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values | problem stringlengths 19 6.18k | generation stringlengths 345 68.1k |
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cn_contest | 6. For the equation in terms of $x$
$$
\frac{1}{x^{2}-x}+\frac{k-5}{x^{2}+x}=\frac{k-1}{x^{2}-1}
$$
if it has an extraneous root $x=1$. Then the value of $k$ is ( ).
(A) 2
(B) 3
(C) 6
(D) 10 | <think>
Okay, so I have this equation here:
$$
\frac{1}{x^{2}-x} + \frac{k-5}{x^{2}+x} = \frac{k-1}{x^{2}-1}
$$
And I need to find the value of k such that x=1 is an extraneous root. The options are A) 2, B) 3, C) 6, D) 10. Hmm, let's start by recalling what an extraneous root is. If I remember right, an extraneous ... |
olympiads | 1.121 A wheel has a rubber tire with an outer diameter of 25 inches. When the radius is reduced by $\frac{1}{4}$ inch, the number of revolutions in one mile will
(A) increase by about $2 \%$.
(B) increase by about $1 \%$.
(C) increase by about $20 \%$.
(D) increase by $\frac{1}{2} \%$.
(E) remain unchanged.
(7th Americ... | <think>
Okay, so here's this problem about a wheel with a rubber tire. The outer diameter is 25 inches. Then, the radius is reduced by 1/4 inch. We need to figure out how this affects the number of revolutions the wheel makes in one mile. The options are about increasing by 2%, 1%, 20%, 0.5%, or remaining unchanged. Hm... |
olympiads | Example 2 (2006 National High School Mathematics Competition Question) Express 2006 as the sum of 5 positive integers $x_{1}, x_{2}, x_{3}, x_{4}$, $x_{5}$. Let $S=\sum_{1 \leqslant i<j \leqslant 5} x_{i} x_{j}$. Questions:
(1) For what values of $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ does $S$ attain its maximum value;
(2... | <think>
Okay, let's tackle this problem step by step. So, we need to express 2006 as the sum of 5 positive integers x₁, x₂, x₃, x₄, x₅. Then, we have to find the maximum and minimum values of S, which is the sum of all products x_i x_j where i < j.
Starting with part (1): Maximizing S. Hmm, I remember from algebra th... |
olympiads | 2. If $a_{1}, a_{2}, \cdots, a_{n}$ is a set of real numbers, and $a_{1}+a_{2}+\cdots+a_{n}=k$. Find the minimum value of $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}$. | <think>
Okay, let's see. I need to find the minimum value of the sum of the squares of some real numbers, given that their sum is a constant k. So, we have n real numbers a₁, a₂, ..., aₙ, and their sum is k. The problem is to find the minimum of a₁² + a₂² + ... + aₙ². Hmm, how do I approach this?
I remember that in op... |
cn_contest | Example $1$, $, $ Given line segments $b, c$. Find the line segment $a$ such that
$$
\frac{1}{a}=\frac{1}{b}+\frac{1}{c} .
$$ | <think>
Okay, let's see. I need to find a line segment a such that 1/a equals 1/b plus 1/c. Hmm, that equation looks familiar. It's similar to resistors in parallel or something. But here it's line segments, so maybe a geometric construction?
First, let's restate the equation. If 1/a = 1/b + 1/c, then solving for a gi... |
olympiads | 一、Fill in the Blanks (Total 3 questions, 10 points each)
1. In a 1000-meter race, when A reaches the finish line, B is 50 meters from the finish line; when B reaches the finish line, C is 100 meters from the finish line. So when A reaches the finish line, C is $\qquad$ meters from the finish line. | <think>
Okay, let's see. I need to figure out how far C is from the finish line when A finishes the race. Hmm. The problem says that in a 1000-meter race, when A finishes, B is 50 meters behind. Then, when B finishes, C is 100 meters behind. So, I guess I have to relate their speeds or something.
First, let's think ab... |
olympiads | 1.4. Find all real numbers $x$ for which the equality
$$
|| x|-1|+|| x|+2|=3
$$
holds. | <think>
Okay, so I need to solve the equation ||x| - 1| + ||x| + 2| = 3. Hmm, this looks a bit complicated with all those absolute value signs. Let me take it step by step.
First, I notice that the equation involves |x| inside other absolute values. Maybe if I let y = |x|, the equation becomes simpler. Since |x| is al... |
olympiads | 5. In a right triangle $ABC$ ( $\angle C$ - right), a point $D$ is taken on side $BC$ such that $\angle CAD=30^{\circ}$. From point $D$, a perpendicular $DE$ is dropped to $AB$. Find the distance between the midpoints of segments $CE$ and $AD$, given that $AC=3 \sqrt{3}, DB=4$. | <think>
Okay, let's try to solve this geometry problem. It's a right triangle ABC with a right angle at C. A point D is on BC such that angle CAD is 30 degrees. Then from D, we drop a perpendicular DE to AB. We need to find the distance between the midpoints of segments CE and AD, given that AC is 3√3 and DB is 4. Hmm,... |
olympiads |
G2 Let $A D, B F$ and $C E$ be the altitudes of $\triangle A B C$. A line passing through $D$ and parallel to $A B$ intersects the line $E F$ at the point $G$. If $H$ is the orthocenter of $\triangle A B C$, find the angle $\widehat{C G H}$.
| <think>
Okay, so I have this geometry problem here. Let me read it again to make sure I understand all the parts.
We have triangle ABC, and AD, BF, and CE are the altitudes. So, these are the three heights of the triangle, each dropping from a vertex perpendicular to the opposite side. Their intersection point would ... |
olympiads | ## Problem Statement
Find the derivative.
$$
y=\frac{x^{3}}{3} \cdot \arccos x-\frac{2+x^{2}}{9} \cdot \sqrt{1-x^{2}}
$$ | <think>
Okay, so I need to find the derivative of this function: y equals (x cubed over 3) times arccos x minus (2 plus x squared over 9) times the square root of (1 minus x squared). Hmm, let me write that down again to make sure I have it right.
So, y = (x³/3) * arccos(x) - [(2 + x²)/9] * sqrt(1 - x²). Yeah, that's... |
aops_forum | Find all number triples $(x,y,z)$ such that when any of these numbers is added to the product of the other two, the result is 2. | <think>
Alright, so I need to find all triples of numbers (x, y, z) such that when any of these numbers is added to the product of the other two, the result is 2. Let me try to parse this problem step by step.
First, the problem says: "when any of these numbers is added to the product of the other two, the result is 2... |
olympiads | 6. The sequence $\left\{x_{n}\right\}: 1,3,3,3,5,5,5,5,5, \cdots$ is formed by arranging all positive odd numbers in ascending order, and each odd number $k$ appears consecutively $k$ times, $k=1,3,5, \cdots$. If the general term formula of this sequence is $x_{n}=a[\sqrt{b n+c}]+d$, then $a+b+c$ $+d=$ | <think>
Okay, so I need to figure out the general term formula for this sequence where each positive odd number k appears k times in ascending order. The formula is given as x_n = a[√(bn + c)] + d, and I need to find the sum a + b + c + d. Let me start by understanding the sequence better.
The sequence is: 1, 3, 3, 3,... |
olympiads | The three triangles in the figure intersect at 12 different points. What is the maximum number of intersection points of any three triangles?
 | <think>
Okay, so I have this problem here: "The three triangles in the figure intersect at 12 different points. What is the maximum number of intersection points of any three triangles?" Hmm, interesting. Let me try to figure this out step by step.
First, I know that when two triangles intersect, the maximum number of... |
olympiads | ## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{3} \frac{e^{\sqrt{(3-x) /(3+x)}} d x}{(3+x) \sqrt{9-x^{2}}}
$$ | <think>
Okay, let me try to figure out how to solve this integral. The integral is from 0 to 3 of [e^(sqrt((3 - x)/(3 + x))) divided by (3 + x) times sqrt(9 - x²)] dx. Hmm, that looks a bit complicated. Let me break it down.
First, let's write the integrand again to see it clearly:
e^(sqrt[(3 - x)/(3 + x)}) / [(3 + x... |
aops_forum | A particle starts at $(0,0,0)$ in three-dimensional space. Each second, it randomly selects one of the eight lattice points a distance of $\sqrt{3}$ from its current location and moves to that point. What is the probability that, after two seconds, the particle is a distance of $2\sqrt{2}$ from its original location? ... | <think>
Okay, so I need to find the probability that after two seconds, the particle is at a distance of 2√2 from the origin. The particle starts at (0,0,0), and each second it moves to one of the eight neighboring lattice points that are a distance of √3 away. Let me try to visualize this.
First, in three-dimensional... |
olympiads | Find all four-digit numbers that, after scratching out the middle two digits, decrease by 120 times.
(S. Bednářová) | <think>
Okay, let's see. I need to find all four-digit numbers such that when I scratch out the middle two digits, the resulting two-digit number is 120 times smaller than the original four-digit number. Hmm, let's parse that carefully.
First, a four-digit number can be represented as ABCD, where A, B, C, D are digits... |
olympiads | 1.1. (4 points) When going from the first to the third floor, Petya walks 36 steps. When going from the first floor to his own floor in the same building, Vasya walks 72 steps. On which floor does Vasya live? | <think>
Okay, let's see. The problem is about Petya and Vasya climbing stairs in a building. We need to figure out on which floor Vasya lives based on the number of steps they take. Hmm, let me try to break it down.
First, when it says "going from the first to the third floor, Petya walks 36 steps." So, moving from th... |
olympiads | (Infinite Solitaire)(^) You all know the rules of solitaire: there are marbles on a board, and you eliminate marbles by jumping over them with another marble, etc. Now consider the plane and place marbles on all integer points of the negative half-plane. The goal of the game is to place a marble as high as possible in ... | <think>
Alright, let's try to figure out this infinite solitaire problem. So, the setup is that we have marbles on all integer points in the negative half-plane. That probably means all points (x, y) where y ≤ 0, right? The goal is to get a marble as high as possible in a finite number of moves. The example shows how t... |
olympiads | ## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{\operatorname{arctan}(2 / 3)} \frac{6+\operatorname{tan} x}{9 \sin ^{2} x+4 \cos ^{2} x} d x
$$ | <think>
Okay, so I have this integral to solve: the integral from 0 to arctan(2/3) of (6 + tan x) divided by (9 sin²x + 4 cos²x) dx. Hmm, let me see. First, I need to figure out how to approach this. The denominator is 9 sin²x + 4 cos²x. Maybe I can simplify that somehow. Let me recall that sin²x and cos²x can sometime... |
olympiads | Example 5 Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ modulo 3 can be divided by $x^{2}+1$.
untranslated text:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
translated text:
Example 5 Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ modulo 3 can be divided by $x^... | <think>
Okay, let's tackle this problem: Find the smallest positive integer n such that the polynomial (x+1)^n - 1 modulo 3 is divisible by x² + 1. Hmm, so we need to find the smallest n where when we expand (x+1)^n, subtract 1, and then take coefficients modulo 3, the resulting polynomial is divisible by x² + 1. That ... |
olympiads | 12. Let the equation $x y=6(x+y)$ have all positive integer solutions $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right)$, then $\sum_{k=1}^{n}\left(x_{k}+y_{k}\right)=$
$\qquad$ . | <think>
Okay, so I need to solve this problem where the equation is xy = 6(x + y), and I have to find all positive integer solutions (x, y) and then sum up all the x's and y's. Hmm, let me think step by step.
First, the equation given is xy = 6x + 6y. Since we're dealing with positive integers, both x and y have to be... |
olympiads | 1. Given the sets $U=\{1,2,3,4,5,6,7,8\}, A=\{1,2,3,4,5\}, B=\{4,5,6,7,8\}$, the number of subsets of set $U$ that are not subsets of set $A$ and are not subsets of set $B$ is | <think>
Okay, so I need to find the number of subsets of the universal set U that are not subsets of A and not subsets of B. Hmm, let me think. Let's break down the problem step by step.
First, I remember that the total number of subsets of a set with n elements is 2^n. Since U has 8 elements, the total number of subs... |
olympiads | João has a deck of 52 cards numbered from 1 to 52. A set of three cards is called lucky if the sum of the digits on each card is the same. What is the minimum number of cards João has to take from the deck, without looking, so that among the cards he takes there are necessarily three cards that form a set of lucky card... | <think>
Okay, so João has a deck of 52 cards numbered from 1 to 52. We need to find the minimum number of cards he has to take so that among those cards, there must be at least three that form a "lucky" set. A lucky set is defined as three cards where the sum of the digits on each card is the same.
First, I need to u... |
olympiads | 3. If $A, B, C$ are the three interior angles of $\triangle ABC$, and $A<B<C .\left(C \neq \frac{\pi}{2}\right)$, then which of the following conclusions is correct?
A. $\sin A<\sin C$
B. $\cot A<\cot C$
C. $\tan A<\tan C$
D. $\cos A<\cos C$ | <think>
Okay, so I need to solve this trigonometry problem about a triangle where the angles A, B, C are in ascending order, meaning A < B < C, and also C is not equal to π/2 (which is 90 degrees). The question is asking which of the given conclusions is correct among options A, B, C, D. The options involve comparing d... |
cn_contest | 3. In the afternoon, 5 classes need to be scheduled: Physics, Chemistry, Biology, and two self-study periods. If the first class cannot be Biology and the last class cannot be Physics, then the number of different scheduling methods is ( ) kinds.
(A) 36
(B) 39
(C) 60
(D) 78 | <think>
Okay, let's see. The problem is about scheduling 5 classes: Physics, Chemistry, Biology, and two self-study periods. The constraints are that the first class cannot be Biology, and the last class cannot be Physics. We need to figure out how many different scheduling methods there are under these conditions. The... |
olympiads | [ Fermat's Little Theorem ]
Find the remainder when $3^{102}$ is divided by 101.
# | <think>
Okay, so I need to find the remainder when 3^102 is divided by 101. Hmm, let's see. The problem mentions Fermat's Little Theorem, which I remember has something to do with prime numbers and exponents. Let me try to recall how that works.
Fermat's Little Theorem states that if p is a prime number, then for any ... |
olympiads | A triangle has sides $a, b, c$, and its area is $\frac{(a+b+c)(a+b-c)}{4}$. What is the measure of the largest angle of the triangle? | <think>
Okay, so I have this problem here: A triangle has sides a, b, c, and its area is ((a + b + c)(a + b - c))/4. I need to find the measure of the largest angle of the triangle. Hmm, let's see. I should start by recalling some formulas related to the area of a triangle. The most common one is the Heron's formula, w... |
olympiads | 2. The express passenger train, two days ago in the morning, departed from Gevgelija to Skopje, with several carriages in which there was a certain number of passengers. At the station in Veles, one passenger disembarked from the first carriage, two passengers disembarked from the last carriage, and no passengers board... | <think>
Okay, let's try to figure out this problem step by step. Hmm, it's about a train with passengers and some getting off at a station, then later another trip with fewer carriages and passengers. The question is asking for the number of passengers that started the journey in the morning from Gevgelija. Alright, le... |
olympiads | Find the number of different sequences with $2 n+1$ terms, where each term is $\pm 1$ and the absolute value of the sum of any odd number of consecutive terms does not exceed 1.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | <think>
Okay, let's see. The problem is to find the number of different sequences with 2n+1 terms, where each term is either +1 or -1. The condition is that the absolute value of the sum of any odd number of consecutive terms doesn't exceed 1. So, for any odd-length consecutive subsequence, the sum must be either -1, 0... |
olympiads | 13. (10 points) In the equation below, $A, B, C, D, E, F, G, H, I$ each represent different digits from $1 \sim 9$.
$$
\overline{\mathrm{ABCD}}+\overline{\mathrm{EF}} \times \overline{\mathrm{GH}}-I=X
$$
Then the minimum value of $X$ is . $\qquad$ | <think>
Okay, let's tackle this problem step by step. So, we need to find the minimum value of X in the equation:
ABCD + EF × GH - I = X
where each letter A, B, C, D, E, F, G, H, I represents a different digit from 1 to 9. Our goal is to assign these digits such that X is as small as possible. Since all digits must b... |
olympiads | 5. Inside an isosceles triangle $A B C (A B=A C)$, a point $K$ is marked. Point $L$ is the midpoint of segment $B K$. It turns out that $\angle A K B=\angle A L C=90^{\circ}, A K=C L$. Find the angles of triangle $A B C$. | <think>
Okay, let me try to solve this geometry problem. Hmm, so we have an isosceles triangle ABC with AB = AC. There's a point K inside the triangle, and L is the midpoint of BK. We know that angle AKB and angle ALC are both 90 degrees, and AK equals CL. We need to find the angles of triangle ABC. Alright, let's star... |
olympiads | 2.1. Determine the number of natural divisors of the number $11!=1 \cdot 2 \cdot \ldots \cdot 10 \cdot 11$ that are multiples of three. Answer. 432. | <think>
Okay, so I need to figure out how many natural divisors of 11! are multiples of three. Hmm. Let's start by recalling that 11! means the product of all integers from 1 to 11. So, 11! = 1×2×3×...×10×11. To find the number of divisors that are multiples of three, I think I need to first find the prime factorizatio... |
aops_forum | Find all pairs $(m, n)$ of positive integers, for which number $2^n - 13^m$ is a cube of a positive integer.
[i]Proposed by Oleksiy Masalitin[/i] | <think>
Okay, so I need to find all pairs of positive integers (m, n) such that 2^n - 13^m is a perfect cube. Hmm, let's start by understanding the problem. We have two exponential terms here: one is a power of 2 and the other is a power of 13. Their difference should be equal to some cube, say k^3 where k is a positiv... |
olympiads | Example 5 Let $x, y, z, w$ be four real numbers, not all zero. Find:
$S=\frac{x y+2 y z+z w}{x^{2}+y^{2}+z^{2}+w^{2}}$'s maximum value. | <think>
Okay, so I need to find the maximum value of the expression S = (xy + 2yz + zw) / (x² + y² + z² + w²) where x, y, z, w are real numbers and not all zero. Hmm, this seems like an optimization problem where I have to maximize the numerator while keeping the denominator as small as possible. Maybe using calculus o... |
olympiads | ## Task 4 - 070914
Four teams $A, B, C$, and $D$ are participating in a football tournament. Each team plays exactly one match against each of the others, and teams are awarded 2, 1, or 0 "bonus points" for a win, draw, or loss, respectively.
The day after the tournament concludes, Peter hears the end of a radio repo... | <think>
Okay, let me try to figure out this problem step by step. So, there are four teams: A, B, C, D. Each plays against each other once, so each team plays 3 matches. The points are awarded as 2 for a win, 1 for a draw, and 0 for a loss. But wait, hold on—the problem says "bonus points," but it's specified as 2, 1, ... |
olympiads | 2. Positive integers $x, y$, for which $\gcd(x, y)=3$, are the coordinates of a vertex of a square with its center at the origin and an area of $20 \cdot \text{lcm}(x, y)$. Find the perimeter of the square. | <think>
Okay, so I need to solve this problem where we have positive integers x and y with gcd(x, y) = 3. They form a vertex of a square centered at the origin, and the area of the square is 20 times the least common multiple of x and y. The goal is to find the perimeter of the square.
First, let me parse the problem ... |
olympiads | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{(n+4)!-(n+2)!}{(n+3)!}
$$ | <think>
Okay, so I need to find the limit as n approaches infinity of [(n + 4)! - (n + 2)!] divided by (n + 3)!. Hmm, factorials can get really big, so maybe there's a way to simplify the expression before taking the limit. Let me think step by step.
First, let's write out what each factorial term looks like. Remember... |
olympiads | For example, $13 x_{1}, x_{2}, \cdots, x_{1993}$ satisfy $\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{1992}-x_{1993}\right|=1993$, $y_{k}=\frac{x_{1}+x_{2}+\cdots+x_{k}}{k}(k=1,2, \cdots, 1993)$, then what is the maximum possible value of $\left|y_{1}-y_{2}\right|+\left|y_{2}-y_{3}\right|+\cdots+\... | <think>
Okay, let's try to tackle this problem step by step. So, we have these variables x₁, x₂, ..., x₁₉₉₃, and the sum of the absolute differences between consecutive terms is 1993. Then, we define yₖ as the average of the first k x's. We need to find the maximum possible value of the sum of absolute differences betw... |
olympiads | Example 4. A discrete random variable $X$ has the following distribution:
| $X$ | 3 | 4 | 5 | 6 | 7 |
| :--- | :--- | :--- | :--- | :--- | :--- |
| $P$ | $p_{1}$ | 0.15 | $p_{3}$ | 0.25 | 0.35 |
Find the probabilities $p_{1}=P(X=3)$ and $p_{3}=P(X=5)$, given that $p_{3}$ is 4 times $p_{1}$. | <think>
Okay, let me try to solve this problem. So, we have a discrete random variable X with possible values 3, 4, 5, 6, 7 and their respective probabilities p1, 0.15, p3, 0.25, 0.35. We need to find p1 and p3, given that p3 is 4 times p1. Hmm, let's see.
First, I remember that the sum of all probabilities in a proba... |
olympiads | 2. What is the minimum number of factors that need to be crossed out from the number 99! (99! is the product of all numbers from 1 to 99) so that the product of the remaining factors ends in 2? | <think>
Okay, so I need to figure out the minimum number of factors to cross out from 99! so that the remaining product ends in 2. Hmm, let me start by understanding what the problem is asking.
First, 99! is the product of all integers from 1 to 99. The question is asking, if I remove some factors from this product, ... |
olympiads | 1.6. Find the distance between two intersecting medians of the faces of a regular tetrahedron with edge 1. (Consider all possible arrangements of the medians.)
## § 2. Angles between lines and planes | <think>
Okay, so I need to find the distance between two intersecting medians of the faces of a regular tetrahedron with edge length 1. Hmm, let me start by visualizing a regular tetrahedron. It has four triangular faces, all equilateral triangles. Each face has three medians, which in an equilateral triangle are also ... |
olympiads | # Problem 7. ( **points** )
A number written on the board is allowed to be multiplied by 8, have 14 added to it, or have 14 subtracted from it. In each case, the old number is erased and the new number is written in its place. Initially, a single digit was written on the board. After several operations, the number 777... | <think>
Okay, let's see. So the problem is that we start with a single-digit number on the board, and we can perform three operations: multiply by 8, add 14, or subtract 14. After several operations, we end up with 777772. We need to figure out what the initial single-digit number was. If there are multiple possibiliti... |
olympiads | 455. Based on a sample of size $n=41$, a biased estimate $D_{\text {v }}=3$ of the population variance was found. Find the unbiased estimate of the population variance. | <think>
Okay, let's see. I need to find the unbiased estimate of the population variance based on a sample of size n=41, and they gave a biased estimate D_v=3. Hmm, right. So, first, I remember that when we talk about biased versus unbiased estimates of variance, it usually comes down to whether we're dividing by n or ... |
olympiads | 1. Find the smallest natural number, which when multiplied by 127008 gives a cube of a natural number. Explain your answer! | <think>
Okay, so I need to find the smallest natural number that when multiplied by 127008 results in a cube of a natural number. Hmm, let's see. I remember that for a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3. So, maybe I should start by factoring 127008 into its ... |
olympiads | 2. Two bottles of equal volume are filled with a mixture of water and juice. In the first bottle, the ratio of the quantities of water and juice is $2: 1$, and in the second bottle, it is $4: 1$. If we pour the contents of both bottles into a third bottle, what will be the ratio of the quantities of water and juice in ... | <think>
Okay, so there's this problem about two bottles with water and juice. Both bottles have the same volume. The first one has water and juice in a ratio of 2:1, and the second one is 4:1. We need to find the ratio of water to juice when they're mixed together in a third bottle. Hmm, ratios can sometimes be tricky,... |
olympiads | 9. (2004 Shanghai High School Competition Problem) The sequence $\left\{a_{n}\right\}$ satisfies $(n-1) a_{n+1}=(n+1) a_{n}-2(n-1), n=$ $1,2, \cdots$ and $a_{100}=10098$. Find the general term formula of the sequence $\left\{a_{n}\right\}$. | <think>
Okay, let's see. I need to find the general term formula for the sequence {a_n} given the recurrence relation (n - 1)a_{n+1} = (n + 1)a_n - 2(n - 1) for n = 1, 2, ... and knowing that a_{100} = 10098. Hmm. Alright, let me start by understanding the recurrence relation.
First, the equation is (n - 1)a_{n+1} = ... |
olympiads | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \sqrt{\operatorname{arctg} x \cdot \sin ^{2}\left(\frac{1}{x}\right)+5 \cos x}$ | <think>
Okay, so I need to find the limit as x approaches 0 of the square root of [arctg(x) times sin squared (1/x) plus 5 cos x]. Hmm, let me break this down step by step.
First, the function inside the square root is arctg(x) * sin²(1/x) + 5 cos x. The limit is as x approaches 0. Since there's a square root, I gues... |
olympiads | $[$ Theory of algorithms (miscellaneous) $]$ $[$ Extreme principle (miscellaneous) ]
Vasya solved problems for 10 days - at least one problem each day. Each day (except the first), if the wea... | <think>
Okay, let's try to figure out this problem. So, Vasya solved problems for 10 days, at least one each day. The weather affected how many problems he did each subsequent day: if it was cloudy, he solved one more than the previous day; if sunny, one less. In the first 9 days, he solved 13 problems. We need to find... |
olympiads | 5. (20 points) The clock shows the time as fifteen minutes past five. Determine the angle between the minute and hour hands at this moment.
---
Note: The translation maintains the original text's formatting and structure. | <think>
Okay, so I need to figure out the angle between the minute and hour hands when the time is fifteen minutes past five. Let me start by recalling how clock angles work.
First, I know that a clock is a circle, which has 360 degrees. There are 12 hours on the clock, so each hour represents 30 degrees because 360 ... |
olympiads | 5. In the number $2 * 0 * 1 * 6 * 0 *$, each of the 5 asterisks needs to be replaced with any of the digits $0,1,2,3,4,5,6,7,8$ (digits can repeat) so that the resulting 10-digit number is divisible by 18. In how many ways can this be done? | <think>
Okay, so I need to figure out how many ways I can replace the five asterisks in the number 2*0*1*6*0* with digits from 0 to 9 (but wait, the problem says digits 0 through 8, actually. Let me check that again. The original problem says: each of the 5 asterisks needs to be replaced with any of the digits 0,1,2,3,... |
olympiads | # Task 9.2
Factorize $x^{4}+2021 x^{2}+2020 x+2021$.
## Number of points 7
# | <think>
Okay, let's see. I need to factorize the polynomial x⁴ + 2021x² + 2020x + 2021. Hmm, quartic polynomial. Factoring quartics can be tricky. Let me start by checking if there are any obvious rational roots using the Rational Root Theorem. The possible roots would be factors of 2021 over factors of 1, so ±1, ±2021... |
olympiads | 14. Insert parentheses in the equation: $1: 2: 3: 4: 5: 6: 7: 8: 9: 10=7$, to make it true. | <think>
Okay, so I need to figure out where to put parentheses in this equation: 1:2:3:4:5:6:7:8:9:10=7. The colons represent division, right? So it's like 1 ÷ 2 ÷ 3 ÷ 4 ÷ 5 ÷ 6 ÷ 7 ÷ 8 ÷ 9 ÷ 10 = 7. But obviously, if you just divide all those numbers sequentially, the result would be a tiny fraction, way less than 1. ... |
olympiads | 2. (3 points) A student, while solving a calculation problem, was supposed to divide by 20 in the last step, but mistakenly added 20, thus obtaining an incorrect result of 180. What should the correct answer to this calculation problem be? $\qquad$ . | <think>
Okay, let me try to figure out this problem step by step. So, the problem says that a student was solving a calculation where they were supposed to divide by 20 in the last step, but instead they added 20, and got an incorrect result of 180. We need to find what the correct answer should be.
First, let me para... |
olympiads | 8. (10 points) In the inscribed quadrilateral $A B C D$, the degree measures of the angles are in the ratio $\angle A: \angle B: \angle C=2: 3: 4$. Find the length of $A C$, if $C D=8, B C=7.5 \sqrt{3}-4$. | <think>
Okay, so I have this problem about an inscribed quadrilateral ABCD. The angles at A, B, and C are in the ratio 2:3:4. I need to find the length of AC, given that CD is 8 and BC is 7.5√3 - 4. Alright, let's start by recalling some properties of cyclic quadrilaterals.
First off, in a cyclic quadrilateral, the o... |
olympiads | 5. Let $a, b$ be real numbers, and the function $f(x)=a x+b$ satisfies: for any $x \in[0,1]$, we have $|f(x)| \leq 1$. Then the maximum value of $a b$ is $\qquad$ . | <think>
Okay, so I need to find the maximum value of the product \( ab \) given that the linear function \( f(x) = ax + b \) satisfies \( |f(x)| \leq 1 \) for all \( x \) in the interval \([0, 1]\). Hmm, let's start by understanding the constraints here.
First, since \( f(x) \) is a linear function, its graph is a str... |
olympiads | Problem 2. The numbers $d$ and $e$ are the roots of the quadratic trinomial $a x^{2}+b x+c$. Could it be that $a, b, c, d, e$ are consecutive integers in some order? | <think>
Okay, so I need to figure out if there's a quadratic equation ax² + bx + c where the roots d and e, along with the coefficients a, b, c, are consecutive integers in some order. Hmm. Let me start by recalling some basic facts about quadratic equations and their roots.
First, if d and e are the roots of the quad... |
olympiads | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt[3]{n^{3}-7}+\sqrt[3]{n^{2}+4}}{\sqrt[4]{n^{5}+5}+\sqrt{n}}$ | <think>
Okay, so I need to find the limit as n approaches infinity of this sequence: the numerator is the cube root of (n³ -7) plus the cube root of (n² +4), and the denominator is the fourth root of (n⁵ +5) plus the square root of n. Hmm, limits at infinity often involve comparing the growth rates of the numerator and... |
olympiads | 43rd Putnam 1982 Problem B1 ABC is an arbitary triangle, and M is the midpoint of BC. How many pieces are needed to dissect AMB into triangles which can be reassembled to give AMC? Solution | <think>
Okay, so I have this problem here from the 43rd Putnam exam, problem B1. It says: ABC is an arbitrary triangle, and M is the midpoint of BC. How many pieces are needed to dissect AMB into triangles which can be reassembled to give AMC? Hmm, interesting. Let me try to visualize this first.
Alright, so triangle ... |
olympiads | 8. (10 points) Among the three given phrases “尽心尽力”, “力可拔山”, and “山穷水尽”, each Chinese character represents a number between 1 and 8, with the same character representing the same number and different characters representing different numbers. If the sum of the numbers represented by the characters in each phrase is 19,... | <think>
Okay, so I need to solve this Chinese character number puzzle. Let me try to break it down step by step.
First, the problem says there are three phrases: “尽心尽力”, “力可拔山”, and “山穷水尽”. Each Chinese character represents a unique number between 1 and 8. The same character has the same number, and different charact... |
olympiads | # Task No. 1.4
## Condition:
Five friends - Kristina, Nadya, Marina, Liza, and Galia - meet in the park every day after buying ice cream from the shop around the corner. One day, the girls had a conversation.
Kristina: There were five people in front of me.
Marina: I was the first in line!
Liza: No one was behind ... | <think>
Alright, let's try to figure out this problem. So, we have five friends: Kristina, Nadya, Marina, Liza, and Galia. They each made statements about their positions in the queue for ice cream. Our job is to determine how many people were between Kristina and Nadya based on their truthful statements.
First, let's... |
amc_aime | In an isosceles trapezoid, the parallel bases have lengths $\log 3$ and $\log 192$, and the altitude to these bases has length $\log 16$. The perimeter of the trapezoid can be written in the form $\log 2^p 3^q$, where $p$ and $q$ are positive integers. Find $p + q$. | <think>
Okay, let's try to figure out this problem. So, we have an isosceles trapezoid with two parallel bases. The lengths of the bases are log 3 and log 192. The altitude (which is the distance between the two bases) is log 16. We need to find the perimeter of the trapezoid and express it as log 2^p 3^q, then find p ... |
olympiads | ## Task $9 / 89$
Determine all non-negative real numbers $x$ that satisfy the equation:
$$
\sqrt[3]{x}+\sqrt[4]{x}=6 \sqrt[6]{x}
$$ | <think>
Okay, let's try to solve this equation: the cube root of x plus the fourth root of x equals six times the sixth root of x. Hmm, so in symbols, that's ∛x + ∜x = 6⁶√x. All right, first, I need to find all non-negative real numbers x that satisfy this. Since they mention non-negative, maybe we don't have to worry ... |
olympiads | ### 3.476 Find the maximum value of the expression
$A=\sin ^{2}\left(\frac{15 \pi}{8}-4 \alpha\right)-\sin ^{2}\left(\frac{17 \pi}{8}-4 \alpha\right)$ for $0 \leq \alpha \leq \frac{\pi}{8}$. | <think>
Okay, let's see. I need to find the maximum value of the expression A, which is given by:
A = sin²(15π/8 - 4α) - sin²(17π/8 - 4α) where α is between 0 and π/8. Hmm. Alright, so first, I remember that there's a trigonometric identity for the difference of squares of sine functions. Let me recall that.
The ide... |
olympiads | An ant starts from a vertex of a cube, walking only along the edges, until it returns to the initial vertex, without passing through any vertex twice. What is the longest walk that this ant can make? | <think>
Okay, so there's this problem about an ant walking on the edges of a cube. The ant starts at a vertex, moves along the edges, can't pass through any vertex twice, and wants to return to the starting point. The question is asking for the longest possible walk the ant can make under these conditions. Hmm, let's s... |
cn_contest | 8. Ryan's car averages 40 miles per gallon of gasoline, Tom's car averages 10 miles per gallon, Ryan and Tom drive their respective cars the same distance. Considering both cars together, on average, they can travel ( ) miles per gallon of gasoline.
(A) 10
(B) 16
(C) 25
(D) 30
(E) 40 | <think>
Okay, let's see. The problem is about Ryan's car and Tom's car. Ryan's car averages 40 miles per gallon, and Tom's car averages 10 miles per gallon. They both drive the same distance. We need to find the combined average miles per gallon for both cars together. The options are given from A to E, with B being 16... |
olympiads | ## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{x^{2}+\ln x^{2}}{x} d x
$$ | <think>
Okay, let's see. I need to find the indefinite integral of (x² + ln x²)/x dx. Hmm, first, maybe I should simplify the integrand to make it easier to integrate. Let me break it down into two separate terms divided by x. So, that would be (x²)/x + (ln x²)/x. Simplifying each part individually.
Starting with the ... |
aops_forum | The sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ are extended to meet at $E$. Let $H$ and $G$ be the midpoints of $BD$ and $AC$, respectively. Find the ratio of the area of the triangle $EHG$ to that of the quadrilateral $ABCD$. | <think>
Okay, so I have this problem here about a convex quadrilateral ABCD. The sides AD and BC are extended to meet at point E. Then, H and G are the midpoints of BD and AC, respectively. I need to find the ratio of the area of triangle EHG to the area of quadrilateral ABCD. Hmm, okay, let me try to visualize this.
... |
olympiads | ## Task 4 - 260834
In a plane, there are 100 different points arranged such that the following conditions are met:
(1) There is exactly one line on which more than two of the 100 points lie.
(2) On this line, exactly three of the 100 points lie.
Determine, under these conditions, the number of lines that pass throu... | <think>
Okay, let's see. So, the problem is about 100 points in a plane arranged with two conditions. The first condition is that there's exactly one line that has more than two points on it. The second condition says that on that particular line, exactly three of the 100 points lie. The question is to determine the nu... |
olympiads | A8. How many three-digit integers less than 1000 have exactly two different digits in their representation (for example, 232 , or 466 )? | <think>
Okay, let's try to figure out how many three-digit numbers less than 1000 have exactly two different digits. Hmm, so we're talking about numbers from 100 to 999, right? Because three-digit numbers start at 100 and go up to 999. The problem says "less than 1000," which includes all three-digit numbers.
First, ... |
olympiads | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-8 ; 8]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | <think>
Okay, so I need to find the maximum value of the expression E = a + 2b + c + 2d - ab - bc - cd - da, where each variable a, b, c, d is in the interval [-8, 8]. Hmm, let's see. This is an optimization problem with four variables, all bounded between -8 and 8. The expression is linear in each variable, but there ... |
olympiads | ## Task 5 - 120735
Determine all non-negative rational numbers $x$ that satisfy the equation $x+|x-1|=1$! | <think>
Okay, let's see. I need to find all non-negative rational numbers x that satisfy the equation x + |x - 1| = 1. Hmm, alright. So first, since there's an absolute value here, I remember that the expression inside the absolute value, which is (x - 1), can be either positive or negative, and the absolute value will... |
olympiads | Determine the number of pairs $(x, y)$ of positive integers for which $0<x<y$ and $2 x+3 y=80$. | <think>
Okay, let's see. I need to find the number of pairs (x, y) where both x and y are positive integers, x is less than y, and they satisfy the equation 2x + 3y = 80. Hmm, okay. Let me start by understanding the problem better.
First, the equation is 2x + 3y = 80. Both x and y have to be positive integers, so I ca... |
olympiads | . What is the value of the sum of binomial coefficients
$$
\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\cdots+\binom{n}{n-1}+\binom{n}{n} ?
$$ | <think>
Okay, so I need to find the value of the sum of binomial coefficients from 0 to n. That is, I have to calculate:
$$
\binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \cdots + \binom{n}{n-1} + \binom{n}{n}
$$
Hmm, binomial coefficients... I remember they are the numbers in Pascal's triangle, right? Each term $\bino... |
olympiads | 17. The sum of two integers $A$ and $B$ is 2010 . If the lowest common multiple of $A$ and $B$ is 14807 , write down the larger of the two integers $A$ or $B$. | <think>
Okay, so I need to find two integers A and B such that their sum is 2010 and their least common multiple (LCM) is 14807. Then, I have to determine which one is larger. Hmm, let's start by recalling some math concepts.
First, I remember that for any two numbers, the product of their LCM and their greatest commo... |
olympiads | ## Problem Statement
Calculate the areas of figures bounded by lines given in polar coordinates.
$$
r=\frac{5}{2} \sin \phi, r=\frac{3}{2} \sin \phi
$$ | <think>
Okay, so I need to calculate the areas bounded by these two polar curves: r = (5/2) sin φ and r = (3/2) sin φ. Hmm, let's think. Both of these are equations of circles in polar coordinates. I remember that the general form for a circle in polar coordinates is r = a sin φ, which represents a circle with diameter... |
olympiads | 2. Thirteen millionaires arrived at the economic forum and settled in the "Super Luxury+" hotel. The hotel has rooms of 3 different types: 6-star, 7-star, and 8-star. The millionaires need to be accommodated in such a way that all three types of rooms are used (i.e., at least one person must be placed in a 6-star room,... | <think>
Okay, so there's this problem about thirteen millionaires staying at a super luxury hotel with 6-star, 7-star, and 8-star rooms. The goal is to figure out how many ways they can be accommodated, given two conditions: all room types must be used, and a richer millionaire can't be in a room with fewer stars than ... |
olympiads | 40. Pete bought eight rice and cabbage buns and paid 1 ruble for them. Vasya bought nine buns and paid 1 ruble 1 kopek. How much does a rice bun cost, if it is known that it is more expensive than a cabbage bun and the buns cost more than 1 kopek? | <think>
Okay, let's see. So, Pete bought eight rice and cabbage buns and paid 1 ruble. Vasya bought nine buns and paid 1 ruble 1 kopek. We need to find out how much a rice bun costs, given that it's more expensive than a cabbage bun, and both types of buns cost more than 1 kopek.
First, I need to recall the currency. ... |
olympiads | Task B-2.3. (20 points) In a right triangle $A B C$, point $D$ is the foot of the altitude dropped from vertex $C$ to the hypotenuse $\overline{A B}$, point $E$ is the midpoint of segment $\overline{C D}$, and point $F$ is the intersection of lines $A E$ and $B C$. If $|A D|=4$ and $|B D|=9$, determine the length of se... | <think>
Okay, so I need to find the length of segment AF in this right triangle setup. Let me start by visualizing the problem. We have a right triangle ABC, with the right angle at C, I assume, since the altitude is dropped from C to hypotenuse AB. Point D is the foot of that altitude, so CD is perpendicular to AB. Th... |
olympiads | # Problem № 5 (15 points)
A person was building a house. Only the roof was left to be done. The builder wanted to achieve the goal that raindrops falling on the roof would slide off it as quickly as possible. Determine the angle of inclination of the roof necessary to achieve this goal. Neglect the friction of the rai... | <think>
Okay, so I need to figure out the angle of inclination for the roof that allows raindrops to slide off as quickly as possible, right? The problem says to neglect friction, which probably simplifies things. Let me start by visualizing the scenario. The roof is a sloped surface, and when a raindrop lands on it, i... |
olympiads | Example. Calculate the triple integral
$$
\iiint_{\Omega} \frac{x^{2}}{x^{2}+y^{2}} d x d y d z
$$
where the region $\Omega$ is bounded by the surfaces
$$
z=\frac{9}{2} \sqrt{x^{2}+y^{2}}, \quad z=\frac{11}{2}-x^{2}-y^{2}
$$ | <think>
Okay, so I need to calculate this triple integral of x²/(x² + y²) over a region Ω bounded by two surfaces. The surfaces are z = (9/2)√(x² + y²) and z = 11/2 - x² - y². Let me start by understanding the problem step by step.
First, the integrand is x²/(x² + y²). That simplifies to (x²)/(x² + y²) which is the sa... |
olympiads | ## PROBLEM 2
Knowing that $3^{6 n+12}+9^{3 n+6}+27^{2 n+4}=3^{4(n+3)+255}, n \in \mathbb{N}$.
Find the remainder of the division of A by 5, where $A=2^{n}+3^{n}+4^{n}+7^{n}$. | <think>
Okay, let's see. The problem is asking for the remainder when A is divided by 5, where A is 2^n + 3^n + 4^n + 7^n. But first, we need to find n from the given equation: 3^{6n+12} + 9^{3n+6} + 27^{2n+4} = 3^{4(n+3)+255}. Hmm, that seems a bit complicated, but maybe I can simplify all the terms to have the same b... |
olympiads | 1. Find all values of $p$, for each of which the numbers $p-2$, $3 \cdot \sqrt{p}$, and $-8-p$ are respectively the first, second, and third terms of some geometric progression. | <think>
Okay, so I need to find all values of p such that p - 2, 3 times the square root of p, and -8 - p are the first, second, and third terms of a geometric progression. Hmm, let me remember what a geometric progression is. It's a sequence where each term after the first is found by multiplying the previous term by ... |
olympiads | 1. Solve the equation
$$
(\sqrt{3-\sqrt{8}})^{x}+(\sqrt{3+\sqrt{8}})^{x}=6
$$ | <think>
Okay, let's see. I need to solve the equation \((\sqrt{3-\sqrt{8}})^{x} + (\sqrt{3+\sqrt{8}})^{x} = 6\). Hmm, this looks a bit complicated with those nested square roots. Maybe I can simplify the terms first. Let me focus on the bases: \(\sqrt{3 - \sqrt{8}}\) and \(\sqrt{3 + \sqrt{8}}\).
Wait, those expressio... |
olympiads | 4. On the board, the numbers $1 / 1, 1 / 2, 1 / 3, 1 / 4, \ldots 1 / 100$ are written.
It is allowed to erase any two numbers a, b and write $a b + a + b$ in their place,
then do the same with any two of the remaining numbers, and so on. What number can be left last? | <think>
Okay, let's see. The problem is about these numbers written on the board: 1/1, 1/2, 1/3, ..., up to 1/100. The allowed operation is to erase any two numbers, say a and b, and replace them with ab + a + b. Then you repeat this process until only one number remains. The question is, what number can be left last?
... |
olympiads | ## Task B-2.2.
In triangle $ABC$, the measures of the angles at vertex $A$ and vertex $C$ are $\alpha=60^{\circ}$ and $\gamma=75^{\circ}$, respectively. Calculate the distance from the orthocenter of triangle $ABC$ to vertex $B$ if $|BC|=8\sqrt{6}$. | <think>
Okay, so I need to find the distance from the orthocenter of triangle ABC to vertex B. The given angles are at A and C: α = 60°, γ = 75°, and the length of BC is 8√6. Hmm, let me recall what an orthocenter is. The orthocenter is the point where the three altitudes of a triangle intersect. So, I need to figure o... |
olympiads | 25. A sequence $\left\{a_{n}\right\}$ of real numbers is defined by $a_{1}=1$ and for all integers $n \geq 1$,
$$
a_{n+1}=\frac{a_{n} \sqrt{n^{2}+n}}{\sqrt{n^{2}+n+2 a_{n}^{2}}} .
$$
Compute the sum of all positive integers $n<1000$ for which $a_{n}$ is a rational number. | <think>
Okay, so I need to solve this problem where there's a sequence defined by a recursive formula, and I have to find the sum of all positive integers n < 1000 for which a_n is a rational number. Let me start by understanding the problem step by step.
First, the sequence is given by a₁ = 1, and for each n ≥ 1, the... |
olympiads | 24. Given $a, b$ are real numbers, and $a b=1$, let $M=\frac{a}{a+1}+\frac{b}{b+1}, N=\frac{n}{a+1}+\frac{n}{b+1}$, when $M=N$, $n=$ . $\qquad$ | <think>
Okay, let's see. The problem is about finding the value of n when M equals N, given that ab = 1. Alright, first, let me parse the problem again to make sure I understand all the parts.
We have two real numbers a and b, and their product ab is 1. Then we have M defined as (a)/(a + 1) + (b)/(b + 1). Then N is (n... |
cn_contest | 10. (20 points) Find the number of all positive integer solutions $(x, y, z)$ to the equation $\arctan \frac{1}{x}+\arctan \frac{1}{y}+\arctan \frac{1}{z}=\frac{\pi}{4}$. | <think>
Okay, so I need to find all positive integer solutions (x, y, z) to the equation arctan(1/x) + arctan(1/y) + arctan(1/z) = π/4. Hmm, arctangent functions adding up to π/4. I remember there are some addition formulas for arctangent. Let me recall.
The formula for adding two arctangent terms: arctan(a) + arctan(... |
olympiads | 6. Given that $P$ is a point on a sphere $O$ with radius $r$, and three mutually perpendicular moving chords $P A$, $P B$, and $P C$ are drawn through $P$. If the maximum distance from point $P$ to the plane $A B C$ is 1, then $r=$ $\qquad$ | <think>
Okay, so I need to find the radius r of a sphere O given that the maximum distance from a point P on the sphere to the plane ABC is 1. Here, PA, PB, and PC are three mutually perpendicular chords passing through P. Hmm, let me start by visualizing the problem.
First, since PA, PB, and PC are mutually perpendic... |
olympiads | Question 24: If $f(x)$ is an odd function with a period of 2, and when $x \in [0,1), f(x)=2^{x}-1$, then $f\left(\log _{2} \frac{1}{24}\right)=$ _.
| <think>
Okay, let's see. The problem says that f(x) is an odd function with a period of 2. And when x is in [0,1), f(x) = 2^x - 1. We need to find f(log₂(1/24)). Hmm, log base 2 of 1/24... That's the same as log₂(24⁻¹) which is -log₂(24). So the input to the function is -log₂(24).
First, let's recall what an odd func... |
olympiads | Problem 7. There is a certain number of identical plastic bags that can be nested inside each other. If all the other bags end up inside one of the bags, we will call this situation a "bag of bags." Calculate the number of ways to form a "bag of bags" from 10 bags.
## Explanation. Denote a bag with parentheses.
If th... | <think>
Okay, let me try to figure out this problem. So, we have 10 identical plastic bags that can be nested inside each other, and we need to find the number of ways to form a "bag of bags," which means all the bags end up inside one of them. The examples given use parentheses to denote the structure. For instance, o... |
inequalities | 4. 122 Solve the system of equations $\left\{\begin{array}{l}\sqrt{x-1}+\sqrt{y-3}=\sqrt{x+y}, \\ \lg (x-10)+\lg (y-6)=1 .\end{array}\right.$ | <think>
Okay, so I need to solve this system of equations. Let me first write them down clearly to make sure I have them right.
The first equation is: √(x - 1) + √(y - 3) = √(x + y)
The second equation is: lg(x - 10) + lg(y - 6) = 1
Hmm, lg usually stands for logarithm base 10, right? So that's the common logarithm.... |
olympiads | 1. (7 points) The first term of the sequence is 934. Each subsequent term is equal to the sum of the digits of the previous term, multiplied by 13. Find the $2019-$th term of the sequence. | <think>
Okay, so I need to find the 2019th term of a sequence where the first term is 934. Each next term is calculated by taking the sum of the digits of the previous term and then multiplying that sum by 13. Hmm, let me start by understanding how this sequence works.
First, let me write down the first few terms to s... |
olympiads | 2. Given is a square $A B C D$ with side length 4. Determine the largest natural number $k$ such that, for any arrangement of $k$ points strictly inside the square $A B C D$, there always exists a square with side length 1, contained within the square $A B C D$ (whose sides do not have to be parallel to the sides of th... | <think>
Okay, so I need to figure out the largest natural number k such that no matter how I place k points inside a square ABCD of side length 4, there will always be a square of side length 1 (not necessarily aligned with ABCD) that doesn't contain any of those points in its strict interior. Hmm. This seems like a pr... |
olympiads | Problem 3. How many odd three-digit numbers exist where the digit in the tens place is 4? | <think>
Alright, so I need to figure out how many odd three-digit numbers have the digit 4 in the tens place. Let me start by breaking down the problem.
First, a three-digit number has three places: hundreds, tens, and ones. The problem specifies that the tens place is 4, so the structure of the number is _ 4 _. Now,... |
olympiads | 3. In a circle of radius 3, triangles $A B C$ and $A M N$ are inscribed. The line $A M$ passes through the midpoint $E$ of segment $B C$, and the line $B N$ passes through the midpoint $F$ of segment $A C$. Find the perimeter of triangle $A B C$, if $A M: A E=2: 1$ and $B N: B F=17: 13$. | <think>
Okay, let me try to work through this problem step by step. So, we have a circle with radius 3, and two inscribed triangles ABC and AMN. The line AM passes through the midpoint E of BC, and the line BN passes through the midpoint F of AC. We need to find the perimeter of triangle ABC given the ratios AM:AE = 2:... |
amc_aime | Let $N$ be the number of positive integers that are less than or equal to $2003$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the [remainder](https://artofproblemsolving.com/wiki/index.php/Remainder) when $N$ is divided by $1000$. | <think>
Okay, so I need to find the number of positive integers ≤2003 whose base-2 (binary) representations have more 1's than 0's. Then, find this number modulo 1000. Hmm, let's break this down step by step.
First, understanding the problem: For each number from 1 to 2003, convert it to binary, count the number of 1'... |
olympiads | 1st Iberoamerican 1985 Problem A1 Find all integer solutions to: a + b + c = 24, a 2 + b 2 + c 2 = 210, abc = 440. | <think>
Alright, so I need to find all integer solutions for a, b, c that satisfy the following three equations:
1. a + b + c = 24
2. a² + b² + c² = 210
3. abc = 440
Hmm, okay. Let me think step by step. Since these are integers, maybe I can use some algebraic methods or factorizations. Let me start by recalling that... |
olympiads | 9.012. Find the natural values of $x$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
\log _{\sqrt{2}}(x-1)<4 \\
\frac{x}{x-3}+\frac{x-5}{x}<\frac{2 x}{3-x}
\end{array}\right.
$$ | <think>
Okay, so I have this system of inequalities to solve for natural values of x. The system is:
1. log base sqrt(2) of (x - 1) is less than 4
2. (x)/(x - 3) + (x - 5)/x < (2x)/(3 - x)
First, I need to find the natural numbers x that satisfy both inequalities. Let me tackle each inequality one by one and then fin... |
olympiads_ref | Determine all positive integers $n$ for which the equation
$$
x^{n}+(2+x)^{n}+(2-x)^{n}=0
$$
has an integer as a solution.
## Answer: $n=1$. | <think>
Okay, so I need to figure out for which positive integers n the equation xⁿ + (2 + x)ⁿ + (2 - x)ⁿ = 0 has an integer solution. Hmm, let's start by trying to understand the equation better. Maybe plug in some small values of n and see what happens.
First, let's consider n=1. Then the equation becomes x + (2 + ... |
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