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olympiads | ## Problem 1
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(f(x) f(1-x))=f(x) \quad \text { and } \quad f(f(x))=1-f(x)
$$
for all real $x$.
| <think>
Okay, let's see. I need to find all functions f from the real numbers to the real numbers that satisfy two conditions:
1. f(f(x) * f(1 - x)) = f(x) for all real x.
2. f(f(x)) = 1 - f(x) for all real x.
Hmm, functional equations can be tricky. Let me start by analyzing the second equation first because it migh... |
olympiads | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(1 ; 2 ;-3)$
$M_{2}(1 ; 0 ; 1)$
$M_{3}(-2 ;-1 ; 6)$
$M_{0}(3 ;-2 ;-9)$ | <think>
Okay, so I need to find the distance from point M0 to the plane that passes through points M1, M2, and M3. Hmm, let me remember how to do this. I think the formula for the distance from a point to a plane is |Ax + By + Cz + D| divided by the square root of (A² + B² + C²), where Ax + By + Cz + D = 0 is the equat... |
olympiads_ref | Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy
$$
f(f(x)+y)+x f(y)=f(x y+y)+f(x)
$$
for all real numbers $x$ and $y$. | <think>
Okay, so I need to solve this functional equation: f(f(x) + y) + x f(y) = f(x y + y) + f(x) for all real numbers x and y. Hmm, functional equations can be tricky, but let me start by trying some standard techniques. Maybe plugging in specific values for x or y could help simplify things.
First, let's see what ... |
olympiads | 6.200. $\left\{\begin{array}{l}(x+y)^{2}+2 x=35-2 y, \\ (x-y)^{2}-2 y=3-2 x\end{array}\right.$ | <think>
Alright, so I have this system of equations to solve. Let me write them down again to make sure I have them right:
First equation: (x + y)² + 2x = 35 - 2y
Second equation: (x - y)² - 2y = 3 - 2x
Hmm. Both equations are quadratic, which might make things a bit tricky. Let me see if I can expand these equatio... |
olympiads | ## Task 5 - 010615
How many different types of second-class passenger train tickets are needed for a route with 15 stations, if there should be a ticket for every possible connection?
How did you determine the number?
| <think>
Okay, so I need to figure out how many different types of second-class passenger train tickets are required for a route with 15 stations, where each possible connection has a ticket. Hmm, let's start by understanding what a "connection" means here. I think it means a ticket from one station to another, right? S... |
olympiads | Exercise 2. Find all pairs of digits $(a, b)$ such that the integer whose four digits are $a b 32$ is divisible by 99. | <think>
Okay, let's try to solve this problem where we need to find all pairs of digits (a, b) such that the four-digit number ab32 is divisible by 99. Hmm, divisibility by 99... I remember that 99 is 9 times 11, so the number must be divisible by both 9 and 11. That means I need to check the conditions for divisibilit... |
olympiads | In the diagram, $\triangle A B C$ has an area of 84 and $A C=12$. Point $D$ is on $A C$ so that $B D$ is perpendicular to $A C$. What is the length of $B D$ ?
 | <think>
Okay, so I have this problem here about triangle ABC. The area is given as 84, and the length of AC is 12. There's a point D on AC such that BD is perpendicular to AC, and I need to find the length of BD. Hmm, let me visualize this. Triangle ABC with AC as one of its sides, which is 12 units long. BD is an alti... |
olympiads | ## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{x^{3}}{(x-1)(x+1)(x+2)} d x
$$ | <think>
Okay, so I need to find the indefinite integral of x³ divided by (x - 1)(x + 1)(x + 2). Hmm, that looks like a rational function. I remember that when integrating rational functions, partial fraction decomposition is usually the way to go. Let me try that.
First, the degree of the numerator is 3, and the denom... |
aops_forum | A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB| = 4$. Let $E $ be the intersection point of lines $BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle. | <think>
Okay, let's try to figure out the radius of the circle in which quadrilateral ABCD is inscribed. So, first, ABCD is a cyclic quadrilateral, meaning all its vertices lie on a circle. The given side lengths are DA = BC = 2, AB = 4, and CD is less than AB, which is 4. The intersection point E of lines BC and DA fo... |
olympiads | 2.222. $\left(\frac{2-n}{n-1}+4 \cdot \frac{m-1}{m-2}\right):\left(n^{2} \cdot \frac{m-1}{n-1}+m^{2} \cdot \frac{2-n}{m-2}\right) ; m=\sqrt[4]{400}, n=\sqrt{5}$. | <think>
Okay, so I need to simplify this big expression here. Let me first write it down again to make sure I have it right:
\[\left(\frac{2 - n}{n - 1} + 4 \cdot \frac{m - 1}{m - 2}\right) : \left(n^2 \cdot \frac{m - 1}{n - 1} + m^2 \cdot \frac{2 - n}{m - 2}\right)\]
where \( m = \sqrt[4]{400} \) and \( n = \sqrt{5}... |
olympiads | 1. If $a b c d=1$, then
$$
\frac{1}{1+a+a b+a b c}+\frac{1}{1+b+b c+b c d}+\frac{1}{1+c+c d+c d a}+\frac{1}{1+d+d a+d a b}=
$$
$\qquad$ | <think>
Alright, so I have this problem here: given that \( abcd = 1 \), I need to find the value of the sum
\[
\frac{1}{1+a+ab+abc} + \frac{1}{1+b+bc+bcd} + \frac{1}{1+c+cd+cda} + \frac{1}{1+d+da+dab}
\]
Hmm, okay. Let me try to figure this out step by step. First, the condition is \( abcd = 1 \), which suggests tha... |
olympiads | Square $A B C D$ has sides of length 2. The midpoint of $B C$ is $E$. Point $F$ is on $A E$ so that $D F$ is perpendicular to $A E$. Determine the length of $D F$.
 | <think>
Okay, so I need to find the length of DF in square ABCD where DF is perpendicular to AE. Let me try to visualize the square first. Let's name the square ABCD with points labeled clockwise: A at the bottom-left corner, B at the bottom-right, C at the top-right, and D at the top-left. That makes sense since it's ... |
olympiads | ## Example 5. Given the function
$$
f(x)= \begin{cases}0 & \text { if } x \leq 0 \\ c e^{-\alpha x} & \text { if } x>0 \quad(\alpha>0)\end{cases}
$$
For what value of the constant $c$ is the function $f(x)$ a probability density function of some random variable $X$? | <think>
Okay, so I need to figure out the value of the constant c that makes the function f(x) a probability density function (pdf) for some random variable X. Alright, let me start by recalling what a pdf requires. A probability density function must satisfy two main conditions: first, it must be non-negative for all ... |
olympiads | 4. (New Zealand Mathematics Competition) Divide the set of numbers $\{1,2, \cdots, 10\}$ into two groups such that the product of the numbers in the first group $p_{1}$ is divisible by the product of the numbers in the second group $p_{2}$. Find the minimum value of $\frac{p_{1}}{p_{2}}$. | <think>
Okay, so I need to divide the numbers from 1 to 10 into two groups. Let me call them Group 1 and Group 2. The product of Group 1 is p1, and the product of Group 2 is p2. The problem says that p1 should be divisible by p2, which means p1/p2 has to be an integer. But we need to find the minimum value of this rati... |
olympiads | 45. Find the scalar product of vectors $\vec{a}=(3 ; 5)$ and $\vec{b}=(-2 ; 7)$. | <think>
Okay, so I need to find the scalar product of two vectors, right? The vectors are given as \(\vec{a} = (3 ; 5)\) and \(\vec{b} = (-2 ; 7)\). Hmm, scalar product is also known as the dot product, if I remember correctly. Let me recall how that works.
From what I learned, the scalar or dot product of two vectors... |
olympiads | 1. (5-7,8,9) There are 2014 boxes on the table, some of which contain candies, while the others are empty.
On the first box, it is written: “All boxes are empty.”
On the second - “At least 2013 boxes are empty.”
On the third - “At least 2012 boxes are empty.”
...
On the 2014th - “At least one box is empty.”
It is... | <think>
Okay, so I have this problem here with 2014 boxes. Each box has a statement written on it. The first box says, "All boxes are empty." The second one says, "At least 2013 boxes are empty," and so on, down to the 2014th box which says, "At least one box is empty." The rule is that empty boxes have false inscripti... |
cn_contest | 4. From $\{1,2,3,4, \cdots, 20\}$, select four different numbers $a, b, c, d$, satisfying $a+c=b+d$. If the order of $a, b, c, d$ is not considered, then the total number of selection methods is ( . .
(A) 1050
(B) 525
(C) 1140
(D) 190 | <think>
Okay, let's see. The problem is asking for the number of ways to select four different numbers a, b, c, d from the set {1, 2, 3, ..., 20} such that a + c = b + d. And since the order isn't considered, we have to count the combinations, not the permutations. The answer choices are given, and we need to figure ou... |
olympiads | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{2}}(\cos x+1)^{\sin x}$ | <think>
Okay, let's see. I need to find the limit of (cos x + 1) raised to sin x as x approaches π/2. Hmm, so first, let me recall what happens to cos x and sin x when x approaches π/2.
Well, cos(π/2) is 0, right? So as x approaches π/2, cos x approaches 0. Therefore, cos x + 1 would approach 1. And sin x at π/2 is 1... |
olympiads | Is there a 6-digit number that becomes 6 times larger when its last three digits are moved to the beginning of the number, keeping the order of the digits? | <think>
Okay, so I need to figure out if there's a 6-digit number that becomes six times larger when you move its last three digits to the front. Let me try to break this down.
First, let's represent the 6-digit number as two parts: the first three digits and the last three digits. Let's call the first three digits "A... |
olympiads | 26. Xiao Qiaoba arranges the numbers 1 to 25 in a random order, then subtracts $1, 2, 3, \cdots$, 25 from them respectively, and always subtracts the smaller number from the larger one. Adding up the 25 differences, the maximum result is $\qquad$ | <think>
Okay, let me try to figure out this problem. So, Xiao Qiaoba is arranging the numbers 1 to 25 in a random order. Then, for each position, he subtracts the number in that position from the original number (which is 1 to 25 in order) but always subtracts the smaller one from the larger one. Then he adds up all th... |
cn_contest | 3. Given quadrilateral $A B C D$ is an isosceles trapezoid, $A B / / C D, A B=6, C D=16 . \triangle A C E$ is a right triangle, $\angle A E C=90^{\circ}, C E=B C=A D$. Then the length of $A E$ is . $\qquad$ | <think>
Okay, let me try to figure out this problem step by step. So we have an isosceles trapezoid ABCD with AB parallel to CD. AB is 6 units long, and CD is 16 units. There's also a right triangle ACE where angle AEC is 90 degrees, and CE is equal to both BC and AD. We need to find the length of AE.
First, let me re... |
aops_forum | Can we put intengers $1,2,\cdots,12$ on a circle, number them $a_1,a_2,\cdots,a_{12}$ in order. For any $1\leq i<j\leq12$, $|a_i-a_j|\neq|i-j|$? | <think>
Okay, so the problem is asking if we can arrange the integers from 1 to 12 on a circle such that for any two distinct numbers, the absolute difference between them isn't equal to the absolute difference between their positions. That is, if we number the positions on the circle as \(a_1, a_2, \ldots, a_{12}\), t... |
olympiads | 10.210. Find the radius of the circle circumscribed around an isosceles trapezoid with bases 2 and 14 and a lateral side of $10 \mathrm{~cm}$. | <think>
Okay, so I need to find the radius of the circle circumscribed around an isosceles trapezoid. The trapezoid has bases of 2 cm and 14 cm, and each lateral side is 10 cm. Hmm, first things first, I remember that a trapezoid can be cyclic (meaning it can be inscribed in a circle) if and only if it is isosceles. Wa... |
olympiads_ref | 9. (FRG) A sequence $\left(a_{n}\right)$ is defined by means of the recursion
$$
a_{1}=1, \quad a_{n+1}=\frac{1+4 a_{n}+\sqrt{1+24 a_{n}}}{16}
$$
Find an explicit formula for $a_{n}$. | <think>
Alright, so I need to find an explicit formula for the sequence defined by the recursion \( a_1 = 1 \) and \( a_{n+1} = \frac{1 + 4a_n + \sqrt{1 + 24a_n}}{16} \). Hmm, let's see. I remember that sometimes recursions can be transformed into linear recursions or something simpler by making a substitution. The squ... |
olympiads | 13. If the eight digits $1,2,3,4,5,6,7,8$ are randomly permuted, what is the probability that the resulting eight-digit number is divisible by 8 ?
若把 $1 、 2 、 3 、 4 、 5 、 6 、 7 、 8$ 這八個數字任意排列, 所得八位數可被 8 整除的概率是多少? | <think>
Okay, so I need to figure out the probability that a random permutation of the digits 1 through 8 forms an eight-digit number divisible by 8. Hmm. Let's start by recalling the divisibility rule for 8. From what I remember, a number is divisible by 8 if its last three digits form a number that's divisible by 8. ... |
olympiads | Problem 2. Misha, Anton, Katya, and Natasha organized a table tennis tournament. When asked who took which place, they answered:
Misha: - I was neither first nor last.
Anton: - I was not last.
Katya: - I was first.
Natasha: - I was last.
It is known that one of the kids lied, while the other three told the truth. ... | <think>
Okay, so we have this table tennis tournament with four participants: Misha, Anton, Katya, and Natasha. They each made statements about their rankings, and we know that one of them lied while the other three told the truth. We need to figure out who took third place, and it's given that the person in third plac... |
olympiads | 2. Vasya constructed a robot, but he messed up the program, so the robot makes random moves that somehow always lead it to the right point.
A move is a one-kilometer displacement either north (N), east (E), south (S), or west (W).
The robot reached its goal in 22 moves:
# EENENNWNENESEEEEESSSWN.
How many moves woul... | <think>
Okay, so Vasya's robot made 22 moves, but some of those must have been unnecessary because it ended up at the right point anyway. The problem is asking how many moves the robot would need if it didn't make any extra moves. That means we need to figure out the minimum number of moves required to get from the sta... |
olympiads | ## Task A-1.1.
Determine the real number $a$ so that $x=\frac{1}{2}$ is a solution to the equation
$$
\left(\frac{x+1}{1-x}-\frac{1-x}{x+1}-\frac{4 x^{2}}{x^{2}-1}\right)\left(\frac{1}{a^{3}+a^{2}}-\frac{1-a}{a^{2}}-1\right)=\frac{1}{2}
$$ | <think>
Alright, let's tackle this problem step by step. The task is to find the real number \( a \) such that when \( x = \frac{1}{2} \) is substituted into the given equation, it holds true. The equation is a product of two expressions on the left side equals \( \frac{1}{2} \).
First, I need to understand the struct... |
olympiads | 28. Consider a sequence of real numbers $\left\{a_{n}\right\}$ defined by $a_{1}=1$ and $a_{n+1}=\frac{a_{n}}{1+n a_{n}}$ for $n \geq 1$.
Find the value of $\frac{1}{a_{2005}}-2000000$. | <think>
Okay, so I need to find the value of 1/a_{2005} minus 2000000 for this sequence defined by a_1 = 1 and a_{n+1} = a_n / (1 + n a_n) for n ≥ 1. Hmm, let's start by trying to understand the recurrence relation here.
First, maybe I should compute the first few terms manually to see if there's a pattern. Starting ... |
olympiads | 4. Teacher Wang decorated the classroom with stickers, arranging them in a row. She first stuck 15 Bing Dwen Dwen stickers, and then inserted 2 Shuey Rhon Rhon stickers between each adjacent pair of Bing Dwen Dwen stickers. Teacher Wang stuck a total of $\qquad$ stickers. | <think>
Okay, let's see. So Teacher Wang is decorating the classroom with stickers. She arranges them in a row. First, she sticks 15 Bing Dwen Dwen stickers. Then, between each adjacent pair of these Bing Dwen Dwen stickers, she inserts 2 Shuey Rhon Rhon stickers. The question is asking for the total number of stickers... |
cn_contest | Given that there are only three positive integers between the fractions $\frac{112}{19}$ and $\frac{112+x}{19+x}$. Find the sum of all possible integer values of $x$. | <think>
Alright, let's try to figure out this problem. So, we're given two fractions: 112/19 and (112 + x)/(19 + x). And there are exactly three positive integers between them. We need to find all possible integer values of x and sum them up. Hmm, okay.
First, let me understand the problem. Between the two fractions, ... |
olympiads | 1. (16 points) Fresh mushrooms contain $80 \%$ water by mass, while dried mushrooms contain $20 \%$ water. How many kg of dried mushrooms can be obtained from 20 kg of fresh mushrooms? | <think>
Okay, so I need to figure out how many kilograms of dried mushrooms can be obtained from 20 kg of fresh mushrooms. The problem says that fresh mushrooms have 80% water by mass, and dried mushrooms have 20% water. Hmm, I remember that when something dries, the water content decreases, but the actual solid part (... |
olympiads | Ana and Luíza train every day for the Big Race that will take place at the end of the year at school, each always at the same speed. The training starts at point $A$ and ends at point $B$, which are $3000 \mathrm{~m}$ apart. They start at the same time, but when Luíza finishes the race, Ana still has $120 \mathrm{~m}$ ... | <think>
Okay, let's try to figure out this problem step by step. So, Ana and Luíza are training for a race from point A to point B, which is 3000 meters apart. They both start at the same time, but when Luíza finishes the race, Ana still has 120 meters left. Now, Luíza wants to give Ana a head start by starting a few m... |
olympiads | In one American company, every employee is either a Democrat or a Republican. After one Republican decided to become a Democrat, the number of each became equal. Then, three more Republicans decided to become Democrats, and the number of Democrats became twice the number of Republicans. How many employees are there in ... | <think>
Okay, let's see. So, there's this problem about a company where everyone is either a Democrat or a Republican. Initially, there's a certain number of Democrats and Republicans, and then after some people switch parties, the numbers change in a couple of steps. The question is asking for the total number of empl... |
cn_contest | 3. Let the function $f: \mathbf{R} \rightarrow \mathbf{R}$, for any $x, y \in \mathbf{R}$, we have $f(x+y)=f(x)+f(y)$, and when $x<0$, $f(x)<0$. If the graph of $y=f(x)$ intersects or is tangent to the line $2007 x + 2008 y = 2009$, then the number of intersection or tangent points is ( ).
(A) 1
(B) 2
(C) 3
(D) Uncerta... | <think>
Okay, let me try to work through this problem step by step. First, let's parse the question carefully.
We have a function \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfies two conditions:
1. For any \( x, y \in \mathbb{R} \), \( f(x + y) = f(x) + f(y) \). This is Cauchy's functional equation. Typically... |
olympiads | Problem 4. In triangle $X Y Z$, the lengths of the sides are 2, 7, and $5 \sqrt{3}$. Find the area of the figure consisting of those and only those points $A$ inside triangle $X Y Z$ for which the condition $A X^{2}+A Y^{2}+A Z^{2} \leqslant 43$ is satisfied. | <think>
Okay, so I need to find the area of a certain region inside triangle XYZ where all points A satisfy the condition AX² + AY² + AZ² ≤ 43. The triangle has sides of lengths 2, 7, and 5√3. Hmm, let me start by recalling some geometry concepts related to this.
First, the expression AX² + AY² + AZ² reminds me of the... |
olympiads | Example 1. Find the volume of the body bounded by the surfaces
$$
x=17 \sqrt{2 y}, \quad x=2 \sqrt{2 y}, \quad z=1 / 2-y, \quad z=0
$$ | <think>
Okay, so I need to find the volume bounded by these four surfaces: x = 17√(2y), x = 2√(2y), z = 1/2 - y, and z = 0. Hmm, let me start by visualizing what these surfaces look like in 3D space.
First, the surfaces x = 17√(2y) and x = 2√(2y) are both parabolic cylinders opening along the y-axis, right? Because x ... |
olympiads | Example 1 Find all positive integer solutions $(x, y)$ for $\left|12^{x}-5^{y}\right|=7$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | <think>
Okay, so I need to solve this equation: |12^x - 5^y| = 7, where x and y are positive integers. Hmm, let's start by understanding what this equation means. The absolute value tells me that either 12^x - 5^y = 7 or 5^y - 12^x = 7. Both cases need to be considered.
First, let's consider small values of x and y si... |
olympiads | 1. (5 points) The left figure below is the recently discovered Archimedes' "Stomachion" puzzle, dividing a square into 14 polygons: After research, experts found that the 14 pieces can be correctly drawn on a square with a side length of $12 \mathrm{~cm}$, as shown in the figure. Question: The area of the gray piece is... | <think>
Okay, so I need to find the area of the gray piece in the Stomachion puzzle. The square has a side length of 12 cm, so first off, the total area of the square should be 12 cm multiplied by 12 cm, which is 144 square centimeters. The puzzle is divided into 14 polygons, and I guess the gray piece is one of them. ... |
olympiads | 6. A circle with diameter $A B$ intersects segments $A C$ and $B C$ at points $M$ and $N$, respectively, and the length of segment $M N$ is equal to the radius of the circle. Find the area of quadrilateral $A B N M$, given that $A C=12$ and $B C=8$.
 There is a magical tree with 58 fruits on it. On the first day, 1 fruit will fall from the tree. Starting from the second day, the number of fruits that fall each day is 1 more than the previous day. However, if the number of fruits on the tree is less than the number that should fall on a certain day, t... | <think>
Okay, let's see. There's this magical tree with 58 fruits. On the first day, 1 fruit falls. Then each subsequent day, the number of fruits that fall increases by 1 compared to the previous day. But if there are fewer fruits left than the number that should fall on a certain day, then it starts over by dropping ... |
olympiads | 9. (10 points) If the digits of a natural number can be divided into two groups such that the sum of the digits in each group is equal, then the number is called a "balanced number". For example, 25254 is a "balanced number" because $5+2+2=4+5$. If two consecutive natural numbers are both "balanced numbers", then this ... | <think>
Okay, so I need to find the smallest pair of consecutive natural numbers where both are balanced numbers. Then, I have to add those two numbers together to get the answer. Let me start by understanding what exactly a balanced number is.
A balanced number is one where you can split its digits into two groups, ... |
olympiads | 6. (10 points) The sum of 13 different natural numbers is 996, and the sum of the digits of these numbers are all equal. Find these 13 numbers.
The text above is translated into English, preserving the original text's line breaks and format. | <think>
Okay, so I need to find 13 different natural numbers such that their sum is 996, and all of these numbers have the same sum of their digits. Hmm, let me break this down step by step.
First, natural numbers are positive integers, right? So each of the 13 numbers must be unique, positive integers. Their total su... |
olympiads | 7.199. $\log _{2} \log _{3}\left(x^{2}-16\right)-\log _{1 / 2} \log _{1 / 3} \frac{1}{x^{2}-16}=2$. | <think>
Okay, so I have this logarithmic equation to solve: log base 2 of log base 3 of (x² - 16) minus log base 1/2 of log base 1/3 of 1/(x² - 16) equals 2. Hmm, that looks a bit complicated with all the logs and different bases. Let me try to break it down step by step.
First, let me write the equation again to make... |
olympiads | 5. Xiaoming's honor stand has 3 gold medals, 2 silver medals, and 1 bronze medal, arranged from left to right in the order of gold, silver, and bronze. At the sports meet, he won 1 more gold medal and 1 more silver medal, and he needs to place these two medals on the honor stand while ensuring all medals are still arra... | <think>
Okay, let's see. So Xiaoming has an honor stand with medals arranged in a specific order: gold, silver, bronze. Originally, he has 3 gold, 2 silver, and 1 bronze medal. Then he wins 1 more gold and 1 more silver medal. He needs to add these two new medals to the stand while keeping the order from left to right ... |
aops_forum | Given a natural number $k$, find the smallest natural number $C$ such that
$$\frac C{n+k+1}\binom{2n}{n+k}$$is an integer for every integer $n\ge k$. | <think>
Okay, let's try to tackle this problem step by step. The question is asking for the smallest natural number \( C \) such that when you take \( C \) divided by \( n + k + 1 \) and multiply it by the binomial coefficient \( \binom{2n}{n + k} \), the result is an integer for every integer \( n \geq k \).
First, ... |
olympiads | 7.2. Find the largest natural number in which each digit, starting from the third, is equal to the sum of all the previous digits of the number. | <think>
Okay, so I need to find the largest natural number where each digit, starting from the third one, is equal to the sum of all the previous digits. Hmm, let me try to understand the problem first.
Let me rephrase it: if the number has digits d1, d2, d3, ..., dn, then for each digit from the third position onwar... |
olympiads | In a single-round-robin tournament, 10 chess players are participating. What is the minimum number of rounds after which a sole winner can be determined prematurely? (In each round, the participants are paired.
Win - 1 point, draw - 0.5 points, loss - 0).
# | <think>
Okay, so there's this problem about a chess tournament. Let me try to wrap my head around it. The question is: In a single-round-robin tournament with 10 players, what's the minimum number of rounds after which we can determine a sole winner prematurely? Each round, the participants are paired. Wins give 1 poin... |
olympiads | 1. In a pile of 200 coins, $2 \%$ are gold coins and the rest are silver. Simple Simon removes one silver coin every day until the pile contains $20 \%$ gold coins. How many silver coins does Simon remove? | <think>
Okay, let's try to figure out this problem. So, there's a pile of 200 coins. 2% of them are gold coins, and the rest are silver. Simon removes one silver coin every day until the pile contains 20% gold coins. We need to find out how many silver coins Simon removes.
First, let me break down the initial numbers.... |
olympiads | Example 4. Knowing the fundamental system of solutions $y_{1}=\ln x, y_{2}=x$ of the corresponding homogeneous equation, find a particular solution of the equation
$$
x^{2}(1-\ln x) y^{\prime \prime}+x y^{\prime}-y=\frac{(1-\ln x)^{2}}{x}
$$
satisfying the condition $\lim _{x \rightarrow+\infty} y=0$. | <think>
Okay, let's tackle this differential equation problem. So, we have a nonhomogeneous equation, and we know the fundamental system of solutions for the corresponding homogeneous equation: y₁ = ln x and y₂ = x. The goal is to find a particular solution that approaches zero as x approaches infinity. Alright, let me... |
olympiads | 27.10. (Belgium, 81). Two players $A$ and $B$ observe a boy who continuously flips a coin. The results of the flips are recorded sequentially using letters: on the $k$-th position in the sequence, the letter $O$ or the letter $P$ is placed depending on whether "heads" or "tails" appears on the $k$-th flip, respectively... | <think>
Okay, so there's this problem about two players, A and B, watching a boy flip a coin. The coin flips are recorded as a sequence of O's and P's—O for heads and P for tails, I think? The problem states that player A thinks the triplet 000 (three O's in a row) will appear before the triplet OPO. Player B is bettin... |
olympiads | 7. (6 points) Cars A and B depart from locations $A$ and $B$ respectively at the same time and travel towards each other. Car A travels at 40 kilometers per hour, and Car B travels at 60 kilometers per hour. After reaching $B$ and $A$ respectively, they immediately return. On the return trip, Car A's speed increases by... | <think>
Okay, let's tackle this problem step by step. So, we have two cars, A and B, starting from points A and B respectively, heading towards each other. Car A goes at 40 km/h, and Car B at 60 km/h. After they reach their destinations (B for A and A for B), they immediately turn around. On the return trip, Car A's sp... |
olympiads | 7.1.1. (12 points) Find the greatest negative root of the equation
$$
\frac{\sin \pi x - \cos 2 \pi x}{(\sin \pi x - 1)^2 + \cos^2 \pi x - 1} = 0
$$ | <think>
Okay, so I need to find the greatest negative root of the equation given. Let me first write it down again to make sure I have it right:
$$
\frac{\sin \pi x - \cos 2 \pi x}{(\sin \pi x - 1)^2 + \cos^2 \pi x - 1} = 0
$$
Alright, the equation is a fraction equal to zero. For a fraction to be zero, the numerator... |
olympiads | 13.079. The first tourist, after riding a bicycle for 1.5 hours at a speed of $16 \mathrm{~km} / \mathrm{h}$, makes a stop for 1.5 hours, and then continues the journey at the initial speed. After 4 hours from the departure of the first tourist, the second tourist sets off in pursuit on a motorcycle at a speed of $56 \... | <think>
Okay, so I have this problem here about two tourists. The first one rides a bicycle for 1.5 hours at 16 km/h, then stops for 1.5 hours, and then continues at the same speed. The second tourist starts 4 hours after the first one, going on a motorcycle at 56 km/h. I need to find out the distance they cover before... |
olympiads | [ [Motion Problems ]
Author: Shnol D. $\cdot$.
Daddy, Masha, and Yasha are walking to school together. While Daddy takes 3 steps, Masha takes 5 steps. While Masha takes 3 steps, Yasha takes 5 steps. Masha and Yasha counted that together they made 400 steps. How many steps did Daddy take? | <think>
Okay, let me try to figure out this problem. So, Daddy, Masha, and Yasha are walking to school. The problem says that while Daddy takes 3 steps, Masha takes 5 steps. Then, while Masha takes 3 steps, Yasha takes 5 steps. Together, Masha and Yasha made 400 steps. We need to find out how many steps Daddy took. Hmm... |
olympiads | ## Task Condition
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{(3-n)^{2}+(3+n)^{2}}{(3-n)^{2}-(3+n)^{2}}
$$ | <think>
Okay, so I need to find the limit of this sequence as n approaches infinity:
$$
\lim _{n \rightarrow \infty} \frac{(3-n)^{2}+(3+n)^{2}}{(3-n)^{2}-(3+n)^{2}}
$$
Hmm, let me start by expanding both the numerator and the denominator to see if I can simplify the expression. Maybe that will make it easier to find... |
cn_contest | 9.5. On an infinitely large chessboard, the distance between any two squares is defined as the minimum number of steps a King needs to move from one square to another. Given three squares that are pairwise 100 steps apart, find the number of squares that are 50 steps away from each of these three squares. | <think>
Okay, so I have this problem about an infinite chessboard where the distance between two squares is defined as the minimum number of steps a king needs to move from one to another. A king can move one square in any direction, including diagonally, right? So that means the distance between two squares is the Che... |
olympiads | 6. If $a>0, b>0$, and $\arcsin a+\arcsin b=\frac{\pi}{2}, m=\log _{2} \frac{1}{a}+\log _{2} \frac{1}{b}$, then the range of $m$ is $\qquad$ . | <think>
Okay, let's try to tackle this problem step by step. So, we have two positive numbers a and b, and the sum of their arcsin values is π/2. We need to find the range of m, where m is the sum of the log base 2 of 1/a and 1/b. Hmm, let me break this down.
First, let's recall that arcsin(x) is defined for x in the ... |
olympiads | 7.4. On the island, there live knights and liars. Knights always tell the truth, and liars always lie. Some of the inhabitants claimed that there is an even number of knights on the island, while the others claimed that there is an odd number of liars on the island. Is the number of inhabitants on the island even or od... | <think>
Okay, let's try to figure out this knights and liars problem. So, the setup is that on the island, there are knights who always tell the truth and liars who always lie. Some inhabitants say there's an even number of knights, and others say there's an odd number of liars. We need to determine if the total number... |
olympiads | In a convex hexagon, two random diagonals are chosen independently of each other. Find the probability that these diagonals intersect inside the hexagon (inside means not at a vertex).
# | <think>
Okay, so I need to find the probability that two randomly chosen diagonals in a convex hexagon intersect inside the hexagon. Hmm, let me start by understanding the problem better.
First, a convex hexagon. So, all its interior angles are less than 180 degrees, and all the diagonals lie inside the hexagon. A dia... |
olympiads | Example 6. Solve the equation
$$
\sqrt[3]{2 x-1}+\sqrt[3]{x-1}=1
$$ | <think>
Alright, let me try to solve this equation: the cube root of (2x - 1) plus the cube root of (x - 1) equals 1. Hmm, cube roots can be tricky, but maybe if I let each cube root be a variable, I can simplify the equation. Let me see.
So, let me set a = cube_root(2x - 1) and b = cube_root(x - 1). Then the equation... |
olympiads | Consider the sequence $\left(a_{n}\right)$ with the first term $a_{1}=1$ and defined by the following relation: For any positive integer $n$,
$$
a_{n}=a_{\left\lfloor\frac{n}{2}\right\rfloor}+a_{\left\lceil\frac{n}{2}\right\rceil}
$$
Express $a_{n}$ as a function of $n$. | <think>
Alright, let's try to tackle this problem. So, we have a sequence where the first term is a₁ = 1, and each term is defined by aₙ = a⌊n/2⌋ + a⌈n/2⌉. The task is to express aₙ as a function of n. Hmm, okay. Let me start by understanding the recurrence relation here.
First, let me write down the first few terms t... |
olympiads | 1. A bus leaves the station at exactly 7:43 a.m. and arrives at its destination at exactly 8:22 a.m. on the same day. How long, in minutes, was the bus trip? | <think>
Okay, let's see. The problem is about calculating the duration of a bus trip. The bus leaves the station at 7:43 a.m. and arrives at 8:22 a.m. on the same day. I need to figure out how many minutes that is. Hmm, time problems can sometimes be tricky because of the different units, but let's take it step by step... |
olympiads | Problem 9.5. Point $M$ is the midpoint of side $B C$ of triangle $A B C$, where $A B=17$, $A C=30, B C=19$. A circle is constructed with side $A B$ as its diameter. An arbitrary point $X$ is chosen on this circle. What is the minimum value that the length of segment $M X$ can take?
;
draw((0,0)--(2,0)--(2,2)--(0,... | <think>
Okay, so the problem is about a square inscribed in a circle with radius 6. Then there's a quarter circle inscribed in the square, and we need to find the area inside the larger circle but outside the quarter circle, which is given as nπ. We have to find n.
First, let me visualize this. A square inscribed in a... |
aops_forum | Find
$$ \inf_{\substack{ n\ge 1 \\ a_1,\ldots ,a_n >0 \\ a_1+\cdots +a_n <\pi }} \left( \sum_{j=1}^n a_j\cos \left( a_1+a_2+\cdots +a_j \right)\right) . $$
| <think>
Okay, so I need to find the infimum of this sum: the sum from j=1 to n of a_j times cosine of the sum from k=1 to j of a_k. And this is under the conditions that n is at least 1, each a_j is positive, and the sum of all a_j is less than pi. Alright. Let me try to parse this.
First, the infimum here is the grea... |
olympiads | 1.041. $\frac{\left(4-3.5 \cdot\left(2 \frac{1}{7}-1 \frac{1}{5}\right)\right): 0.16}{X}=\frac{3 \frac{2}{7}-\frac{3}{14}: \frac{1}{6}}{41 \frac{23}{84}-40 \frac{49}{60}}$. | <think>
Okay, let me try to solve this problem. Hmm, so we have an equation here where a fraction involving some complicated expressions equals another fraction, and we need to find X. Let me write down the equation first to get a clearer picture.
The equation is:
\[
\frac{\left(4 - 3.5 \cdot \left(2 \frac{1}{7} - 1 ... |
olympiads | 3. Find the reduced polynomial $P(x)$ (the coefficient of the highest degree of the variable is 1), for which the identity $x P(x-1)=(x-6) P(x)$ holds. | <think>
Alright, let's tackle this problem. So, we need to find a reduced polynomial P(x) such that the identity x P(x-1) = (x - 6) P(x) holds for all x. And the polynomial should be monic, meaning the leading coefficient is 1. Hmm, okay. Let me start by understanding what the problem is asking.
First, let's parse the... |
olympiads | 11.5. On each of ten cards, a real number is written. For each non-empty set of these cards, the sum of all numbers written on the cards in this set was found. It is known that not all of the obtained sums are integers. What is the maximum possible number of integer sums that could have resulted? | <think>
Okay, so I have this problem here: There are ten cards, each with a real number written on them. For every non-empty subset of these cards, we calculate the sum of the numbers on that subset. We know that not all of these sums are integers. The question is asking for the maximum possible number of integer sums ... |
olympiads | 3.075. $(\cos \alpha-\cos 2 \beta)^{2}+(\sin \alpha+\sin 2 \beta)^{2}$. | <think>
Okay, so I need to simplify this expression: (cos α − cos 2β)² + (sin α + sin 2β)². Hmm, let's see. I remember there are some trigonometric identities that can help simplify expressions like this. Maybe expanding the squares first could be a good start. Let me try that.
First, let's expand each square separate... |
olympiads | 385. $(\tan \alpha+\cot \alpha)^{2}-(\tan \alpha-\cot \alpha)^{2}$. | <think>
Okay, so I have this expression to simplify: (tan α + cot α)² - (tan α - cot α)². Hmm, let me think. I remember that when you have something like (a + b)² - (a - b)², there's a formula for that. Maybe it's similar to expanding squares? Let me recall. The difference of squares formula is a² - b² = (a - b)(a + b)... |
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