source stringclasses 7
values | problem stringlengths 19 6.18k | generation stringlengths 345 68.1k |
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olympiads_ref | Determine the largest real number $M$ such that for every infinite sequence $x_{0}, x_{1}, x_{2}, \ldots$ of real numbers that satisfies
a) $x_{0}=1$ and $x_{1}=3$,
b) $x_{0}+x_{1}+\cdots+x_{n-1} \geq 3 x_{n}-x_{n+1}$,
it holds that
$$
\frac{x_{n+1}}{x_{n}}>M
$$
for all $n \geq 0$. | <think>
Okay, so I need to find the largest real number M such that for every infinite sequence x₀, x₁, x₂, ... that satisfies the given conditions, the ratio x_{n+1}/x_n is always greater than M for all n ≥ 0.
First, let's parse the problem. We have two conditions:
a) x₀ = 1 and x₁ = 3.
b) For every n, the sum o... |
olympiads | Task B-3.3. (20 points) From a cube with an edge length of $3 \mathrm{~cm}$, all eight vertices have been cut off so that the new body has all edges of equal length. What is the volume of the newly formed body?
 \cdot x y=30 \\ (x+y) \cdot x y=120\end{array}\right.$ | <think>
Okay, let's see. I need to solve this system of equations:
\[
\begin{cases}
(x - y) \cdot xy = 30 \\
(x + y) \cdot xy = 120
\end{cases}
\]
Hmm, so both equations have the term \( xy \). Maybe I can simplify this by letting \( xy = k \) or something. Let me try that substitution. Let \( k = xy \). Then the equ... |
olympiads | LVII OM - III - Problem 2
Determine all positive integers $ k $ for which the number $ 3^k+5^k $ is a power of an integer with an exponent greater than 1. | <think>
Okay, so I need to figure out all positive integers k such that 3^k + 5^k is a perfect power, meaning it's equal to m^n where n is an integer greater than 1. Hmm, let's start by checking small values of k to see if I can spot a pattern or maybe find all possible solutions.
First, let's try k = 1. Then 3^1 + 5^... |
amc_aime | On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling ... | <think>
Okay, so I need to figure out the maximum number of cars that can pass a photoelectric eye in one hour on this single-lane highway. The cars all travel at the same speed and follow a safety rule where the distance between the front of one car and the back of the next is exactly one car length for each 15 km/h o... |
aops_forum | Let the major axis of an ellipse be $AB$, let $O$ be its center, and let $F$ be one of its foci. $P$ is a point on the ellipse, and $CD$ a chord through $O$, such that $CD$ is parallel to the tangent of the ellipse at $P$. $PF$ and $CD$ intersect at $Q$. Compare the lengths of $PQ$ and $OA$. | <think>
Okay, so I need to compare the lengths PQ and OA in this ellipse problem. Let me start by recalling some properties of ellipses. The standard equation of an ellipse is (x²/a²) + (y²/b²) = 1, where 2a is the length of the major axis, and 2b is the minor axis. The center O is the midpoint of both axes. The foci a... |
olympiads | A parabola has equation $y=a x^{2}+b x+c$ and passes through the points $(-3,50),(-1,20)$ and $(1,2)$. Determine the value of $a+b+c$. | <think>
Okay, so I need to find the value of a + b + c for the parabola equation y = ax² + bx + c. The parabola passes through three points: (-3, 50), (-1, 20), and (1, 2). Hmm, since the equation has three unknowns (a, b, c), I can set up a system of three equations using these three points. Then solve the system to f... |
olympiads | A group consisting of 5 girls and 7 boys plays "wedding": they select (in the following order) a couple, a registrar, two flower girls each with a gentleman, and two witnesses. How many ways can this "wedding party" be formed if 3 of the girls have 1 brother each present, and the couple cannot be siblings, and the regi... | <think>
Okay, let's try to tackle this problem step by step. It seems a bit complicated at first, but if I break it down into the different roles they need to assign, maybe it will make more sense.
First, let's list out all the roles that need to be filled:
1. A couple (which I assume is one girl and one boy)
2. A re... |
olympiads | 42 Let $k$ be a natural number. Try to determine the smallest natural number $n$ such that: in any $n$ integers, there must be two numbers whose sum or difference is divisible by $2 k+1$.
| <think>
Okay, so I need to find the smallest natural number n such that in any n integers, there must be at least two numbers whose sum or difference is divisible by 2k+1. Hmm, let's try to break this down step by step.
First, let's understand the problem. We have a natural number k, and we need to determine the minim... |
olympiads | 7. Given moving points $P$, $M$, and $N$ are on the $x$-axis, circle $\Gamma_{1}$: $(x-1)^{2}+(y-2)^{2}=1$, and circle $\Gamma_{2}$: $(x-3)^{2}+$ $(y-4)^{2}=3$, respectively. Then the minimum value of $|P M|+|P N|$ is $\qquad$. | <think>
Okay, so I need to find the minimum value of |PM| + |PN| where P is a point on the x-axis, M is on the circle Γ₁: (x-1)² + (y-2)² = 1, and N is on the circle Γ₂: (x-3)² + (y-4)² = 3. Hmm, let me try to visualize this.
First, let's recall that Γ₁ has center at (1, 2) and radius 1, and Γ₂ has center at (3, 4) wi... |
olympiads | 16. Given the function $f(x)=\left(x^{2}-4\right)^{-\frac{1}{2}}(x>2), f^{-1}(x)$ is the inverse function of $f(x)$.
(1) If the sequence $\left\{a_{n}\right\}, a_{1}=1, \frac{1}{a_{n+1}}=f^{-1}\left(a_{n}\right)\left(n \in \mathbf{N}_{+}\right)$, find $a_{n}$;
(2) Let $S_{n}=a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}, b_{n}=... | <think>
Okay, let's tackle this problem step by step. First, part (1) asks us to find the sequence \( a_n \) defined by \( a_1 = 1 \) and \( \frac{1}{a_{n+1}} = f^{-1}(a_n) \) for \( n \in \mathbb{N}_+ \). The function \( f(x) = (x^2 - 4)^{-1/2} \) for \( x > 2 \), so we need to find its inverse first.
Starting with p... |
olympiads | 12. In the Cartesian coordinate system $X O Y$, given two points $M(-1,2)$ and $N(1,4)$, point $P$ moves on the $X$-axis. When $\angle M P N$ takes its maximum value, the x-coordinate of point $P$ is $\qquad$ | <think>
Alright, let's tackle this problem step by step. So, we have points M(-1, 2) and N(1, 4) in the Cartesian coordinate system, and point P is moving along the X-axis. We need to find the x-coordinate of point P when the angle MPN is maximized. Hmm, okay, angles and coordinates. Let me recall some geometry here.
... |
aops_forum | Find a polynomial $P$ of lowest possible degree such that
(a) $P$ has integer coefficients,
(b) all roots of $P$ are integers,
(c) $P(0) = -1$,
(d) $P(3) = 128$. | <think>
Okay, let's see. I need to find a polynomial P with the lowest possible degree that meets all these conditions: integer coefficients, all roots are integers, P(0) = -1, and P(3) = 128. Hmm, starting to think about how to approach this.
First, since all roots are integers and the polynomial has integer coeffici... |
aops_forum | In the morning, Esther biked from home to school at an average speed of $x$ miles per hour. In the afternoon, having lent her bike to a friend, Esther walked back home along the same route at an average speed of 3 miles per hour. Her average speed for the round trip was 5 miles per hour. What is the value of $x$? | <think>
Okay, let me try to figure out this problem. So Esther bikes to school in the morning at an average speed of x miles per hour, and then walks back home along the same route at 3 miles per hour. The average speed for the entire round trip is 5 miles per hour. We need to find the value of x.
Hmm, average speed c... |
olympiads | 9. Let the sequence $a_{1}, a_{2}, a_{3}, \cdots, a_{21}$ satisfy: $\left|a_{n+1}-a_{n}\right|=1(n=1,2,3, \cdots, 20), a_{1}, a_{7}, a_{21}$ form a geometric progression. If $a_{1}=1, a_{21}=9$, then the number of different sequences that meet the conditions is $\qquad$ | <think>
Okay, let's try to tackle this problem step by step. Hmm, so we have a sequence of 21 terms: a₁, a₂, ..., a₂₁. The conditions are that the absolute difference between consecutive terms is 1. So, each term is either 1 more or 1 less than the previous one. That makes me think of a path that moves up or down by 1 ... |
amc_aime | In the set of equations $z^x = y^{2x},\quad 2^z = 2\cdot4^x, \quad x + y + z = 16$, the integral roots in the order $x,y,z$ are:
$\textbf{(A) } 3,4,9 \qquad \textbf{(B) } 9,-5,-12 \qquad \textbf{(C) } 12,-5,9 \qquad \textbf{(D) } 4,3,9 \qquad \textbf{(E) } 4,9,3$ | <think>
Okay, so I need to solve this system of equations:
1. \( z^x = y^{2x} \)
2. \( 2^z = 2 \cdot 4^x \)
3. \( x + y + z = 16 \)
And the answer choices are given. Let me think step by step. Since the problem mentions integral roots, x, y, z are integers, so maybe I can use that to test the options or solve algebra... |
olympiads | 1.5. Two players take turns taking balls from two boxes. When it's their turn, a player takes an arbitrary number of balls from one of the boxes (at least one ball is mandatory). The winner is the player who takes the last ball. How should the first player play to win if the first box contains 19 and the second box con... | <think>
Okay, so there's this problem where two players take turns taking balls from two boxes. The first box has 19 balls and the second has 96. On your turn, you can take any number of balls from one box, but you have to take at least one. The person who takes the last ball wins. The question is, how should the first... |
olympiads | 12. There is a class of students going boating. They calculated that if they added one more boat, there would be exactly 6 people per boat; if they reduced one boat, there would be exactly 9 people per boat. Question: How many students are there in the class? | <think>
Okay, let's see. There's a class going boating, and they have some number of boats. When they add one more boat, each boat has exactly 6 people. If they reduce one boat, each boat has exactly 9 people. We need to find out how many students there are. Hmm, let me think step by step.
First, let's define some var... |
olympiads | 4. Calculate $\sqrt{6+\sqrt{20}}-\sqrt{6-\sqrt{20}}$. | <think>
Okay, so I need to calculate the expression √(6 + √20) minus √(6 - √20). Hmm, let me think. First, these nested square roots look a bit complicated, but maybe I can simplify them somehow. I remember that expressions like √(a ± √b) can sometimes be rewritten as √c ± √d if they fit a certain form. Let me recall t... |
olympiads | 3.274. $\frac{\sin ^{2} \alpha}{\sin (\alpha-\beta)}+\frac{\sin ^{2} \beta}{\sin (\beta-\alpha)}$. | <think>
Okay, so I need to simplify the expression: (sin²α)/(sin(α - β)) + (sin²β)/(sin(β - α)). Hmm, let's start by looking at the denominators. I notice that sin(β - α) is the same as -sin(α - β) because sine is an odd function. So maybe I can rewrite the second term using that identity.
Let me write that down. The... |
olympiads | 12. Teacher Li brought a stack of art paper, which was just enough to be evenly distributed among 24 students. Later, 8 more students came, so each person received 2 fewer sheets than originally. How many sheets of art paper did Teacher Li bring in total? | <think>
Okay, so Teacher Li has a stack of art paper that can be evenly distributed among 24 students. Then 8 more students come, making the total number of students 32. Because of the extra students, each person gets 2 fewer sheets than originally. We need to find the total number of sheets of art paper Teacher Li bro... |
olympiads | 3. [4] Let $A B C D E F$ be a regular hexagon. Let $P$ be the circle inscribed in $\triangle B D F$. Find the ratio of the area of circle $P$ to the area of rectangle $A B D E$. | <think>
Okay, so I need to find the ratio of the area of circle P inscribed in triangle BDF to the area of rectangle ABDE in a regular hexagon ABCDEF. Let me try to visualize this first. A regular hexagon has all sides equal and all internal angles equal to 120 degrees. Let me recall that in a regular hexagon, the dist... |
olympiads | 4. Starting from 1, alternately add 4 and 3, to get the following sequence of numbers $1,5,8,12,15,19,22 \ldots \ldots$ The number in this sequence that is closest to 2013 is $\qquad$ . | <think>
Okay, so I need to find the number in this sequence that's closest to 2013. The sequence starts at 1, then alternately adds 4 and 3 each time. Let me write down the sequence again to make sure I understand the pattern:
1, 5, 8, 12, 15, 19, 22, ...
Let me check the differences between the numbers. Starting fro... |
olympiads | 4. Let $f(x)=x^{3}+3 x^{2}+5 x+7$. Find the polynomial $g(x)$ of the smallest degree such that
$$
f(3)=g(3), \quad f(3-\sqrt{3})=g(3-\sqrt{3}), \quad f(3+\sqrt{3})=g(3+\sqrt{3}) .
$$ | <think>
Okay, let's see. I have to find a polynomial g(x) of the smallest degree such that it matches f(x) at three points: x=3, x=3−√3, and x=3+√3. The original function f(x) is a cubic polynomial, degree 3. So, the question is, what's the smallest degree polynomial that can pass through these three points? Hmm.
Wait... |
olympiads | 2. It is known that the numbers $x, y, z$ form an arithmetic progression in the given order with a common difference $\alpha=\arccos \frac{2}{3}$, and the numbers $\frac{1}{\sin x}, \frac{6}{\sin y}, \frac{1}{\sin z}$ also form an arithmetic progression in the given order. Find $\sin ^{2} y$. | <think>
Okay, let's try to tackle this problem step by step. So, we have three numbers x, y, z forming an arithmetic progression with a common difference α, which is equal to arccos(2/3). Then, the numbers 1/sin x, 6/sin y, 1/sin z also form an arithmetic progression. We need to find sin² y.
First, let me recall what ... |
olympiads | 3.242. $\sqrt{(1-\sin \alpha \sin \beta)^{2}-\cos ^{2} \alpha \cos ^{2} \beta}$. | <think>
Okay, let me try to simplify this expression: √[(1 - sin α sin β)² - cos² α cos² β]. Hmm, first, I need to expand the squared term inside the square root. Let me do that step by step.
So, expanding (1 - sin α sin β)², that would be 1² - 2*1*sin α sin β + (sin α sin β)², which simplifies to 1 - 2 sin α sin β +... |
aops_forum | A student read the book with $480$ pages two times. If he in the second time for every day read $16$ pages more than in the first time and he finished it $5$ days earlier than in the first time. For how many days did he read the book in the first time? | <think>
Okay, let's see. The problem is about a student who read a book with 480 pages twice. In the second time, every day he read 16 pages more than the first time, and he finished it 5 days earlier. We need to find out for how many days he read the book in the first time.
Hmm. So, let's break this down. Let me deno... |
olympiads | In how many ways can we distribute 14 identical candies among three children so that each child receives at least three candies? | <think>
Alright, so I need to figure out how many ways there are to distribute 14 identical candies among three children, with each child getting at least three candies. Hmm, okay. Let me start by recalling some combinatorics principles here.
First, distributing identical items (like candies) to distinct recipients (... |
olympiads | Three candidates ran for the election of a class representative at a school: João, Rosa, and Marcos. João received $2 / 7$ of the votes and Rosa $2 / 5$ of the votes. Who won the election? | <think>
Okay, so there's this problem about three candidates in a school election: João, Rosa, and Marcos. The question is asking who won the election based on the fractions of votes they received. João got 2/7 of the votes, Rosa got 2/5, and we need to figure out who came out on top. Let me think through this step by ... |
olympiads | Problem 10.5. One winter day, 43 children were throwing snowballs at each other. Each of them threw exactly one snowball at someone else. It is known that:
- the first threw a snowball at the one who threw a snowball at the second,
- the second threw a snowball at the one who threw a snowball at the third,
- the forty... | <think>
Okay, let's try to figure out this snowball problem. So, there are 43 kids, each throwing exactly one snowball at someone else. The problem gives us these three statements:
1. The first threw a snowball at the one who threw a snowball at the second.
2. The second threw a snowball at the one who threw a snowbal... |
olympiads | 3. Let $x, y, z$ be positive real numbers such that $x+y+z=3$. Find the maximum value reached by
$$
\sqrt{x}+\sqrt{2 y+2}+\sqrt{3 z+6}
$$
For what values of $x, y, z$ is this maximum achieved? | <think>
Okay, so I need to find the maximum value of the expression √x + √(2y + 2) + √(3z + 6) given that x, y, z are positive real numbers and x + y + z = 3. Hmm, let's see. Since we're dealing with positive variables and a constraint, maybe I can use methods from calculus like Lagrange multipliers? Or maybe some ineq... |
olympiads | 6. [5 points] Two circles of the same radius 9 intersect at points $A$ and $B$. On the first circle, a point $C$ is chosen, and on the second circle, a point $D$ is chosen. It turns out that point $B$ lies on the segment $C D$, and $\angle C A D=90^{\circ}$. On the perpendicular to $C D$ passing through point $B$, a po... | <think>
Okay, let's tackle this geometry problem step by step. I need to find the length of segment CF given the conditions. First, let me parse the problem again to make sure I understand all the details.
We have two circles with the same radius 9 intersecting at points A and B. On the first circle, there's a point C... |
olympiads | 9. Given $\cos (\alpha+\beta)=\cos \alpha+\cos \beta$, find the maximum value of $\cos \alpha$.
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 find the maximum value of cos α given that cos(α + β) = cos α + cos β. Hmm, let's see. First, I remember some trigonometric identities. Maybe I can expand the left side using the cosine addition formula. The formula for cos(α + β) is cos α cos β - sin α sin β. So substituting that into the eq... |
olympiads | 9. Using the 24 -hour clock, the time $16: 11: 22$ on the date $16 / 11 / 22$ (i.e. the 16 th of November 2022) has hours, minutes and seconds equal to date, month and (two-digit) year respectively. Let $\mathrm{S}$ be the number of seconds which will elapse after that date and time until this phenomenon next occurs.
W... | <think>
Okay, let me try to figure out this problem. So, we have the date and time 16/11/22 at 16:11:22, where the hours, minutes, and seconds (16:11:22) correspond to the date, month, and two-digit year (16/11/22). The question is asking for the number of seconds S that will elapse until this phenomenon next occurs, a... |
cn_contest | 6. Let $a, b \in \mathbf{N}_{+}$, and $a+b \sqrt{2}=(1+\sqrt{2})^{100}$. Then the last two digits of $a b$ are $\qquad$ . | <think>
Okay, let's see. The problem says that for natural numbers a and b, we have a + b√2 = (1 + √2)^100. We need to find the last two digits of the product ab. Hmm, interesting. I remember that expressions like (1 + √2)^n can be expanded using the binomial theorem, but since √2 is irrational, the terms with √2 will ... |
olympiads_ref | Let $P$ be a point inside a triangle $A B C$ and let $a, b, c$ be the side lengths and $p$ the semi-perimeter of the triangle. Find the maximum value of
$$
\min \left(\frac{P A}{p-a}, \frac{P B}{p-b}, \frac{P C}{p-c}\right)
$$
over all possible choices of triangle $A B C$ and point $P$. | <think>
Okay, so I have this problem here where I need to find the maximum value of the minimum among three ratios: PA/(p - a), PB/(p - b), and PC/(p - c), where P is a point inside triangle ABC, a, b, c are the side lengths, and p is the semi-perimeter. Hmm, that sounds a bit complex, but let me break it down.
First,... |
olympiads | 2. Two cars started from the same point on a circular track 150 km long, heading in opposite directions. After some time, they met and continued moving in the same directions. Two hours after the start, the cars met for the second time. Find the speed of the second car if the speed of the first car is 60 km/h. | <think>
Okay, so we have two cars starting from the same point on a circular track that's 150 km long. They head in opposite directions, meet at some point, and then keep going. Then, two hours after the start, they meet for the second time. We need to find the speed of the second car, given that the first car is going... |
olympiads | Condition of the problem
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{4}}(\operatorname{tg} x)^{\operatorname{ctg} x}$ | <think>
Okay, let me try to figure out how to calculate the limit of (tan x)^{cot x} as x approaches π/4. Hmm, so first, I need to remember some limit laws and maybe properties of exponential functions. Since the function is in the form f(x)^{g(x)}, it might be helpful to take the natural logarithm to simplify the expr... |
aops_forum | The differentiable function $F:\mathbb{R}\to\mathbb{R}$ satisfies $F(0)=-1$ and \[\dfrac{d}{dx}F(x)=\sin (\sin (\sin (\sin(x))))\cdot \cos( \sin (\sin (x))) \cdot \cos (\sin(x))\cdot\cos(x).\] Find $F(x)$ as a function of $x$. | <think>
Okay, so I need to find the function F(x) given its derivative and the initial condition F(0) = -1. The derivative is this big chain of sines and cosines. Let me write it down again to make sure I have it right:
d/dx F(x) = sin(sin(sin(sin(x)))) * cos(sin(sin(x))) * cos(sin(x)) * cos(x)
Wow, that's a mouthful... |
olympiads | Pat and Mat took a swim in the pond and then had a race to their cottage. At a certain moment, it was true that if Mat had covered half the distance he had already run, he would still be short of the cottage by three times that half distance. At the same moment, it was true that if Pat had covered twice the distance he... | <think>
Okay, so Pat and Mat went swimming and then raced to their cottage. At a certain moment, both of them have some distance covered and some remaining. The problem gives us two conditional statements about their distances, and we need to figure out who was closer to the cottage at that specific moment.
Let me par... |
olympiads | 33. Find the minimum value of $\sum_{k=1}^{100}|n-k|$, where $n$ ranges over all positive integers. | <think>
Okay, so I need to find the minimum value of the sum from k=1 to 100 of |n - k| where n is a positive integer. Hmm, let's see. This is like finding the integer n that minimizes the total distance to all numbers from 1 to 100. I remember something about medians minimizing the sum of absolute deviations. Is that ... |
cn_contest | Example 6 There are 1994 matches on the table, two children, A and B, take turns to take 1, 2 or 3 matches each time, the one who can take the last match wins. Now A takes first, which child will win? How should he play to win this game? | <think>
Okay, so there's this problem with 1994 matches on the table. Two kids, A and B, take turns picking up 1, 2, or 3 matches each time. Whoever takes the last match wins. A goes first. The question is, which child will win, and how should they play to ensure victory?
Hmm, let me think. This sounds like a classic ... |
olympiads | Variant 11.1.2. In a basket, there are 38 apples: 9 green, 12 yellow, and 17 red. Alyona sequentially takes one apple at a time from the basket. If at any point she has pulled out fewer green apples than yellow ones, and fewer yellow ones than red ones, she will stop taking apples from the basket.
(a) (1 point) What i... | <think>
Okay, let's try to figure out these problems step by step.
First, the problem says there are 38 apples in the basket: 9 green, 12 yellow, and 17 red. Alyona takes apples one at a time, and if at any point she has fewer green apples than yellow ones, and fewer yellow ones than red ones, she stops.
Starting w... |
olympiads | (2) Let $\mathbf{N}$ denote the set of natural numbers, and $f: \mathbf{N} \rightarrow \mathbf{N}$ satisfies the conditions: $f(1)=1$, and for any natural number $n$,
$$
\left\{\begin{array}{l}
3 f(n) f(2 n+1)=f(2 n)(1+3 f(n)) \\
f(2 n)<6 f(n)
\end{array}\right.
$$
Find all solutions to the equation
$$
f(k)+f(l)=293, ... | <think>
Okay, so I need to solve this problem where there's a function f from natural numbers to natural numbers with certain conditions, and then find all pairs (k, l) with k < l such that f(k) + f(l) = 293. Let me start by understanding the function f first. The problem gives me f(1) = 1, and two recursive conditions... |
olympiads | 6. In two baskets, there are apples. If we take out $\frac{5}{6}$ from the first basket and $\frac{3}{4}$ of the number of apples in the second basket, 20 apples will remain in each basket. What is the total number of apples, and how many will remain in both baskets together if we take out $\frac{2}{3}$ of the apples f... | <think>
Okay, let's see. I need to solve this problem about apples in two baskets. Let me try to understand the problem first.
So, there are two baskets with apples. If we take out 5/6 from the first basket and 3/4 from the second basket, then 20 apples remain in each basket. Then, the question is asking for the tota... |
olympiads | 8. Given positive integers $a_{1}, a_{2}, \cdots, a_{2018}$ satisfying
$$
a_{1}<a_{2}<\cdots<a_{2018} \text {. }
$$
For $i=1,2, \cdots, 2018, b_{i}$ is the number of positive integers in $a_{1}, a_{2}, \cdots, a_{2018}$ that do not exceed $i$. Then
$$
\frac{\sum_{k=1}^{2018} a_{k}+\sum_{k=1}^{a_{2018}} b_{k}}{a_{2018}... | <think>
Okay, let me try to figure out this problem. So, we have a sequence of positive integers a₁, a₂, ..., a₂₀₁₈ that are strictly increasing. That means each aₖ is bigger than the one before. For each i from 1 to 2018, bᵢ is defined as the number of terms in the a sequence that do not exceed i. Then we need to comp... |
olympiads | Example 6. In five boxes, there are balls of the same weight and size. In two boxes - 6 blue and 4 red balls each (this is box composition $H_{1}$). In two other boxes (composition $H_{2}$) - 8 blue and 2 red balls each. In one box (composition $H_{3}$) - 2 blue and 8 red balls. A box is randomly chosen and a ball is d... | <think>
Okay, so I need to solve this probability problem. Let me read it again to make sure I understand. There are five boxes with different compositions of blue and red balls. Two boxes have 6 blue and 4 red each (H₁). Two other boxes have 8 blue and 2 red each (H₂). And one box has 2 blue and 8 red (H₃). A box is r... |
cn_contest | In $\triangle A B C$, let
$$
\begin{array}{l}
x=\cos A+\cos B+\cos C, \\
y=\sin \frac{A}{2}+\sin \frac{B}{2}+\sin \frac{C}{2} .
\end{array}
$$
Then the relationship between $x$ and $y$ is ( ).
(A) $x=y$
(B) $x \geqslant y$
(C) $x \leqslant y$
(D) cannot be determined | <think>
Okay, so I need to figure out the relationship between x and y in triangle ABC. Let's see, x is the sum of the cosines of the angles, and y is the sum of the sines of half the angles. The options are whether x equals y, x is greater than or equal to y, x is less than or equal to y, or it can't be determined. Hm... |
olympiads | Solve the following equation:
$$
1.2065^{x}+1.2065^{x+1}+1.2065^{x+2}+1.2065^{x+3}=1.2065^{10}
$$ | <think>
Okay, let's see. I need to solve the equation 1.2065^x + 1.2065^{x+1} + 1.2065^{x+2} + 1.2065^{x+3} = 1.2065^{10}. Hmm, all the terms on the left side have the same base, which is 1.2065, but they have different exponents. Maybe I can factor out the common term to simplify the equation.
Let me write down each... |
cn_contest | 9. Arrange the $n$ positive integers from $1 \sim n (n>1)$ in a row so that the sum of any two adjacent numbers is a perfect square. What is the minimum value of $n$? | <think>
Okay, so I need to figure out the minimum value of n (where n is greater than 1) such that the numbers from 1 to n can be arranged in a row where the sum of any two adjacent numbers is a perfect square. Hmm, interesting. Let me think about how to approach this.
First, I know that a perfect square is a number l... |
olympiads | 7. If a natural number can be written as the sum of two consecutive natural numbers and also as the sum of three consecutive natural numbers, it is called a "good number." Please find the "good number" among 2007, 2008, 2009, 2010, 1011. | <think>
Alright, let's try to figure out which of these numbers—2007, 2008, 2009, 2010, or 1011—is a "good number." From the problem statement, a good number must be expressible both as the sum of two consecutive natural numbers and as the sum of three consecutive natural numbers.
First, let me make sure I understand... |
olympiads | Tompsongo A.K.
In a certain kingdom, there were 32 knights. Some of them were vassals of others (a vassal could have only one suzerain, and the suzerain was always richer than his vassal). A knight who had at least four vassals bore the title of baron. What is the maximum number of barons that could be under these con... | <think>
Okay, so I need to figure out the maximum number of barons possible in this kingdom with 32 knights. Let me start by understanding the problem again.
Each knight can be a vassal to only one suzerain, and the suzerain must be richer than their vassal. A baron is a knight who has at least four vassals. Also, the... |
olympiads | 4. On the side $AC$ of triangle $ABC$, a point $K$ is taken which divides $AC$ in the ratio $1:3$, and on the side $BC$ a point $L$ is taken which divides $BC$ in the ratio $1:4$. Let $M$ be the intersection point of segments $AL$ and $BK$. Determine the ratio of segments $KM$ and $MB$. | <think>
Okay, so I need to find the ratio of KM to MB where M is the intersection point of AL and BK in triangle ABC. Points K and L divide sides AC and BC in the ratios 1:3 and 1:4 respectively. Hmm, let's see.
First, maybe I should draw a diagram. Let me visualize triangle ABC. Point K is on AC, dividing it into a ... |
olympiads | 4. The numbers $x$ and $y$ are such that $x+y=xy=17$. Find the value of the expression:
$$
\left(x^{2}-17 x\right)\left(y+\frac{17}{y}\right)
$$ | <think>
Okay, let's see. I need to find the value of the expression \(\left(x^{2} - 17x\right)\left(y + \frac{17}{y}\right)\) given that \(x + y = 17\) and \(xy = 17\). Hmm, that seems interesting. Let me break it down step by step.
First, let's recall what we know. The problem states that \(x + y = 17\) and \(xy = 17... |
olympiads | 3. Two thousand dominoes are arranged in a row at equal (and small) distances from each other. Then, the dominoes are numbered from 1 to 2000. Several more dominoes are lined up in the same way in front of the domino with number 1, and at the other end, $k$ times more $(k>8)$. At the same time and with equal force, the... | <think>
Alright, let me try to work through this domino problem. Okay, so we have 2000 dominoes numbered 1 to 2000 in a row. Then, there are some more dominoes arranged in front of domino 1, and at the other end beyond domino 2000, there are k times more dominoes than in front of domino 1. Both ends are knocked over to... |
olympiads | Problem 5.8. Inside a large triangle with a perimeter of 120, several segments were drawn, dividing it into nine smaller triangles, as shown in the figure. It turned out that the perimeters of all nine small triangles are equal to each other. What can they be equal to? List all possible options.
The perimeter of a fig... | <think>
Okay, so I need to figure out what the perimeters of the nine small triangles can be equal to, given that the large triangle has a perimeter of 120. The problem says that several segments were drawn, dividing the large triangle into nine smaller triangles, and all of these small triangles have equal perimeters.... |
olympiads | 6. As shown in the figure, in $\triangle A B C$, $A B=30, A C=20, S_{\triangle A B C}=210$, $D, E$ are the midpoints of sides $A B, A C$ respectively, and the angle bisector of $\angle B A C$ intersects $D E, B C$ at points $F, G$ respectively, then the area of quadrilateral $B G F D$ is . $\qquad$ | <think>
Okay, let's see. I need to find the area of quadrilateral BG FD in triangle ABC. Given that AB is 30, AC is 20, and the area of ABC is 210. D and E are midpoints of AB and AC respectively. The angle bisector of angle BAC intersects DE and BC at F and G. Hmm, okay. Let me try to visualize this.
First, I'll sket... |
olympiads | 11. (20 points) For any real numbers $a_{1}, a_{2}, \cdots, a_{5}$ $\in[0,1]$, find the maximum value of $\prod_{1 \leq i<j \leq 5}\left|a_{i}-a_{j}\right|$. | <think>
Okay, so I have this problem where I need to find the maximum value of the product of the absolute differences between all pairs of five real numbers a₁ to a₅, each between 0 and 1. The product is over all pairs where i < j, right? So first, let me make sure I understand the problem correctly.
Given five numb... |
olympiads | 5. In $\triangle A B C$, $\angle C=90^{\circ}, \angle B=30^{\circ}$, $A C=1$, $M$ is the midpoint of $A B$. Fold $\triangle A C M$ along $C M$ so that the distance between points $A$ and $B$ is $\sqrt{2}$. Then the distance from point $A$ to the plane $B C M$ is $\qquad$ . | <think>
Okay, so I need to solve this geometry problem. Let me start by understanding the given information and what is being asked.
We have triangle ABC where angle C is 90 degrees, angle B is 30 degrees, and AC = 1. M is the midpoint of AB. Then, we're supposed to fold triangle ACM along CM such that the distance b... |
olympiads | 4. (28th Russian Mathematical Olympiad) Find the smallest positive integer that can be represented as the sum of 2002 positive integers with equal sums of digits, and also as the sum of 2003 positive integers with equal sums of digits. | <think>
Okay, so I need to find the smallest positive integer that can be written as the sum of 2002 positive integers, each having the same sum of digits, and also as the sum of 2003 positive integers with the same sum of digits. Hmm, this seems a bit tricky, but let me try breaking it down step by step.
First, let's... |
olympiads | 7.78 If $a>0, a \neq 1, \quad F(x)$ is an odd function, then the parity of the following function
$$
G(x)=F(x) \cdot\left[\frac{1}{a^{x}-1}+\frac{1}{2}\right]
$$
is
(A) Odd function.
(B) Even function.
(C) Neither odd nor even function.
(D) The parity depends on the specific value of $a$.
(China High School Mathematic... | <think>
Alright, so I have this problem here where I need to determine the parity of the function G(x), which is defined as F(x) multiplied by [1/(a^x - 1) + 1/2]. The options are whether it's odd, even, neither, or depends on the value of a. Let me try to work through this step by step.
First, let me recall what odd ... |
olympiads | 2. The solution to the equation $(x+2)^{5}+x^{5}+2 x+2=0$ with respect to $x$ is
untranslated part remains the same as it is a mathematical expression. | <think>
Okay, so I need to solve the equation \((x + 2)^5 + x^5 + 2x + 2 = 0\). Hmm, let me think about how to approach this. It looks like a polynomial equation of degree 5, which can be pretty tough because there's no general formula for quintic equations. Maybe there's a clever substitution or factorization that can... |
olympiads | Riquinho distributed 1000.00 reais among his friends Antônio, Bernardo, and Carlos in the following manner: he gave, successively, 1 real to Antônio, 2 reais to Bernardo, 3 reais to Carlos, 4 reais to Antônio, 5 reais to Bernardo, etc. What was the amount received by Bernardo? | <think>
Alright, let's try to figure out how much Bernardo received. The problem says Riquinho distributed 1000 reais among Antônio, Bernardo, and Carlos by giving money in successive increments: 1 real to Antônio, 2 to Bernardo, 3 to Carlos, then 4 to Antônio, 5 to Bernardo, 6 to Carlos, and so on. So the amounts give... |
olympiads | 5. As shown in Figure 2, in $\triangle A B C$, $\angle A B C=120^{\circ}$,
$A B=B C=2$. Take a point $D$ on side
$A C$ (excluding $A$ and $C$), and fold $\triangle A B D$
along line segment $B D$, to get
$\triangle P B D$. When plane
$P B D \perp$ plane $A B C$, the maximum distance from point $P$ to plane $A B C$ is $... | <think>
Okay, let me try to figure out this geometry problem. So, we have triangle ABC where angle ABC is 120 degrees, and sides AB and BC are both 2 units long. We need to take a point D on side AC (not including A and C), fold triangle ABD along BD to get triangle PBD. When the plane PBD is perpendicular to plane ABC... |
olympiads | 666. Which of the points $A(5 / 8)$ and $B(8 / 13)$ is to the left? (The coordinates of the points are given in parentheses; the positive direction is to the right.) | <think>
Okay, so I have this problem here: I need to figure out which of the two points A(5/8) and B(8/13) is to the left on a number line. The positive direction is to the right, so the number line goes from smaller numbers on the left to larger numbers on the right. That means if one number is smaller, it's to the le... |
olympiads | 8. Let $f(x)$ be an odd function defined on $\mathbf{R}$, and when $x \geqslant 0$, $f(x)=x^{2}$. If for any $x \in[a, a+2]$, we have $f(x+a) \geqslant 2 f(x)$, then the range of the real number $a$ is . $\qquad$ | <think>
Okay, so I need to find the range of real numbers a such that for any x in [a, a+2], the inequality f(x + a) ≥ 2f(x) holds. The function f(x) is an odd function defined on all real numbers, and when x is non-negative, f(x) = x². Hmm, let me start by recalling what an odd function is. An odd function satisfies f... |
olympiads | 11. (This question is worth 20 points) In the plane rectangular coordinate system $x O y$, let $A B$ be a chord of the parabola $y^{2}=4 x$ passing through the point $F(1,0)$, and the circumcircle of $\triangle A O B$ intersects the parabola at point $P$ (different from points $O, A, B$). If $P F$ bisects $\angle A P B... | <think>
Okay, let's tackle this problem step by step. It seems a bit involved, but I'll try to break it down. First, let me understand the given information and what we need to find.
We have a parabola \( y^2 = 4x \) in the plane rectangular coordinate system \( xOy \). There's a chord \( AB \) of this parabola that p... |
olympiads | Question 75, Given real numbers $a \geq b \geq c \geq d, a+b+c+d=9, a^{2}+b^{2}+c^{2}+d^{2}=21$, find the minimum possible value of $\mathrm{ab}-\mathrm{cd}$. | <think>
Okay, so I need to find the minimum possible value of ab - cd given these four real numbers a, b, c, d with the constraints that a is greater than or equal to b, which is greater than or equal to c, then d, and the sum of them is 9, and the sum of their squares is 21. Hmm, let's break this down step by step.
F... |
olympiads | 3. During the shooting practice, each soldier fired 10 times. One of them completed the task successfully and scored 90 points. How many times did he score 8 points, if there were 4 tens, and the results of the hits were sevens, eights, and nines. There were no misses at all. | <think>
Okay, so there's this problem about a soldier who scored 90 points in shooting practice. Each soldier fired 10 times, and this guy managed to get 90 points. The question is asking how many times he scored 8 points. They also mentioned that he had 4 tens, and the results were only sevens, eights, and nines. No m... |
olympiads | 2. A three-digit number, all digits of which are different, will be called balanced if it is equal to the sum of all possible two-digit numbers formed from the different digits of this number. Provide an example of any balanced number. Justify your answer. | <think>
Okay, so I need to find a three-digit number where all the digits are different, and the number itself is equal to the sum of all possible two-digit numbers formed from its different digits. Hmm, let me break this down step by step.
First, let's understand the problem. Let's say the three-digit number is ABC, ... |
amc_aime | A [convex](https://artofproblemsolving.com/wiki/index.php/Convex) [polyhedron](https://artofproblemsolving.com/wiki/index.php/Polyhedron) has for its [faces](https://artofproblemsolving.com/wiki/index.php/Face) 12 [squares](https://artofproblemsolving.com/wiki/index.php/Square), 8 [regular](https://artofproblemsolving.... | <think>
Okay, so I need to figure out how many segments joining vertices of this convex polyhedron lie in the interior rather than along an edge or a face. The polyhedron has 12 squares, 8 regular hexagons, and 6 regular octagons as faces. At each vertex, one square, one hexagon, and one octagon meet. Hmm, this sounds ... |
olympiads | 8.1 The average age of the teaching staff of a school, consisting of 20 teachers, was 49 years. When another teacher joined the school, the average age became 48 years. How old is the new teacher? | <think>
Okay, let me try to figure out this problem. So, the school has 20 teachers with an average age of 49 years. Then, when a new teacher joins, the average age drops to 48 years. We need to find out how old the new teacher is. Hmm, let's break it down step by step.
First, I remember that average age is calculated... |
olympiads | 2. The sum of the absolute values of the terms of a finite arithmetic progression is 100. If all its terms are increased by 1 or all its terms are increased by 2, then in both cases the sum of the absolute values of the terms of the resulting progression will also be equal to 100. What values can the quantity $n^{2} d$... | <think>
Alright, let me try to tackle this problem step by step. So, the problem is about a finite arithmetic progression where the sum of the absolute values of its terms is 100. Then, if we add 1 or 2 to all the terms, the sum of the absolute values remains 100 in both cases. We need to find the possible values of \(... |
olympiads | ## Task B-1.6.
A bus departs from the initial station with a constant number of passengers. At the first station, $20 \%$ of the passengers get off, and 24 passengers get on. At the next station, $\frac{2}{3}$ of the passengers get off, and no one gets on. At the last station, the remaining 16 passengers get off. How ... | <think>
Okay, let's see. So there's a bus that starts with some passengers, and then at each station, some people get off and sometimes people get on. We need to figure out how many passengers were there initially. Hmm. Let me try to break this down step by step.
First, the bus departs from the initial station with a ... |
aops_forum | 10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front and in the back as well, hence $11$ possible places) with a uniform $\tfrac{1}{11}$ probability of each location. Then, either the frontmost or the backmost student is removed from the row (each with a $\... | <think>
Okay, so I need to figure out the probability that I, being the eighth student in a row of 10, will exit from the front rather than the back. The process is that every minute, a new student is inserted into the row in one of 11 possible positions (including front and back) with equal probability, and then eithe... |
aops_forum | Let $A$, $B$, and $C$ be distinct points on a line with $AB=AC=1$. Square $ABDE$ and equilateral triangle $ACF$ are drawn on the same side of line $BC$. What is the degree measure of the acute angle formed by lines $EC$ and $BF$?
[i]Ray Li[/i] | <think>
Alright, let's tackle this geometry problem step by step. First, let me try to visualize the setup. We have three distinct points A, B, and C on a line, with AB equal to AC, both being 1 unit. That suggests that A is between B and C, right? Because if AB and AC are both 1, then B and C must be on either side of... |
olympiads | 2. (17 points) Given a rectangle $A B C D$. On two sides of the rectangle, different points are chosen, six points on $A B$ and seven - on $B C$. How many different triangles exist with vertices at the chosen points? | <think>
Okay, so the problem is about a rectangle ABCD. They've chosen six points on side AB and seven points on side BC. We need to find out how many different triangles can be formed with vertices at these chosen points. Hmm, let me think.
First, let me visualize the rectangle. Let's say AB is one side, BC is the ad... |
olympiads | Task 2. (10 points) A circle touches the extensions of two sides $A B$ and $A D$ of square $A B C D$ with side $2 \sqrt{3} \mathrm{~cm}$. Two tangents are drawn from point $C$ to this circle. Find the radius of the circle if the angle between the tangents is $30^{\circ}$, and it is known that $\sin 15^{\circ}=\frac{\sq... | <think>
Okay, let's try to solve this geometry problem. Hmm, so we have a square ABCD with side length 2√3 cm. A circle touches the extensions of two sides AB and AD. From point C, which is the opposite corner of the square from A, we draw two tangents to this circle, and the angle between these tangents is 30 degrees.... |
olympiads | 3. The tennis player and the badminton player got tired of sports tournaments and decided to have a mathematical duel. They wrote down the numbers from 25 to 57 inclusive in a row and agreed to place either “+” or “-” (in any order, but only these signs) between them. If the result of the expression turns out to be eve... | <think>
Okay, so I need to figure out whether the result of placing "+" or "-" signs between the numbers from 25 to 57 inclusive will be even or odd. Depending on that, either the tennis player or the badminton player wins. Let me think step by step.
First, let's understand the problem. We have numbers from 25 to 57 w... |
olympiads | Example 4-10 In Figure 4-12(a), $v_{1} v_{2} v_{3}$ are 3 equally divided points on a circle, decorated with beads of red, blue, and green. How many different schemes are there? | <think>
Okay, let's see. The problem is about three points, v1, v2, v3, which are equally divided on a circle. They're decorated with beads of red, blue, and green. We need to find out how many different schemes there are. Hmm, different schemes probably mean different color arrangements. Since it's a circle, I think r... |
olympiads | Example 1 Let $M=\{1,2,3, \cdots, 1995\}, A$ be a subset of $M$ and satisfy the condition: if $x \in A, 15 x \notin A$, then the maximum number of elements in $A$ is
(1995 National High School League Question) | <think>
Okay, so I need to solve this problem where I have the set M = {1, 2, 3, ..., 1995}, and I need to find the maximum number of elements in a subset A such that if x is in A, then 15x is not in A. Hmm, okay. Let me think through this step by step.
First, let me understand the problem. The subset A can't have bot... |
olympiads | A regular tetrahedron $s$ contains three spheres. The first sphere passes through the vertices of the tetrahedron, the second through the midpoints of the edges; the third touches the faces of the tetrahedron. How do the surface areas of these spheres relate to each other? | <think>
Okay, so I need to figure out the relationship between the surface areas of three spheres inside a regular tetrahedron. The first sphere passes through the vertices, which I think is the circumscribed sphere or the circumsphere. The second goes through the midpoints of the edges, which might be the midsphere. T... |
olympiads | 3. In tetrahedron $ABCD$, $\triangle ADB$ is an isosceles right triangle, $AD=1, \angle ADB=90^{\circ}, \angle BDC=$ $\angle ADC=60^{\circ}$. Then the distance between the skew lines $AB$ and $CD$ is $\qquad$ | <think>
Okay, let's see. I need to find the distance between the skew lines AB and CD in tetrahedron ABCD. Hmm, skew lines are lines that don't intersect and aren't parallel, right? So they're in different planes. The problem gives some information about the tetrahedron: triangle ADB is an isosceles right triangle with... |
olympiads | 75. Students A, B, and C are competing in a chess tournament, where each pair of them plays only 1 game. When A has played 2 games, and B has played 1 game, C has played $\qquad$ games. | <think>
Alright, let's see. So, there are three students: A, B, and C. They're competing in a chess tournament, and each pair plays only one game. The question is: When A has played 2 games, and B has played 1 game, how many games has C played?
Hmm, okay. Let me try to visualize this. Since there are three students, e... |
olympiads | $6 \cdot 28 x$ is any real number, which of the following equations can be true?
(A) $\sqrt{x^{2}-1}=\sqrt{x+1} \cdot \sqrt{x-1}$.
(B) $10^{\lg x}=x$.
(C) $x^{0}=1$.
(D) $\sqrt[2 n+1]{x}=-\sqrt[2 n+1]{-x}$ ( $n$ is a natural number $)$.
(China Guangzhou, Wuhan, Fuzhou and other five cities junior high school mathematic... | <think>
Okay, let's tackle this problem step by step. The question is asking which of the given equations can be true when \( x \) is any real number. Let's go through each option one by one.
**Option A: \( \sqrt{x^{2} - 1} = \sqrt{x + 1} \cdot \sqrt{x - 1} \).**
Hmm, so the left side is the square root of \( x^2 - 1... |
olympiads | 1. The numerical sequence $\left\{a_{n}\right\}_{n=1}^{\infty}$ is defined such that $a_{1}=\log _{2}\left(\log _{2} f(2)\right), \quad a_{2}=$ $\log _{2}\left(\log _{2} f(f(2))\right), \ldots, a_{n}=\log _{2}(\log _{2} \underbrace{f(f(\ldots f}_{n}(2)))), \ldots$, where $f(x)=x^{x}$. Determine the index $n$ for which ... | <think>
Okay, let me try to figure out this problem step by step. The problem is about a sequence defined using a function f(x) = x^x. The terms of the sequence involve iterating this function and taking logarithms. The goal is to find the index n for which a_n equals 2059 + 2^2059. Hmm, that seems complicated, but let... |
aops_forum | There are $2n$ different numbers in a row. By one move we can interchange any two numbers or interchange any $3$ numbers cyclically (choose $a,b,c$ and place $a$ instead of $b$, $b$ instead of $c$, $c$ instead of $a$). What is the minimal number of moves that is always sufficient to arrange the numbers in increasing or... | <think>
Okay, so there's this problem about sorting a sequence of 2n different numbers using two types of operations: swapping any two numbers or cyclically permuting any three numbers. The question is asking for the minimal number of moves that is always sufficient to sort the numbers in increasing order, no matter ho... |
aops_forum | Find all pairs of odd integers $a$ and $b$ for which there exists a natural number$ c$ such that the number $\frac{c^n+1}{2^na+b}$ is integer for all natural $n$. | <think>
Okay, so I need to find all pairs of odd integers a and b such that there exists a natural number c where (c^n + 1)/(2^n a + b) is an integer for all natural numbers n. Hmm, let's start by unpacking this problem step by step.
First, the problem states that a and b are odd integers, and c is a natural number. T... |
olympiads_ref | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ for which
$$
f(x)=\max _{y \in \mathbb{R}}(2 x y-f(y))
$$
for all $x \in \mathbb{R}$.
(In general, the expression $a=\max _{s \in S} g(s)$ means: $a \geq g(s)$ for all $s \in S$ and there is an $s \in S$ for which $a=g(s)$.) | <think>
Okay, so I need to find all functions f from the real numbers to the real numbers such that for every real number x, f(x) is equal to the maximum over all y of (2xy - f(y)). Hmm, that seems a bit abstract. Let me try to unpack it step by step.
First, the equation given is f(x) = max_{y ∈ ℝ} [2xy - f(y)]. So fo... |
olympiads |
Opgave 3. Gegeven is een onbekende rij $a_{1}, a_{2}, a_{3}, \ldots$ van gehele getallen die voldoet aan de volgende eigenschap: voor elk priemgetal $p$ en elk positief geheel getal $k$ geldt
$$
a_{p k+1}=p a_{k}-3 a_{p}+13
$$
Bepaal alle mogelijke waarden van $a_{2013}$.
| <think>
Okay, let's see. I need to find all possible values of \( a_{2013} \) given that the sequence \( a_1, a_2, a_3, \ldots \) of integers satisfies the condition that for every prime number \( p \) and every positive integer \( k \), it holds that:
\[
a_{p k + 1} = p a_k - 3 a_p + 13
\]
Hmm, so the value of the t... |
olympiads | 2. For what largest natural $\mathrm{k}$ can the cells of a $5 \times 5$ table be filled with zeros and ones (each cell must contain exactly one number) so that there are $k$ rows, in each of which the sum of the numbers is at least 3, and $k$ columns, in each of which the sum of the numbers is at most 2? (O. Nechaeva,... | <think>
Okay, let's try to figure out this problem. So, we need to find the largest natural number k such that we can fill a 5x5 grid with 0s and 1s. The conditions are that there are k rows where each of those rows has a sum of at least 3, and k columns where each of those columns has a sum of at most 2. Hmm, okay. Le... |
olympiads | [ Arithmetic. Mental calculation, etc.]
The alarm clock is ahead by 9 minutes per day. When going to bed at 22:00, the exact time was set on it. At what time should the alarm be set so that it rings exactly at 6:00? Explain your answer. | <think>
Okay, let's see. The problem is about an alarm clock that gains 9 minutes each day. When I go to bed at 22:00 (which is 10 PM), I set the exact time on it. Now, I need to figure out what time I should set the alarm so that it rings exactly at 6:00 AM the next morning. Hmm, let me break this down step by step.
... |
olympiads | 4.1. The segment connecting the lateral sides of the trapezoid and parallel to its bases, which are 3 and 21, divides the trapezoid into two parts of equal area. Find the length of this segment. | <think>
Okay, so I need to find the length of a segment that connects the lateral sides of a trapezoid. This segment is parallel to the two bases, which are 3 and 21 units long. The key thing here is that this segment divides the trapezoid into two parts of equal area. Hmm, trapezoid area problems can sometimes be tric... |
olympiads | Problem 4. In a singing competition, a Rooster, a Crow, and a Cuckoo participated. Each member of the jury voted for one of the three performers. The Woodpecker calculated that there were 59 judges in total, and that the Rooster and the Crow received a total of 15 votes, the Crow and the Cuckoo received 18 votes, and t... | <think>
Okay, so we have a singing competition with three participants: a Rooster, a Crow, and a Cuckoo. Each judge votes for one of them. The Woodpecker counted the votes and said there were 59 judges total. Then he also mentioned the pairwise totals: Rooster and Crow got 15 votes together, Crow and Cuckoo got 18, and... |
olympiads | 5. Point $M$ divides the diagonal $A C$ of square $A B C D$ in the ratio $A M: M C=2: 1$. A line passing through point $M$ divides the square into two parts, the areas of which are in the ratio $9: 31$. In what ratio does this line divide the perimeter of the square? | <think>
Okay, let's try to figure out this geometry problem. So, we have a square ABCD. Point M is on the diagonal AC, and it divides AC in the ratio AM:MC = 2:1. There's a line passing through M that splits the square into two parts with areas in the ratio 9:31. We need to find the ratio in which this line divides the... |
olympiads | Problem 2. Calculate the value of the expression $\frac{x^{4}-2005 x^{3}+2005 x^{2}-x}{x-1}$ for $x=2005$. | <think>
Okay, so I need to calculate the value of this expression: (x⁴ - 2005x³ + 2005x² - x) divided by (x - 1) when x is 2005. Hmm, let me think. First, maybe I can plug in x = 2005 directly, but wait, if I do that, the denominator becomes 2005 - 1, which is 2004. But the numerator might be complicated. Let me check ... |
olympiads | 3. [5] Dragoons take up $1 \times 1$ squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the $x$-coordinates of their centers and the difference in the $y$-coordinates of their centers are bot... | <think>
Okay, so I need to figure out the maximum number of dragoons that can fire at a single dragoon simultaneously. Let me parse the problem again to make sure I understand all the conditions.
First, dragoons are 1x1 squares in the plane, placed with sides parallel to the coordinate axes. Their interiors do not int... |
olympiads_ref | Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ with the property
$$
f(f(x)+y)=2 x+f(f(y)-x)
$$
for all $x, y \in \mathbb{R}$. | <think>
Okay, so I have this functional equation to solve: f(f(x) + y) = 2x + f(f(y) - x) for all real numbers x and y. Hmm, functional equations can be tricky, but let me start by trying different strategies. Maybe I can substitute specific values for x or y to simplify the equation and see if I can find a pattern or ... |
olympiads | # Problem 10.
Set $A$ on the plane $O x y$ is defined by the equation $x^{2}+y^{2}=2 x+2 y+23$. Set $B$ on the same plane is defined by the equation $|x-1|+|y-1|=5$. Set $C$ is the intersection of sets $A$ and $B$. What is the maximum value that the product of the lengths of $n$ segments $X Y_{1} \cdot X Y_{2} \cdot X... | <think>
Okay, let's try to tackle this problem step by step. First, I need to understand what sets A and B are, find their intersection C, and then figure out how to maximize the product of distances from a point X in A to multiple points Y₁, Y₂, ..., Yₙ in C. Hmm, sounds a bit involved, but let's break it down.
Start... |
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