source stringclasses 7
values | problem stringlengths 19 6.18k | generation stringlengths 345 68.1k |
|---|---|---|
olympiads | Problem 1. Calculate $\lim _{x \rightarrow 0} \frac{(\cos x)^{\cos x}-\cos 2 x}{x^{2}}$. | <think>
Okay, so I need to find the limit as x approaches 0 of [(cos x)^(cos x) - cos 2x] divided by x squared. Hmm, let me think about how to approach this. Limits involving trigonometric functions often use L'Hospital's Rule or Taylor series expansions. Since both the numerator and denominator are approaching 0 as x ... |
olympiads | 5. Given the vector $\boldsymbol{a}=(\cos \theta, \sin \theta)$, vector $\boldsymbol{b}=(\sqrt{3},-1)$, then the maximum value of $|2 \boldsymbol{a}-\boldsymbol{b}|$ is | <think>
Okay, so I have this problem here: Given vector a, which is (cos θ, sin θ), and vector b, which is (√3, -1). I need to find the maximum value of the magnitude of 2a - b. Hmm, let's see.
First, I remember that when dealing with vectors, the magnitude of a vector (x, y) is calculated using the formula √(x² + y²... |
olympiads | 5. (Problem 1500 from "Mathematics Bulletin") Given positive integers $x_{1}<x_{2}<\cdots<x_{n}, x_{1}+x_{2}+\cdots+$ $x_{n}=2003, n \geqslant 2$. Find the minimum value of $f(n)=n\left(x_{1}+x_{n}\right)$. | <think>
Okay, so I need to find the minimum value of f(n) = n(x₁ + xₙ) where x₁ < x₂ < ... < xₙ are positive integers that add up to 2003, and n is at least 2. Hmm, let's start by understanding the problem step by step.
First, we have a sequence of distinct positive integers in increasing order. Their sum is 2003, and... |
olympiads | A new student joined the class, and it was known that besides English, he also spoke another foreign language very well. Three classmates were guessing which language it was.
The first one thought: "It's not French."
The second guessed: "It's Spanish or German."
The third reasoned: "It's Spanish."
Soon they found o... | <think>
Okay, let's try to figure out which foreign language the new student speaks besides English. The three classmates made different guesses, and we know that at least one is right and at least one is wrong. The possible languages mentioned are French, Spanish, and German. So, the answer must be one of these three.... |
olympiads | 4.50. Calculate the solid angle of a cone with an angle of $2 \alpha$ at the vertex. | <think>
Okay, so I need to calculate the solid angle of a cone that has an angle of 2α at its vertex. Hmm, solid angle... right, solid angle is like the 3D equivalent of a regular angle. Instead of degrees or radians in a plane, it's measured in steradians. The total solid angle around a point in 3D space is 4π steradi... |
olympiads | An urn contains $k$ balls marked with $k$, for all $k=1,2, \ldots, 2016$. What is the minimum number of balls we must withdraw, without replacement and without looking at the balls, to be sure that we have 12 balls with the same number? | <think>
Okay, so there's this urn with balls numbered from 1 to 2016, right? And for each number k, there are exactly k balls in the urn. The question is asking for the minimum number of balls we need to pull out to guarantee that we have 12 balls with the same number. Hmm, that sounds like a pigeonhole principle probl... |
cn_contest | Example 2 If $f(x) (x \in \mathbf{R})$ is an even function with a period of 2, and when $x \in [0,1]$, $f(x) = x^{\frac{1}{19988}}$, then the ascending order of $f\left(\frac{98}{19}\right), f\left(\frac{101}{17}\right), f\left(\frac{104}{15}\right)$ is | <think>
Okay, so I need to figure out the ascending order of these three function values: f(98/19), f(101/17), and f(104/15). The function f(x) is even and has a period of 2. When x is in [0,1], f(x) = x^(1/19988). Alright, let's break this down step by step.
First, since the function is periodic with period 2, I know... |
olympiads | 4. From the 2022 natural numbers 1 to 2022, select several distinct numbers such that the sum of any 2 of them is not a multiple of 5. The maximum number of numbers that can be selected is $\qquad$. | <think>
Okay, so the problem is: From the natural numbers 1 to 2022, select several distinct numbers such that the sum of any two of them is not a multiple of 5. We need to find the maximum number of numbers that can be selected. Hmm, interesting. Let me try to figure this out step by step.
First, I remember that prob... |
olympiads | 2. Condition A: $\sqrt{1+\sin \theta}=a$. Condition B: $\sin \frac{\theta}{2}+\cos \frac{\theta}{2}=a$.
(A) A is a necessary and sufficient condition for B;
(B) A is a necessary condition for B;
(C) A is a sufficient condition for B;
(D) A is neither a necessary condition nor a sufficient condition for B. | <think>
Okay, so I have this problem here with two conditions, A and B. Condition A is the square root of (1 + sin θ) equals a, and Condition B is sin(θ/2) + cos(θ/2) equals a. I need to figure out the relationship between these two conditions. The options are whether A is necessary and sufficient for B, necessary but ... |
olympiads | 18 Given real numbers $x_{1}, x_{2}, \cdots, x_{10}$ satisfy
$$
\sum_{i=1}^{10}\left|x_{i}-1\right| \leqslant 4, \sum_{i=1}^{10}\left|x_{i}-2\right| \leqslant 6,
$$
find the average $\bar{x}$ of $x_{1}, x_{2}, \cdots, x_{10}$. | <think>
Okay, so I have this problem where there are 10 real numbers, x₁ through x₁₀. They satisfy two conditions: the sum of the absolute differences between each xᵢ and 1 is less than or equal to 4, and the sum of the absolute differences between each xᵢ and 2 is less than or equal to 6. I need to find the average of... |
olympiads | 48. A More Interesting Case
- Let's move on to a more interesting case, said the King.
Alice perked up her ears.
- This time, three accused individuals, A, B, and C, stood before the court, began the King. - The court knew that one of them was a knight, one a liar, and one a spy. But who was who, the court did not k... | <think>
Okay, let's try to figure out who the spy is here. So we have three people: A, B, and C. One is a knight (always tells the truth), one is a liar (always lies), and one is a spy (can lie or tell the truth). The court knew this setup but didn't know who was who.
First, A accused B of being the spy. Then B accus... |
olympiads | 1. Find all natural numbers $a, b$, and $c$ that satisfy the equation
$$
\frac{a+b}{a+c}=\frac{b+c}{b+a}
$$
and for which $a b+a c+b c$ is a prime number. | <think>
Okay, let's see. I need to find all natural numbers a, b, and c that satisfy the equation (a + b)/(a + c) = (b + c)/(b + a), and also make sure that ab + ac + bc is a prime number. Hmm. Alright, let's start by understanding what this equation is saying.
First, since a, b, c are natural numbers, they are all po... |
olympiads | G1.2 On the coordinate plane, there are $T$ points $(x, y)$, where $x, y$ are integers, satisfying $x^{2}+y^{2}<10$, find the value of $T$. (Reference: 2002 FI4.3) | <think>
Okay, so I need to find the number of integer coordinate points (x, y) such that x² + y² is less than 10. Hmm, let me think. Alright, first, since both x and y are integers, their squares will be non-negative integers. The inequality x² + y² < 10 means that the sum of their squares has to be less than 10.
So,... |
olympiads | ## Problem Statement
Calculate the definite integral:
$$
\int_{6}^{9} \sqrt{\frac{9-2 x}{2 x-21}} d x
$$ | <think>
Okay, let's see. I need to calculate the definite integral of sqrt((9 - 2x)/(2x - 21)) from 6 to 9. Hmm, that looks a bit complicated. Let me think about how to approach this.
First, maybe I can simplify the expression inside the square root. The numerator is (9 - 2x) and the denominator is (2x - 21). Let me w... |
olympiads | Example 5.18. Find the region of convergence of the power series
$$
x+2!x^{2}+3!x^{3}+\ldots+n!x^{n}+\ldots
$$ | <think>
Okay, so I need to find the region of convergence for the power series x + 2!x² + 3!x³ + … + n!xⁿ + … Hmm, let me recall what I know about power series and convergence.
First, I remember that for any power series, the region of convergence is usually a disk or interval where the series converges, and it might... |
olympiads | 5. In a $4 \times 4$ grid, fill each cell with 0 or 1, such that the sum of the four numbers in each $2 \times 2$ subgrid is odd. There are $\qquad$ different ways to do this. | <think>
Alright, so I need to figure out how many ways there are to fill a 4x4 grid with 0s and 1s such that every 2x2 subgrid has an odd sum. Hmm, okay. Let me start by understanding the problem better. Each 2x2 subgrid must have a sum that's odd. That means in each of these 2x2 sections, there must be either 1 or 3 o... |
olympiads | The 27th question: Find the smallest real number $\lambda$, such that $\left(a_{1}{ }^{n}+\prod_{i=1}^{n} a_{i}, a_{2}{ }^{n}+\prod_{i=1}^{n} a_{i}, \ldots, a_{n}{ }^{n}+\prod_{i=1}^{n} a_{i}\right) \leq$ $\lambda\left(a_{1}, a_{2}, \ldots, a_{n}\right)^{n}$ holds for any positive odd number $n$ and $n$ positive intege... | <think>
Okay, so I need to find the smallest real number λ such that for any positive odd number n and any n positive integers a₁, a₂, ..., aₙ, the vector (a₁ⁿ + product of a_i's, a₂ⁿ + product of a_i's, ..., aₙⁿ + product of a_i's) is less than or equal to λ times the vector (a₁, a₂, ..., aₙ)ⁿ.
Hmm, first, I need to... |
olympiads | 8.70 In chess, the king traverses an $8 \times 8$ chessboard, visiting each square exactly once and returning to the starting point (the king's moves follow the usual rules of chess), and when the centers of the squares the king has visited are connected by line segments in sequence, the resulting path is a non-interse... | <think>
Okay, let me try to tackle this problem step by step. It's about a king's tour on a chessboard, right? The king has to visit every square exactly once, return to the starting point, and the path formed by connecting the centers of the squares shouldn't intersect itself. There are three parts to the problem. Let... |
olympiads | Let $d(n)$ denote the number of positive divisors of the positive integer $n$. Determine those numbers $n$ for which
$$
d\left(n^{3}\right)=5 \cdot d(n)
$$ | <think>
Okay, so I need to find all positive integers n such that the number of divisors of n cubed is five times the number of divisors of n. The function d(n) counts the number of positive divisors of n. Hmm, right. Let me recall that the divisor function, d(n), for a number n with prime factorization n = p₁^a₁ * p₂^... |
olympiads | 2.2. Find all values of $a$ for which the quadratic function $f(x)=a x^{2}-4 a x+1$ takes values, the modulus of which does not exceed 3, at all points of the interval $[0 ; 4]$. In your answer, specify the total length of the intervals to which the found values of $a$ belong. | <think>
Okay, so I have this problem here: I need to find all values of 'a' for which the quadratic function f(x) = a x² - 4a x + 1 has its modulus (absolute value) not exceeding 3 at all points in the interval [0, 4]. Then, I have to specify the total length of the intervals where these 'a' values belong. Hmm, okay. L... |
olympiads | 4. On the plane, there are points $A, B, C$ and $D$ such that $|A B|=|B C|=$ $|A C|=|C D|=10 \mathrm{~cm}$ and $|A D|=17 \mathrm{~cm}$. What is the measure of angle $\Varangle A D B$? | <think>
Okay, so I need to find the measure of angle ADB given these points A, B, C, D on a plane with specific distances between them. Let me start by trying to visualize the problem.
We have four points: A, B, C, and D. The distances between some of them are given. Specifically, AB, BC, AC, and CD are all 10 cm eac... |
olympiads | 5. The teacher wrote thirteen natural numbers on the blackboard and asked Xiaoming to calculate the average (to two decimal places). Xiaoming's answer was 12.43, but the teacher said the last digit was wrong, while the other digits were correct. What should the correct answer be? | <think>
Okay, so let me try to figure out this problem step by step. The teacher wrote thirteen natural numbers, and Xiaoming calculated their average as 12.43, but the teacher said the last digit was wrong while the others were correct. We need to find the correct average. Hmm, let's see.
First, let me recall that a ... |
olympiads | 21.2.10 ** Divide a circle with a circumference of 24 into 24 equal segments. Choose 8 points from the 24 points such that the arc length between any two points is not equal to 3 or 8. How many different ways are there to select such a group of 8 points? | <think>
Okay, so I need to figure out how many ways there are to select 8 points from a circle divided into 24 equal segments, such that the arc length between any two selected points isn't 3 or 8. The circumference is 24, so each segment is 1 unit long. The problem is asking for the number of different ways to choose ... |
olympiads | Problem 4. Positive numbers $a, b, c, d$ are greater than 1. Find the smallest possible value of the expression
$$
\log _{a}\left(a b^{2}\right)+\log _{b}\left(b^{2} c^{3}\right)+\log _{c}\left(c^{5} d^{6}\right)+\log _{d}\left(d^{35} a^{36}\right)
$$ | <think>
Okay, so I have this problem here where I need to find the smallest possible value of an expression involving logarithms with different bases. The variables a, b, c, d are all positive numbers greater than 1. The expression is:
$$
\log _{a}\left(a b^{2}\right)+\log _{b}\left(b^{2} c^{3}\right)+\log _{c}\left(c... |
olympiads | 16. (6 points) As shown in the figure, the book "Classic Fairy Tales" has a total of 382 pages. Then, the number of digit 0 in the page numbers of this book is $\qquad$ . | <think>
Okay, so I need to figure out how many times the digit 0 appears in the page numbers of a book that has 382 pages. Hmm, let's see. I think the best way to approach this is to break it down by each digit position in the page numbers. Pages can be 1-digit, 2-digit, or 3-digit numbers. Since the book has 382 pages... |
olympiads | ## Task A-4.6.
Let $M$ and $N$ be the feet of the altitudes from vertices $A$ and $B$ of an acute-angled triangle $ABC$. Let $Q$ be the midpoint of segment $\overline{M N}$, and $P$ be the midpoint of side $\overline{A B}$. If $|M N|=10$ and $|A B|=26$, determine the length of $|P Q|$. | <think>
Okay, so I need to find the length of PQ in this acute-angled triangle ABC. Let me start by understanding the problem.
First, M and N are the feet of the altitudes from vertices A and B, respectively. That means, if I draw the altitude from A to BC, it meets BC at point M. Similarly, the altitude from B to AC... |
olympiads | $2 \cdot 45$ A sequence of integers that is monotonically increasing, if its first term is odd, the second term is even, the third term is odd, the fourth term is even, $\cdots$, and so on, is called an alternating sequence. The empty set is also considered an alternating sequence. Let $A(n)$ denote the number of alter... | <think>
Alright, let's tackle this problem step by step. The question is about finding the number of alternating sequences A(n) where each term is from the set {1, 2, ..., n}. An alternating sequence is defined as a monotonically increasing sequence where the terms alternate between odd and even, starting with an odd n... |
olympiads | Problem 5. In $\triangle A B C$, a point $E$ is marked on side $B C$ such that $\overline{A E}=\overline{B E}$. Calculate the perimeter of $\triangle A E C$, if $\overline{A C}=5 \mathrm{~cm}$ and $\overline{B C}=9 \mathrm{~cm}$. | <think>
Alright, so I need to find the perimeter of triangle AEC given that AE equals BE, AC is 5 cm, and BC is 9 cm. Hmm, okay. Let me try to visualize this. Triangle ABC with BC being 9 cm. Point E is on BC such that AE equals BE. So, AE and BE are congruent. The perimeter of triangle AEC would be AC + AE + EC. We kn... |
olympiads | 1. Solve the equations $2 \log _{3}(x+2)-\log _{3} x=2$ and $25^{x}=0.008$.
a) Solve the equations.
b) The solutions of the equations are the zeros of a polynomial $p$ of the third degree. The graph of this polynomial also passes through the point (2,7). Write the formula of the polynomial $p$ in its factored form. | <think>
Okay, let's tackle these two equations step by step. First, I need to solve both equations separately and then figure out how their solutions relate to the polynomial in part b. Let's start with the first equation: \(2 \log _{3}(x+2)-\log _{3} x=2\).
Hmm, logarithmic equations can sometimes be tricky because o... |
aops_forum | Define a sequence by $a_0=2019$ and $a_n=a_{n-1}^{2019}$ for all positive integers $n$. Compute the remainder when \[a_0+a_1+a_2+\dots+a_{51}\] is divided by $856$.
[i]Proposed by Tristan Shin[/i] | <think>
Okay, let's tackle this problem. So, we have a sequence defined by a₀ = 2019, and each subsequent term is the previous term raised to the 2019th power. We need to find the remainder when the sum of the first 52 terms (from a₀ to a₅₁) is divided by 856. Hmm, that seems pretty big, especially considering how quic... |
olympiads | In a bookshelf, four volumes of Astrid Lindgren's collected works stand in order, with 200 pages in each volume. A worm living in this collection gnawed a path from the first page of the first volume to the last page of the fourth volume. How many pages did the worm gnaw through?
# | <think>
Okay, let's see. So there's this problem about a worm gnawing through pages in four volumes of books. Each volume has 200 pages. The worm starts at the first page of the first volume and goes to the last page of the fourth volume. The question is, how many pages did the worm go through?
Hmm. First, I need to v... |
olympiads | [ Parallelepipeds (miscellaneous). ] [ Parallelism of lines and planes ] Through the midpoints $M$ and $N$ of the edges $A D$ and $C C 1$ of the parallelepiped $A B C D A 1 B 1 C 1 D 1$, a plane is drawn parallel to the diagonal $D B 1$. Construct the section of the parallelepiped by this plane. In what ratio does it ... | <think>
Alright, so I need to figure out how to construct the section of a parallelepiped when a plane is drawn through points M and N, which are midpoints of edges AD and CC1 respectively, and this plane is parallel to the diagonal DB1. Then, I also need to determine the ratio in which this plane divides the edge BB1.... |
olympiads | 3.108. $\frac{4 \sin \left(\frac{5}{2} \pi+\alpha\right)}{\operatorname{tg}^{2}\left(\frac{3}{2} \pi-\frac{\alpha}{2}\right)-\operatorname{ctg}^{2}\left(\frac{3}{2} \pi+\frac{\alpha}{2}\right)}$.
3.108. $\frac{4 \sin \left(\frac{5}{2} \pi+\alpha\right)}{\tan^{2}\left(\frac{3}{2} \pi-\frac{\alpha}{2}\right)-\cot^{2}\le... | <think>
Okay, let's tackle this trigonometric expression. The problem is:
\[ \frac{4 \sin \left(\frac{5}{2} \pi+\alpha\right)}{\tan^{2}\left(\frac{3}{2} \pi-\frac{\alpha}{2}\right)-\cot^{2}\left(\frac{3}{2} \pi+\frac{\alpha}{2}\right)} \]
First, I need to simplify both the numerator and the denominator. Let's start w... |
olympiads | 【Question 3】 Divisors of 2015 are called factors of 2015. 1 and 2015 obviously divide 2015, and are called trivial factors of 2015. In addition to the trivial factors, 2015 has some non-trivial factors. Therefore, the sum of all non-trivial factors of 2015 is $\qquad$ . | <think>
Okay, let's see. I need to find the sum of all non-trivial factors of 2015. Hmm, the problem says that trivial factors are 1 and 2015, so non-trivial factors are all the other ones. So first, I should find all the factors of 2015 except 1 and 2015, and then add them up.
But how do I find the factors of 2015? ... |
olympiads | 3.2. Let's say that number A hides number B if you can erase several digits from A to get B (for example, the number 123 hides the numbers 1, 2, 3, 12, 13, and 23). Find the smallest natural number that hides the numbers 121, 221, 321, ..., 1921, 2021, 2121.
Natural numbers are numbers used for counting objects.
Answ... | <think>
Okay, so I need to find the smallest natural number that hides all the numbers from 121, 221, 321, up to 2121. Each of these numbers is a four-digit number where the last three digits are "21" and the first digit ranges from 1 to 21. For example, 121 would actually be 0121 if we consider it as a four-digit numb... |
olympiads | 9. (10 points) The Fifth Plenary Session of the 18th Central Committee of the Communist Party of China, which concluded on October 29, 2015, determined the policy allowing universal second children. Xiao Xiao's father, after seeing the news on that day, told Xiao Xiao: The total age of our family this year is 7 times y... | <think>
Okay, let's see. The problem is about figuring out Xiao Xiao's current age based on the information given about their family's total age and the new sibling being added next year. Hmm, let's break it down step by step.
First, the problem states that this year (2015), the total age of the family is 7 times Xiao... |
olympiads | 1. Find the real tuple $(a, b, c, d)$ such that:
$$
\left\{\begin{array}{l}
a b+c+d=3, \\
b c+d+a=5, \\
c d+a+b=2, \\
d a+b+c=6 .
\end{array}\right.
$$
(2013, Dutch National Team Selection Exam) | <think>
Okay, so I need to solve this system of equations with four variables: a, b, c, d. Let me write them down again to make sure I have them right:
1. ab + c + d = 3
2. bc + d + a = 5
3. cd + a + b = 2
4. da + b + c = 6
Hmm, four equations with four variables. Let me see how to approach this. Since all equations ... |
olympiads | 1. In a row, the numbers $1,2,3, \ldots, 2014,2015$ are written. We will call a number from this row good if, after its removal, the sum of all the remaining 2014 numbers is divisible by 2016. Find all the good numbers. | <think>
Okay, so I need to figure out which numbers from 1 to 2015 are "good." A number is good if when you remove it, the sum of the remaining numbers is divisible by 2016. Hmm, let's start by understanding the problem step by step.
First, the original set of numbers is 1, 2, 3, ..., 2015. There are 2015 numbers in t... |
olympiads_ref | SAU
What is the greatest number of integers that can be selected from a set of 2015 consecutive numbers so that no sum of any two selected numbers is divisible by their difference? | <think>
Okay, so I have this problem: I need to find the greatest number of integers that can be selected from a set of 2015 consecutive numbers such that no sum of any two selected numbers is divisible by their difference. Hmm, that's a bit of a mouthful. Let me try to break it down.
First, let's make sure I understa... |
olympiads | 1. Find all real numbers $x$ for which the inequality
$$
|||2-x|-x|-8| \leq 2008
$$
holds. | <think>
Okay, so I have this inequality to solve: |||2 - x| - x| - 8| ≤ 2008. Hmm, that's a lot of absolute value signs. Let me try to break it down step by step.
First, I need to understand the innermost part, which is |2 - x|. Depending on whether 2 - x is positive or negative, the absolute value will change. So, m... |
olympiads | Problem 3. The students from $\mathrm{IV}^{\mathrm{a}}$ class are members of the environmental, literary, or math section. Five students are members of all three sections, and nine students are members of two sections. Eight students are members of both the environmental and literary sections, and the same number are m... | <think>
Okay, let's see. I need to figure out how many students are in the IVa class. They’re part of three sections: environmental, literary, or math. Hmm, this sounds like a problem involving sets and maybe Venn diagrams. Let me recall how those work.
So, there are three sections: let's call them E (environmental),... |
cn_contest | gode 4. In the Cartesian coordinate system, the curve represented by
$$
\frac{|x+y|}{2 a}+\frac{|x-y|}{2 b}=1
$$
$(a, b$ are two distinct positive numbers) is ( ).
(A) Triangle
(B) Square
(C) Non-square Rectangle
(D) Non-square Rhombus | <think>
Okay, so I have this problem here about a curve defined by the equation |x + y|/(2a) + |x - y|/(2b) = 1, where a and b are distinct positive numbers. The question is asking what shape this curve is, with the options being Triangle, Square, Non-square Rectangle, or Non-square Rhombus. Let me try to figure this o... |
olympiads | [ [tangents drawn from one point] [ Midline of a triangle $\quad$]
A circle of radius 1 is inscribed in triangle $ABC$, where $\cos \angle B=0.8$. This circle touches the midline of triangle $ABC$, parallel to side $AC$. Find the side $AC$. | <think>
Okay, so I need to find the length of side AC in triangle ABC where a circle of radius 1 is inscribed. They also mention that the cosine of angle B is 0.8, and the incircle touches the midline of the triangle that's parallel to AC. Hmm, okay, let's break this down step by step.
First, let me recall some concep... |
cn_contest | 10.1. Summation
$$
\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{2^{2}}{3}\right]+\left[\frac{2^{3}}{3}\right]+\cdots+\left[\frac{2^{1000}}{3}\right]
$$ | <think>
Okay, so I need to find the sum of the floor of 2^k divided by 3, starting from k=0 up to k=1000. That is, the sum is [1/3] + [2/3] + [4/3] + [8/3] + ... + [2^1000/3]. Hmm, floor function here means we take the integer part just below the number, right? So, for example, [1/3] would be 0, [2/3] is also 0, [4/3] ... |
olympiads | Problem 3. A rectangle with an area of $99 \mathrm{~cm}^{2}$ has a width of $9 \mathrm{~cm}$. Calculate the area of the square whose perimeter is equal to the perimeter of the given rectangle. | <think>
Okay, let me try to figure out this problem. So, there's a rectangle with an area of 99 cm² and a width of 9 cm. I need to find the area of a square whose perimeter is equal to the perimeter of this rectangle. Hmm, okay. Let me break this down step by step.
First, let's recall some formulas. The area of a rect... |
olympiads | 2. Two circles with radii $R$ and $r$ touch at the point $M$. On the circle with radius $r$, there is a point $N$, diametrically opposite to $M$, and a tangent is drawn at this point. Find the radius of the circle that touches both circles and the tangent passing through point $N$. | <think>
Okay, so I need to find the radius of a circle that touches both the two original circles and the tangent at point N. Let me try to visualize this first. There are two circles that touch at point M. One has radius R and the other r. The point N is diametrically opposite to M on the smaller circle (radius r), an... |
olympiads | 11.065. Find the lateral surface area of a regular triangular pyramid if the plane angle at its vertex is $90^{\circ}$, and the area of the base is $S$. | <think>
Okay, so I need to find the lateral surface area of a regular triangular pyramid where the plane angle at its vertex is 90 degrees, and the base area is S. Hmm, let's start by understanding the problem.
First, a regular triangular pyramid. That should mean the base is an equilateral triangle, right? Because "r... |
olympiads | B1 Given a triangle $A B C, X, Y$ are points on side $A B$, with $X$ closer to $A$ than $Y$, and $Z$ is a point on side $A C$ such that $X Z$ is parallel to $Y C$ and $Y Z$ is parallel to $B C$. Suppose $A X=16$ and $X Y=12$. Determine the length of $Y B$. | <think>
Okay, let's see. I need to find the length of YB in triangle ABC. The problem says that X and Y are on side AB, with X closer to A than Y. So the order on AB must be A, X, Y, B. Then there's a point Z on AC such that XZ is parallel to YC, and YZ is parallel to BC. Given AX is 16 and XY is 12. Hmm, so AX = 16, X... |
olympiads | 3. The equation $x^{2}+a x+5=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}^{2}+\frac{250}{19 x_{2}^{3}}=x_{2}^{2}+\frac{250}{19 x_{1}^{3}}
$$
Find all possible values of $a$. | <think>
Okay, so I need to find all possible values of 'a' for the quadratic equation x² + a x + 5 = 0 such that the equation given involving the roots x₁ and x₂ is satisfied. The equation is x₁² + 250/(19 x₂³) = x₂² + 250/(19 x₁³). Alright, let me try to break this down step by step.
First, since x₁ and x₂ are roots ... |
olympiads | 11. Solve the equation $27^{x}-7 \sqrt[3]{7 \cdot 3^{x}+6}=6$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | <think>
Alright, let's tackle this equation: 27^x - 7 times the cube root of (7*3^x + 6) equals 6. Hmm, okay. So first, 27^x. I know that 27 is 3 cubed, so maybe I can rewrite that as 3^(3x). That could simplify things later. Let me write that down: 3^(3x) - 7 * cube root(7*3^x + 6) = 6.
Now, the cube root is a bit tr... |
olympiads | The students of two schools achieved the following results on a test:
The average score of students from the first school is 74 points, with boys scoring an average of 71 and girls 76 points. The average score of students from the second school is 84 points, with boys scoring an average of 81 and girls 90 points. If t... | <think>
Alright, let's tackle this problem step by step. It's about finding the average score of the girls from both schools combined, given some averages for each school and the overall average for boys. Hmm, okay. Let me try to break this down.
First, let's restate the problem in my own words to make sure I understa... |
olympiads | \section*{Problem 2 - 171232}
For every integer \(a\), determine all real solutions \(x\) of the equation
\[
x^{4}+x^{3}+a^{2} x^{2}+x+1=0
\] | <think>
Okay, let's see. The problem is to find all real solutions x for the equation x⁴ + x³ + a²x² + x + 1 = 0, where a is any integer. Hmm, quartic equation. That's a fourth-degree polynomial. Solving quartic equations can be tricky, but maybe there's a way to factor this or simplify it somehow. Let me think.
First... |
olympiads | 6. As shown in Figure 2, in the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $A B=4, B C=C C_{1}=2 \sqrt{2}$, $M$ is the midpoint of $B C_{1}$, and $N$ is the midpoint of $M C_{1}$. If the angle between the skew lines $A N$ and $C M$ is $\theta$ and the distance is $d$, then $d \sin \theta=$ _ | <think>
Okay, let's try to figure out this problem step by step. So, we have a rectangular prism, which is like a box, right? The vertices are labeled ABCD-A₁B₁C₁D₁. The edges AB = 4, BC = CC₁ = 2√2. M is the midpoint of BC₁, and N is the midpoint of MC₁. We need to find d sinθ, where θ is the angle between the skew li... |
olympiads | Estefânia has five cards marked with the letters $A, B, C, D$ and $E$, stacked in that order from top to bottom. She shuffles the cards by taking the top two and placing them, in reversed order, at the bottom of the stack. The figure shows what happens in the first two times she shuffles the cards.
$, there is a point $P$ on it such that the lines connecting $P$ to the two foci $F_{1}$ and $F_{2}$ are perpendicular to each other. Then the area of $\mathrm{Rt} \triangle P F_{1} F_{2}$ is $(\quad)$.
(A) $|n|$
(B) $m$
(C) $n^{2}$
(D) Uncerta... | <think>
Okay, let's see. I need to solve this problem about a central conic section given by the equation \(\frac{x^2}{m} + \frac{y^2}{n} = 1\), where \(m > |n| > 0\). The question is about finding the area of the right triangle formed by a point \(P\) on the conic and the two foci \(F_1\) and \(F_2\), such that the li... |
olympiads | ## Task 3 - 230823
Let $k$ be a circle with center $M$. Three points $A, B$, and $C$ on $k$ are positioned such that the point $M$ lies inside the triangle $A B C$. Furthermore, $\overline{\angle C A M}=20^{\circ}$ and $\overline{\angle A M B}=120^{\circ}$.
Determine the size of the angle $\angle C B M$ from these co... | <think>
Okay, so I need to find the measure of angle CBM given the conditions. Let me start by visualizing the problem. There's a circle with center M, and triangle ABC is inscribed in this circle such that the center M is inside the triangle. We know two angles: angle CAM is 20 degrees, and angle AMB is 120 degrees. T... |
cn_contest | 3. As shown in Figure $10, A B C D$ is a square with a side length of 1, $U$ and $V$ are points on $A B$ and $C D$ respectively, $A V$ intersects $D U$ at point $P$, and $B V$ intersects $C U$ at point $Q$. Find the maximum value of the area of quadrilateral $P U Q V$.
(2000, Shanghai Junior High School Mathematics Com... | <think>
Okay, so I need to find the maximum area of quadrilateral PUQV in square ABCD with side length 1. Let me visualize this first. ABCD is a square, right? So let me label the square: A is top-left, B is top-right, C is bottom-right, D is bottom-left. Then AB is the top side, BC is the right side, CD is the bottom ... |
olympiads | 3.100. The base of the pyramid is a rectangle, where the angle between the diagonals is $\alpha$. A sphere of radius $R$ is circumscribed around this pyramid. Find the volume of the pyramid if all its lateral edges form an angle $\beta$ with the base. | <think>
Okay, let's try to tackle this geometry problem step by step. So, the problem is about a pyramid with a rectangular base. The angle between the diagonals of the base is α, and there's a circumscribed sphere around the pyramid with radius R. Also, all the lateral edges form an angle β with the base. We need to f... |
olympiads | 2. The arithmetic mean of several consecutive natural numbers is 5 times greater than the smallest of them. How many times is the arithmetic mean less than the largest of these numbers? | <think>
Alright, let's try to tackle this problem step by step. The question says that the arithmetic mean of several consecutive natural numbers is 5 times greater than the smallest of them. We need to find out how many times the arithmetic mean is less than the largest number in the sequence.
First, let me make sure... |
olympiads | 7. From the sides and diagonals of a regular 12-sided polygon, three different segments are randomly selected. The probability that their lengths are the side lengths of a triangle is $\qquad$ | <think>
Okay, let's tackle this probability problem. So, we have a regular 12-sided polygon, and we're supposed to find the probability that three randomly selected different segments (either sides or diagonals) can form a triangle.
First, let me make sure I understand the problem correctly. We need to select three d... |
olympiads | Find the set of positive integers $n$ such that $5^{n}+4$ is a perfect square
untranslated text kept the same format and line breaks. | <think>
Okay, let's see. The problem is to find all positive integers n such that 5^n + 4 is a perfect square. Hmm. Alright, let me start by understanding what the question is asking. We need to find all n where if you take 5 raised to the power of n and then add 4, the result is a perfect square. So, 5^n + 4 = k^2 for... |
cn_contest | 9. (16 points) Let $f(x)=x+\frac{1}{x}(x>0)$. If for any positive number $a$, there exist $m+1$ real numbers $a_{1}, a_{2}, \cdots, a_{m+1}$ in the interval $\left[1, a+\frac{2013}{a}\right]$, such that the inequality
$$
f\left(a_{1}\right)+f\left(a_{2}\right)+\cdots+f\left(a_{m}\right)<f\left(a_{m+1}\right)
$$
holds, ... | <think>
Alright, let me try to tackle this problem. So, we have the function f(x) = x + 1/x defined for x > 0. The question is saying that for any positive number a, there exist m + 1 real numbers a₁, a₂, ..., a_{m+1} in the interval [1, a + 2013/a] such that the sum of the first m f(a_i) is less than f(a_{m+1}). We ne... |
cn_contest | Example 1 Given a quadratic equation $x^{2} + m x - m + 1 = 0$ ($m$ is an integer) has two distinct positive integer roots. Find the value of $m$.
(1996, Sichuan Province Math Competition)
Analysis: For a quadratic equation to have integer roots, it must first satisfy that the discriminant is a perfect square. If we fo... | <think>
Okay, let me try to figure out this problem. The quadratic equation is x² + m x - m + 1 = 0, and we need to find the integer m such that the equation has two distinct positive integer roots. Hmm, okay, so let's start by recalling some basics about quadratic equations.
First, if a quadratic equation ax² + bx + ... |
cn_contest | 4. The maximum value of the function $y=\frac{4-\sin x}{3-\cos x}$ is
保留了源文本的换行和格式。 | <think>
嗯，我现在要解决的问题是找到函数y=(4−sinx)/(3−cosx)的最大值。这个题目看起来是关于求函数极值的，可能需要用微积分的知识来解决。不过，我还不太确定具体应该怎么做，先回忆一下相关的解题方法吧。
首先，这个函数是分式形式的，分子是4−sinx，分母是3−cosx。我记得对于这样的分数函数，求极值通常有两种方法：一种是使用导数，找到导数为零的点，然后判断是否为极大值；另一种可能的方法是利用三角函数的某些特性，或者转化为某种几何意义的问题，比如单位圆上的点之类的。不过可能导数方法更直接些。
好的，那我先试试用导数的方法。首先，我需要求出y的导数，然后找到临界点，也就是导数为零或者导数不存在的点。不过这里分... |
olympiads | 14. How many ordered pairs $(x, y)$ of integers satisfy the equation $y^{4}-8 y^{2}+7=8 x^{2}-2 x^{2} y^{2}-x^{4} ?$ | <think>
Okay, let's try to solve this problem. The question is asking for the number of ordered pairs of integers (x, y) that satisfy the equation:
y⁴ - 8y² + 7 = 8x² - 2x²y² - x⁴.
Hmm, that looks a bit complicated, but maybe I can rearrange terms to make it simpler. Let me first try to move all terms to one side so ... |
olympiads | Prince Zrychlený invited Princess Zpomalená to his castle. When she took too long, he set out to meet her. After two days of travel, he met her at one fifth of her journey. Together, they continued their journey twice as fast as when the princess was traveling alone. They arrived at Prince's castle the second Saturday ... | <think>
Alright, let's try to figure out this problem step by step. So, here's the situation: Prince Zrychlený invited Princess Zpomalená to his castle. The princess started her journey on Friday. But she took too long, so after two days of traveling, the prince decided to go meet her. They met at one fifth of her jour... |
olympiads | 6. Let the complex number $z$ satisfy $|z|=1$. Then the value of $|(z+1)+\mathrm{i}(7-z)|$ cannot be ( ).
(A) $4 \sqrt{2}$
(B) $4 \sqrt{3}$
(C) $5 \sqrt{2}$
(D) $5 \sqrt{3}$ | <think>
Okay, so I need to solve this complex number problem. Let me read it again. The question says: Let the complex number z satisfy |z| = 1. Then the value of |(z + 1) + i(7 - z)| cannot be one of the options given. The options are A) 4√2, B) 4√3, C) 5√2, D) 5√3. My task is to figure out which of these values is no... |
olympiads | 7. In the equation below, all denominators are four-digit numbers. Please fill in each box with a digit to make the equation true.
The translation is provided while preserving the original text's line breaks and format. | <think>
Okay, so I have this equation where all the denominators are four-digit numbers, and I need to fill in each box with a digit to make the equation true. The equation is:
\[
\frac{1}{●●●●} + \frac{1}{●●●●} + \frac{1}{●●●●} = \frac{1}{1999}
\]
Hmm, let's see. Each of those denominators is a four-digit number, wh... |
olympiads | 11. $[7]$ Define $\phi^{\prime}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when
$$
\sum_{\substack{2 \leq n \leq 50 \\ \operatorname{gcd}(n, 50)=1}} \phi^{!}(n)
$$
is divided by 50 . | <think>
Okay, let me try to figure out this problem. So, we need to compute the sum of φ!(n) for all n between 2 and 50 inclusive, where gcd(n, 50) = 1, and then find the remainder when that sum is divided by 50. First, I need to make sure I understand what φ!(n) is. The problem says it's the product of all positive in... |
olympiads | I2.3 Given that $T=\sin 50^{\circ} \times\left(S+\sqrt{3} \times \tan 10^{\circ}\right)$, find the value of $T$. | <think>
Alright, so I have this problem here: T equals sin 50 degrees multiplied by (S plus the square root of 3 times tan 10 degrees). And I need to find the value of T. Hmm, okay. Wait, the problem mentions S, but it doesn't give me any information about what S is. Is that a typo? Maybe it's supposed to be a number i... |
olympiads | 4. Given a tetrahedron $P-ABC$ with three lateral edges $PA, PB, PC$ that are pairwise perpendicular, let the dihedral angles $P-AB-C, P-BC-A$, $P-CA-B$ be $\alpha, \beta, \gamma$ respectively,
then $\frac{\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma}{\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma}=$ $\qquad$ . | <think>
Okay, so I have this tetrahedron P-ABC where the lateral edges PA, PB, and PC are pairwise perpendicular. That means PA is perpendicular to PB, PA is perpendicular to PC, and PB is perpendicular to PC. The problem is asking about the ratio of the sum of the squares of the sines of the dihedral angles α, β, γ to... |
olympiads | 2. [6 points] Solve the equation $\sqrt{x+4}-\sqrt{6-x}+4=2 \sqrt{24+2 x-x^{2}}$. | <think>
Okay, so I need to solve this equation: √(x + 4) - √(6 - x) + 4 = 2√(24 + 2x - x²). Hmm, radical equations can be tricky because of the square roots. Let me start by recalling that to solve equations with square roots, you often need to isolate a radical and then square both sides to eliminate it. But this one ... |
olympiads | B3. In the city hospital, there are 120 patients. After the morning heart rate measurement, the head nurse classified the measurements into 8 frequency classes. Calculate the arithmetic mean of the measurements and the standard deviation.
| Number of heartbeats per minute | Frequency |
| :--- | :---: |
| $60-64.9$ | 2... | <think>
Okay, so I need to calculate the arithmetic mean and the standard deviation for the heart rate measurements of 120 patients in the city hospital. The data is grouped into 8 classes with their respective frequencies. Hmm, let me remember how to do this with grouped data.
First, for the arithmetic mean. When dea... |
olympiads | 1. Find the sum of the numbers:
$$
3+33+333+3333+\cdots+\underbrace{33333 \ldots 3}_{2018 \text { of them }} .
$$ | <think>
Okay, so I need to find the sum of this series: 3 + 33 + 333 + 3333 + ... up to a term with 2018 threes. Hmm, each term seems to have one more three than the previous one. Let me think about how each term is constructed. The first term is 3, which is just 3. The second term is 33, which is 3*10 + 3. The third t... |
olympiads_ref | In a country with $n$ cities, all direct airlines are two-way. There are $r>2014$ routes between pairs of different cities that include no more than one intermediate stop (the direction of each route matters). Find the least possible $n$ and the least possible $r$ for that value of $n$. | <think>
Okay, let's tackle this problem step by step. So, we have a country with n cities, and all direct airlines are two-way. That probably means that if there's a flight from city A to city B, there's also a flight from B to A. The problem states there are r > 2014 routes between pairs of different cities that inclu... |
olympiads | 7. (5 points) Two mischievous children walk against the direction of the moving escalator, from one end to the other. The boy took 100 seconds, and the girl took 300 seconds. It is known that when the escalator is stationary, the boy walks 3 meters per second, and the girl walks 2 meters per second. Then the length of ... | <think>
Okay, let's see. So there's this problem about two kids walking against the direction of a moving escalator. The boy takes 100 seconds, and the girl takes 300 seconds. When the escalator is stationary, the boy walks 3 meters per second and the girl walks 2 meters per second. We need to find the length of the es... |
olympiads | We colored the faces of a regular dodecahedron with red, blue, yellow, and green such that any two adjacent faces have different colors. How many edges does the dodecahedron have where one of the two adjacent faces is blue and the other is green? | <think>
Okay, so I have this problem where I need to figure out how many edges of a regular dodecahedron have one blue face and one green face, given that all the faces are colored with red, blue, yellow, and green, and no two adjacent faces share the same color. Let me start by recalling what a dodecahedron is. A regu... |
cn_contest | Initially 243 Hexagon $A B C D E F$ is inscribed in $\odot O$, $A F // D C, F E // C B, E D // B A, A B+B C=$ $2 C D$. Constructing squares on each of the six sides, the sum of the areas of these six squares is 2008. Find the perimeter of hexagon $A B C D E F$. | <think>
Okay, let me try to figure this problem out step by step. So, we have a regular hexagon named ABCDEF inscribed in a circle with center O. Wait, the problem says "243 Hexagon ABCDEF". Hmm, maybe that's a typo or mistranslation? Maybe it's supposed to be "regular hexagon"? But then again, if it's regular, all sid... |
olympiads | 10. Assume that $a_{i} \in\{1,-1\}$ for all $i=1,2, \cdots, 2013$. Find the least positive number of the following expression
$$
\sum_{1 \leq i<j \leq 2013} a_{i} a_{j} .
$$ | <think>
Okay, so I have this problem here where I need to find the least positive number of the sum ∑_{1 ≤ i < j ≤ 2013} a_i a_j, where each a_i is either 1 or -1. Hmm, let me think about how to approach this.
First, I remember that sums involving products of a_i and a_j can sometimes be related to the square of the ... |
olympiads | 8. Let $a, b, c$ be three distinct positive integers. If $\{a+b, b+c, c+a\}=\left\{n^{2},(n+1)^{2},(n+\right.$ $\left.2)^{2}\right\}$, then the minimum value of $a^{2}+b^{2}+c^{2}$ is $\qquad$ . | <think>
Okay, so I need to solve this problem where we have three distinct positive integers a, b, c. The set containing their pairwise sums, {a+b, b+c, c+a}, is equal to the set {n², (n+1)², (n+2)²} for some integer n. The goal is to find the minimum value of a² + b² + c². Alright, let me break this down step by step.... |
olympiads | Given a point $P(3,1)$ and two lines $l_{1}: x+2 y+3=0, l_{2}: x+2 y-7=0$, find the equation of the circle passing through $P$ and tangent to both $l_{1}$ and $l_{2}$. | <think>
Alright, let's tackle this geometry problem step by step. So, we need to find the equation of a circle that passes through the point P(3,1) and is tangent to both lines l₁: x + 2y + 3 = 0 and l₂: x + 2y - 7 = 0. Hmm, okay. Let me start by recalling some concepts about circles and tangents.
First, the general e... |
olympiads | Task 10.4. In the central cell of a $21 \times 21$ board, there is a chip. In one move, the chip can be moved to an adjacent cell by side. Alina made 10 moves. How many cells can the chip end up in? | <think>
Okay, so I need to figure out how many different cells a chip can end up on after 10 moves, starting from the center of a 21x21 board. Each move is to an adjacent cell by side, which means up, down, left, or right. Hmm, let's start by visualizing the board. A 21x21 grid means there are 21 rows and 21 columns. T... |
olympiads | 8. Let $a_{1}, a_{2}, \cdots, a_{21}$ be a permutation of $1,2, \cdots, 21$, satisfying
$$
\left|a_{20}-a_{21}\right| \geqslant\left|a_{19}-a_{21}\right| \geqslant\left|a_{18}-a_{21}\right| \geqslant \cdots \geqslant\left|a_{1}-a_{21}\right| \text {. }
$$
The number of such permutations is $\qquad$ | <think>
Okay, let me try to figure out this problem. So, we have a permutation of the numbers 1 through 21, let's call them a₁, a₂, ..., a₂₁. The condition given is that the absolute differences between a₂₀ and a₂₁, a₁₉ and a₂₁, all the way down to a₁ and a₂₁ must be in non-increasing order. So, |a₂₀ - a₂₁| ≥ |a₁₉ - a₂... |
olympiads | 4. Let $a_{n}=6^{n}+8^{n}$. Then the remainder when $a_{2018}$ is divided by 49 is $\qquad$ . | <think>
Okay, so I need to find the remainder when \( a_{2018} = 6^{2018} + 8^{2018} \) is divided by 49. Hmm, modular arithmetic, right? Let me think about how to approach this. Maybe using properties of exponents modulo 49?
First, I remember that Euler's theorem says that if two numbers are coprime, then \( a^{\phi(... |
olympiads | 5. [5 points] Given the numbers $\log _{\sqrt{2 x-8}}(x-4), \log _{(x-4)^{2}}(5 x-26), \log _{\sqrt{5 x-26}}(2 x-8)$. For which $x$ are two of these numbers equal, and the third one greater than them by 1? | <think>
Okay, so I need to figure out the value of x where two of the given logarithms are equal, and the third one is greater than them by 1. The logarithms provided are:
1. log base √(2x - 8) of (x - 4)
2. log base (x - 4)^2 of (5x - 26)
3. log base √(5x - 26) of (2x - 8)
Hmm, first, I should make sure that all the... |
olympiads | 2. Let $a, b, c$ be pairwise distinct positive integers such that $a+b, b+c$ and $c+a$ are all square numbers. Find the smallest possible value of $a+b+c$.
(1 mark)
Let $a, b, c$ be pairwise distinct positive integers such that $a+b, b+c$ and $c+a$ are all square numbers. Find the smallest possible value of $a+b+c$. | <think>
Okay, so I have this problem where I need to find three distinct positive integers a, b, and c such that the sums a+b, b+c, and c+a are all perfect squares. And I need to find the smallest possible value of a+b+c. Hmm, interesting. Let me try to break this down.
First, since a, b, and c are distinct positive i... |
olympiads | Point $A$ lies in the plane $\alpha$, the orthogonal projection of segment $A B$ onto this plane is $1, A B=2$. Find the distance from point $B$ to the plane $\alpha$. | <think>
Okay, let's see. I need to find the distance from point B to the plane α. Hmm, the problem says that point A is in plane α, and the orthogonal projection of segment AB onto the plane is 1, while the actual length of AB is 2. Alright, so let me visualize this.
First, since point A is on the plane α, and we're t... |
olympiads | I3.1 Given that $\frac{2 x-3}{x^{2}-x}=\frac{A}{x-1}+\frac{B}{x}$, where $A$ and $B$ are constants. If $S=A^{2}+B^{2}$, find the value of $S$ | <think>
Okay, so I need to solve this problem where I have to find the constants A and B in the partial fraction decomposition of the expression (2x - 3)/(x² - x). Then, once I have A and B, I need to compute S which is the sum of their squares. Hmm, let me think about how to approach this.
First, the given equation i... |
olympiads | 3. Let the sides of the cyclic quadrilateral $ABCD$ be $AB=3, BC=4, CD=5, DA=6$, then the area of quadrilateral $ABCD$ is $\qquad$ . | <think>
Okay, so I need to find the area of a cyclic quadrilateral ABCD with sides AB=3, BC=4, CD=5, and DA=6. Hmm, cyclic quadrilaterals have some special properties. I remember that for cyclic quadrilaterals, the opposite angles add up to 180 degrees. But how does that help me find the area?
Wait, maybe there's a fo... |
inequalities | 4. 212 The river water is flowing, entering a still lake at point $Q$, a swimmer travels downstream from $P$ to $Q$, then across the lake to $R$, taking a total of 3 hours. If he travels from $R$ to $Q$ and then to $P$, it takes a total of 6 hours. If the lake water also flows, at the same speed as the river, then trav... | <think>
Okay, let's try to figure out this problem step by step. Hmm, so there's a river flowing into a still lake at point Q. A swimmer goes downstream from P to Q, then across the lake to R, taking 3 hours. When he goes the reverse route, from R to Q and then upstream to P, it takes 6 hours. Then, if the lake is also... |
olympiads | 7. In how many different ways can 7 different prizes be awarded to 5 students in such a way that each student has at least one prize? | <think>
Okay, so I have this problem here: In how many different ways can 7 different prizes be awarded to 5 students in such a way that each student has at least one prize? Hmm, let me try to figure this out.
First off, the problem is about distributing 7 distinct prizes to 5 distinct students, with the condition th... |
olympiads | 5. Find the number of pairs of integers $(a ; b)$ such that $1 \leqslant a \leqslant 70, 1 \leqslant b \leqslant 50$, and at the same time,
the area $S$ of the figure defined by the system of inequalities
$$
\left\{\begin{array}{l}
\frac{x}{a}+\frac{y}{b} \geqslant 1 \\
x \leqslant a \\
y \leqslant b
\end{array}\right... | <think>
Alright, let's tackle this problem step by step. So, we need to find the number of integer pairs (a, b) where 1 ≤ a ≤ 70, 1 ≤ b ≤ 50, and the area S of the figure defined by the given system of inequalities satisfies that 2S is divisible by 5. Hmm, okay. Let's break this down.
First, let me visualize the syste... |
aops_forum | What is the maximum possible value for the sum of the squares of the roots of $x^4+ax^3+bx^2+cx+d$ where $a$, $b$, $c$, and $d$ are $2$, $0$, $1$, and $7$ in some order? | <think>
Okay, so I need to find the maximum possible value for the sum of the squares of the roots of the quartic polynomial \(x^4 + ax^3 + bx^2 + cx + d\), where the coefficients \(a\), \(b\), \(c\), and \(d\) are 2, 0, 1, and 7 in some order. Hmm, let's break this down step by step.
First, I recall that for a polyno... |
olympiads | 2. In 6th grade class a, there are 24 students. Half of the students in this class play an instrument. Also, $\frac{5}{8}$ of the students in this class sing in the choir. If it is known that more than a third, but less than $\frac{5}{12}$ of the students in 6th grade class a both play an instrument and sing in the cho... | <think>
Okay, let me try to figure out this problem step by step. So, we have a 6th grade class with 24 students. Half of them play an instrument, which means 12 students play an instrument. Then, 5/8 of the students sing in the choir. Let me calculate that: 5/8 of 24 is... 24 divided by 8 is 3, so 5 times 3 is 15. So,... |
olympiads | Kalinin D.A.
In the room, there are 20 chairs of two colors: blue and red. On each chair sits either a knight or a liar. Knights always tell the truth, and liars always lie. Each of the seated individuals claimed that they were sitting on a blue chair. Then they somehow rearranged themselves, after which half of those... | <think>
Okay, let's try to tackle this problem step by step. First, let me make sure I understand it correctly.
We have 20 chairs, some blue and some red. On each chair, there's either a knight (who always tells the truth) or a liar (who always lies). Initially, every person, whether knight or liar, claimed they were... |
olympiads | 2. Each of the 10 people is either a knight, who always tells the truth, or a liar, who always lies. Each of them thought of some natural number. Then the first said: “My number is greater than 1”, the second said: “My number is greater than 2”, ..., the tenth said: “My number is greater than 10”. After that, they, in ... | <think>
Okay, let's try to figure out this problem step by step. So, we have 10 people, each either a knight (always tells the truth) or a liar (always lies). Each person thought of a natural number (so positive integers: 1, 2, 3, ...).
First, each person made a statement in order: the first said, "My number is great... |
olympiads | 2. (3 points) Let two sequences of numbers $\left(x_{0}, x_{1}, \ldots, x_{2009}\right)$, $\left(y_{0}, y_{1}, \ldots, y_{2009}\right)$ be constructed according to the following rules:
a) $x_{0}=12, x_{1}=\frac{1}{3}, y_{0}=4, y_{1}=\frac{1}{18}$
b) $x_{i+1}=x_{i-1}+4 x_{i}$ and $y_{i+1}=y_{i-1}-4 y_{i}$ for $i=1, \ld... | <think>
Alright, let's tackle this problem step by step. So, we have two sequences, x and y, each with 2010 terms (from index 0 to 2009). They start with given initial values and follow certain recurrence relations. The task is to compute the value of x_{2009} y_{2008} + x_{2008} y_{2009}. Hmm, okay. Let me parse this.... |
olympiads | 4. Confused Teacher is teaching the students long division, and he gave the students a question: The smallest four-digit number without repeated digits that can be divided by 13 is $\qquad$ . | <think>
Okay, so the problem is to find the smallest four-digit number without any repeated digits that's divisible by 13. Hmm, let me start by breaking down what exactly is needed here. First, it's a four-digit number, so it has to be between 1000 and 9999. Next, no repeated digits, meaning all four digits must be uni... |
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