source stringclasses 7
values | problem stringlengths 19 6.18k | generation stringlengths 345 68.1k |
|---|---|---|
olympiads | 11. Given that the internal angles $A, B, C$ of $\triangle ABC$ have opposite sides $a, b, c$ respectively, and $\sqrt{3} b \cos \frac{A+B}{2}=c \sin B$.
(1) Find the size of $\angle C$;
(2) If $a+b=\sqrt{3} c$, find $\sin A$. | <think>
Okay, let's tackle this problem step by step. First, the problem is about a triangle ABC with angles A, B, C opposite sides a, b, c respectively. The given equation is √3 * b * cos[(A + B)/2] = c * sin B. We need to find the measure of angle C in part (1), and if a + b = √3 * c, find sin A in part (2).
Startin... |
olympiads | 2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation. | <think>
Okay, let's tackle this problem. So, N is an integer represented as 777 in base b, and we need to find the smallest positive integer b such that N is a fourth power of an integer in decimal. Hmm, interesting. Let me break this down step by step.
First, I need to recall how numbers are represented in different ... |
olympiads | 7.1. Solve the equation $\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \cdot 2 \cdot \ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a solution is a triplet $(n, k, l)$; if soluti... | <think>
Okay, let's try to solve this equation: n! / 2 = k! + l! where n, k, l are natural numbers. The problem asks to find all possible solutions, and if there are multiple, sum the n values. If none, write 0. Alright, let's start by understanding the equation.
First, n! divided by 2 equals the sum of two factorials... |
olympiads | 1. A line $l$ intersects a hyperbola $c$, then the maximum number of intersection points is ( ).
A. 1
B. 2
C. 3
D. 4 | <think>
Okay, so I need to figure out the maximum number of points where a line can intersect a hyperbola. The options are 1, 2, 3, or 4. Let me start by recalling what a hyperbola is.
A hyperbola is a type of conic section, right? It's defined as the set of all points where the difference of the distances from two f... |
olympiads | 2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$. | <think>
Okay, let's see. The problem is asking how many integers \( b \) exist such that the quadratic equation \( x^2 + b x - 9600 = 0 \) has an integer solution that is a multiple of both 10 and 12. Also, we need to specify the largest possible \( b \).
First, let's parse the question. We need to find integers \( b ... |
olympiads | 10.319. The diagonals of an isosceles trapezoid are perpendicular to each other, and its area is $a^{2}$. Determine the height of the trapezoid. | <think>
Okay, let's tackle this problem step by step. So, we have an isosceles trapezoid with perpendicular diagonals, and the area is given as \( a^2 \). We need to find the height of the trapezoid. Hmm, alright, let's start by recalling some properties of isosceles trapezoids and then see how the diagonals being perp... |
olympiads | 8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).
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 last four digits of 7 raised to the power of 7^(-7), but there are 100 sevens mentioned. Wait, maybe I misread that. The original problem says "8. Find the last four digits of \(7^{7^{-7}}\) (100 sevens)." Hmm. So the notation here is a bit confusing. The user mention... |
olympiads | 9.27 In the metro train at the initial stop, 100 passengers entered. How many ways are there to distribute the exit of all these passengers at the next 16 stops of the train? | <think>
Okay, so there's this problem about a metro train where 100 passengers get on at the initial stop. The question is asking how many ways there are to distribute the exits of all these passengers at the next 16 stops. Hmm, let me think about how to approach this.
First, let me paraphrase the problem to make sure... |
olympiads | 8. If $n$ is a natural number less than 50, find all values of $n$ such that the values of the algebraic expressions $4n+5$ and $7n+6$ have a common divisor greater than 1. | <think>
Okay, so I need to find all natural numbers n less than 50 where the expressions 4n + 5 and 7n + 6 share a common divisor greater than 1. Hmm, let's break this down step by step.
First, let me recall that if two numbers have a common divisor greater than 1, then their greatest common divisor (GCD) must be grea... |
olympiads | Example 5. Expand the function
$$
f(z)=\frac{z}{z^{2}-2 z-3}
$$
into a Taylor series in the neighborhood of the point $z_{0}=0$ using expansion (12), and find the radius of convergence of the series. | <think>
Okay, so I need to expand the function f(z) = z/(z² - 2z - 3) into a Taylor series around z₀ = 0 and find the radius of convergence. Hmm, let me think. First, I remember that Taylor series expansions around 0 are like Maclaurin series. The general approach for rational functions like this is probably partial fr... |
aops_forum | Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \in \{1,2,3,4,5\}$ and $i \neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$? | <think>
Okay, so I need to figure out the smallest possible number of distinct sums you can get by adding two distinct numbers from a set of five distinct real numbers. Hmm, the problem is asking for the minimal number of distinct sums when you consider all possible pairs. Let me think step by step.
First, let me unde... |
olympiads | 149. Two equally matched opponents are playing chess. Find the most probable number of wins for any chess player if $2 N$ decisive (without draws) games will be played. | <think>
Okay, so we have two equally matched opponents playing chess, and they're going to play 2N decisive games. We need to find the most probable number of wins for any player. Hmm, let me think.
First, since they're equally matched, the probability of either player winning a game should be the same, right? So, li... |
cn_contest | 7. A circle with a radius of 1 has six points, these six points divide the circle into six equal parts. Take three of these points as vertices to form a triangle. If the triangle is neither equilateral nor isosceles, then the area of this triangle is ( ).
(A) $\frac{\sqrt{3}}{3}$
(B) $\frac{\sqrt{3}}{2}$
(C) 1
(D) $\sq... | <think>
Okay, let's see. There's a circle with radius 1, divided into six equal parts by six points. So, those points are equally spaced around the circumference, right? Each arc between consecutive points must be 60 degrees because 360 divided by 6 is 60. That makes sense. So, the circle is divided into six 60-degree ... |
olympiads | # Problem 3.
A polynomial $P(x)$ with integer coefficients has the properties
$$
P(1)=2019, \quad P(2019)=1, \quad P(k)=k,
$$
where the number $k$ is an integer. Find this number $k$.
# | <think>
Okay, let's tackle this polynomial problem. So, we have a polynomial P(x) with integer coefficients. The conditions given are P(1) = 2019, P(2019) = 1, and there's some integer k where P(k) = k. We need to find what k is.
First, since P(x) has integer coefficients, that should remind me of the integer root the... |
olympiads | 15.15 A paper punch can be placed at any point in the plane, and when it operates, it can punch out points at an irrational distance from it. What is the minimum number of paper punches needed to punch out all points in the plane?
(51st Putnam Mathematical Competition, 1990) | <think>
Alright, let me try to figure out this problem. So, the question is about a paper punch that can punch out all points at an irrational distance from where it's placed. We need to find the minimum number of these punches required to punch out every point in the plane. Hmm, interesting. Let me break this down ste... |
olympiads | 9. Point $Y$ lies on line segment $X Z$ such that $X Y=5$ and $Y Z=3$. Point $G$ lies on line $X Z$ such that there exists a triangle $A B C$ with centroid $G$ such that $X$ lies on line $B C, Y$ lies on line $A C$, and $Z$ lies on line $A B$. Compute the largest possible value of $X G$. | <think>
Okay, so I need to find the largest possible value of XG, given that Y is on XZ with XY=5 and YZ=3. Point G is on line XZ and is the centroid of some triangle ABC. The conditions are that X lies on line BC, Y lies on line AC, and Z lies on line AB. Hmm, centroids... right, the centroid divides each median in a ... |
olympiads | 5. Let the complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=\left|z_{1}+z_{2}\right|=3,\left|z_{1}-z_{2}\right|=3 \sqrt{3}$, then $\log _{3}\left|\left(z_{1} \overline{z_{2}}\right)^{2000}+\left(\overline{z_{1}} z_{2}\right)^{2000}\right|=$ $\qquad$ . | <think>
Okay, so I have this problem involving complex numbers z₁ and z₂. The given conditions are the magnitudes: |z₁| = 3, |z₁ + z₂| = 3, and |z₁ - z₂| = 3√3. I need to find the logarithm base 3 of the absolute value of (z₁ times the conjugate of z₂) raised to the 2000th power plus (the conjugate of z₁ times z₂) rais... |
olympiads | A unit side regular hexagonal lattice point is reflected to the midpoint of the six sides. Calculate the area of the resulting hexagon.
Consider an arbitrary internal point of a regular hexagon with a side length of one unit. This point is reflected to the midpoint of each of the six sides. Calculate the area of the r... | <think>
Okay, so I have this problem where there's a regular hexagon with side length 1 unit. Inside this hexagon, there's an arbitrary point, and we're supposed to reflect this point over the midpoints of each of the six sides. Then, we need to calculate the area of the resulting hexagon formed by these reflections. H... |
olympiads | 22. A year is called a leap year if it is either divisible by 4 but not divisible by 100 , or divisible by 400 . Hence, the years 2000, 2004 and 2400 are leap years while the years 2006, 2100 and 2200 are not. Find the number of leap years between 2000 and 4000 inclusive. | <think>
Okay, so I need to find the number of leap years between 2000 and 4000 inclusive. Let me recall the rules for leap years. A leap year is divisible by 4, but not by 100 unless it's also divisible by 400. So, for example, 2000 was a leap year because it's divisible by 400, but 2100 won't be a leap year because it... |
number_theory | Example 7 How many positive integer factors does 20! have? | <think>
Okay, so I need to figure out how many positive integer factors 20! has. Hmm, let me remember. I think the number of factors of a number is related to its prime factorization. Right, if a number can be written as a product of primes raised to exponents, like \( n = p_1^{e_1} \times p_2^{e_2} \times \dots \times... |
olympiads | 2. On a line, several points were marked, including points $A$ and $B$. All possible segments with endpoints at the marked points are considered. Vasya calculated that point $A$ is inside 50 of these segments, and point $B$ is inside 56 segments. How many points were marked? (The endpoints of a segment are not consider... | <think>
Okay, so I have this problem here: There's a line with several marked points, including points A and B. All possible segments with endpoints at these marked points are considered. Vasya found that point A is inside 50 of these segments, and point B is inside 56 segments. The question is, how many points were ma... |
olympiads_ref | Find all triples $(p, q, r)$ of prime numbers such that all of the following numbers are integers
$$
\frac{p^{2}+2 q}{q+r}, \quad \frac{q^{2}+9 r}{r+p}, \quad \frac{r^{2}+3 p}{p+q}
$$ | <think>
Alright, so I need to find all triples of prime numbers (p, q, r) such that the three given expressions are all integers. Let me write them out again:
1. (p² + 2q)/(q + r)
2. (q² + 9r)/(r + p)
3. (r² + 3p)/(p + q)
Hmm, primes involved. Primes are numbers greater than 1 that have no divisors other than 1 and t... |
olympiads | ## 3. Math Puzzle $8 / 65$
A crane is being transported to a construction site. To measure the length of the crane, Peter walks from the tip of the crane to the end and takes 15 steps (step length 80 $\mathrm{cm}$).
To walk back from the end to the tip of the crane while the train is moving at a constant speed, he ne... | <think>
Okay, so there's this problem about a crane on a train, and Peter is measuring its length by walking. Let me try to wrap my head around it.
First, the problem says that Peter walks from the tip of the crane to the end and takes 15 steps, each step being 80 cm. Then, when the train is moving at a constant spee... |
olympiads | 5. Let $a$ and $n$ be natural numbers, and it is known that $a^{n}-2014$ is a number with $2014$ digits. Find the smallest natural number $k$ such that $a$ cannot be a $k$-digit number. | <think>
Okay, let me try to figure out this problem. So, we have natural numbers \( a \) and \( n \), and it's given that \( a^n - 2014 \) is a number with 2014 digits. We need to find the smallest natural number \( k \) such that \( a \) cannot be a \( k \)-digit number. In other words, we need to determine the minimu... |
olympiads | What is the $21^{\varrho}$ term of the sequence
$$
1 ; 2+3 ; 4+5+6 ; 7+8+9+10 ; 11+12+13+14+15 ; \ldots ?
$$ | <think>
Okay, so I need to find the 21st term of this sequence. Let me look at the given sequence again: it starts with 1, then 2+3, then 4+5+6, then 7+8+9+10, then 11+12+13+14+15, and so on. Each term seems to be a group of consecutive numbers, where the first term has 1 number, the second term has 2 numbers, the thir... |
olympiads | Example 4 Find the integer solution of the equation $\left[\frac{x}{1!}\right]+\left[\frac{x}{2!}\right]+\left[\frac{x}{3!}\right]+\cdots+$ $\left[\frac{x}{10!}\right]=1001$. | <think>
Alright, so I need to find an integer solution x such that the sum of the floor functions from [x/1!] up to [x/10!] equals 1001. Hmm, okay. Let me think about how to approach this.
First, floor functions can complicate things because they involve taking the integer part of a division. So, for each term [x/n!],... |
olympiads | 321. $\sqrt{2 x+5}+\sqrt{x-1}=8$.
Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.
321. $\sqrt{2 x+5}+\sqrt{x-1}=8$. | <think>
Alright, so I have this equation to solve: √(2x + 5) + √(x - 1) = 8. Hmm, radicals can be tricky, but I think I need to get rid of them by squaring both sides. Let me write that down.
First, let me isolate one of the radicals if possible. But here, both radicals are added together. Maybe squaring the entire eq... |
olympiads | Problem 2. Find the number of the subsets $B$ of the set $\{1,2, \ldots, 2005\}$ having the following property: the sum of the elements of $B$ is congruent to 2006 modulo 2048 .
Emil Kolev | <think>
Okay, let me try to figure out this problem. So we need to find the number of subsets B of the set {1, 2, ..., 2005} such that the sum of the elements in B is congruent to 2006 modulo 2048. Hmm, that sounds like a problem involving modular arithmetic and combinatorics. Let me break it down step by step.
First,... |
olympiads | For example, in the sequence $x_{1}, x_{2}, \cdots, x_{n}, \cdots$, the sum of any three consecutive terms is 20, and $x_{1}=9, x_{12}=7$. Find the value of $x_{2000}$. | <think>
Okay, let me try to figure out this problem. So, we have a sequence where the sum of any three consecutive terms is 20. We know the first term, x₁ = 9, and the twelfth term, x₁₂ = 7. We need to find x₂₀₀₀. Hmm, sequences with recurring patterns or periodic behavior might be involved here. Let me think step by s... |
olympiads | 2. Let $x$ be a positive integer, and $y$ is obtained from $x$ when the first digit of $x$ is moved to the last place. Determine the smallest number $x$ for which $3 x=y$. | <think>
Okay, so I need to find the smallest positive integer x such that when you move the first digit to the end of the number, you get y, and y is equal to 3 times x. Hmm, that sounds interesting. Let me break it down step by step.
First, let's understand the problem. If x is a number, say, for example, 1234, then ... |
olympiads | ## Task B-4.6.
Determine the area of the set of all points associated with complex numbers $z$ for which
$$
|z| \leqslant\left|\frac{1}{z}\right| \quad \text{and} \quad \frac{28277 \pi}{12} \leqslant \arg \left(\frac{z^{2020}}{1+i}\right) \leqslant \frac{40397 \pi}{12}
$$ | <think>
Okay, so I need to find the area of the set of complex numbers z that satisfy two conditions: first, the magnitude of z is less than or equal to the magnitude of 1/z, and second, the argument of (z^2020 divided by (1+i)) is between 28277π/12 and 40397π/12. Hmm, this seems a bit complicated, but let me break it ... |
olympiads | 2.1 Find $f(2013)$, if for any real $x$ and $y$ the equality holds
$$
f(x-y)=f(x)+f(y)-2xy
$$ | <think>
Okay, let's try to figure out how to find f(2013) given that for any real numbers x and y, the equation f(x - y) = f(x) + f(y) - 2xy holds. Hmm, functional equations can be tricky, but maybe I can find a pattern or figure out the general form of the function f.
First, let's recall some common functional equati... |
olympiads | 7. (10 points) Two wheels rotate, meshed with each other, around fixed axes passing through the centers of wheels $A$ and $B$. The radii of the wheels differ by a factor of three. The smaller wheel makes 30 revolutions per minute. Determine how many seconds the larger wheel spends on one revolution?
. Hmm, that looks pretty complicated. Both terms under the square roots are quartic polynomials. Let me start by understanding the structure of this function.
First, since both terms a... |
olympiads | Zaslavsky A.A.
Two ants each crawled along their own closed path on a $7 \times 7$ board. Each ant only crawled along the sides of the cells and visited each of the 64 vertices of the cells exactly once. What is the minimum possible number of such sides that both the first and the second ants crawled along? | <think>
Okay, so I need to figure out the minimum number of overlapping sides that two ants could have crawled along when each ant has traveled a closed path on a 7x7 grid, visiting every vertex exactly once. Hmm, let me break this down.
First, each ant's path is a closed tour that covers all 64 vertices. That means e... |
olympiads | B. As shown in Figure 2, in the square $A B C D$ with side length 1, $E$ and $F$ are points on $B C$ and $C D$ respectively, and $\triangle A E F$ is an equilateral triangle. Then the area of $\triangle A E F$ is | <think>
Okay, so I need to find the area of triangle AEF in square ABCD with side length 1. The problem says that triangle AEF is equilateral. Hmm, let's start by visualizing the square. Let me sketch it out mentally. Square ABCD, so the vertices are A, B, C, D in order. Let me assume it's a standard square with A at t... |
olympiads | Task B-2.1. (20 points) For complex numbers $z, w$ such that $|z|=|w|=|z-w|$, calculate $\left(\frac{z}{w}\right)^{99}$. | <think>
Okay, let me try to solve this problem. So, we have two complex numbers z and w where the absolute values (or magnitudes) of z, w, and z - w are all equal. We need to calculate (z/w)^99. Hmm, complex numbers with equal magnitudes... Let me think.
First, since |z| = |w|, that means both z and w lie on the same ... |
olympiads_ref | A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find all possible values of $n$.
A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find all possible values of $n$. | <think>
Okay, let's see. I need to find all possible values of n such that a rectangle can be divided into n equal squares and also into n+76 equal squares. Hmm, interesting problem. Let me try to break this down step by step.
First, if a rectangle can be divided into equal squares, that means the rectangle's dimensio... |
olympiads | 8. Let $a>0, b>0$. Then the inequality that does not always hold is ( ).
(A) $\frac{2}{\frac{1}{a}+\frac{1}{b}} \geqslant \sqrt{a b}$
(B) $\frac{1}{a}+\frac{1}{b} \geqslant \frac{4}{a+b}$
(C) $\sqrt{|a-b|} \geqslant \sqrt{a}-\sqrt{b}$
(D) $a^{2}+b^{2}+1>a+b$ | <think>
Okay, let's see. The problem is asking which inequality does not always hold given that a and b are positive. There are four options: A, B, C, D. I need to check each one and see if there are cases where the inequality might not hold.
Starting with option A: \(\frac{2}{\frac{1}{a} + \frac{1}{b}} \geq \sqrt{ab}... |
aops_forum | Find all increasing sequences $a_1,a_2,a_3,...$ of natural numbers such that for each $i,j\in \mathbb N$, number of the divisors of $i+j$ and $a_i+a_j$ is equal. (an increasing sequence is a sequence that if $i\le j$, then $a_i\le a_j$.) | <think>
Okay, let me try to figure out this problem. So, we need to find all increasing sequences of natural numbers \( a_1, a_2, a_3, \ldots \) such that for any natural numbers \( i \) and \( j \), the number of divisors of \( i + j \) and \( a_i + a_j \) is the same. Hmm, interesting.
First, let's parse the problem... |
aops_forum | For some positive integer $n$, a coin will be flipped $n$ times to obtain a sequence of $n$ heads and tails. For each flip of the coin, there is probability $p$ of obtaining a head and probability $1-p$ of obtaining a tail, where $0<p<1$ is a rational number.
Kim writes all $2^n$ possible sequences of $n$ heads and tails... | <think>
Okay, let's try to figure out this problem step by step. So, the question is about determining all pairs (n, p) where n is a positive integer and p is a rational number between 0 and 1, such that Kim can divide all 2^n possible sequences of n heads and tails into two columns (left and right). The requirement is... |
olympiads | Tomongho A.K.
Find all increasing arithmetic progressions with a finite number of terms, the sum of which is 1, and each term has the form $1 / k$, where $k$ is a natural number. | <think>
Alright, let's tackle this problem step by step. The goal is to find all increasing arithmetic progressions with a finite number of terms where each term is of the form 1/k for some natural number k, and the sum of the terms is 1.
First, let's make sure I understand the problem correctly. An arithmetic progre... |
olympiads | ## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(5 ; 3 ;-1)$
$B(0 ; 0 ;-3)$
$C(5 ;-1 ; 0)$ | <think>
Okay, so I need to find the equation of the plane that passes through point A and is perpendicular to the vector BC. Let me see... First, I should recall the general equation of a plane. If I remember correctly, the equation of a plane can be written as a(x - x₀) + b(y - y₀) + c(z - z₀) = 0, where (a, b, c) is ... |
olympiads | \section*{Problem 1 - 101211}
In a parent-teacher meeting, exactly 18 fathers and exactly 24 mothers were present, with at least one parent of each student in the class attending.
Of exactly 10 boys and exactly 8 girls, both parents were present for each. For exactly 4 boys and exactly 3 girls, only the mother was pr... | <think>
Okay, let's see. So, the problem is about figuring out the number of children in a class who have siblings in the same class. The information given involves parents present at a parent-teacher meeting. Let me try to parse this step by step.
First, there are 18 fathers and 24 mothers present. Each student has a... |
olympiads | 8. In $\triangle A B C$, the medians $A E, B F$, and $C D$ intersect at $M$. Given that $E, C, F$, and $M$ are concyclic, and $C D=n$. Find the length of segment $A B$.
(18th All-Russian Competition Problem) | <think>
Okay, let me try to figure out this geometry problem. So, we have triangle ABC with medians AE, BF, and CD intersecting at point M. We're told that points E, C, F, and M are concyclic, which means they lie on a circle. Also, the length of median CD is given as n. We need to find the length of segment AB.
First... |
olympiads | 4. (6 points) A number when divided by 3 leaves a remainder of 2, when divided by 4 leaves a remainder of 3, and when divided by 5 leaves a remainder of 4. This number is $\qquad$ | <think>
Okay, so I need to find a number that when divided by 3 leaves a remainder of 2, when divided by 4 leaves a remainder of 3, and when divided by 5 leaves a remainder of 4. Hmm, let's think about how to approach this.
First, maybe I can write down these conditions as equations. Let me call the number N. Then:
1... |
olympiads | 11. (15 points) Let real numbers $a, b$ be such that the equation $a x^{3}-x^{2} + b x-1=0$ has three positive real roots. For all real numbers $a, b$ that satisfy the condition, find the minimum value of $P=\frac{5 a_{2}^{2}-3 a b+2}{a^{2}(b-a)}$.
| <think>
Okay, so I need to find the minimum value of this expression P given that the cubic equation has three positive real roots. Let me start by recalling some algebra related to polynomials and their roots.
First, since the equation is a cubic with three positive real roots, I can use Vieta's formulas to relate t... |
olympiads | I1.1 Let $a$ be a real number and $\sqrt{a}=\sqrt{7+\sqrt{13}}-\sqrt{7-\sqrt{13}}$. Find the value of $a$.
I1.2 In Figure 1, the straight line $\ell$ passes though the point $(a, 3)$, and makes an angle $45^{\circ}$ with the $x$-axis. If the equation of $\ell$ is $x+m y+n=0$ and $b=|1+m+n|$, find the value of $b$.
I1.... | <think>
Okay, let's try to solve these four problems step by step. I need to take my time with each one to make sure I understand what's being asked and how to approach it. Let's start with I1.1.
**Problem I1.1:**
We have a real number \( a \) such that \( \sqrt{a} = \sqrt{7 + \sqrt{13}} - \sqrt{7 - \sqrt{13}} \). We ... |
olympiads | 7.1. Pasha and Sasha made three identical toy cars. Sasha made a third of the first car, a fifth of the second car, and a fifteenth of the third car. What part of the entire work did Sasha do? | <think>
Okay, let's try to figure out this problem step by step. So, Pasha and Sasha made three identical toy cars. The question is asking what part of the entire work did Sasha do. Hmm, let me parse the details again.
Sasha made a third of the first car, a fifth of the second car, and a fifteenth of the third car. So... |
amc_aime | Let $n$ be the number of pairs of values of $b$ and $c$ such that $3x+by+c=0$ and $cx-2y+12=0$ have the same graph. Then $n$ is:
$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ \text{finite but more than 2}\qquad \textbf{(E)}\ \infty$ | <think>
Okay, let's see. The problem is asking for the number of pairs (b, c) such that the two linear equations 3x + by + c = 0 and cx - 2y + 12 = 0 represent the same graph. Then, we need to determine if the number of such pairs is 0, 1, 2, finite but more than 2, or infinite. The answer choices are given from A to E... |
aops_forum | Let $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generated by a rotation of the figure enclosed
by the curve $C: y=\frac{x}{x+k}\ (x\geq 0)$, the line $x=a$ and the $x$-axis around the $x$-axis, and denote by $V_2$ that of
the solid by a rotation of the figure enclosed by the curve $C$, t... | <think>
Okay, so I need to find the ratio V2/V1 where V1 is the volume obtained by rotating the area under the curve y = x/(x + k) from x=0 to x=a around the x-axis, and V2 is the volume obtained by rotating the area bounded by the same curve, the line y = a/(a + k), and the y-axis around the y-axis. Hmm, let me start ... |
olympiads | Francisco has 3 daughters: Alina, Valentina, and Civela. An interesting fact is that all three daughters were born on March 18. Today, March 18, 2014, is their birthday. Noting another curious fact, Francisco says:
- Alina, your age is now double the age of Valentina.
a) Show that this could never have happened befor... | <think>
Okay, let's try to tackle part a) first. So Francisco has three daughters: Alina, Valentina, and Civela. All three were born on March 18, and today is March 18, 2014, their birthday. Francisco says that Alina's age is now double Valentina's age. We need to show that this could never have happened before and tha... |
olympiads | 7. Highway (from 7th grade, 3 points). A highway running from west to east intersects with $n$ equal roads, numbered from 1 to $n$ in order. Cars travel on these roads from south to north and from north to south. The probability that a car will approach the highway from each of these roads is $\frac{1}{n}$. Similarly, ... | <think>
Okay, let me try to figure out this probability problem. So, there's a highway that goes from west to east, and it intersects with n equal roads. These roads are numbered from 1 to n in order, right? Cars are coming from both south to north and north to south on these roads. The probability that a car enters th... |
olympiads | 10. On a plane, 2011 points are marked. We will call a pair of marked points $A$ and $B$ isolated if all other points are strictly outside the circle constructed on $A B$ as its diameter. What is the smallest number of isolated pairs that can exist? | <think>
Okay, so I have this problem here: On a plane, 2011 points are marked. We need to find the smallest number of isolated pairs. An isolated pair A and B is defined as a pair where all the other points are strictly outside the circle constructed on AB as its diameter. Hmm, interesting.
First, let me make sure I u... |
cn_contest | 10. If for all positive numbers $x, y$, the inequality $\sqrt{x}+\sqrt{y}$ $\leqslant a \sqrt{x+y}$ holds, then the minimum value of $a$ is $\qquad$ | <think>
Okay, so I need to find the minimum value of 'a' such that for all positive numbers x and y, the inequality √x + √y ≤ a√(x + y) holds. Hmm, let's see. I think this might be related to some kind of inequality theorem or maybe Cauchy-Schwarz? I'm not sure, but let me work through it step by step.
First, the prob... |
cn_contest | Example 5 Let $f(n)$ be a function defined on $\mathbf{N}_{+}$ taking non-negative integer values, and for all $m, n \in \mathbf{N}_{+}$, we have
$$
\begin{array}{l}
f(m+n)-f(m)-f(n)=0 \text{ or } 1, \\
f(2)=0, f(3)>0, f(6000)=2000 . \\
\text{Find } f(5961) .
\end{array}
$$ | <think>
Okay, let me try to work through this problem step by step. First, let me make sure I understand what's being asked. We have a function f defined on the positive integers, taking non-negative integer values. For any two positive integers m and n, the expression f(m + n) - f(m) - f(n) is either 0 or 1. Additiona... |
olympiads | What is the least number of moves it takes a knight to get from one corner of an $n \times n$ chessboard, where $n \geqslant 4$, to the diagonally opposite corner? | <think>
Okay, so I need to figure out the least number of moves a knight takes to go from one corner of an n x n chessboard to the diagonally opposite corner. Let's start by recalling how a knight moves. A knight moves in an L-shape: two squares in one direction and then one square perpendicular, right? So from any giv... |
olympiads | Example 10 Arrange all positive integers that are coprime with 105 in ascending order. Try to find the 1000th term of this sequence. | <think>
Okay, so I need to find the 1000th term in the sequence of all positive integers that are coprime with 105, arranged in ascending order. Hmm, let me start by understanding what the problem is asking.
First, numbers that are coprime with 105. That means numbers that don't share any common factors with 105 excep... |
olympiads | The figure below represents a bookshelf with two shelves and five stacks of books, three of them with two books and two of them with only one book. Alice and Luiz invented a game in which each of them, alternately, removes one or two books from one of the stacks of books. The one who takes the last book wins. Alice sta... | <think>
Okay, so there's this bookshelf game between Alice and Luiz. Let me try to figure out who has the winning strategy. Hmm. The setup is two shelves with five stacks. Three of the stacks have two books each, and two have only one book. So total books would be (3*2) + (2*1) = 6 + 2 = 8 books. They take turns removi... |
olympiads | We are given a regular polygon with 200 sides. How many triangles can be formed whose three vertices are vertices of this polygon? | <think>
Okay, so I need to figure out how many triangles can be formed from the vertices of a regular 200-sided polygon. Hmm, let's see. I remember that when dealing with combinations of vertices, triangles are just combinations of three vertices. So maybe I can use the combination formula here. The formula for combina... |
olympiads | 13. 6 different points are given on the plane, no three of which are collinear. Each pair of points is to be joined by a red line or a blue line subject to the following restriction: if the lines joining $A B$ and $A C$ (where $A, B, C$ denote the given points) are both red, then the line joining $B C$ is also red. How... | <think>
嗯，这个题目看起来有点挑战性，不过让我慢慢来思考。题目是说，平面上有6个不同的点，没有三点共线。每两个点之间用红色或蓝色连线，但有一个限制：如果AB和AC都是红色的，那么BC也必须是红色的。要求有多少种不同的颜色组合。
首先，我需要理解题目的条件。这个条件类似于一种传递性吗？比如，如果两点通过第三个点都连了红色边，那么这两点之间也必须连红色边。这种情况下，红色边可能构成某种闭包的性质，或者说红色边的集合需要满足某种条件，避免出现某种结构？
或许可以考虑将红色边的集合看作满足某种闭包条件的图，也就是说，如果两点有共同的红色邻居，那么这两点之间也必须有红色边。或者说，这样的红色边构成的子图不能存在某个特定的结构，比如... |
olympiads | 6. (15 points) The radar received the signal reflected from the target after 6 microseconds. Determine the distance to the target, given that the speed of the radar signal is $300000 \mathrm{k} \nu / \mathrm{c}$. Note that one microsecond is one millionth of a second. | <think>
Okay, so I need to figure out the distance to the target based on the time it took for the radar signal to bounce back. Hmm, the problem says the radar received the reflected signal after 6 microseconds. And the speed of the radar signal is given as 300,000 km per second. Wait, let me make sure I got the units ... |
olympiads | 3. Let $a, b \in \mathbf{C}, a^{2}+b^{2}=7, a^{3}+b^{3}=10$, and let $m, n$ be the maximum and minimum real values that $a+b$ can take, respectively. Then $m+n=$ $\qquad$ | <think>
Okay, let's see. I have this problem where a and b are complex numbers, and they satisfy two equations: a² + b² = 7 and a³ + b³ = 10. I need to find the maximum and minimum real values that a + b can take, and then add those two together to get m + n. Hmm, okay. Let me think about how to approach this.
First, ... |
cn_contest | Four. (20 points) Let $n$ be a positive integer greater than 1. If there exists a positive integer $i (i < n)$ such that
$$
\sum_{a=1}^{i} a^{2}-\sum_{b=i+1}^{n} b^{2}
$$
is a perfect square, then $n$ is called a "TI number." For example, by $\left(1^{2}+2^{2}+\cdots+9^{2}\right)-\left(10^{2}+11^{2}\right)=8^{2}$, we ... | <think>
Okay, let's tackle this problem step by step. We need to determine if 46, 36, and 26 are TI numbers. A TI number is defined as a positive integer n > 1 for which there exists some positive integer i < n such that the difference between the sum of squares from 1 to i and the sum of squares from i+1 to n is a per... |
olympiads | If $x=\frac{1}{4}$, which of the following has the largest value?
(A) $x$
(B) $x^{2}$
(C) $\frac{1}{2} x$
(D) $\frac{1}{x}$
(E) $\sqrt{x}$ | <think>
Alright, so I need to figure out which of the options A through E is the largest when x is 1/4. Let's start by recalling the value of x here. The problem says x equals 1/4. That's 0.25 in decimal form. Maybe converting each option to a decimal will help me compare them more easily. Let me write down each option... |
olympiads | [ Inscribed and Circumscribed Circles ] [ Inscribed Angle Subtended by a Diameter ]
A circle with radius 1 is circumscribed around triangle $A P K$. The extension of side $A P$ beyond vertex $P$ intercepts a segment $B K$ from the tangent to the circle at vertex $K$, and $B K$ is equal to 7. Find the area of triangle ... | <think>
Okay, let's try to tackle this geometry problem. Hmm, so we have a triangle APK with a circumscribed circle of radius 1. The extension of side AP beyond point P intercepts a segment BK from the tangent at K, and BK is 7 units long. Also, angle ABK is arctg(2/7). We need to find the area of triangle APK.
First,... |
aops_forum | To [i]dissect [/i] a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals:
[img]https://cdn.artofproblemsolving.com/attachments/0/a/378e477bcbcec26fc90412c3eada855ae52b45.png... | <think>
Okay, so I need to figure out for which integers n it's possible to dissect a regular n-sided polygon into integer-ratio right triangles. Hmm. Let me start by understanding the problem thoroughly.
First, a regular n-gon. Dissecting it means cutting it into several regions using straight line segments. The key ... |
olympiads | Example 5 Find the maximum constant $k$, such that $\frac{k a b c}{a+b+c} \leqslant(a+b)^{2}+(a+b+4 c)^{2}$ holds for all positive real numbers $a, b, c$.
| <think>
Okay, let me try to figure out this problem. So, the question is asking for the maximum constant k such that the inequality (k a b c)/(a + b + c) ≤ (a + b)^2 + (a + b + 4c)^2 holds for all positive real numbers a, b, c. Hmm, that sounds a bit complicated, but maybe I can break it down step by step.
First, I ne... |
olympiads | 4. Find all real numbers $p$ such that the cubic equation $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ has three positive integer roots.
untranslated text remains the same as requested. | <think>
Okay, let me try to solve this problem. We need to find all real numbers p such that the cubic equation 5x³ -5(p+1)x² + (71p -1)x +1 = 66p has three positive integer roots. Hmm, okay. Let me break this down step by step.
First, let me rewrite the equation to standard form by moving the 66p to the left side:
5... |
olympiads | Tokarev S.i.
The path from platform A to platform B was traveled by the electric train in $X$ minutes ($0<X<60$). Find $X$, given that at the moment of departure from $A$ and upon arrival at $B$, the angle between the hour and minute hands was $X$ degrees. | <think>
Okay, let's see. The problem is about an electric train traveling from platform A to B in X minutes, where X is between 0 and 60. We need to find X such that both when the train departs A and arrives at B, the angle between the hour and minute hands of a clock is X degrees. Hmm, interesting.
First, I need to r... |
olympiads | 7. Given a moving point $P(x, y)$ satisfies the quadratic equation $10 x-2 x y-2 y+1=0$. Then the eccentricity of this quadratic curve is
$\qquad$ - | <think>
Okay, so I need to find the eccentricity of the quadratic curve defined by the equation 10x - 2xy - 2y + 1 = 0. Hmm, quadratic curves can be circles, ellipses, parabolas, hyperbolas, or degenerate forms. The eccentricity tells me which type it is and how "stretched" it is. Let me recall that eccentricity for co... |
olympiads | Let $A$ be the following.
A numerical sequence is defined by the conditions: $a_{1}=1, a_{n+1}=a_{n}+\left[\sqrt{a_{n}}\right]$.
How many perfect squares occur among the first terms of this sequence, not exceeding
1000000? | <think>
Okay, so I need to figure out how many perfect squares are in the sequence defined by a₁ = 1 and aₙ₊₁ = aₙ + [√aₙ], where the brackets mean the floor function. And we need to check up to 1,000,000. Hmm, let's break this down step by step.
First, let me make sure I understand the sequence. The first term is 1. ... |
cn_contest | Long) and 1 public F: foundation code. A customer wants to buy two kilograms of candy. The salesperson places a 1-kilogram weight on the left pan and puts the candy on the right pan to balance it, then gives the candy to the customer. Then, the salesperson places the 1-kilogram weight on the right pan and puts more can... | <think>
Okay, let me try to figure out this problem. So, the customer wants to buy two kilograms of candy. The salesperson uses a balance scale with a 1-kilogram weight. Hmm, but the problem mentions "Long) and 1 public F: foundation code." Wait, maybe that's a typo or something mistranslated? Let me ignore that part f... |
olympiads | 3A. Given the set of quadratic functions $f(x)=a x^{2}+b x+c$, for which $a<b$ and $f(x) \geq 0$ for every $x \in \mathbb{R}$. Determine the smallest possible value of the expression $A=\frac{a+b+c}{b-a}$. | <think>
Okay, so I need to find the smallest possible value of A, which is (a + b + c) divided by (b - a). The conditions are that for the quadratic function f(x) = ax² + bx + c, it's always non-negative for every real x, and also that a < b. Hmm, let me start by breaking down the problem.
First, since f(x) is a quadr... |
olympiads | In an equilateral triangle $A B C$ with a side length of $8 \mathrm{~cm}$, point $D$ is the midpoint of side $B C$ and point $E$ is the midpoint of side $A C$. Point $F$ lies on segment $B C$ such that the area of triangle $A B F$ is equal to the area of quadrilateral $A B D E$.
Calculate the length of segment $B F$.
... | <think>
Okay, let's tackle this geometry problem. So, we have an equilateral triangle ABC with each side being 8 cm. Point D is the midpoint of BC, and point E is the midpoint of AC. We need to find the length of segment BF, where point F is on BC such that the area of triangle ABF is equal to the area of quadrilateral... |
olympiads | From a barrel, we pour out 4 liters of wine and replace it with 4 liters of water. From the resulting mixture, we again pour out 4 liters and replace it with 4 liters of water. We repeat this process a total of three times, and in the final result, there is 2.5 liters more water than wine. How many liters of wine were ... | <think>
Okay, so I have this problem where we start with a barrel of wine. Then, we pour out 4 liters of wine and replace it with 4 liters of water. We do this three times total, and after the third time, there's 2.5 liters more water than wine. The question is, how many liters of wine were originally in the barrel?
H... |
olympiads | XLVII OM - I - Problem 2
A palindromic number is defined as a natural number whose decimal representation read from left to right is the same as when read from right to left. Let $ (x_n) $ be the increasing sequence of all palindromic numbers. Determine all prime numbers that are divisors of at least one of the differ... | <think>
Okay, so I have this problem about palindromic numbers and their differences. The question is asking for all prime numbers that can divide at least one of the differences between consecutive palindromic numbers. Hmm, let's start by recalling what a palindromic number is. It's a number that reads the same forwar... |
olympiads | ## Task A-1.5.
How many four-digit numbers divisible by 3 do not contain the digits $2, 4, 6$ or 9 in their decimal representation? | <think>
Okay, so I need to figure out how many four-digit numbers are divisible by 3 and don't contain the digits 2, 4, 6, or 9. Let me start by breaking down the problem step by step.
First, a four-digit number ranges from 1000 to 9999. The first digit can't be zero, right? So the thousands place has to be from 1 to ... |
olympiads | $\frac{1}{4}$ of 100 is equal to
(A) $20 \%$ of 200
(B) $10 \%$ of 250
(C) $15 \%$ of 100
(D) $25 \%$ of 50
(E) $5 \%$ of 300 | <think>
Okay, let's see. The problem is asking me to figure out which of the options (A) through (E) is equal to 1/4 of 100. Alright, let's start by calculating what 1/4 of 100 is.
So, 1/4 of 100... Well, 100 divided by 4 is 25. Right? Because 4 times 25 is 100. So, 1/4 of 100 is 25. Got that down. Now, I need to che... |
olympiads | ## 162. Math Puzzle $11 / 78$
Assume that a fly lays 120 eggs at the beginning of summer, on June 21st, and after 20 days, fully developed insects emerge from these eggs, each of which then lays 120 eggs. How many "descendants" would this fly have in total by the beginning of autumn? | <think>
Okay, so I need to figure out how many descendants the original fly has by the beginning of autumn, which is September 21st, right? Let me break this down step by step.
First, the timeline. The fly lays 120 eggs on June 21st. Then, after 20 days, those eggs hatch into adult flies, each of which also lays 120 ... |
olympiads | Given an integer $\mathrm{n} \geq 3$, let $\mathrm{A}_{1}, \mathrm{~A}_{2}, \ldots, \mathrm{~A}_{2 \mathrm{n}}$ be pairwise distinct non-empty subsets of the set $\{1,2, \ldots, \mathrm{n}\}$, and let $A_{2 n+1}=A_{1}$. Find the maximum value of $\sum_{i=1}^{2 n} \frac{\left|A_{i} \cap A_{i+1}\right|}{\left|A_{i}\right... | <think>
Okay, so I need to find the maximum value of this sum: the sum from i=1 to 2n of (|A_i ∩ A_{i+1}|)/(|A_i| * |A_{i+1}|), where each A_i is a distinct non-empty subset of {1,2,...,n}, and A_{2n+1} is set to A_1. Hmm. Alright, let me break this down.
First, the problem is about arranging 2n distinct non-empty sub... |
cn_contest | 3. In the regular triangular frustum $A^{\prime} B^{\prime} C^{\prime}-A B C$, $A^{\prime} B^{\prime} : A B = 5 : 7$, the sections $A B C^{\prime}$ and $A^{\prime} B C^{\prime}$ divide the frustum into three pyramids $C^{\prime}-A B C$, $B-A^{\prime} B^{\prime} C^{\prime}$, and $C^{\prime}-A A B A^{\prime}$. The volume... | <think>
Okay, so I need to find the volume ratio of three pyramids formed by sections in a regular triangular frustum. The frustum is given as A'B'C'-ABC, with the ratio of A'B' to AB being 5:7. The sections mentioned are ABC' and A'BC', which divide the frustum into three pyramids: C'-ABC, B-A'B'C', and C'-AABA'. Hmm,... |
olympiads | 1 Convex quadrilateral $E F G H$ has vertices $E, F, G, H$ on the sides $A B, B C, C D, D A$ of convex quadrilateral $A B C D$, respectively, satisfying
$$
\frac{A E}{E B} \cdot \frac{B F}{F C} \cdot \frac{C G}{G D} \cdot \frac{D H}{H A}=1
$$
and points $A, B, C, D$ are on the sides $H_{1} E_{1}, E_{1} F_{1}, F_{1} G_... | <think>
Okay, let's try to tackle this problem. It seems a bit complicated at first glance, but maybe breaking it down step by step will help.
First, let me parse the given information. We have two convex quadrilaterals: ABCD and EFGH. The vertices E, F, G, H are on the sides AB, BC, CD, DA of ABCD respectively. Ther... |
olympiads | 12.309. The base of the prism is a rectangle. The lateral edge forms equal angles with the sides of the base and is inclined to the plane of the base at an angle $\alpha$. Find the angle between the lateral edge and the side of the base. | <think>
Okay, so I need to find the angle between the lateral edge of a prism and the side of the base. The base is a rectangle, and the lateral edge forms equal angles with the sides of the base. Also, the lateral edge is inclined to the plane of the base at an angle α. Hmm, let me try to visualize this.
First, let m... |
olympiads | 7. Given positive real numbers $a, b$ satisfy $a b(a+b)=4$, then the minimum value of $2 a+b$ is $\qquad$
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | <think>
Alright, let's tackle this problem. We have positive real numbers a and b that satisfy the equation ab(a + b) = 4. We need to find the minimum value of 2a + b.
First, I should recall some optimization techniques. Since there's a constraint given by ab(a + b) = 4, maybe I can use the method of Lagrange multipli... |
olympiads | 9. (15 points) As shown in Figure 2, circle $\odot M$ is externally tangent to circle $\odot O$ at point $C$, with radii $r$ and $R$ for $\odot M$ and $\odot O$ respectively. Line $TPQ$ is tangent to $\odot M$ at point $T$, and intersects $\odot O$ at points $P$ and $Q$. Find the value of $\frac{CQ - CP}{PQ}$. | <think>
Okay, so I need to find the value of (CQ - CP)/PQ where C is the point of external tangency between two circles, circle M and circle O. The circle M has radius r and circle O has radius R. There's a tangent line TPQ that touches circle M at point T and intersects circle O at points P and Q.
First, I should vi... |
olympiads | 8.2. One side of the rectangle (width) was increased by $10 \%$, and the other (length) - by $20 \%$. a) Could the perimeter increase by more than 20\%? b) Find the ratio of the sides of the original rectangle if it is known that the perimeter of the new rectangle is $18 \%$ larger than the perimeter of the original? | <think>
Okay, so I've got this problem about a rectangle where one side is increased by 10% and the other by 20%. Part a asks if the perimeter could increase by more than 20%. Hmm. Let me think about how perimeters work.
First, the perimeter of a rectangle is calculated as P = 2*(length + width). So, if both sides ar... |
olympiads | 7.5. Vasya had 101 real coins. One of them was replaced with a fake one (it differs in weight, but it is unknown whether it is lighter or heavier than a real coin). Vasya wants to find as many real coins as possible in one weighing on a balance scale without weights. What should he do and how many real coins will he be... | <think>
Okay, let's see. Vasya has 101 coins, and one of them is fake. The fake one is either lighter or heavier, but we don't know which. He wants to find as many real coins as possible using just one weighing on a balance scale. Hmm, interesting problem. So, the key here is that he can only do one weighing, and from ... |
olympiads | 1. Find the minimum value of the expression $2 x+y$, defined on the set of all pairs $(x, y)$ satisfying the condition
$$
3|x-y|+|2 x-5|=x+1
$$ | <think>
Okay, let me try to figure out how to find the minimum value of 2x + y given the condition 3|x - y| + |2x - 5| = x + 1. Hmm, this looks like an absolute value equation constraint. I remember that absolute value equations can often be tackled by considering different cases based on the critical points where the ... |
cn_contest | 7. Let $z=x+y \mathrm{i} (x, y \in \mathbf{R}, \mathrm{i}$ be the imaginary unit), the imaginary part of $z$ and the real part of $\frac{z-\mathrm{i}}{1-z}$ are both non-negative. Then the area of the region formed by the points $(x, y)$ on the complex plane that satisfy the conditions is $\qquad$ . | <think>
Alright, let's tackle this problem step by step. The problem states:
Let \( z = x + y \mathrm{i} \) where \( x, y \) are real numbers, and \( \mathrm{i} \) is the imaginary unit. The imaginary part of \( z \) and the real part of \( \frac{z - \mathrm{i}}{1 - z} \) are both non-negative. We need to find the are... |
olympiads | Problem 5. If $a_{0}=0$ and $a_{n}=a_{\left[\frac{n}{2}\right]}+\left[\frac{n}{2}\right], n \geq 1$, find $\lim _{n \rightarrow+\infty} \frac{a_{n}}{n}$. | <think>
Okay, let's see. I need to find the limit as n approaches infinity of a_n divided by n, where the sequence a_n is defined recursively. The initial term is a_0 = 0, and for n ≥ 1, a_n is equal to a_[n/2] plus [n/2]. Here, the brackets probably mean the floor function, right? So [n/2] is the greatest integer less... |
olympiads | There are functions $f(x)$ with the following properties:
- $f(x)=a x^{2}+b x+c$ for some integers $a, b$ and $c$ with $a>0$, and
- $f(p)=f(q)=17$ and $f(p+q)=47$ for some prime numbers $p$ and $q$ with $p<q$. For each such function, the value of $f(p q)$ is calculated. The sum of all possible values of $f(p q)$ is $S... | <think>
Okay, let's try to figure out this problem. So, we have a quadratic function f(x) = ax² + bx + c, where a, b, c are integers and a is positive. There are two primes p and q, with p < q, such that f(p) = f(q) = 17 and f(p + q) = 47. We need to find all possible values of f(pq), sum them up to get S, and then fin... |
olympiads | ## Problem Statement
Calculate the area of the figure bounded by the lines given by the equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=2 \sqrt{2} \cdot \cos ^{3} t \\
y=\sqrt{2} \cdot \sin ^{3} t
\end{array}\right. \\
& x=1(x \geq 1)
\end{aligned}
$$ | <think>
Okay, let's see. I need to calculate the area of the figure bounded by the parametric equations given and the line x=1 (for x ≥ 1). Hmm, parametric equations... So, first, let me try to understand what the curve looks like. The parametric equations are x = 2√2 cos³t and y = √2 sin³t. That seems like a parametri... |
olympiads | Russian problems Problem 39 We place labeled points on a circle as follows. At step 1, take two points at opposite ends of a diameter and label them both 1. At step n>1, place a point at the midpoint of each arc created at step n-1 and label it with the sum of the labels at the two adjacent points. What is the total su... | <think>
Okay, let's try to tackle this problem step by step. So, we're supposed to figure out the total sum of the labels after each step n on a circle where points are added following specific rules. The example given is after step 4, the labels are a sequence of numbers adding up to... well, the sum isn't given, but ... |
olympiads | Subject 2. Determine the integer values of $x$ and $y$ such that
$$
x-3 y+4=0 \quad \text { and } \quad \sqrt{x^{2}+7 y^{2}+8 x+8 y+4} \in \mathbb{Q}
$$ | <think>
Okay, let's see. I need to find integer values of x and y that satisfy two conditions: first, the equation x - 3y + 4 = 0, and second, that the square root of x² + 7y² + 8x + 8y + 4 is a rational number. Hmm. Let me start by breaking down each part.
First, the linear equation: x - 3y + 4 = 0. Maybe I can solve... |
olympiads | 1. Find the number of points in the $x O y$ plane that have natural coordinates $(x, y)$ and lie on the parabola $y=-\frac{x^{2}}{4}+3 x+\frac{253}{4}$. | <think>
Okay, so I need to find the number of points with natural coordinates (x, y) that lie on the parabola given by the equation y = -x²/4 + 3x + 253/4. Hmm, natural coordinates mean that both x and y have to be positive integers, right? So, x and y are elements of N, which is 1, 2, 3, and so on. Let me try to figur... |
olympiads | As shown in the figure, a rectangular table has 8 columns. Numbers $1, 2, \cdots$ are filled into the table in a certain order (filled from left to right, and when a row is full, move to the next row, still filling from left to right). A student first colors the cell with the number 1 black, then skips 1 cell, and colo... | <think>
Okay, let's try to figure out this problem step by step. So, we have a rectangular table with 8 columns. Numbers are filled in from left to right, row by row. The student starts coloring cells in a specific pattern: color 1, skip 1, color 3, skip 2, color 6, skip 3, and so on. This continues until every column ... |
olympiads | (4) Given that $\vec{a}$ and $\vec{b}$ are non-zero and non-collinear vectors, let condition $M: \vec{b} \perp (\vec{a}-\vec{b})$;
condition $N$: for all $x \in \mathbf{R}$, the inequality $|\vec{a}-x \vec{b}| \geqslant |\vec{a}-\vec{b}|$ always holds. Then $M$ is ( ) of $N$.
(A) a necessary but not sufficient conditio... | <think>
Okay, so I need to figure out the relationship between condition M and condition N. Let me first understand what each condition means.
Condition M is that vector b is perpendicular to the vector (a - b). So mathematically, that should mean the dot product of b and (a - b) is zero. So if I write that out, it's:... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.