x^{7}} |
Shapovalov A.V.
What is the maximum number of white and black chips that can be placed on a chessboard so that in each row and each column there are exactly twice as many white chips as black ones? | Okay, so I need to figure out the maximum number of white and black chips that can be placed on a chessboard. The condition is that in each row specifically Latin squares or something similar, but with unequal distributions.
Wait, maybe I should model this as a matrix where each entry is either 0 (empty), 1 (black), o... | 0 | 48 |
2・22 Let $S=\{1,2, \cdots, 1990\}$. If the sum of the elements of a 31-element subset of $S$ is divisible by 5, it is called a good subset of $S$. Find the number of good subsets of $S$. | Alright, so I have this problem where I need to find the number of good subsets of S, where S is the set {1, 2, ..., 1990}. A good subset is defined as a 31-element subset where the sum of its elements is divisible by 5. Hmm, okay, let me think about how to approach this.
First, I know that dealing with subsets and di... | 0 | \frac{1}{5}C_{1990}^{31} |
$A$ and $B$ are on a circle of radius $20$ centered at $C$, and $\angle ACB = 60^\circ$. $D$ is chosen so that $D$ is also on the circle, $\angle ACD = 160^\circ$, and $\angle DCB = 100^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$. What is $DE$? | Alright, so I just came across this geometry problem, and I wanted to figure it out step by step. Let me walk through my thought process here.
First, let me try to visualize the problem. We have a circle with center C and radius 20. Points A and B are on this circle, and the angle ACB is 60 degrees. Then there's anoth... | 1 | 20 |
3. (3 points) Solve the system of equations: $\left\{\begin{array}{l}x y=6(x+y) \\ x z=4(x+z) \\ y z=2(y+z)\end{array}\right.$ | Alright, so I've got this system of equations to solve, and I need to figure out how to approach it step by step. Let me write them down first:
1. \( xy = 6(x + y) \)
2. \( xz = 4(x + z) \)
3. \( yz = 2(y + z) \)
Hmm, okay. So there are three equations with three variables: x, y, z. I need can factor the left side:
... | 0 | 0 |
4. The function $y=a \cos x+b(a, b$ are constants $)$ has a maximum value of 1 and a minimum value of $-7, y=$ $3+a b \sin x$ has a maximum value of $\qquad$. | Alright, let me try to figure out this problem step by step. So, we have two functions here, and I need to find the maximum value of the second one based on the information given about the first one.
First, the problem states: "The function \( y = a \cos x + b \) y = 3 + a b \sin x \) has a maximum value of _______." ... | 1 | 15 |
The teacher gave the children a task to multiply the number written on the board by three and add the number seven to the result. Kamča solved the task correctly. Her friend Růža, however, calculated a different problem: she multiplied the given number by seven and added the number three to the result. Růža's result wa... | result. So her calculation was 7n + 3. Got it.
Kamča's answer equals 84. Let me write that equation down:
(7n + 3) - (3n + 7) = 84
Okay, now I need to solve for 'n'. Let me simplify the left side of the equation step by step.
First, distribute the negative sign to both terms in the parentheses:
7n + 3 - 3n - 7 ... | 1 | 22 |
7.2. Losharik is going to visit Sovunya along the river at a speed of 4 km/h. Every half hour, he launches paper boats that float to Sovunya at a speed of 10 km/h. With what time interval do the boats arrive at Sovunya? (Provide a complete solution, not just the answer.) | minutes}
\]
3. **Determining Losharik's Travel Time:**
- Losharik takes 1.5 hours to reach Sovunya.
4. **Calculating the Time Interval Between Boats:**
- Since Losharik takes 1.5 hours to reach Sovunya, the first boat he launches will arrive 1.5 hours later.
- He launches a new boat every half hour (30... | 0 | 18 |
12. Restore the digits. On the board, there was the product of three consecutive even numbers. During the break, Vasya erased some digits. As a result, on the board remained $87 * * * * * 8$. Help Petya find the missing digits in the product. | Alright, let me try to figure out this problem. So, we have three consecutive even numbers multiplied together, and their product is displayed as 87****...****8. Vasya erased some digits, leaving only the first two digits as 87 and the last digit as 8. Our goal is to find the missing digits in between.
First, let's br... | 0 | 87526608 |
6.177. $\sqrt[3]{x+7}-\sqrt{x+3}=0$.
6.177. $\sqrt[3]{x+7}-\sqrt{x+3}=0$. | Alright, so I've got this equation to solve: the cube root of ( sides to the 6th power:
(∛(x + 7))^6 = (√(x + 3))^6.
Simplifying both sides, the left side becomes (x + 7)^2 because (∛a)^6 = a^(6/3) = a^2. Similarly, the right side becomes (x + 3)^3 because (√ x^2 - 14x - 49.
Simplify the right side:
x^3 + (9x^2 - x... | 1 | 1 |
Given that $a,b,c$ are integers with $abc = 60$, and that complex number $\omega \neq 1$ satisfies $\omega^3=1$, find the minimum possible value of $| a + b\omega + c\omega^2|$.
| Okay, so I have this problem where I need to find the minimum possible value of |a + bω + cω²|, where a, b, c are integers with abc = 60, and ω is a complex number not equal to 1 that satisfies ω³ = 1. Hmm, ω is a cube root of unity, right? So, ω is one of the primitive cube roots of 1. I remember that cube roots of un... | 1 | \sqrt{3} |
3. Compute:
$$
\left\lfloor\frac{2005^{3}}{2003 \cdot 2004}-\frac{2003^{3}}{2004 \cdot 2005}\right\rfloor .
$$ | Okay, so I need to compute this expression:
$$
\left\lfloor\frac{2005^{3}}{2003 \cdot 2004}-\frac{2003^{3}}{2004 \cdot 2005}\right\rfloor .
$$
Hmm, let me break it down. The expression has two main parts subtracted from each other, and then I take the floor of the result. The floor function just means I take the grea... | 1 | 8 |
5. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $A B=$ $2, A A_{1}=A D=1$, points $E$, $F$, and $G$ are the midpoints of edges $A A_{1}$, $C_{1} D_{1}$, and $B C$ respectively. The volume of the tetrahedron $B_{1}-E F G$ is $\qquad$ | Okay, so I have this problem about a rectangular prism, which is like a two segments of 0.5 each.
The question is asking for the volume of the tetrahedron B₁-EGF. So, this tet₁D₁. C₁ is (2,1,1) and D₁ is (0,1,1). The midpoint F is ((2+, the four vertices of the tetrahedron are:
- B₁: (2, 0.5, 0)
I need to compute th... | 1 | \frac{3}{8} |
6.2. The shape of a certain castle is a heptagon, and there is a bell tower at each of its vertices. Each side of the castle wall is guarded by the soldiers in the bell towers at the two endpoints of that side. How many soldiers are needed in total to ensure that each side of the wall is guarded by at least 7 soldiers? | Okay, so I've got this problem here about a castle shaped like a heptagon, which I remember is a seven-sided polygon. There's a bell tower at each vertex, and each side of the castle is guarded by soldiers in the two bell towers at the endpoints. The question is asking how many soldiers are needed in total to make sure... | 1 | 25 |
1. At Andrei's birthday, Yana was the last to arrive, giving him a ball, and Eduard was the second to last, giving him a calculator. Testing the calculator, Andrei noticed that the product of the total number of his gifts and the number of gifts he had before Eduard arrived is exactly 16 more than the product of his ag... | Okay, so I have this problem about Andrei's birthday gifts, and I Yana arrived.
Hmm, okay. Let's parse that sentence carefully.
Let me denote some variables to make this clearer.
Let me define:
- Let T gifts he had before Yana arrived.
- Let A be Andrei's age.
So, according to the problem, the product of T and E ... | 1 | 18 |
7. Given a fixed point $A(4, \sqrt{7})$. If a moving point $P$ is on the parabola $y^{2}=4 x$, and the projection of point $P$ on the $y$-axis is point $M$, then the maximum value of $|P A|-|P M|$ is $\qquad$. | Okay, so I have this problem here where there's a fixed point A at (4, sqrt(7)), and a moving point P on the parabola y² = 4x. The projection of P onto the y-axis is point M. I need to find the maximum value of |PA| - |PM|. Hmm, that sounds a bit tricky, but let's try to break it down step by step.
First, let me visua... | 1 | 5 |
I3.1 Given that the unit digit of $1+2+3+\ldots+1997+1998+1999+1998+1997+\ldots+3+2+1$ is $P$, find the value of $P$. | Alright, let me try to figure out this problem step by step. So, the question is about finding the unit digit (which is the last digit) of a big sum. The sum goes from 1 up to 1999 and then back down to 1. So it's like 1 + 2 + 3 + ... + 1997 + 1998 + 1999 + 1998 + 1997 + ... + 3 + 2 + 1. And they want the unit digit of... | 1 | 1 |
For any real number $x$, let the function $f(x)$ be the smaller of $2-x^{2}$ and $x$, then the maximum value of $f(x)$ is | Okay, so I have this problem where for any real number x, the function that interval, f(x) = x.
Now, outside of this interval, which function is smaller? Let's pick a point greater than 1, say. Therefore, for x > 1, f(x) = 2 -) = 2 - x².
So, to summarize:
- For x between -2 and 1, f(x) = x.
- For x < -2 and x > 1, f(... | 1 | 1 |
## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(1 ; 3 ; 6)$
$M_{2}(2 ; 2 ; 1)$
$M_{3}(-1 ; 0 ; 1)$
$M_{0}(5 ;-4 ; 5)$ | Alright, so I have this problem where I need to find the distance from a point M₀ to a plane defined by three other points M₁, M₂, and M₃. The coordinates are given, and I need to figure this out step by step. Let me try to remember how to do this.
1, 3, 6). Subtracting component-wise, v₁ = (2 - 1, 2 - 3, 1 - 0 = -15.... | 1 | 2\sqrt{14} |
5. The equation $x+y+z=2011$ satisfies $x<y<z$ to prove: the number of solutions $(x, y, z)$ is $\qquad$ groups. | Alright, so I have this problem where I need to find the number of solutions to the equation \( x + y + z = 2011 \) with the condition that \( x < y < z \). Hmm, okay. Let me try to figure this out step by step.
First off, I remember that without any restrictions, the number of non-negative integer solutions to \( x +... | 0 | 336005 |
## 51. Mixed-Up Hands.
- Yesterday, between two and three o'clock, - said Colonel Crackham, - I looked at my watch and, mistaking the hour hand for the minute hand, I was wrong in my estimate of the time. The erroneous time was 55 minutes less than the actual time. What was the actual time? | Okay, so I'm trying to solve this problem where Colonel Crackham looked at time. I need to figure out what the actual time was.
First, let actual time is H:M.
But he saw the hour hand at 60 + 0.5*M degrees and thought it was the minute hand. So6.
Therefore, we have:
5 + M = (60 + 5M = 60
Subtract 30:
5.5M = 30
S... | 0 | 2 |
$\underline{\text { Fomin S. }}$.
Two people toss a coin: one tossed it 10 times, the other - 11 times.
What is the probability that the second person got heads more times than the first? | Okay, so I have this probability problem here where two people are tossing a coin. One person tosses it 10 times, and the other tosses it 11 times. I need to find the probability that the person who tossed the coin 11 times gets more heads than the person who tossed it 10 times. Hmm, interesting.
Let me think about ho... | 1 | \frac{1}{2} |
Let $M$ and $N$ be the midpoints of sides $C D$ and $D E$ of a regular hexagon $A B C D E F$. Find the angle between the lines $A M$ and $B N$.
# | Alright, so I need to figure out the angle between lines AM and BN in√3/2)
Okay, so now I can find M, which is the midpoint of CD. Coordinates of C are (-0.5, √, I need to find vectors AM and BN.
Vector AM goes from A((-0.75, -√3/4). So, components are:
BN_x = -0.75 - 0.5 = -1.25
BN_y = -√3/4 - √) = (-3√3)/4
Now, ... | 1 | 60 |
3. Given $a>1$. Then the minimum value of $\log _{a} 16+2 \log _{4} a$ is $\qquad$ . | Okay, so I have this problem here: Given that \( a > 1 simplifies to:
\( \frac{4}{\log_{2} a} + \log_{2} a \).
Let me write that down:
\( \frac{4}{x} + x \), where I let \( x = \log_{2} a \).
Since \( a > 1, \( x \) must be greater than 0. So we're looking to minimize \( f(x) = \frac{4}{x} + x \) for \( x > 0 \).
Hm... | 1 | 4 |
19. (5 points)
Han Xin led his troops, forming a solid rectangular array, with infantry standing inside the array and archers surrounding the outermost layer. If there are 130 archers, then the maximum number of infantry is $\qquad$ people. | Okay, so I need to figure out the maximum number of infantry that Han Xin number of infantry, which is the total soldiers minus the archers. So, the number of infantry would be (L*W) - 130.
Our goal is to maximize this number. So, we need to maximize L*W given that L + W = 67.
From algebra, I integers close to that.
... | 1 | 992 |
2. Connecting the intersection points of $x^{2}+y^{2}=10$ and $y=\frac{4}{x}$ in sequence, a convex quadrilateral is formed. The area of this quadrilateral is $\qquad$ | Alright, so I have this problem where I need to find the area of a convex quadrilateral formed by connecting the intersection points of the circle \(x^2 + y^2 = 10\) and the hyperbola \(y = \frac{4}{x}\). Hmm, okay, let me break this down step by step.
First, I think I need to find the points where these two equations... | 1 | 12 |
15. (6 points) A little squirrel stored some pine cones for the winter. The little squirrel originally planned to eat 6 pine cones per day, but actually ate 2 more than planned each day, and as a result, finished the pine cones 5 days earlier. How many pine cones did the little squirrel store in total? | Let me try to figure out how to solve this problem.
So, there's so he took D - 5 days.
Therefore, the total number of pine cones can also be expressed as 8 * (D - 5).
Since both expressions represent the total number of pine cones, we can set them equal to each other:
6D = 8(D - 5)
Now, let's solve for D:
6D = 8D... | 1 | 120 |
3. Let $\lfloor x\rfloor$ denote the greatest integer not exceeding the real number $x$. Find the integer solution to the equation
$$
\left\lfloor\frac{x}{1!}\right\rfloor+\left\lfloor\frac{x}{2!}\right\rfloor+\cdots+\left\lfloor\frac{x}{10!}\right\rfloor=2019
$$ | Alright, so I have this equation to solve: the sum from n=1 to n=10 of the floor of x divided by n factorial equals 2019. Mathematically, that's written as
$$
My goal is to find the integer solution x. Since this involves floor functions,! = 1 → 2000/1 = 2000
- 2! = 2 → 2000/2 = 1000
- 3! = 6, then:
1000/1 = 1000
100... | 1 | 1176 |
The hour, minute, and second hands of a clock are on a common axis. At 12 o'clock, they overlap. When will they next overlap? | Okay, so I need to figure out when the hour, minute, and second hands of a clock will next overlap after 12 o'clock. Hmm, at 12 o'clock, all three hands are on top of each other, right? But after that, the second hand moves quickly, the minute hand moves a bit slower, and the hour hand moves even slower. So, I guess th... | 1 | 12 |
37. There are two-digit numbers where the sum of the tens digit and the units digit is 8. There are $\qquad$ such two-digit numbers. | To determine the number of two-digit numbers where the sum of the tens digit and the units digit is 8, I'll approach the problem systematically.
First, let's define the digits. Let the tens digit be \( x \) and the units digit be \( y \). Since it's a two-digit number, \( x \) must be at least 1 (ranging from 1 to 9) ... | 1 | 8 |
Problem 4 If
$$
f(x)=|x-a|-|x-b|(a<b) \text {, }
$$
find the minimum value of this function. | Okay, so I have this problem where I need to find the minimum value of, that simplified nicely. So, for x < a, f(x) is just a constant value, a - b. Since a < b, this means a - b is negative. So, in this interval, f(x) is constant at a negative value.
Moving on to the second interval: a ≤ x a linear function with a sl... | 1 | a-b |
1. Given the sets $A=\left\{2,4, a^{3}-2 a^{2}-a+7\right\}, B=\left\{-4, a+3, a^{2}-2 a+2, a^{3}+a^{2}+3 a+7\right\}$ $(a \in \mathbf{R})$. If $A \cap B=\{2,5\}$, then the real number $a=$ $\qquad$ . | Okay, so I have this problem where there are two sets, A and B and B are:
A = {2, 4, a³ - it can be 2. Then a³ + a² + 3a + 7 = 2, so a³ + a² + 3 2 = 0. So, 0. So, a = 1 is a root. Therefore, (a - 1) is a factor a = 1, 2, or -1.
So, possible a1.
Now, remember, 2 is also in set B as an element. So, let's see which a'... | 1 | 2 |
5. A ladybird is free to fly between the $1 \times 1$ cells of a $10 \times 10$ square grid. She may begin in any $1 \times 1$ cell of the grid. Every second she flies to a different $1 \times 1$ cell that she has not visited before.
Find the smallest number of cells the ladybird must visit, including her starting cell... | Okay, so I'm trying to figure out this ladybird problem on a 10x10 grid. Let me restate the problem to make sure I understand it correctly. The ladybird can fly from cell to cell, starting anywhere, and each second she moves to a different cell she hasn't visited before. We need to find the smallest number of cells she... | 0 | 76 |
If in the city there were 3000 more residents; then, each resident would have paid 10 forints less in tribute to the enemy; if, however, there were 1000 fewer residents; then each would have paid 10 forints more. - How many residents were there, how much did one pay, and how much did the entire city pay? | Okay, so I have this problem about residents in a city and how much they pay in tribute. Let me try to break it down step by step.
First, the problem says: If there were 3000 more residents, each would have paid 10 forints less. On the other hand, if there were 1000 fewer residents, each would have paid 10 forints mor... | 0 | 3000 |
5. Find the largest three-digit number that is divisible by the sum of its digits and in which the first digit matches the third, but does not match the second. | Okay, so I have this problem: Find the largest three-digit number that is divisible by the sum of its digits and in which the first digit matches the third, but does not match the second. Hmm, let me break this down step by step.
First, let's understand the requirements. The number is a three-digit number, so it's bet... | 1 | 828 |
$4 \cdot 21$ Find the sum of all divisors $d=2^{a} \cdot 3^{b}(a, b$ being natural numbers $)$ of $N=19^{88}-1$.
(Austrian Mathematical Competition, 1988) | Okay, so I have this problem here: Find the sum of all divisors \( d = 2^{a} \cdot 3^{b} \) (where \( a \) and \( b \) are natural numbers) of \( N = 19^{88} - 1 \). Hmm, that sounds a bit challenging, but let me try to break it down step by step.
First, let me understand what exactly is being asked. I need to find al... | 0 | 744 |
9.4. In triangle $ABC$ with sides $AB=c, BC=a, AC=b$, the median $BM$ is drawn. Incircles are inscribed in triangles $ABM$ and $BCM$. Find the distance between the points of tangency of these incircles with the median $BM$. | Okay, so I'm trying to solve this geometry problem about triangle ABC with sides AB = c, BC = a, and AC = b. The median BM is drawn, and we need to find the distance between the points of tangency of the incircles of triangles ABM and BCM with the median BM. Hmm, that sounds a bit complicated, but let's break it down s... | 0 | \frac{|-|}{2} |
A fenced, rectangular field measures $24$ meters by $52$ meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the fie... | Okay, so I have this problem about partitioning a rectangular field into square test plots. The field is 24 meters by 52 meters, and the researcher has 1994 meters of fence for the internal fencing. The goal is to find the largest number of square test plots possible using all or some of the 1994 meters of fence. The s... | 1 | 702 |
A figure on a plane has exactly two axes of symmetry. Find the angle between these axes.
# | Okay, so I have this problem: A figure on a plane has exactly two axes of symmetry. I need to find the angle between these axes. Hmm, maybe I'm misunderstanding the problem.
Alternatively, perhaps the figure has exactly two axes of symmetry, and we need to find the angle between them in the most general case. But that... | 1 | 90 |
Exercise 9. Let $A B C$ be a triangle such that $\widehat{C A B}=20^{\circ}$. Let $D$ be the midpoint of the segment $[A B]$. Suppose that $\widehat{C D B}=40^{\circ}$. What is the value of the angle $\widehat{A B C}$? | Alright, so I have this geometry problem here, and I need to figure out know angle at A is 20 degrees. If I can find angle at C or angle at B, I can determine the other angles since the sum of angles in a triangle is 180 degrees.
But I don't have 20 - θ, which is 160 - θ ACB.
Hmm, but if I denote angle ACB as 160 - θ... | 1 | 70 |
Problem 10-1. For real numbers $a, b$, and $c$, it is known that $a b c + a + b + c = 10$, and $a b + b c + a c = 9$. For which numbers $x$ can it be asserted that at least one of the numbers $a, b, c$ is equal to $x$? (Find all such numbers $x$ and prove that there are no others.)
Answer: 1. | Okay, so I have this problem here where I need to find all real numbers x such that at least one of a, b, or c equals x, given the equations:
1. \( abc + a + b + c = 10 \)
2. \( ab + bc + ac = 9 \)
Hmm, okay. I need to figure out what possible values x can take, right? So, I guess I need to analyze these equations an... | 1 | 1 |
A4. How many positive integers $N$ of two digits are there with the following property: The sum of $N$ and the reverse of $N$ is a square of an integer (the reverse of $N$ is the number you get by writing $N$ backwards)? | Okay, so I need to figure out how many two-digit positive integers N satisfy the condition that the sum of N and its reverse is a perfect square. Let(a + b) = k².
Now, let me analyze this equationm, where m is a positive integer.
Substituting back, we get(a + b) = 121m² => a + b = 11m².
Now, a and b are digits, so ... | 1 | 8 |
14. Given that $\tan \alpha+\cot \alpha=4$, find $\sqrt{\sec ^{2} \alpha+\csc ^{2} \alpha-\frac{1}{2} \sec \alpha \csc \alpha}$. | Okay, so I'm trying to solve this problem where I'm given that tan α + cot α = 4, and I need to find the value of the expression √(sec² α + csc² α - (1/2) sec α csc α). Hmm, that looks a bit complicated, but maybe I can break it down step by step.
First, let's1/4.
Alright, that's something. Now, I need to find √(sec²... | 1 | \sqrt{14} |
Task 4. (20 points) In the center of a circular field stands a geologists' cabin. From it, 6 straight roads extend, dividing the field into 6 equal sectors. Two geologists set out on a journey from their cabin at a speed of 4 km/h along a road each arbitrarily chooses. Determine the probability that the distance betwee... | Okay, so I have this problem where there's a circular field with a geologists' cabin right in the center. From this cabin, there are 6 straight roads going out, each dividing the field into 6 equal sectors. So, if I imagine it, it's like a clock with 12 numbers but only the numbers at 12, 2, 4, 6, 8, 10, but not the ot... | 1 | 0.5 |
1. (15 points) In a bookstore, there is a rule for "summing" discounts: "if different types of discounts apply to an item, they are applied sequentially one after another." For example, if two discounts A% and B% apply to an item, the first discount is applied to the original price, and the second discount is applied t... | Okay, so I need to figure out the size of the "Random" discount all discounts. Wait, the person has three discounts: Autumn, Random, and Loyalty. But does the problem specify the order? It doesn't, so Loyalty. Let me see: each discount is a percentage, so applying one after another. The order is important because perce... | 1 | 50 |
23. Consider the 800 -digit integer
$$
234523452345 \cdots 2345 .
$$
The first $m$ digits and the last $n$ digits of the above integer are crossed out so that the sum of the remaining digits is 2345 . Find the value of $m+n$. | Alright, let's see. I have this problem where there's an 8 consider the position modulo 4, since each block is 4 digits. So, positions 1, 5, 9, ... are 2; positions 2, 6, 10, ... are 3; positions 3, 7, 11, ... are 4; and positions 4, 8, 12, ... are 5.
So, for any position k, the value is:
- 2 if k ≡ 1 mod 4,
- 3 if k ... | 1 | 130 |
2. A truck traveling at $60 \mathrm{~km} / \mathrm{h}$ started from city $A$ to city $B$. After some time, a car traveling at $90 \mathrm{~km} / \mathrm{h}$ also started from city $A$ to city $B$. It was planned for the car to catch up with the truck in city $B$. However, after covering $\frac{2}{3}$ of the distance, t... | Alright, so I have this problem where a truck and a car are traveling from city A to city B. The truck starts first at 60 km/h, and later the car starts at 90 km/h. The car was supposed to catch up with the truck in city B, but due to some issues, the truck reduced its speed to 30 km/h after covering 2/3 of the distanc... | 0 | 200 |
Let $p$ be a prime. It is given that there exists a unique nonconstant function $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$ such that $\chi(1) = 1$ and $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \not\equiv 0 \pmod p$ (here the product $mn$ is taken mod $p$). For how many positive primes $p$ less than $100$ is it true that \[... | Alright, so I have this problem here, and it's a bit involved, but I want to break it down step by step. Let me try to understand what's being asked.
We have a prime number \( p \). There's this unique nonconstant function \( \chi \) from the set \( \{1, 2, \ldots, p-1\} \) to \( \{-1, 1\} \) such that \( \chi(1) = 1 ... | 0 | 24 |
2. Find the value of the expression $\frac{a}{b}+\frac{b}{a}$, where $a$ and $b$ are the largest and smallest roots of the equation $x^{3}-7 x^{2}+7 x=1$, respectively. | Alright, so I have this problem here: I need to find the value of the expression (a/b) + (b/a), where a and b are1.
Multiply 1 by 1 = 1, add to -7: -6.
Multiply -6 by 1 = -6, add to 7: 1.
Multiply 1 by 1 = 1,.
Now, a is the largest root, and b is the smallest root. Let me identify these.
Looking at the roots:
3 +... | 1 | 34 |
[ Regular Pyramid ]
Find the volume of a regular hexagonal pyramid with a lateral edge $b$ and the area $Q$ of a lateral face.
# | Alright, so I have this problem about finding the volume of a regular hexagonal pyramid. The given information is the lateral edge length, which is denoted as \( b \), and the area of a lateral face, which is \( Q \). Hmm, okay. Let me try to visualize this first.
A regular hexagonal pyramid has a base that's a regula... | 0 | \frac{\sqrt{3}}{2}(2b^2-\sqrt{4b^{2}-16Q^{2}})\sqrt{\sqrt{4b^{2}-16Q^{2}}-b^{2}} |
7,8,9 |
What is the maximum number of kings that can be placed on a chessboard so that no two of them attack each other
# | Alright, so I came across this chess problem today, and it's been on my mind ever since. The question is, what's the maximum number of kings you can place on a chessboard without any two attacking each other? Hmm, interesting. I remember that kings in chess can move one square in any direction, so they can attack any a... | 1 | 16 |
We draw spheres with a radius of $\frac{1}{2}$ around the vertices of a unit cube. There are two such spheres that touch all eight of these spheres. Calculate the difference in volume between the two spheres.
---
The translation maintains the original text's format and structure. | Okay, so I'm trying to figure out this geometry problem where we have a unit cube, and we're drawing spheres of radius 1/2 around each of its vertices. The question is, there are two spheres that touch all eight of these spheres, and we need to calculate the difference in volume between these two spheres. Hmm, that sou... | 1 | \frac{10}{3}\pi |
5. Given a natural number $x=6^{n}+1$, where $n-$ is an odd natural number. It is known that $x$ has exactly three distinct prime divisors, one of which is 11. Find $x$. | Okay, so I have this problem here where I need to find a natural number \( x = 6^n + 1 \) where \( n \) is an odd natural number. It's given that \( x \) has exactly three distinct prime divisors, one of which is 11. I need to find \( x \).
Alright, let's break this down step by step. First, \( n \) is odd, so \( n = ... | 1 | 7777 |
Among the numbers 712, 1262, and 1680, which is the only one that can be written as a product of four consecutive natural numbers? | Alright, so I have this problem here: Among the numbers 712, 1262, and 1680, which one can be written as a product of four consecutive natural numbers? Hmm, okay. natural number. So their product would be n(n+1)(n+2 n. Hmm, maybe I can compute the product for different values of n and that the product of four consecuti... | 1 | 1680 |
3. A three-stage launch vehicle consists of stages in the form of cylinders. All these cylinders are similar to each other. The length of the middle stage is two times less than the sum of the lengths of the first and third stages. In the fueled state, the mass of the middle stage is $13 / 6$ times less than the total ... | Alright, let's try to tackle this problem step by step. It's about a three-stage launch vehicle with similar cylindrical stages. The problem gives some relationships between the lengths and masses of these stages and asks for the ratio of the lengths of the first and third stages.
First, let me parse the problem again... | 1 | \frac{7}{5} |
2. What is the degree measure of angle $\angle A$, if its bisector forms an angle with one of its sides that is three times smaller than the angle adjacent to $\angle A$? | Alright, so I need to find the degree measure of angle ∠A. The problem says that the angle bisector of ∠A forms an angle with one of its sides that is three times smaller than the angle adjacent to ∠A. Hmm, okay, let's break this down.
First, let me and another side? Or is it adjacent in the sense of forming a linear ... | 1 | 72 |
11 Given that point $P$ is on the curve $y=\mathrm{e}^{x}$, and point $Q$ is on the curve $y=\ln x$, then the minimum value of $|P Q|$ is $\qquad$ . | Okay, so I have this problem here: Point P is on the curve y = e^x, and point Q is on the curve y = ln x. I need to find the minimum distance between P and Q. Hmm, that sounds interesting. Let me see how I can approach this.
First, I remember and then find its minimum.
Since P is on y = e^x,, and solve the resulting ... | 1 | \sqrt{2} |
1. Divide the set of positive even numbers $\{2,4,6, \cdots\}$ into groups in ascending order, with the $n$-th group containing $3 n-2$ numbers:
$$
\{2\},\{4,6,8,10\}, \quad\{12,14,16,18,20,22,24\}, \cdots
$$
Then 2018 is in the $\qquad$ group. | Okay, so I have this problem where I need to figure out which group the number 2018 is in. The groups are defined in a specific way. Let me break it down step by step.
First, the set of positive even numbers is given as {2, 4, 6, 8, ...}. These are just all the even numbers starting from 2 and going on infinitely. Now... | 0 | 27 |
Problem 4. The area of a courtyard, which has the shape of a rectangle, is 10 ares. The length of one side is 25 meters. To fence the courtyard, a post needs to be placed every 5 meters. Calculate how many posts are needed to fence the courtyard and how many posts will be on each side? | First, I need to understand the problem. The courtyard is a rectangle with an can calculate the perimeter of the rectangle to determine how many posts are needed. The5 meters, I divide the perimeter by 5 to find the number of posts: 130 m ÷ 5 m/post = 26 posts.
Finally, to find how many posts are on each side, I divid... | 0 | 26 |
12. (10 points) When $N$ takes all the numbers in $1,2,3, \cdots, 2015$, how many numbers of the form $3^{n}+n^{3}$ are divisible by 7? | Alright, so I have this problem here: I need to find how many numbers of the form \(3^n + n^3\) are divisible by 7, where \(n\) ranges from 1 to 2015. Hmm, okay. Let me try to break this down step by step.
First off, divisibility by 7. That means I need \(3^n + n^3 \equiv 0 \mod 7\). So, essentially, \(3^n \equiv -n^3... | 1 | 288 |
6. Let the set $M=\{1,2, \cdots, 1000\}$, and for any non-empty subset $X$ of $M$, let $a_{x}$ denote the sum of the largest and smallest numbers in $X$. Then, the arithmetic mean of all such $a_{x}$ is $\qquad$ . | Okay, so I have this problem where I need to find the arithmetic mean of all such \( a_X \) where \( X \) is a non-empty subset of the set \( M = \{1, 2, \ldots, 1000\} \). For each subset \( X \), \( a_X \) is defined as the sum of the largest and smallest numbers in \( X \).
First, I should clarify what exactly is ... | 1 | 1001 |
3. Calculate $\left[\sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\ldots+\sqrt[2009]{\frac{2009}{2008}}\right]$ (here $[x]$ is the integer part of the number $x$, i.e., the greatest integer not exceeding the number $x$). | Okay, so I have this problem where I need to calculate the integer part of a sum. The sum is given by:
\[
\sqrt{2} + \sqrt[3]{\frac{3}{2}} + \sqrt[4]{\frac{4}{3}} + \ldots + \sqrt[2009]{\frac{2009}{2008}}
\]
And the integer part, denoted by [x], is the greatest integer less than or equal to x. So, I need to compute t... | 1 | 2008 |
5. The natural number $n$ satisfies $1 \leqslant n \leqslant 2022$, and $2^{n}+n^{3}$ is divisible by 7, such natural numbers $n$ have $\qquad$ | Alright, so I've got this problem here that says: Find the number of natural numbers \( n \) such that \( 1 \leq n \leq 2022 \) and \( 2^n + n^3 \) is divisible by 7. Hmm, okay, let's break this down step by step.
First, I know that if \( 2^3 \mod 7 \). Hmm, that might be useful later.
both of these expressions will... | 1 | 290 |
4. (7 points) When copying a problem, Lin Lin mistakenly wrote a repeating decimal as 0.123456. If the digits are correct but the dots indicating the repeating section were omitted, there are $\qquad$ possible original decimals. | Alright, so I have this problem where Lin Lin mistakenly wrote a repeating decimal as 0.123456. The digits are correct, but the dots indicating the repeating section were omitted. I need to figure out how many possible original decimals there could be. Hmm, okay, let me try to break this down.
First, I know that a rep... | 0 | 6 |
# Problem 6.
Find the smallest natural number $n$, for which the number $n+2018$ is divisible by 2020, and the number $n+2020$ is divisible by 2018. | Alright, so I've got this problem here: I need to find the smallest natural number \( n \) such that two conditions are satisfied. The first condition is that \( n + 2018 \) is divisible by 2020, and the second condition is that \( n + 2020 \) is divisible by 2018. Hmm, okay, let me break this down step by step.
First... | 1 | 2034142 |
7. 5 people participate in 4 groups, each group has two people, and each person must participate in at least one group. Then, the number of different groupings is
$\qquad$ kinds. | Okay, so I have this problem here: 5 people participate in 4 groups, each group has two people, and each person must participate in at least one group. I need to find the number of different groupings. Hmm, okay, let me try to figure this out.
First, let me make sure I understand the problem correctly. There are 5 peo... | 1 | 135 |
Let $t$ be TNYWR.
Gray has $t$ dollars consisting of $\$ 1$ and $\$ 2$ coins. If she has the same number of $\$ 1$ and $\$ 2$ coins, how many $\$ 1$ coins does she have?
## | Okay, so I need to figure out how many $1 coins Gray has. Let me see what information I have.
Gray has a total of $t$ sides by 3. That gives x = t/3. So, the number of $1 coins Gray has is t divided by 3.
Wait,, or perhaps it's implied.
Alternatively, maybe the problem assumes that x has, there's no solution. But th... | 0 | 4 |
20. (6 points) At 12 o'clock, the angle between the hour hand and the minute hand is 0 degrees. Thereafter, the moment when the hour hand and the minute hand form a 90-degree angle for the 6th time is $\qquad$ . (12-hour clock) | Alright, so I have this problem here about clock angles. It says that at 12 o'clock, the angle between the hour and minute hands is 0 degrees, and then it asks for the moment when they form a 90-degree angle for the 6th time. Hmm, okay. Let me think about this step by step.
First, I know that in a clock, between them ... | 0 | 3 |
1. There are 100 students who want to sign up for the class Introduction to Acting. There are three class sections for Introduction to Acting, each of which will fit exactly 20 students. The 100 students, including Alex and Zhu, are put in a lottery, and 60 of them are randomly selected to fill up the classes. What is ... | Alright, so I've got this problem here about 100 students trying to get into a class that has three sections, each holding exactly 20 students. That means there are 60 spots in total. The students are in a lottery, so 60 are randomly selected, and then assigned to the sections. The question is asking for the probabilit... | 1 | \frac{19}{165} |
2. Determine the maximum possible value of the expression
$$
\frac{a_{1} a_{2} \ldots a_{n}}{\left(1+a_{1}\right)\left(a_{1}+a_{2}\right) \ldots\left(a_{n-1}+a_{n}\right)\left(a_{n}+2^{n+1}\right)}
$$
where $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ are positive real numbers. | Alright, so I've got this problem here where I need to find the maximum possible value of the expression:
$$
\frac{a_{1} a_{2} \ldots a_{n}}{\left(1+a_{1}\right)\left(a_{1}+a_{2}\right) \ldots\left(a_{n-1}+a_{n}\right)\left(a_{n}+2^{n+1}\right)}
$$
where each \( a_i \) is a positive real number. Hmm, okay, so I need ... | 1 | \frac{1}{3^{n+1}} |
13. How many ways are there to arrange four 3 s and two 5 s into a six-digit number divisible by 11 ? | Okay, so I need to figure out how many six-digit numbers can be formed using four 3s and two 5s that are divisible by 11. Alright, let me start by recalling the divisibility rule for 1 is 0. So, the difference between the sum of digits in odd positions and the sum of digits in even positions must be 0. That simplifies ... | 1 | 9 |
Transform the fraction
$$
\frac{5 \sqrt[3]{6}-2 \sqrt[3]{12}}{4 \sqrt[3]{12}+2 \sqrt[3]{6}}
$$
so that the denominator is a rational number. | Okay, so I have this fraction here that I need to transform so that the denominator becomes a rational number. The fraction is:
$$
\frac{5 \sqrt[3]{6} - 2 \sqrt[3]{12}}{4 \sqrt[3]{12} + 2 \sqrt[3]{6}}
$$
Alright, first, I remember that to roots.
Let me note down the denominator first: it's 4 times6. Hmm. So, denomi... | 0 | \frac{24\sqrt[3]{4}-12\sqrt[3]{2}-11}{34} |
21. [11] Call a number overweight if it has at least three positive integer divisors (including 1 and the number), and call a number obese if it has at least four positive integer divisors (including 1 and the number). How many positive integers between 1 and 200 are overweight, but not obese? | Okay, so I need to figure out how many positive integers between 1 and 200 are overweight but not obese. Hmm, let's break this to be overweight but not obese, it must have exactly three divisors. Therefore, these numbers are squares of primes. So, I need to find how many squares of primes are less than or equal to 200.... | 1 | 6 |
3 Let $x, y$ be real numbers, $2x + y \geqslant 1$, then the minimum value of the bivariate function $u = x^{2} + 4x + y^{2} - 2y$ is $\qquad$ . | Okay, so I have this problem where I need to find the minimum value of the function \( u = x^2 + 4x + y^2 - 2y \) given the constraint \( 2x + y \geq 1 \). Hmm, let me try to figure this out step terms minus 5. That means the minimum value occurs when both squared terms are wait, I need to check if this point \( (-2, 1... | 1 | -\frac{9}{5} |
## Task 4 - 150614
A pioneer group collects waste paper. During the settlement, it was found that the collection result of the last two days accounted for a quarter of the total amount collected, and on the last day, $27 \mathrm{~kg}$ were collected, which was $6 \mathrm{~kg}$ more than the day before.
How many kilog... | kilograms.
Since these two days represent a quarter of the total amount collected, we can let the total amount be \( y \) kilograms. Therefore:
\[ \frac{1}{4}y = 48 \]
Solving for \( y \), we multiply both sides by 4:
\[ y = 48 \times 4 \]
\[ y = 192 \] kilograms.
</think>
Let's break down the problem step by step ... | 1 | 192 |
Define a positive integer $n^{}_{}$ to be a factorial tail if there is some positive integer $m^{}_{}$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1992$ are not factorial tails? | Okay, so I have this problem here: I need to figure out how many positive integers less than 1992 are not factorial tails. A factorial tail is defined as a number n such that there exists some positive integer m where m! ends with exactly n zeros. So, essentially, n is a factorial tail if m! has exactly n trailing zero... | 1 | 396 |
55. When programming on a computer to print out 1000 natural numbers: $1,2,3,4, \cdots \cdots, 999,1000$, unfortunately, the printer malfunctioned and printed "ji" every time it was supposed to print the digit 3. The numbers that were incorrectly printed total $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
---
55. Wh... | 好，我现在要解决这个问题。题目是说，从1到1000的自然数，每个数字都会被一个打印机打印出来。但是打印机 malfunction了，每当它应该打印数字3的时候，就会打印“ji”。那么，所有被错误地打印过的数字总共有多少个呢？
从1到1000这1000个数字中，每个数字中包含3的次数，也就是有多少个数字中至少有一个3，或者有多少个数字中包含多个3的情况。
接下来，我需要考虑如何计算1到1000中包含数字3的次数。通常从1到1000，个位上有多少个3呢？上有100个3。
然后是十位上的3。十位上每100个数字会出现10个3，比如30-939。因此，每100个数字里十位上有10个3，那么从1到1000，十位都有3，会被分别计算一次。... | 0 | 271 |
6. (10 points) $n$ pieces of paper, each written with 3 different positive integers not greater than $n$, any 2 pieces of paper have exactly one number in common, find the sum of all numbers written on the pieces of paper.
| Alright, so I've got this problem here: there are n pieces of paper, each with three different positive integers not greater than n. Any two pieces of paper share exactly one number in common. I need to find the sum of all the numbers written on all the pieces of paper. Hmm, let me see how to approach this.
First, let... | 1 | 84 |
## Task 4 - 040814
The number $62^{* *} 427$ is divisible by 99.
Determine the missing digits, and explain how you found them! How many solutions are there? | Alright, so I have this problem here where the number 62****427 is divisible by 99, and I need to find the missing digits. Hmm, okay, let me think about how to approach this.
First, I remember that if a number is divisible by 99, it means is 9 multiplied by 11. So, I can use the divisibility rules for 9 and 11 to figu... | 0 | 6224427 |
## Problem Statement
Derive the equations of the tangent and normal to the curve at the point corresponding to the parameter value $t=t_{0}$.
\[
\left\{\begin{array}{l}
x=\frac{3 a t}{1+t^{2}} \\
y=\frac{3 a t^{2}}{1+t^{2}}
\end{array}\right.
\]
$t_{0}=2$ | Okay, so I have this problem where I need to find the equations of the is 6a t.
Denominator: 1 + t², derivative is 2t.
So,
dy/dt = [6a t*(. **Evaluate the slope at t₀ = 2:**
Plugging t = 2 into the slope formula:
Slope Tangent = 2*2 / (1 - (2)²) = 4 / (1 - 4) = 4 / (-3) = -4/3
So, the tangent line has a slope of ... | 0 | -\frac{4}{3}\cdotx-4 |
6. (3 points) On a $3 \times 3$ chessboard, there are knights, who always tell the truth, and liars, who always lie. Each of them said: "Among my neighbors, there are exactly three liars." How many liars are on the board?
Neighbors are considered to be people on cells that share a common side. | Okay, so I have this problem about a 3x3 chessboard where. Now, each cell claims that exactly three of its neighbors are liars. For a knight, this is true; for a liar, it's false.
ars (L) among its neighbors is exactly three. If a cell is a liar (L), then the number of liars among its neighbors is not three, meaning i... | 1 | 5 |
One, (50 points) As shown in Figure 4, in the acute triangle $\triangle ABC$, the altitude $CE$ from $A B$ intersects with the altitude $BD$ from $A C$ at point $H$. The circle with diameter $DE$ intersects $AB$ and $AC$ at points $F$ and $G$, respectively. $FG$ intersects $AH$ at point $K$. Given that $BC=25, BD=20, B... | Alright, so I've got this geometry problem here, and I need to find the length of AK in this acute triangle ABC. Let me try to unpack what's given and figure out a plan.
First, the triangle is acute, which means all its angles are less than 90 degrees. That's good to know because it tells me that the orthocenter lies ... | 1 | 8.64 |
(12) $O, A, B$ are three non-collinear points on a plane, vector $\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=\vec{b}$, let $P$ be any point on the perpendicular bisector of line segment $A B$, vector $\overrightarrow{O P}=\vec{p}$. If $|\vec{a}|=5,|\vec{b}|=3$, then the value of $\vec{p} \cdot(\vec{a}-\vec{b})$... | Okay, so I have this problem here where I need to find the value of equal. Maybe I can express this condition mathematically.
Let’s denote the\[
|\vec{a} - \vec{p}|^2 = |\vec{b} - \vec{p}|^2
\]
Expanding both sides using the dot product:
\[
(\vec{a} - \vec{p}) \cdot (\vec{a} - \vec{p}) = (\vec{b} - \vec{p}) \ out:
... | 1 | 8 |
18. Suppose $x_{1}, x_{2}, \ldots, x_{49}$ are real numbers such that
$$
x_{1}^{2}+2 x_{2}^{2}+\cdots+49 x_{49}^{2}=1 .
$$
Find the maximum value of $x_{1}+2 x_{2}+\cdots+49 x_{49}$. | Okay, so I have this problem here where I need to find the maximum value of a linear expression given a quadratic constraint. Let me try to understand it step by step.
First, the problem says: we have real numbers \( x_ constraint, each term is \( k x_k^2 \), so it's like a weighted inner product. That is, if I write ... | 1 | 35 |
7. There are 100 chess pieces, and two people take turns to take the pieces. Each time, you are allowed to take 1 or 2 pieces. The one who takes the last piece wins. If you go first, then the first time you should take ( ) pieces to ensure victory. | Okay, so I have this chess piece problem where two players take turns picking up pieces. There are 100 pieces in total, and each player can or 2 pieces on my first move to ensure victory.
Hmm, let me move.
So, let's define a position as the number of pieces left. pieces on my opponent's turn, meaning I can take the l... | 1 | 1 |
$O$ and $I$ are the circumcentre and incentre of $\vartriangle ABC$ respectively. Suppose $O$ lies in the interior of $\vartriangle ABC$ and $I$ lies on the circle passing through $B, O$, and $C$. What is the magnitude of $\angle B AC$ in degrees? | Okay, let me try to figure this problem out. So, we have triangle angle BAC. So, formula-wise, angle BIC = 90 + (1/2) angle BAC. Is that correct? Let me me find angle BAC.
But since I is on the circumcircle of triangle BOC, what can I say about angle BIC? In a circle, angles subtended by the same chord are equal. So, ... | 1 | 60 |
# Task № 6.4
## Condition:
Dima and Vlad are playing a game: first, they take turns naming a number from 1 to 97 (Dima names the first number, the numbers must be different). Then each of them counts the number of different rectangles with integer sides, the perimeter of which is equal to the named number. The winner... | Okay, so I have this problem where Dima and Vlad are playing a game. They take turns naming a number from 1 to 97, and then each counts the number of different rectangles with integer sides that have a perimeter equal to the named number. The goal is to find the number Dima should name first to have a higher count of s... | 1 | 96 |
11. (10 points) There are three consecutive even numbers, the product of which is a five-digit number. The unit digit of this five-digit number is 0, the ten-thousand digit is 2, and the tens, hundreds, and thousands digits are three different digits. What is the sum of these three consecutive even numbers?
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Pleas... | Okay, so I have this problem here: there are three consecutive even numbers,9,999.
Wait, but earlier I thought \( N \) being 2, it's more precise: \( N \) is between 20,000 and 29,999.
So places are all different.
Now, I need to find three consecutive even numbers \() such that their product \( N = x(x + 2)(x + 4) \... | 1 | 84 |
3. (5) In the space divided into unit cubes with integer-coordinate vertices, the plane $x+y+z=0$ has been drawn. Find the values of the areas of the pieces into which it is divided by the faces of the cubes. | Okay, so I've got this problem here: In a space divided into unit cubes with integer-coordinate vertices, there's a plane given by the equation x + y + z = 0. I need to find the areas of the different pieces that this plane cuts the cubes into. Hmm, interesting. Let me try to visualize this.
First, unit cubes with int... | 0 | \frac{\sqrt{3}}{2} |
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