source stringclasses 7
values | problem stringlengths 19 6.18k | generation stringlengths 345 68.1k |
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olympiads | Kanel-Belov A.Y.
A unit square is divided into a finite number of smaller squares (the sizes of which may vary). Can the sum of the perimeters of the squares intersecting the main diagonal be greater than 1993? (If a square intersects the diagonal at a single point, this is also considered an intersection.) | <think>
Okay, so I need to figure out if the sum of the perimeters of the squares that intersect the main diagonal of a unit square can be greater than 1993. The unit square is divided into a finite number of smaller squares, which can be of different sizes. If a square touches the diagonal even at just one point, it c... |
olympiads | [ Pigeonhole Principle (continued).]
What is the maximum number of kings that can be placed on a chessboard so that no two of them attack each other
# | <think>
Okay, so I need to figure out the maximum number of kings I can place on a chessboard without any two attacking each other. Hmm, let's start by recalling how a king moves in chess. A king can move one square in any direction: horizontally, vertically, or diagonally. That means if two kings are adjacent in any o... |
olympiads | (1) Given the sequence $\left\{a_{n}\right\}$ with the sum of the first $n$ terms $S_{n}=n^{2}+3 n+4\left(n \in \mathbf{N}^{*}\right)$, then $a_{1}+a_{3}+a_{5}+\cdots+a_{21}=$ $\qquad$ . | <think>
Okay, let's see. I need to find the sum of the odd-indexed terms of the sequence {a_n} up to the 21st term. The problem gives me the sum of the first n terms, S_n = n² + 3n + 4. So, first, maybe I should recall how sequences and their sums relate. The sum S_n is the sum of the first n terms, so a_1 + a_2 + ... ... |
olympiads | 5. We will call a number greater than 25 semi-prime if it is the sum of some two different prime numbers. What is the maximum number of consecutive semi-prime numbers that can be semi-prime? | <think>
Okay, so I have this problem here: A number greater than 25 is called semi-prime if it's the sum of two different prime numbers. The question is asking for the maximum number of consecutive semi-prime numbers that can be semi-prime. Hmm, that's a bit of a mouthful. Let me try to break it down.
First, let's cla... |
olympiads | 5. How many pairs of positive integers $(a, b)$ are there, both not exceeding 10 , such that $a \leq \operatorname{gcd}(a, b) \leq b$ and $a \leq \operatorname{lcm}(a, b) \leq b$ ? | <think>
Okay, so I need to figure out how many pairs of positive integers (a, b) there are, both not exceeding 10, such that two conditions are satisfied:
1. a ≤ gcd(a, b) ≤ b
2. a ≤ lcm(a, b) ≤ b
Hmm, let me start by recalling what gcd and lcm mean. The greatest common divisor (gcd) of two numbers is the largest nu... |
olympiads | 4. Find all roots of the equation
$1-\frac{x}{1}+\frac{x(x-1)}{2!}-\frac{x(x-1)(x-2)}{3!}+\frac{x(x-1)(x-2)(x-3)}{4!}-\frac{x(x-1)(x-2)(x-3)(x-4)}{5!}+$
$+\frac{x(x-1)(x-2)(x-3)(x-4)(x-5)}{6!}=0 . \quad($ (here $n!=1 \cdot 2 \cdot 3 . . . n)$
In the Answer, indicate the sum of the found roots. | <think>
Okay, so I need to find all the roots of this equation:
\(1 - \frac{x}{1} + \frac{x(x-1)}{2!} - \frac{x(x-1)(x-2)}{3!} + \frac{x(x-1)(x-2)(x-3)}{4!} - \frac{x(x-1)(x-2)(x-3)(x-4)}{5!} + \frac{x(x-1)(x-2)(x-3)(x-4)(x-5)}{6!} = 0\)
And then sum them up. Hmm, this seems like a polynomial equation. Let me try to ... |
olympiads | [Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part [Pythagorean Theorem (direct and inverse).
A circle is tangent to side $B C$ of triangle $A B C$ at its midpoint $M$, passes through point $A$, and intersects segments $A B$ and $A C$ at points $D$ and $E$ respect... | <think>
Okay, let's try to tackle this geometry problem step by step. First, I need to visualize the problem and maybe draw a diagram to help me understand the given information. The problem states:
We have triangle ABC, with BC = 12. The midpoint M of BC is the point where a circle is tangent to BC. This circle passe... |
olympiads | ## Task 2 - 300712
Five students from class 7a collected waste paper. Of the total amount they collected together, Marco contributed a quarter, Frank a sixth, Matthias a fifth, and Steffen a tenth. Dirk collected $2 \mathrm{~kg}$ more than Marco.
a) How many kilograms of waste paper did each of these five students co... | <think>
Okay, let's try to solve part a) of this problem. So, we have five students: Marco, Frank, Matthias, Steffen, and Dirk. They all collected waste paper, and we need to find out how much each of them contributed in kilograms.
The problem states that Marco contributed a quarter of the total amount, Frank a sixth... |
olympiads | 5. If $P$ is a moving point on the side $A F$ of a regular hexagon $A B C D E F$ with side length 1, then the minimum value of $\overrightarrow{P D} \cdot \overrightarrow{P E}$ is $\qquad$ . | <think>
Okay, so I need to find the minimum value of the dot product of vectors PD and PE as point P moves along side AF of a regular hexagon with side length 1. Let me try to visualize this first. A regular hexagon has all sides equal and all internal angles equal to 120 degrees. The vertices are labeled A, B, C, D, E... |
olympiads | 2. (3 points) In a puddle, there live three types of amoebas: red, blue, and yellow. From time to time, any two amoebas of different types can merge into one amoeba of the third type. It is known that in the morning, there were 47 red, 40 blue, and 53 yellow amoebas in the puddle, and by evening, only one amoeba remain... | <think>
Okay, let's try to figure out this amoeba problem. So we have three types of amoebas: red, blue, and yellow. Whenever two different types merge, they become the third type. The initial counts are 47 red, 40 blue, and 53 yellow. In the end, there's only one amoeba left. We need to find out its color.
Hmm, first... |
olympiads | Cátia leaves school every day at the same time and returns home by bicycle. When she cycles at $20 \mathrm{~km} / \mathrm{h}$, she arrives home at 4:30 PM. If she cycles at $10 \mathrm{~km} / \mathrm{h}$, she arrives home at 5:15 PM. At what speed should she cycle to arrive home at 5:00 PM? | <think>
Okay, let's see. Cátia leaves school at the same time every day and cycles home. If she goes faster, she gets home earlier, and if she goes slower, she gets home later. The problem is asking what speed she needs to cycle at to arrive home exactly at 5:00 PM.
First, I need to figure out the relationship betwee... |
olympiads | 2 Let the medians of $\triangle A B C$ corresponding to sides $a, b, c$ be $m_{a}, m_{b}, m_{c}$, and the angle bisectors be $w_{a}, w_{b}, w_{c}$. Suppose $w_{a} \cap m_{b}=P, w_{b} \cap m_{c}=Q, w_{c} \cap m_{a}=R$. Let the area of $\triangle P Q R$ be $\delta$, and the area of $\triangle A B C$ be $F$. Find the smal... | <think>
Okay, so I need to find the smallest positive constant λ such that the ratio of the area δ of triangle PQR to the area F of triangle ABC is always less than λ. The problem involves medians and angle bisectors intersecting at points P, Q, and R. Hmm, interesting.
First, let me recall some properties of medians ... |
olympiads | Let's determine all functions $f$ defined on the set of positive real numbers that satisfy the equation $f(x)+2 f\left(\frac{1}{x}\right)=3 x+6$ for all $x$ in their domain.
---
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
... | <think>
Alright, so I need to find all functions f defined on the positive real numbers that satisfy the equation f(x) + 2f(1/x) = 3x + 6 for all x > 0. Hmm, functional equations can be tricky, but maybe I can use substitution here. Let me think...
The equation involves both f(x) and f(1/x). If I substitute x with 1/... |
cn_contest | 1. Given the function
$$
f(x)=\arcsin (\cos x) \text {. }
$$
then the smallest positive period of $f(f(f(x)))$ is $\qquad$ . | <think>
Okay, let's try to figure out the smallest positive period of f(f(f(x))) where f(x) = arcsin(cos x). Hmm, this seems a bit tricky, but I'll take it step by step.
First, I need to understand what the function f(x) = arcsin(cos x) looks like. Arcsin is the inverse sine function, right? Its domain is [-1, 1] and ... |
olympiads | 907. Find the correlation function of a stationary random function, given its spectral density $\delta_{x}(\omega)=D \alpha / \pi\left(\alpha^{2}+\omega^{2}\right)$. | <think>
Okay, so I need to find the correlation function of a stationary random function when given its spectral density. The spectral density provided is δ_x(ω) = D α / [π(α² + ω²)]. Hmm, let me recall what I know about correlation functions and spectral densities.
First, I remember that for a stationary random proc... |
olympiads | ## Task 1 - 110721
Determine all three-digit natural numbers that are simultaneously divisible by 2, 3, 4, 6, 7, 8, 9, 12, and 14! | <think>
Okay, so I need to find all three-digit natural numbers that are divisible by 2, 3, 4, 6, 7, 8, 9, 12, and 14. Hmm, that's a lot of numbers to check. Let me think. Wait, instead of checking each number against all these divisors, maybe I can find the least common multiple (LCM) of these numbers first. Because i... |
olympiads | Problem 2. In a football tournament, seven teams played: each team played once with each other. In the next round, teams that scored twelve or more points qualify. Three points are awarded for a win, one point for a draw, and zero points for a loss. What is the maximum number of teams that can advance to the next round... | <think>
Okay, so I need to figure out the maximum number of teams that can advance to the next round in a football tournament where seven teams play each other once. The teams that score twelve or more points qualify. Each win gives three points, a draw gives one point, and a loss gives zero.
First, let me understand... |
olympiads | 1. Given that $x$ is a real number. Then
$$
\sqrt{2016-x}+\sqrt{x-2000}
$$
the maximum value is $\qquad$ | <think>
Okay, so I need to find the maximum value of the expression √(2016 - x) + √(x - 2000) where x is a real number. Hmm, let's start by figuring out the domain of this function because square roots can't be negative.
First, the expressions inside both square roots need to be non-negative. So, for √(2016 - x) to b... |
olympiads | 4. (5 points) The last three digits of the expression $1 \times 1+11 \times 11+111 \times 111+\cdots+111 \cdots 111$ (2010 1’s) $\times 111 \cdots 111$ (2010 1’s) are $\qquad$ . | <think>
Okay, so I have this problem here where I need to find the last three digits of the sum: 1×1 + 11×11 + 111×111 + ... + a number with 2010 ones multiplied by itself. The question is asking for the last three digits of this entire expression. Let me try to break this down step by step.
First, let's understand th... |
olympiads | 1. Petar and Toso came to visit Misha in city $A$. After some time, they decided to return to their city $B$ and organized the transport in the following way: Toso cycled from city $A$ to city $B$. Misha drove Petar towards city $B$ for some time and then left him to continue on foot, while he immediately turned back t... | <think>
Alright, let's try to figure out this problem. So, we have Petar, Toso, and Misha. They all start from city A at the same time. Toso is cycling to city B at 22 km/h. Misha is driving Petar towards city B for some time, then drops him off to continue on foot. After dropping Petar off, Misha immediately turns bac... |
olympiads | 13.355. A car, having traveled a distance from $A$ to $B$, equal to 300 km, turned back and after 1 hour 12 minutes from leaving $B$, increased its speed by 16 km/h. As a result, it spent 48 minutes less on the return trip than on the trip from $A$ to $B$. Find the original speed of the car. | <think>
Okay, let's try to tackle this problem step by step. So, we have a car that travels from A to B, a distance of 300 km. Then it turns back and heads towards A. During the return trip, after 1 hour and 12 minutes from leaving B, the car increases its speed by 16 km/h. Because of this speed increase, the total tim... |
olympiads | The height of a regular quadrilateral pyramid $S A B C D$ ($S$ - the vertex) is $\sqrt{3}$ times the length of the base edge. Point $E$ is the midpoint of the apothem lying in the face $A S B$. Find the angle between the line $D E$ and the plane $A S C$.
# | <think>
Okay, let's tackle this geometry problem step by step. Hmm, so we have a regular quadrilateral pyramid SABCD with base ABCD. The height of the pyramid is √3 times the length of the base edge. Point E is the midpoint of the apothem in the face ASB. We need to find the angle between the line DE and the plane ASC.... |
olympiads | ## Task B-4.2.
Determine the equation of the circle that is tangent to the line $x-3y+4=0$, and its center lies on the lines $2x-3y-9=0$ and $y+1=0$. | <think>
Okay, so I need to find the equation of a circle that's tangent to the line x - 3y + 4 = 0, and its center lies on both of the lines 2x - 3y - 9 = 0 and y + 1 = 0. Hmm, let me start by breaking down the problem step by step.
First, the center of the circle lies on two lines: 2x - 3y - 9 = 0 and y + 1 = 0. That... |
olympiads | 10. (12 points) The figure is composed of 5 identical squares. The number of triangles that can be formed using the 12 points in the figure as vertices is.
保留了原文的换行和格式，以下是翻译结果：
```
10. (12 points) The figure is composed of 5 identical squares. The number of triangles that can be formed using the 12 points in the figur... | <think>
好的，我现在需要解决这个问题：由5个相同的正方形组成的图形中，使用其中的12个点作为顶点，可以组成多少个三角形。题目没有给出具体的图形，不过根据常见的类似题目，可能这五个正方形是排列成一个十字形，或者某种其他的结构。不过我需要先确定图形的大致结构，或者假设一种可能的排列方式，再进行分析。
首先，题目说明图形由5个相同的正方形组成，共有12个点。那可能的排列方式是什么呢？比如说，如果五个正方形排成一个十字形，中间一个正方形，四个方向各一个，这样可能形成的点数是：中间的四个角点会被周围的四个正方形共享，所以可能总点数会比简单的5×4=20少很多。比如中间的正方形四个顶点，每个相邻的正方形可能共享其中的两个点。可能这样总... |
olympiads | 4. Let $f(x)=a x+b$ where $a$ and $b$ are integers. If $f(f(0))=0$ and $f(f(f(4)))=9$, find the value of $f(f(f(f(1))))+f(f(f(f(2))))+\cdots+f(f(f(f(2014))))$.
(1 mark)
設 $f(x)=a x+b$, 其中 $a 、 b$ 為整數。若 $f(f(0))=0$ 而 $f(f(f(4)))=9$, 求 $f(f(f(f(1))))+f(f(f(f(2))))+\cdots+f(f(f(f(2014)))$ 的值。 | <think>
嗯，我现在要解决这个关于线性函数的问题。题目是说，f(x)=ax + b，其中a和b都是整数。已知f(f(0))=0，并且f(f(f(4)))=9，然后要计算从f(f(f(f(1))))加到f(f(f(f(2014))))的和。看起来需要先找到a和b的值，然后再计算这个和。好的，让我一步一步来想。
首先，我需要根据题目给出的两个条件来确定a和b的值。首先处理第一个条件：f(f(0))=0。
先计算f(0)。代入f(x)=ax + b，当x=0时，f(0)=a*0 + b = b。接下来，f(f(0))就是f(b)，即f(b)=a*b + b。根据条件f(f(0))=0，也就是a*b + b = 0。可以把这个方程... |
olympiads | Example. On average, 6 letters are sent to the department per week, of which 2 are from abroad. Find the probability that 13 letters will be sent in 2 weeks, of which 5 are from abroad. | <think>
Okay, let me try to figure out this probability problem. Hmm, so the question is: On average, 6 letters are sent to the department per week, with 2 of those being from abroad. We need to find the probability that in 2 weeks, 13 letters will be sent, with 5 of them being from abroad.
Alright, first, let me par... |
olympiads | 4. Matkov's father goes to work by car. On Monday, he drove to work at a speed of 60 km/h and arrived one minute late. On Tuesday, he left at the same time, on the same route, but drove at a speed of 65 km/h and arrived one minute early. What is the length of the route that Matkov's father travels from home to work and... | <think>
Okay, so here's this problem about Matkov's father driving to work. Let me try to figure it out step by step.
First, the problem says that on Monday, he drove at 60 km/h and arrived one minute late. Then on Tuesday, he drove at 65 km/h and arrived one minute early. Both days he left at the same time. We need ... |
cn_contest | Example 3. The cubic equation $x^{3}-a x^{2}-a^{2} x+2 a^{2}+a=0$ has three distinct real roots. Find the range of positive real values of $\mathrm{a}$. | <think>
Okay, so I need to find the range of positive real values of 'a' for which the cubic equation \(x^{3} - a x^{2} - a^{2} x + 2 a^{2} + a = 0\) has three distinct real roots. Hmm, let's start by recalling that a cubic equation will have three distinct real roots if its discriminant is positive. But calculating th... |
olympiads | |
How many different seven-digit telephone numbers exist (assuming that a number cannot start with zero)?
# | <think>
Okay, let's see... The problem is asking how many different seven-digit telephone numbers exist, and they can't start with zero. Hmm, telephone numbers... so each digit can be from 0 to 9, right? But the first digit has a restriction.
Let me break this down. A seven-digit number has seven positions. Each posit... |
olympiads | 8. Given the equation of circle $C$ as $x^{2}+y^{2}-8 x+15=0$, if there exists at least one point on the line $y=k x-2(k \in \mathbf{R})$ such that a circle with this point as its center and 1 as its radius has a common point with circle $C$, then the maximum value of $k$ is $\qquad$. | <think>
Okay, so I have this problem here. Let me read it again and make sure I understand what's being asked. The equation of circle C is given as x² + y² - 8x + 15 = 0. There's a line y = kx - 2 where k is a real number. The problem states that if there's at least one point on this line such that a circle with this p... |
olympiads | [Tournaments and tournament tables] $[$ Combinations and arrangements $]$
12 chess players played a round-robin tournament. Then each of them wrote 12 lists. The first list contained only themselves, the $(k+1)$-th list contained those who were in the $k$-th list and those they had defeated. It turned out that for eac... | <think>
Okay, so there's this problem about a round-robin tournament with 12 chess players. Each player plays against every other player, right? So each player has 11 games. Then, each player writes 12 lists. The first list is just themselves. The (k+1)-th list includes everyone from the k-th list plus the people they'... |
olympiads | 6. (20 points) A two-kilogram model of a sports car body was made from aluminum at a scale of $1: 8$. What is the mass of the actual body if it is also entirely made of aluminum? | <think>
Okay, so there's this problem about a model car and figuring out the mass of the actual car. Let me try to break it down.
First, the model is made of aluminum, and the actual car is also entirely made of aluminum. That probably means the density of both the model and the actual car is the same. Density is mas... |
olympiads | 3. (CAN) Find the minimum value of
$$
\max (a+b+c, b+c+d, c+d+e, d+e+f, e+f+g)
$$
subject to the constraints
(i) $a, b, c, d, e, f, g \geq 0$.
(ii) $a+b+c+d+e+f+g=1$. | <think>
Okay, so I need to find the minimum value of the maximum of these five expressions: a+b+c, b+c+d, c+d+e, d+e+f, e+f+g. All variables a through g are non-negative and their sum is 1. Hmm. Let me try to break this down.
First, the problem is asking for the minimal possible value of the largest among those five s... |
olympiads_ref | The altitudes of a triangle are 12, 15 and 20. What is the area of the triangle?
Answer: 150. | <think>
Okay, so I need to find the area of a triangle when its three altitudes are given: 12, 15, and 20. Hmm, let me think. I remember that the area of a triangle can be calculated using the base and the corresponding altitude, right? The formula is (1/2)*base*height. But here, all three altitudes are given, not the ... |
olympiads | ## Task 2 - 340732
Imagine the numbers 1, 2, 3, 4, ... up to 100 written consecutively so that a number $z$ of the form $z=12345678910111213 \ldots .9899100$ is formed.
a) How many digits does $z$ have?
b) 100 digits of the number $z$ should be deleted so that the number $z^{\prime}$ represented by the remaining dig... | <think>
Okay, let's tackle part (a) first. The problem is asking for the total number of digits in the number formed by writing all numbers from 1 to 100 consecutively. So, we need to consider how many digits each number contributes and then sum them up.
First, let's break it down by the number of digits in each numbe... |
olympiads | 2. Given $a, b, c > 0$, find the maximum value of the expression
$$
A=\frac{a^{4}+b^{4}+c^{4}}{(a+b+c)^{4}-80(a b c)^{4 / 3}}
$$ | <think>
Alright, so I have this problem here: given that a, b, c are positive real numbers, I need to find the maximum value of the expression A, which is (a⁴ + b⁴ + c⁴) divided by [(a + b + c)⁴ minus 80 times (abc)^{4/3}]. Hmm, okay. Let me start by understanding the problem and then figure out a strategy.
First, the... |
olympiads | Example \ Let $a, b, c$ be 3 distinct real numbers, and the real-coefficient polynomial $p(x)$, when divided by $x-a$, $x-b$, and $x-c$, yields remainders $a$, $b$, and $c$ respectively. Find the remainder when the polynomial $p(x)$ is divided by $(x-a)(x-b)(x-c)$. | <think>
Okay, let's see. The problem is about finding the remainder when a polynomial p(x) is divided by (x - a)(x - b)(x - c), where a, b, c are distinct real numbers. We know that when p(x) is divided by each of (x - a), (x - b), and (x - c), the remainders are a, b, and c respectively. Hmm, that's interesting. So, t... |
amc_aime | Let $S$ be the [set](https://artofproblemsolving.com/wiki/index.php/Set) of points whose [coordinates](https://artofproblemsolving.com/wiki/index.php/Coordinate) $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le2,$ $0\le y\le3,$ and $0\le z\le4.$ Two distinct points are randomly chosen from $S.$ The [probability... | <think>
Okay, so I need to solve this probability problem where we have a set S of points with integer coordinates (x, y, z) such that 0 ≤ x ≤ 2, 0 ≤ y ≤ 3, and 0 ≤ z ≤ 4. We choose two distinct points at random from S, and we want the probability that the midpoint of the segment connecting them is also in S. Then we h... |
olympiads | 2. (6 points) Arrange some small circles with the same radius according to the pattern shown in the figure: the 1st figure has 6 small circles, the 2nd figure has 10 small circles, the 3rd figure has 16 small circles, the 4th figure has 24 small circles, $\cdots$, following this pattern, the 6th figure has $\qquad$ sma... | <think>
Okay, let's try to figure out how many small circles the 6th figure in this pattern has. The problem gives the first few figures: the 1st has 6, the 2nd has 10, the 3rd has 16, the 4th has 24, and so on. We need to find the 6th one. Hmm, let's start by writing down the given numbers and see if we can spot a pat... |
aops_forum | Let $\mathcal{P}$ be a set of monic polynomials with integer coefficients of the least degree, with root $k \cdot \cos\left(\frac{4\pi}{7}\right)$, as $k$ spans over the positive integers. Let $P(x) \in \mathcal{P}$ be the polynomial so that $|P(1)|$ is minimized. Find the remainder when $P(2017)$ is divided by $1000$.... | <think>
Okay, so I need to find a polynomial P(x) that's monic with integer coefficients, of the least degree, which has roots of the form k·cos(4π/7) where k is a positive integer. Then, among all such polynomials in the set 𝒫, I need to find the one where the absolute value of P(1) is minimized. Finally, compute P(2... |
olympiads_ref | Determine the number of pairs of integers $(m, n)$ such that
$$
\sqrt{n+\sqrt{2016}}+\sqrt{m-\sqrt{2016}} \in \mathbb{Q}
$$ | <think>
Okay, so I need to figure out how many pairs of integers (m, n) satisfy the condition that the expression √(n + √2016) + √(m - √2016) is a rational number. Hmm, let's break this down step by step.
First, I know that the sum of two square roots is rational. Since square roots can sometimes be irrational, but th... |
olympiads | 4. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}=1 \text {, }
$$
$S_{n+1}=4 a_{n}+2$ ( $S_{n}$ is the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$).
Then $a_{n}=$ . $\qquad$ | <think>
Okay, so I have this problem here where I need to find the general term \( a_n \) of a sequence. The given information is that the first term \( a_1 = 1 \), and the sum of the first \( n+1 \) terms, denoted as \( S_{n+1} \), is equal to \( 4a_n + 2 \). Hmm, let me try to unpack this step by step.
First, let's ... |
olympiads | 1st Swedish 1961 Problem 3 For which n is it possible to put n identical candles in a candlestick and to light them as follows. For i = 1, 2, ... , n, exactly i candles are lit on day i and burn for exactly one hour. At the end of day n, all n candles must be burnt out. State a possible rule for deciding which candles ... | <think>
Okay, let's see. The problem is about figuring out for which values of n it's possible to have n identical candles in a candlestick, and each day i (from 1 to n), we light exactly i candles. Each candle burns for exactly one hour once lit, and by the end of day n, all n candles must be completely burnt out. Als... |
olympiads | 7. Given the sets
$$
\begin{array}{l}
A=\left\{(x, y) \mid x=m, y=-3 m+2, m \in \mathbf{Z}_{+}\right\}, \\
B=\left\{(x, y) \mid x=n, y=a\left(a^{2}-n+1\right), n \in \mathbf{Z}_{+}\right\} .
\end{array}
$$
Then the number of integers $a$ such that $A \cap B \neq \varnothing$ is $\qquad$. | <think>
Okay, let's see. I need to find the number of integers a such that the intersection of sets A and B is not empty. So, sets A and B are both sets of ordered pairs (x, y). The problem is giving me their definitions in terms of parameters m and n, which are positive integers. Let me parse this step by step.
First... |
olympiads | 2. Andrey, Boris, Vasily, Gennady, and Dmitry played table tennis in pairs such that every two of them played with every other pair exactly once. There were no draws in the tennis matches. It is known that Andrey lost exactly 12 times, and Boris lost exactly 6 times. How many times did Gennady win?
Om vem: Gennady won... | <think>
Okay, let's see. So we have five people: Andrey, Boris, Vasily, Gennady, and Dmitry. They played table tennis in pairs, and every two of them played with every other pair exactly once. Hmm, wait, the problem says "played in pairs such that every two of them played with every other pair exactly once." I need to ... |
olympiads | 2. In the plane, there are $h+s$ lines, among which $h$ lines are horizontal, and $s$ lines satisfy:
(1) None of them are horizontal lines;
(2) No two of them are parallel;
(3) Any 3 of these $h+s$ lines do not intersect at the same point.
If these $h+s$ lines exactly divide the plane into 1992 regions, find all pairs ... | <think>
Okay, so I have this problem here about lines dividing the plane into regions. The problem states that there are h horizontal lines and s non-horizontal lines. The conditions for the s lines are that none are horizontal, no two are parallel, and any three lines (including the horizontal ones) don't all intersec... |
olympiads | ## Task A-1.3.
In the box, there are $k$ balls labeled (k) for all $k=1,2, \ldots, 50$ (thus one ball labeled (1), two balls labeled (2), ..., 50 balls labeled (50).
Balls are drawn from the box without looking. How many balls must be drawn to ensure that at least 10 balls with the same label have been drawn? | <think>
Okay, so the problem here is about drawing balls from a box with various labels. Each label from 1 to 50 has exactly k balls labeled with the number k. So, for example, there's one ball labeled 1, two balls labeled 2, all the way up to fifty balls labeled 50. The question is asking: how many balls must be drawn... |
olympiads | 6. Find all right-angled triangles with integer side lengths into which two identical circles with a prime radius can be inscribed, such that the circles are externally tangent to each other, both touch the hypotenuse, and each touches a different leg.
(Jaromír Šimša) | <think>
Okay, so I need to find all right-angled triangles with integer side lengths where two identical circles with a prime radius can be inscribed. The circles are externally tangent to each other, both touch the hypotenuse, and each touches a different leg. Hmm, let me break this down step by step.
First, let's vi... |
cn_contest | 1. If real numbers $x, y$ satisfy $4 x^{2}+y^{2}=1$, then the minimum value of $\frac{4 x y}{2 x+y-1}$ is . $\qquad$ | <think>
Okay, so I have this problem here: given real numbers x and y that satisfy the equation 4x² + y² = 1, I need to find the minimum value of the expression (4xy)/(2x + y - 1). Hmm, let me think about how to approach this.
First, the constraint is 4x² + y² = 1, which is an ellipse. The expression we need to minimi... |
olympiads | 4-1. Katya attached a square with a perimeter of 40 cm to a square with a perimeter of 100 cm as shown in the figure. What is the perimeter of the resulting figure in centimeters?
 | <think>
Okay, let me try to figure out this problem. So, Katya attached a square with a perimeter of 40 cm to another square with a perimeter of 100 cm, and we need to find the perimeter of the resulting figure. Hmm, since there's a figure mentioned, but I can't see it, I might have to make some assumptions. But maybe ... |
olympiads | 1.5.2 * Let real numbers $a, x, y$ satisfy the following conditions
$$
\left\{\begin{array}{l}
x+y=2 a-1, \\
x^{2}+y^{2}=a^{2}+2 a-3 .
\end{array}\right.
$$
Find the minimum value that the real number $xy$ can take. | <think>
Okay, so I need to find the minimum value of the product xy given the equations x + y = 2a - 1 and x² + y² = a² + 2a - 3. Hmm, let's see. I remember that when dealing with equations involving sums and products of variables, sometimes expanding squares or using substitution can help. Let me try to recall the ide... |
olympiads | The sequence $\mathrm{Az}\left(a_{i}\right)$ is defined as follows: $a_{1}=0, a_{2}=2, a_{3}=3, a_{n}=\max _{1<d<n}\left\{a_{d} \cdot a_{n-d}\right\}(n=4,5,6, \ldots)$. Determine the value of $a_{1998}$. | <think>
Okay, let's try to figure out this problem. So, we have this sequence Az(a_i) where a_1 = 0, a_2 = 2, a_3 = 3, and for n ≥ 4, a_n is the maximum of the products a_d multiplied by a_{n-d} where d is between 2 and n-2 (since 1 < d < n). The question is asking for a_1998. Hmm, seems a bit complicated, but let's br... |
olympiads | 11. (5 points) As shown in the figure, the "L" shaped paper piece with a perimeter of 52 centimeters can be divided into two identical rectangles along the dotted line. If the longest side is 16 centimeters, then the area of the "L" shaped paper piece is $\qquad$ square centimeters. | <think>
Okay, let me try to figure out this problem. So, we have an "L" shaped paper piece with a perimeter of 52 centimeters. It can be divided into two identical rectangles along the dotted line, and the longest side is 16 centimeters. We need to find the area of this "L" shape.
First, let me visualize the "L" shape... |
aops_forum | Find all functions $ f: \mathbb{Q}^{\plus{}} \rightarrow \mathbb{Q}^{\plus{}}$ which satisfy the conditions:
$ (i)$ $ f(x\plus{}1)\equal{}f(x)\plus{}1$ for all $ x \in \mathbb{Q}^{\plus{}}$
$ (ii)$ $ f(x^2)\equal{}f(x)^2$ for all $ x \in \mathbb{Q}^{\plus{}}$. | <think>
Okay, so I need to find all functions f from the positive rationals to the positive rationals that satisfy two conditions: first, when you add 1 to x, the function value increases by 1, and second, when you square x, the function value squares as well. Hmm, interesting. Let me start by breaking down each condit... |
olympiads | Task 4. (20 points) A four-meter gas pipe has rusted in two places. Determine the probability that all three resulting sections can be used as offsets for gas stoves, if according to regulations, the stove should not be located closer than 1 m to the main gas pipe.
# | <think>
Okay, let's see. I need to find the probability that a 4-meter gas pipe, which has rusted in two places, will result in three sections that are all usable as offsets for gas stoves. The regulation says that the stove shouldn't be closer than 1 meter to the main gas pipe. So each section must be at least 1 meter... |
olympiads | Find the largest four-digit number, all digits of which are different and which is divisible by 2, 5, 9, and 11.
# | <think>
Okay, so I need to find the largest four-digit number where all the digits are different, and it's divisible by 2, 5, 9, and 11. Hmm, let's break this down step by step.
First off, since the number has to be divisible by both 2 and 5, that means it must end with 0. Because any number divisible by 2 ends with ... |
olympiads | 8. Let the arithmetic sequence $\left\{a_{n}\right\}$ have all terms as integers, with the first term $a_{1}=2019$, and for any positive integer $n$, there always exists a positive integer $m$ such that $a_{1}+a_{2}+\cdots+a_{n}=a_{m}$. The number of such sequences $\left\{a_{n}\right\}$ is $\qquad$ | <think>
Okay, let's try to figure out this problem. So we have an arithmetic sequence {a_n} where all terms are integers, starting with a₁ = 2019. The key condition here is that for any positive integer n, there exists a positive integer m such that the sum of the first n terms, S_n = a₁ + a₂ + ... + a_n, equals a_m, w... |
olympiads | 7. In rectangle $A B C D$, point $E$ is located on diagonal $A C$ such that $B C=E C$, point $M$ is on side $B C$ such that $E M=M C$. Find the length of segment $M C$, if $B M=6, A E=3$. If the answer is not an integer, round the result to tenths according to rounding rules. | <think>
Alright, let's try to tackle this geometry problem step by step. So, we have a rectangle ABCD. Let me first draw a rough sketch in my mind to visualize the problem. In a rectangle, opposite sides are equal and all angles are 90 degrees. Let's label the rectangle with A at the bottom-left corner, B at the bottom... |
aops_forum | The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq3$.
Let $m,n\geq1$ be integers. Find the minimal degree $d$ for which there exists a polynomial $f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$, which satisfies $f(k)=F_{m+k}$ for all $k=0,1,...,... | <think>
Okay, let me try to figure out this problem. So, we need to find the minimal degree d of a polynomial f(x) such that f(k) = F_{m+k} for all k = 0, 1, ..., n. Hmm. The Fibonacci sequence is defined with F_1 = F_2 = 1 and F_n = F_{n-1} + F_{n-2} for n ≥ 3.
First, let me make sure I understand the problem correc... |
olympiads | ## Zadatak B-2.1.
Odredite vrijednost brojevnog izraza $\sqrt{2}+\left(\frac{3}{\sqrt[3]{4}-\sqrt[3]{2}+1}-1\right)^{3}-\frac{\sqrt{2}}{2+\sqrt{2}}$.
| <think>
Okay, so I need to solve this numerical expression: √2 + [(3 / (∛4 - ∛2 + 1)) - 1]^3 - (√2 / (2 + √2)). Hmm, let's break this down step by step. First, maybe simplify each part separately and then combine them. Let me start with the last term, √2 / (2 + √2). That seems manageable.
I remember that when you have... |
olympiads | 9-3. Arina wrote down all the numbers from 71 to 81 in a row without spaces, forming a large number 717273...81. Sofia started appending the next numbers to it (i.e., she first appended 82, then 83, ...). She stopped when the large number became divisible by 12. The last number she appended was $N$. What is $N$? | <think>
Okay, so here's the problem: Arina wrote down all the numbers from 71 to 81 in a row without spaces, making a big number like 717273...81. Then Sofia started adding the next numbers to it, like 82, 83, and so on. She kept doing this until the big number was divisible by 12. We need to find the last number she a... |
olympiads | ## Task A-1.3.
Given is an isosceles triangle $A B C$ with $\overline{B C}$ as the base. On the outer side of this triangle, isosceles triangles $C B D, A C E$, and $B A F$ similar to triangle $A B C$ are drawn, with bases $\overline{B C}, \overline{C A}$, and $\overline{B F}$, respectively. If $\varangle C A B=38^{\c... | <think>
Okay, so I need to find the measure of angle EDF in this geometric configuration. Let me start by understanding the problem step by step.
First, we have an isosceles triangle ABC with BC as the base. That means sides AB and AC are equal in length. Since it's isosceles with base BC, the base angles at B and C s... |
olympiads | 3. Let's call a natural number special if one of its digits can be replaced by another digit so that all digits in the resulting number are distinct. Numbers in which all digits are already distinct are also considered special. How many special ten-digit numbers exist? (20 points) | <think>
Okay, so I need to figure out how many ten-digit special numbers exist. Let me start by understanding the problem thoroughly.
First, a natural number is called special if either:
1. All its digits are already distinct, or
2. One of its digits can be replaced by another digit such that the resulting number has... |
aops_forum | Let $n$ be a positive integer such that $12n^2+12n+11$ is a $4$-digit number with all $4$ digits equal. Determine the value of $n$. | <think>
Okay, so I need to find a positive integer n such that the expression 12n² + 12n + 11 results in a four-digit number where all four digits are the same. Hmm, let me break this down step by step.
First, the problem states that the result is a four-digit number with all digits equal. That means the number must b... |
olympiads | Example 5 Find the constant $c$, such that $f(x)=\arctan \frac{2-2 x}{1+4 x}+c$ is an odd function in the interval $\left(-\frac{1}{4}, \frac{1}{4}\right)$. | <think>
Okay, so I need to find the constant \( c \) such that the function \( f(x) = \arctan \frac{2 - 2x}{1 + 4x} + c \) is an odd function on the interval \( \left( -\frac{1}{4}, \frac{1}{4} \right) \). Hmm, let me recall what an odd function is. An odd function satisfies the property \( f(-x) = -f(x) \) for all \( ... |
olympiads | 【Question 5】
Two line segments are parallel, forming a pair of parallel line segments. As shown in the figure, among the 12 edges of a rectangular prism, there are $\qquad$ pairs of parallel line segments. | <think>
Okay, so I need to figure out how many pairs of parallel line segments there are among the 12 edges of a rectangular prism. Let me start by visualizing a rectangular prism. It's like a box, right? It has 6 faces, each of which is a rectangle. Each edge is where two faces meet. There are 12 edges in total.
Now... |
olympiads | 3. Given that $a \cos x + b \cos 2x \geqslant -1 (x \in \mathbf{R})$ always holds. Then the range of $a + b$ is $\qquad$ | <think>
Okay, so I have this problem here: Given that \( a \cos x + b \cos 2x \geqslant -1 \) for all real numbers \( x \). Then I need to find the range of \( a + b \). Hmm, let me try to work through this step by step.
First, the problem states that the expression \( a \cos x + b \cos 2x \) is always greater than or... |
olympiads | ## Task 7 - V00907
The sum of the digits of a two-digit number is 12. If you subtract from this number the number that contains the same digits in reverse order, you get 54. What is the number? | <think>
Okay, let's see. I need to find a two-digit number. The sum of its digits is 12. When I subtract the reverse of the number from itself, I get 54. Hmm, let's break this down.
First, let's represent the two-digit number. If it's a two-digit number, then it can be written as 10 times the tens digit plus the units... |
olympiads | 5. In the Cartesian coordinate system $x O y$, if the line $l$ bisects the circle $x^{2}+y^{2}-2 x-4 y=0$ and does not pass through the fourth quadrant, then the range of the slope of $l$ is | <think>
Okay, let's see. The problem is about finding the range of the slope of a line that bisects a given circle and doesn't pass through the fourth quadrant. Hmm, bisecting a circle usually means that the line passes through the center of the circle, right? Because a diameter bisects the circle, so any line through ... |
olympiads | 4. Given that point $P(x, y)$ lies on the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$, find the maximum value of $2 x-y$.
untranslated text remains the same as the source text. | <think>
Alright, let's see. I need to find the maximum value of 2x - y where the point (x, y) lies on the ellipse x²/4 + y²/9 = 1. Hmm, okay. So this is an optimization problem with a constraint. The constraint is the ellipse equation, and the function to maximize is 2x - y.
First, I remember that for optimization pr... |
olympiads | ## Task 1
What time is it at the end of the 5th lesson if school starts at 7 o'clock and a lesson lasts $45 \mathrm{~min}$?
First break: $10 \mathrm{~min}$
Second break (milk break): $15 \mathrm{~min}$
Third break (yard break): $20 \mathrm{~min}$
Fourth break: $10 \mathrm{~min}$ | <think>
Okay, let's see. The problem is asking what time it will be after the 5th lesson ends. School starts at 7 o'clock, each lesson is 45 minutes long, and there are four breaks in between with different durations. So, I need to calculate the total time elapsed from 7:00 AM until the end of the 5th lesson, consideri... |
olympiads | 2. Find the value of the expression $\sin ^{4} \frac{\pi}{24}+\cos ^{4} \frac{5 \pi}{24}+\sin ^{4} \frac{19 \pi}{24}+\cos ^{4} \frac{23 \pi}{24}$. | <think>
Okay, let's try to tackle this problem step by step. The expression we need to evaluate is:
\[
\sin^{4}\left(\frac{\pi}{24}\right) + \cos^{4}\left(\frac{5\pi}{24}\right) + \sin^{4}\left(\frac{19\pi}{24}\right) + \cos^{4}\left(\frac{23\pi}{24}\right)
\]
First, I notice that the angles given are all fractions o... |
olympiads | For what minimum value of $t$ does the inequality $\sqrt{x y} \leq t(2 x+3 y)$ hold for all non-negative real numbers $x, y$? | <think>
Alright, so I need to find the minimum value of t such that the inequality √(xy) ≤ t(2x + 3y) holds for all non-negative x and y. Hmm, okay. Let me start by understanding what the problem is asking. Essentially, for any x and y that are zero or positive, the square root of their product should be less than or e... |
olympiads | ## Task B-1.5.
The internal angles of a triangle are in the ratio 2 : 3 : 7. Determine the length of the longest side of the triangle, if the shortest side has a length of $1 \mathrm{~cm}$. | <think>
Okay, so I need to figure out the length of the longest side of a triangle where the internal angles are in the ratio 2:3:7, and the shortest side is 1 cm. Hmm, let me start by recalling some basic triangle properties.
First, I know that the sum of the internal angles in any triangle is 180 degrees. Since the... |
olympiads | 3. Find the number of natural numbers $k$, not exceeding 291000, such that $k^{2}-1$ is divisible by 291. | <think>
Okay, so I need to find how many natural numbers k ≤ 291,000 such that k² - 1 is divisible by 291. Hmm, let's see. Divisible by 291 means that k² ≡ 1 mod 291. So, k is congruent to ±1 modulo 291? Wait, not necessarily. Because in some moduli, there might be more solutions. Maybe I need to factor 291 first? Let ... |
olympiads | 8. In the cyclic quadrilateral $A B C D$, $\angle A C B=15^{\circ}, \angle C A D=45^{\circ}, B C=\sqrt{6}$, $A C=2 \sqrt{3}, A D=$ $\qquad$ . | <think>
Okay, let's see. I need to find the length of AD in this cyclic quadrilateral ABCD. The given information includes some angles and sides: angle ACB is 15 degrees, angle CAD is 45 degrees, BC is √6, AC is 2√3. Hmm, cyclic quadrilaterals have properties that the opposite angles sum to 180 degrees, but I'm not sur... |
cn_contest | 7. Given $O$ is the circumcenter of acute $\triangle A B C$, $\angle B A C$ $=60^{\circ}$, extend $C O$ to intersect $A B$ at point $D$, extend $B O$ to intersect $A C$ at point $E$. Then $\frac{B D}{C E}=$ $\qquad$ | <think>
Okay, let's try to tackle this geometry problem. Hmm, we have an acute triangle ABC with circumcenter O. Angle BAC is 60 degrees. We need to extend CO to intersect AB at D and BO to intersect AC at E, then find the ratio BD/CE.
First, let me recall that the circumcenter is the intersection of the perpendicula... |
olympiads | 14. Given the parabola $y^{2}=a x(a>0)$ and the line $x=1$ enclose a closed figure with an area of $\frac{4}{3}$. Then the coefficient of the $x^{-18}$ term in the expansion of $\left(x+\frac{a}{x}\right)^{20}$ is $\qquad$ | <think>
Okay, so I have this problem here. Let me read it again. It says: Given the parabola y² = a x (with a > 0) and the line x = 1 enclose a closed figure with an area of 4/3. Then we need to find the coefficient of the x^{-18} term in the expansion of (x + a/x)^20. Hmm, okay. Let's break this down step by step.
Fi... |
olympiads | 3. The number of zeros of the function $f(x)=x^{2} \ln x+x^{2}-2$ is
The translation is provided while preserving the original text's line breaks and format. | <think>
Okay, so I need to figure out how many zeros the function \( f(x) = x^{2} \ln x + x^{2} - 2 \) has. Hmm, zeros mean where the function crosses the x-axis, right? So I need to find the number of solutions to the equation \( x^{2} \ln x + x^{2} - 2 = 0 \).
First, let me think about the domain of this function. S... |
olympiads | 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. | <think>
Okay, so I have this problem where I need to find the maximum 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, let me think. It's an optim... |
olympiads | In the quadrilateral $\mathrm{ABCD}$, the following angles are known:
$B A C \angle=50^{\circ}, A B D \angle=60^{\circ}, D B C \angle=20^{\circ}, B D C \angle=30^{\circ}$. Is it true that the angle $C A D$ is $30^{\circ}$? | <think>
Okay, let's see. I need to figure out if angle CAD is 30 degrees in quadrilateral ABCD. The given angles are BAC = 50°, ABD = 60°, DBC = 20°, and BDC = 30°. Hmm, maybe I should start by drawing a sketch to visualize the problem.
First, let me draw quadrilateral ABCD. Let's label the points A, B, C, D in order.... |
olympiads | (3) In the Cartesian coordinate plane $x O y$, point $A(5,0)$. For some positive real number $k$, there exists a function $f(x)=a x^{2}, a>0$, such that $\angle Q O A=2 \angle P O A$, where $P(1, f(1))$ and $Q(k, f(k))$, then the range of $k$ is ( ).
(A) $(2,+\infty)$
(B) $(3,+\infty)$
(C) $[4,+\infty)$
(D) $[8,+\infty... | <think>
Okay, so I need to figure out the range of k for which the angle QOA is twice the angle POA, given that A is at (5,0), P is at (1, f(1)), Q is at (k, f(k)), and the function is f(x) = a x² where a > 0. The answer choices are given, so eventually I need to match my conclusion to one of them. Let me start by unde... |
olympiads_ref | Find all the three digit numbers $\overline{a b c}$ such that
$$
\overline{a b c}=a b c(a+b+c)
$$ | <think>
Okay, so I need to find all three-digit numbers \(\overline{abc}\) such that the number itself is equal to the product of its digits \(a\), \(b\), \(c\) multiplied by the sum of its digits \(a + b + c\). In other words, the equation is:
\[
\overline{abc} = a \times b \times c \times (a + b + c)
\]
First, let ... |
olympiads | 8. Evaluate $\int_{0}^{(\sqrt{2}-1) / 2} \frac{\mathrm{d} x}{(2 x+1) \sqrt{x^{2}+x}}$. | <think>
Okay, let's see. I need to evaluate the integral from 0 to (sqrt(2)-1)/2 of 1 over (2x + 1) times the square root of (x² + x) dx. Hmm, that looks a bit complicated, but maybe I can simplify it with substitution or something. Let me first look at the integrand: 1/[(2x + 1)sqrt(x² + x)].
First, let's analyze th... |
olympiads | 9.11. The numbers $a_{1}, a_{2}, \ldots, a_{n}$ are such that the sum of any seven consecutive numbers is negative, and the sum of any eleven consecutive numbers is positive. For what largest $n$ is this possible?
118 Chapter 9. Computation of Sums and Products
$$
\text { 9.3. Sums } S_{k}(n)=1^{k}+2^{k}+\ldots+n^{k}... | <think>
Okay, let's see. The problem is about a sequence of numbers a₁, a₂, ..., aₙ where the sum of any seven consecutive numbers is negative, and the sum of any eleven consecutive numbers is positive. We need to find the largest possible n for which this is possible.
Hmm. So, first, let me parse the problem again. T... |
olympiads | What is the value of $\frac{1^{3}+2^{3}+3^{3}}{(1+2+3)^{2}}$ ? | <think>
Alright, let me try to figure out this problem. So, I need to calculate the value of (1³ + 2³ + 3³) divided by (1 + 2 + 3) squared. Hmm, okay. Let me break this down step by step. First, maybe I should compute the numerator and the denominator separately and then divide them. That seems like a straightforward a... |
olympiads | 7. Let $[a]$ denote the greatest integer not exceeding $a$, for example: $[8]=8, [3.6]=3$.
Some natural numbers can be expressed in the form $[x]+[2 x]+[3 x]$, such as 6 and 3:
$$
\begin{array}{c}
6=\left[\frac{5}{4}\right]+\left[2 \times \frac{5}{4}\right]+\left[3 \times \frac{5}{4}\right], \\
3=[0.8]+[2 \times 0.8]+[... | <think>
Okay, so I need to figure out how many numbers from 1 to 2020 can be expressed as [x] + [2x] + [3x], where [a] is the greatest integer not exceeding a, which is basically the floor function. The examples given are 6 and 3, which can be expressed using x = 5/4 and x = 0.8 respectively. Let me start by understand... |
olympiads | 1. Multiplying the first and fourth, and the second and third factors, we get the equation: $\left(x^{2}+5 x+4\right)\left(x^{2}+5 x+6\right)=360$.
By making the substitution $y=x^{2}+5 x+4$, we get $y^{2}+2 y-360=0$, from which $y_{1}=-20$, $y_{2}=18$. Therefore, we have the equations:
$x^{2}+5 x+24=0, x^{2}+5 x-14=... | <think>
Okay, let me try to figure out this problem step by step. So the original equation given is the product of two quadratic expressions: (x² + 5x + 4)(x² + 5x + 6) = 360. Hmm, that looks a bit complicated, but maybe there's a substitution that can simplify things. The problem mentions substituting y = x² + 5x + 4.... |
olympiads | 6. (15 points) A pedestrian is moving towards a crosswalk along a straight path at a constant speed of 3.6 km/h. At the initial moment, the distance from the pedestrian to the crosswalk is 20 m. The length of the crosswalk is $5 \mathrm{~m}$. At what distance from the crosswalk will the pedestrian be after half a minut... | <think>
Okay, let's see here. So, the problem is about a pedestrian moving towards a crosswalk. The pedestrian's speed is given as 3.6 km/h, and initially, they're 20 meters away from the crosswalk. The crosswalk itself is 5 meters long. The question is asking: after half a minute (which is 30 seconds), how far from th... |
olympiads | The cashier at the gallery sells tickets to visitors with a number according to the order in which they came that day. The first visitor gets a ticket with the number 1, the second with the number 2, etc. However, during the day, the yellow paper on which the tickets were printed ran out, so the cashier had to continue... | <think>
Alright, so I need to figure out how many tickets the cashier sold in total that day. The problem says that she sold as many yellow tickets as red ones. The sum of the numbers on the yellow tickets is 1681 less than the sum of the red ones. Hmm, okay. Let me break this down step by step.
First, let me make sur... |
olympiads | ### 6.244. Solve the equation
$$
\begin{aligned}
& x(x+1)+(x+1)(x+2)+(x+2)(x+3)+(x+3)(x+4)+\ldots \\
& \ldots+(x+8)(x+9)+(x+9)(x+10)=1 \cdot 2+2 \cdot 3+\ldots+8 \cdot 9+9 \cdot 10
\end{aligned}
$$ | <think>
Okay, so I need to solve this equation where the left side is a sum of products starting from x(x+1) up to (x+9)(x+10), and the right side is the sum of products from 1·2 up to 9·10. The equation is saying that these two sums are equal. Hmm, let me think about how to approach this.
First, let me make sure I un... |
olympiads | (3) As shown in the figure, given that $L, M, N$ are the midpoints of the three sides $BC, CA, AB$ of $\triangle ABC$, respectively, and $D, E$ are points on $BC, AB$, respectively, such that $AD, CE$ both bisect the perimeter of $\triangle ABC$. $P, Q$ are the points symmetric to $D, E$ with respect to $L, N$, respect... | <think>
Okay, so I have this geometry problem here. Let me try to parse it step by step. The problem is about triangle ABC with midpoints L, M, N on sides BC, CA, AB respectively. Then there are points D on BC and E on AB such that AD and CE both bisect the perimeter of triangle ABC. Then P and Q are the points symmetr... |
olympiads | As shown in the figure, in $\triangle A B C$, $\angle A B C=90^{\circ}, \quad B C=8$. $D$ and $E$ are points on $A B$ and $A C$ respectively, such that $C D=D E$, $\angle D C B=\angle E D A$. If the area of $\triangle E D C$ is 50, then the length of $A B$ is $\qquad$. | <think>
Okay, so I need to find the length of AB in this right triangle ABC where angle ABC is 90 degrees, BC is 8 units. There are points D on AB and E on AC such that CD equals DE, and the angles ∠DCB and ∠EDA are equal. Also, the area of triangle EDC is 50. Hmm, that's a bit to unpack. Let me try to visualize the fi... |
olympiads | Example 8. Find the directional derivative along the radius vector $\mathbf{r}$ for the function $u=\sin r$, where $r=|\mathbf{r}|$. | <think>
Okay, so I need to find the directional derivative of the function u = sin(r) along the radius vector r, where r is the magnitude of the vector r. Hmm, let's start by recalling what a directional derivative is.
From what I remember, the directional derivative of a function in the direction of a vector v is th... |
olympiads | If a right triangle is drawn in a semicircle of radius $1 / 2$ with one leg (not the hypotenuse) along the diameter, what is the triangle's maximum possible area? | <think>
Okay, so I have this problem here: If a right triangle is drawn in a semicircle of radius 1/2 with one leg (not the hypotenuse) along the diameter, what is the triangle's maximum possible area? Hmm, let's try to visualize this first. A semicircle with radius 1/2, so the diameter is 1 unit long. The right triang... |
olympiads_ref | Find all triples $(x, y, z)$ of real positive numbers, which satisfy the system
$$
\left\{\begin{array}{l}
\frac{1}{x}+\frac{4}{y}+\frac{9}{z}=3 \\
x+y+z \leq 12
\end{array}\right.
$$ | <think>
Okay, so I need to find all triples of positive real numbers (x, y, z) that satisfy the system:
1. (1/x) + (4/y) + (9/z) = 3
2. x + y + z ≤ 12
Hmm, let me start by understanding what the problem is asking. We have two equations here, one is an equality and the other is an inequality. So, all positive real num... |
olympiads | 31. Let $Z=\{1,2, \cdots, n\}, k$ be a positive integer, $\frac{n}{2} \leqslant k \leqslant n$, find the number of mappings $f: Z \rightarrow Z$ that satisfy the following conditions:
(i) $f^{2}=f$;
(ii) The image of $f(x)$, $f(Z)$, contains exactly $k$ distinct elements;
(iii) For each $y \in f(Z)$, there are at most ... | <think>
Alright, let me try to figure out this problem. So, we have a set Z which is {1, 2, ..., n}, and we need to find the number of mappings f: Z → Z that satisfy three conditions:
1. f squared equals f, which I think means that applying f twice is the same as applying it once. So, f(f(x)) = f(x) for all x in Z. Th... |
olympiads | 4. In a certain country, there are 47 cities. Each city has a bus station from which buses run to other cities in the country and possibly abroad. A traveler studied the schedule and determined the number of internal bus routes departing from each city. It turned out that if the city of Lake is not considered, then for... | <think>
Okay, so there's this problem about a country with 47 cities, each having a bus station. The traveler looked at the schedules and figured out the number of internal bus routes from each city. When they exclude the city of Lake, the remaining 46 cities all have different numbers of internal routes departing from... |
olympiads | 14. Simplify the fraction: $\frac{37373737}{81818181}$ | <think>
Okay, so I need to simplify the fraction 37373737 over 81818181. Hmm, that's a big fraction. Let me think about how to approach this. Both the numerator and the denominator are repeating numbers. The numerator is 37373737, which seems like the digits 37 repeating four times. Similarly, the denominator is 818181... |
olympiads | How many pairs of positive integers $(x, y)$ are solutions to the equation $3 x+5 y=501 ?$
untranslated text has been directly translated while preserving the formatting and structure. | <think>
Okay, let's see. The problem is to find the number of pairs of positive integers (x, y) that satisfy the equation 3x + 5y = 501. Hmm, linear Diophantine equation. I remember that these are equations where we look for integer solutions. So first, maybe I need to recall how to solve such equations.
Alright, so t... |
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