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value | problem stringlengths 22 3.47k | solution stringlengths 577 6.77k | messages listlengths 2 2 |
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olympiads | Let \( a_0, a_1, \ldots \) be a sequence of positive integers, and let \( (b_n) \) be the sequence defined by \( b_n = \operatorname{gcd}(a_n, a_{n+1}) \) for all \( n \geq 0 \). Show that it is possible to choose the sequence \( (a_n) \) such that every nonzero natural number is equal to exactly one of the terms \( a_... | To construct a sequence \( \left(a_n\right) \) such that every positive integer appears exactly once in the combined sequences \( a_0, b_0, a_1, b_1, \ldots \), let us start with initial terms:
1. Set \( a_0 = 2 \) and \( a_1 = 9 \).
Thus, we have
\[
b_0 = \mathrm{gcd}(a_0, a_1) = \mathrm{gcd}(2, 9) = 1.
... | [
{
"content": "Let \\( a_0, a_1, \\ldots \\) be a sequence of positive integers, and let \\( (b_n) \\) be the sequence defined by \\( b_n = \\operatorname{gcd}(a_n, a_{n+1}) \\) for all \\( n \\geq 0 \\). Show that it is possible to choose the sequence \\( (a_n) \\) such that every nonzero natural number is equa... |
olympiads | Prove that for \( n \geqslant 1 \) and \( m \geqslant 0 \) the following equality holds:
\[
F_{n+m} = F_{n-1} F_{m} + F_{n} F_{m+1}
\]
Try to prove it in two ways: using the method of mathematical induction and using the Fibonacci number interpretation from problem 3.109. Also, prove that Cassini's identity (see pro... |
First, we need to prove the identity using the method of mathematical induction.
**Base Case:**
For \( n = 1 \) and \( m \geq 0 \), we need to verify that:
\[ F_{1+m} = F_{0}F_{m} + F_{1}F_{m+1} \]
Let us check the base cases for some values of \( m \):
1. For \( m = 0 \):
\[ F_{1+0} = F_1 = 1 \]
\[ F_{0}F_{0} + ... | [
{
"content": "Prove that for \\( n \\geqslant 1 \\) and \\( m \\geqslant 0 \\) the following equality holds:\n\\[ \nF_{n+m} = F_{n-1} F_{m} + F_{n} F_{m+1} \n\\]\n\nTry to prove it in two ways: using the method of mathematical induction and using the Fibonacci number interpretation from problem 3.109. Also, pro... |
olympiads | Let \( m, n, k \) be positive integers with \( m \geq n \) and \( 1 + 2 + \ldots + n = mk \). Prove that the numbers \( 1, 2, \ldots, n \) can be divided into \( k \) groups in such a way that the sum of the numbers in each group equals \( m \). |
We are given positive integers \(m\), \(n\), and \(k\) with \(m \geq n\), and the sum of the first \(n\) positive integers equals \(mk\).
Let's recall that
\[
1 + 2 + \cdots + n = \frac{n(n+1)}{2}
\]
Given this sum equals \(mk\), we have
\[
\frac{n(n+1)}{2} = mk \quad \text{(1)}
\]
We need to show that the numbe... | [
{
"content": "Let \\( m, n, k \\) be positive integers with \\( m \\geq n \\) and \\( 1 + 2 + \\ldots + n = mk \\). Prove that the numbers \\( 1, 2, \\ldots, n \\) can be divided into \\( k \\) groups in such a way that the sum of the numbers in each group equals \\( m \\).",
"role": "user"
},
{
"co... |
olympiads | In the cyclic quadrilateral \(ABCD\), \(\odot O_1, \odot O_2, \odot O_3, \odot O_4\) are the incircles of triangles \(\triangle ABD, \triangle BCA, \triangle CDB\), and \(\triangle DAC\) respectively. The touchpoints on \(AB, BC, CD, DA\) are \(E, F, M, N, G, H, P, Q\) respectively. Let \(R_i\) be the radius of \(\odot... | To prove \( EF \cdot MN = R_1 R_3 + R_2 R_4 \):
1. **Establishing Equalities \( EF = GH \) and \( MN = PQ \) Using the Length of Secants (Chords):**
According to the power of a point theorem, labels and setting the intersections as:
- \( A, B, C, D \) on the sides
- \( E, F, M, N, G, H, P, Q \) as interse... | [
{
"content": "In the cyclic quadrilateral \\(ABCD\\), \\(\\odot O_1, \\odot O_2, \\odot O_3, \\odot O_4\\) are the incircles of triangles \\(\\triangle ABD, \\triangle BCA, \\triangle CDB\\), and \\(\\triangle DAC\\) respectively. The touchpoints on \\(AB, BC, CD, DA\\) are \\(E, F, M, N, G, H, P, Q\\) respecti... |
olympiads | The first term and the difference of an arithmetic progression are natural numbers. Prove that there exists a term in the progression that contains the digit 9. |
To solve the problem, let's proceed with the assumption and detailed steps outlined below:
1. **Define the Arithmetic Progression:**
- Let the first term of the arithmetic progression be \( a \).
- Let the common difference be \( d \).
- Therefore, the \( n \)-th term of the arithmetic progression is given b... | [
{
"content": "The first term and the difference of an arithmetic progression are natural numbers. Prove that there exists a term in the progression that contains the digit 9.",
"role": "user"
},
{
"content": "\nTo solve the problem, let's proceed with the assumption and detailed steps outlined below... |
olympiads | Consider an assembly of \( n \) people such that in any group of 4 people, there are either 3 who all know each other, or 3 who do not know each other. Show that it is possible to separate the assembly into two groups such that all members of the first group know each other, and no pair of people in the second group kn... | :
1. **Graph Construction:**
Define a graph \( G = (V, E) \) where the vertices \( V \) represent the \( n \) persons and the edges \( E \) represent the acquaintance relationships between pairs of these persons. Specifically, an edge \( (v_i, v_j) \in E \) means that persons \( v_i \) and \( v_j \) know each other... | [
{
"content": "Consider an assembly of \\( n \\) people such that in any group of 4 people, there are either 3 who all know each other, or 3 who do not know each other. Show that it is possible to separate the assembly into two groups such that all members of the first group know each other, and no pair of peopl... |
olympiads | Prove that any polygon with \( n \) sides can be divided into triangles using non-intersecting diagonals. |
1. **Base Case**: First, let's verify the base case where \( n = 3 \) (i.e., a triangle). For a triangle, there are no diagonals needed as it is already a polygon with three sides. So, it is trivially decomposed into itself.
2. **Inductive Hypothesis**: Assume the statement is true for any \( k \)-gon, where \( 3 \le... | [
{
"content": "Prove that any polygon with \\( n \\) sides can be divided into triangles using non-intersecting diagonals.",
"role": "user"
},
{
"content": "\n1. **Base Case**: First, let's verify the base case where \\( n = 3 \\) (i.e., a triangle). For a triangle, there are no diagonals needed as i... |
olympiads | For a prime \( p \) and a positive integer \( n \), denote by \( \nu_{p}(n) \) the exponent of \( p \) in the prime factorization of \( n! \). Given a positive integer \( d \) and a finite set \(\{p_{1}, \ldots, p_{k}\} \) of primes, show that there are infinitely many positive integers \( n \) such that \( d \mid \nu_... |
1. **Definitions and Notation:**
Let \( p \) be a prime number, and \( n \) be a positive integer. Define \( \nu_p(n) \) as the exponent of \( p \) in the prime factorization of \( n! \). This can be expressed using the \( \operatorname{ord}_p \) function (the exponent of \( p \) in \( n \)) as follows:
$$
\n... | [
{
"content": "For a prime \\( p \\) and a positive integer \\( n \\), denote by \\( \\nu_{p}(n) \\) the exponent of \\( p \\) in the prime factorization of \\( n! \\). Given a positive integer \\( d \\) and a finite set \\(\\{p_{1}, \\ldots, p_{k}\\} \\) of primes, show that there are infinitely many positive i... |
olympiads | Show that for any integers \( a \), \( b \), \( c \), not all zero, and such that \( |a|, |b|, |c| < 10^6 \), we have \( |a + b \sqrt{2} + c \sqrt{3}| > 10^{-21} \). However, show that we can find such \( a \), \( b \), \( c \) with \( |a + b \sqrt{2} + c \sqrt{3}| < 10^{-11} \). |
To show that for any integers \( a, b, c \), not all zero, and such that \( |a|, |b|, |c| < 10^6 \), we have \( |a + b\sqrt{2} + c\sqrt{3}| > 10^{-21} \), and to also find such \( a, b, c \) such that \( |a + b\sqrt{2} + c\sqrt{3}| < 10^{-11} \):
1. **Establish the Main Identity:**
Consider the expression \( a + ... | [
{
"content": "Show that for any integers \\( a \\), \\( b \\), \\( c \\), not all zero, and such that \\( |a|, |b|, |c| < 10^6 \\), we have \\( |a + b \\sqrt{2} + c \\sqrt{3}| > 10^{-21} \\). However, show that we can find such \\( a \\), \\( b \\), \\( c \\) with \\( |a + b \\sqrt{2} + c \\sqrt{3}| < 10^{-11} ... |
olympiads | Given a fixed point \( A \) outside a fixed circle \( \odot O \), points \( B \) and \( C \) are on the tangents \( AJ \) and \( AK \) from \( A \) to \( \odot O \), respectively. If \( BC \) is tangent to \( \odot O \) at point \( D \), show that the circumcircle of \( \triangle ABC \) is tangent to a certain fixed ci... | 1. Consider the circle $\odot O$ with a point $A$ situated outside it. Points $B$ and $C$ are points where the extended tangents $AJ$ and $AK$ from $A$ intersect the circle.
2. Suppose the line segment $BC$ intersects the circle $\odot O$ at point $D$. We need to prove that the circumcircle of $\triangle ABC$ intersect... | [
{
"content": "Given a fixed point \\( A \\) outside a fixed circle \\( \\odot O \\), points \\( B \\) and \\( C \\) are on the tangents \\( AJ \\) and \\( AK \\) from \\( A \\) to \\( \\odot O \\), respectively. If \\( BC \\) is tangent to \\( \\odot O \\) at point \\( D \\), show that the circumcircle of \\( \... |
olympiads | In triangle $\triangle ABC$, $AC > BC$, and $F$ is the midpoint of $AB$. Through $F$, a diameter $DE$ is drawn in the circumcircle of $\triangle ABC$ such that $C$ and $E$ are on the same side of $AB$. A line is drawn through $C$ parallel to $AB$, intersecting $DE$ at $L$. Prove that $(AC + BC)^{2} = 4DL \cdot EF$. | 1. **Construct Perpendiculars:**
- In $\triangle ABC$, $AC > BC$.
- Let $F$ be the midpoint of $AB$.
- Through $F$, draw a diameter $DE$ of the circumcircle such that $C$ and $E$ are on the same side of $AB$.
- Draw $CL \parallel AB$ intersecting $DE$ at $L$.
- Draw perpendiculars from $D$ to $AC$ a... | [
{
"content": "In triangle $\\triangle ABC$, $AC > BC$, and $F$ is the midpoint of $AB$. Through $F$, a diameter $DE$ is drawn in the circumcircle of $\\triangle ABC$ such that $C$ and $E$ are on the same side of $AB$. A line is drawn through $C$ parallel to $AB$, intersecting $DE$ at $L$. Prove that $(AC + BC)^... |
olympiads | ABCD is any convex quadrilateral. Squares center E, F, G, H are constructed on the outside of the edges AB, BC, CD and DA respectively. Show that EG and FH are equal and perpendicular. |
1. **Identify the points in the complex plane:**
Let the points \(A, B, C, D\) be represented as \(a, b, c, d\) respectively in the complex plane.
2. **Determine the positions of the centers of the squares:**
Each square is constructed on the outside of the edges \(AB, BC, CD,\) and \(DA\). The centers of these... | [
{
"content": "ABCD is any convex quadrilateral. Squares center E, F, G, H are constructed on the outside of the edges AB, BC, CD and DA respectively. Show that EG and FH are equal and perpendicular.",
"role": "user"
},
{
"content": "\n1. **Identify the points in the complex plane:**\n Let the poin... |
olympiads | Let \( ABC \) be a non-degenerate triangle (i.e., \( A, B \), and \( C \) are distinct) and let \( H_A, H_B, H_C \) be the feet of the altitudes from \( A, B \), and \( C \) respectively. The orthic triangle of \( ABC \) is the triangle \( H_A H_B H_C \), and we denote \( h \) as the transformation which assigns to eac... |
1. **Initial Observation**:
Let's start by noting that if the triangle \(ABC\) is not a right triangle, then its orthic triangle \(h(ABC)\) is also not degenerate. This means that starting with a non-degenerate triangle \(ABC\), the transformation \(h^n(ABC)\) will produce either a right triangle or an infinite ... | [
{
"content": "Let \\( ABC \\) be a non-degenerate triangle (i.e., \\( A, B \\), and \\( C \\) are distinct) and let \\( H_A, H_B, H_C \\) be the feet of the altitudes from \\( A, B \\), and \\( C \\) respectively. The orthic triangle of \\( ABC \\) is the triangle \\( H_A H_B H_C \\), and we denote \\( h \\) as... |
olympiads | In the infinite sequence $\{a_{n}\}$, $a_{1}$ and $a_{2}$ are positive integers, and $a_{n+2} = |a_{n+1} - a_{n}|$ for $n \in \mathbf{Z}_{+}$. Let $k$ be the greatest common divisor of $a_{1}$ and $a_{2}$. Prove that the sequence $\{a_{n}\}$ contains infinitely many terms equal to $k$. | ### Problem:
In an infinite sequence \( \{a_n\} \), \(a_1\) and \(a_2\) are positive integers, and \(a_{n+2} = |a_{n+1} - a_n|\) for all \(n \in \mathbb{Z}_+\). Let the greatest common divisor (GCD) of \(a_1\) and \(a_2\) be \(k\). Prove that there are infinitely many terms in the sequence \( \{a_n\} \) that are equal ... | [
{
"content": "In the infinite sequence $\\{a_{n}\\}$, $a_{1}$ and $a_{2}$ are positive integers, and $a_{n+2} = |a_{n+1} - a_{n}|$ for $n \\in \\mathbf{Z}_{+}$. Let $k$ be the greatest common divisor of $a_{1}$ and $a_{2}$. Prove that the sequence $\\{a_{n}\\}$ contains infinitely many terms equal to $k$.",
... |
olympiads | Show that for every odd prime number \( p \), there exist infinitely many natural numbers \( n \) such that
\[ p \mid n 2^n + 1 \] | Pour montrer qu'il existe une infinité d'entiers naturels \( n \) tels que \( p \mid n \cdot 2^n + 1 \) pour un nombre premier impair \( p \), nous allons utiliser le petit théorème de Fermat.
Le petit théorème de Fermat stipule que pour un nombre premier \( p \) et tout entier \( a \) qui n'est pas divisible par \( p... | [
{
"content": "Show that for every odd prime number \\( p \\), there exist infinitely many natural numbers \\( n \\) such that\n\n\\[ p \\mid n 2^n + 1 \\]",
"role": "user"
},
{
"content": "Pour montrer qu'il existe une infinité d'entiers naturels \\( n \\) tels que \\( p \\mid n \\cdot 2^n + 1 \\) p... |
olympiads | Prove that for any natural number $n$, the expression $10^{n} + 18n - 1$ is divisible by 27. | To prove that for any natural number \( n \), the expression \( 10^n + 18n - 1 \) is divisible by 27, we will use the following approach:
1. **Rewrite the Expression**: We start by rewriting \( 10^n + 18n - 1 \) in a more manageable form.
2. **Express and Simplify**:
\[
10^n + 18n - 1 = \left( 10^n - 1 \righ... | [
{
"content": "Prove that for any natural number $n$, the expression $10^{n} + 18n - 1$ is divisible by 27.",
"role": "user"
},
{
"content": "To prove that for any natural number \\( n \\), the expression \\( 10^n + 18n - 1 \\) is divisible by 27, we will use the following approach:\n\n1. **Rewrite t... |
olympiads | Quadrilateral \(ABCD\) is inscribed. Prove that the Miquel point for the lines containing its sides lies on the segment connecting the intersection points of the extensions of the sides. |
1. **Identify Intersection Points**
Consider the cyclic quadrilateral \(ABCD\). Let's extend the sides of the quadrilateral as follows:
- Extend \(AB\) and \(DC\) to meet at point \(E\).
- Extend \(BC\) and \(AD\) to meet at point \(F\).
2. **Define the Point \(P\)**
Let \(P\) be the point of inters... | [
{
"content": "Quadrilateral \\(ABCD\\) is inscribed. Prove that the Miquel point for the lines containing its sides lies on the segment connecting the intersection points of the extensions of the sides.",
"role": "user"
},
{
"content": "\n1. **Identify Intersection Points**\n\n Consider the cyclic... |
olympiads | Given that circle \( \Gamma \) is the circumcircle of an acute \( \triangle ABC \), where \( AC > AB \). Let \( D \) be the midpoint of the minor arc \(\overarc{BC}\). On segments \( AB \) and \( AC \), points \( E \) and \( F \) are taken such that \( AE = AF \). The second intersection point of the circumcircle of \(... | ### Problem
Given circle $\Gamma$ is the circumcircle of the triangle $\triangle ABC$ with $AC > AB$. Point $D$ is the midpoint of the arc $\overparen{BC}$ not containing $A$. Points $E$ and $F$ on sides $AB$ and $AC$ respectively, such that $AE = AF$. Circle $\omega$ is the circumcircle of $\triangle AEF$ and it inter... | [
{
"content": "Given that circle \\( \\Gamma \\) is the circumcircle of an acute \\( \\triangle ABC \\), where \\( AC > AB \\). Let \\( D \\) be the midpoint of the minor arc \\(\\overarc{BC}\\). On segments \\( AB \\) and \\( AC \\), points \\( E \\) and \\( F \\) are taken such that \\( AE = AF \\). The second... |
olympiads | Prove: For all positive integers $n$ and $s$, there exists an arbitrarily long sequence of consecutive positive integers such that each number in the sequence has at least $n$ different prime factors, and each of these $n$ prime factors appears at least $s$ times in the prime factorization of the number. | 1. **Define the sequence \(a_j\):**
Let \(p_k\) denote the \(k\)-th prime number. For positive integers \(n\), \(s\), and \(m\), define the sequence \(a_j\) as follows:
\[
a_j = p_{(j-1)n+1}^{s} p_{(j-1)n+2}^{s} \cdots p_{jn}^{s}, \quad j=1, 2, \ldots, m
\]
This means each \(a_j\) is a product of \(n\) d... | [
{
"content": "Prove: For all positive integers $n$ and $s$, there exists an arbitrarily long sequence of consecutive positive integers such that each number in the sequence has at least $n$ different prime factors, and each of these $n$ prime factors appears at least $s$ times in the prime factorization of the ... |
olympiads | A convex \( n \)-gon ( \( n > 4 \) ) has the property that if a diagonal cuts off a triangle from it, then this triangle is isosceles. Prove that among any four sides of this \( n \)-gon, there are at least two equal sides. |
1. **Consider the Properties of the Polygon:**
Let us consider an $n$-sided convex polygon ($n>4$). According to the problem, if a diagonal of this polygon forms a triangle by cutting off a part of it, this triangle is isosceles.
2. **Grouping Sides:**
We analyze the polygon by grouping equal sides that are pla... | [
{
"content": "A convex \\( n \\)-gon ( \\( n > 4 \\) ) has the property that if a diagonal cuts off a triangle from it, then this triangle is isosceles. Prove that among any four sides of this \\( n \\)-gon, there are at least two equal sides.",
"role": "user"
},
{
"content": "\n1. **Consider the Pr... |
olympiads | Two brilliant mathematicians are given natural numbers and told that these numbers differ by one. After that, they take turns asking each other the same question: "Do you know my number?" Prove that sooner or later one of them will answer affirmatively. |
1. **Setup and Hypothesis**:
Let's denote the two natural numbers given to the two mathematicians by \(a\) and \(b\) such that \(a\) and \(b\) differ by one unit, i.e., \(|a - b| = 1\).
2. **Inductive Step and Base Case**:
We will perform an inductive proof on \(N\), where \(N\) represents the number of times... | [
{
"content": "Two brilliant mathematicians are given natural numbers and told that these numbers differ by one. After that, they take turns asking each other the same question: \"Do you know my number?\" Prove that sooner or later one of them will answer affirmatively.",
"role": "user"
},
{
"content... |
olympiads | Three dihedral angles of a tetrahedron are right angles. Prove that this tetrahedron has three right planar angles. |
1. **Given Situation**:
- We have a tetrahedron with three dihedral angles being right angles.
2. **Configuration Analysis**:
- For each edge of the tetrahedron, there exists only one non-adjacent edge. This implies that among any three edges, there will be at least two adjacent ones.
- Observing that thr... | [
{
"content": "Three dihedral angles of a tetrahedron are right angles. Prove that this tetrahedron has three right planar angles.",
"role": "user"
},
{
"content": "\n1. **Given Situation**:\n - We have a tetrahedron with three dihedral angles being right angles.\n \n2. **Configuration Analysis**... |
olympiads | The number 12 is written on the whiteboard. Each minute, the number on the board is either multiplied or divided by one of the numbers 2 or 3 (a division is possible only if the result is an integer). Prove that the number that will be written on the board in exactly one hour will not be equal to 54. |
1. **Initial Conditions:**
The initial number on the whiteboard is \(12\). We start with the prime factorization of \(12\):
\[
12 = 2^2 \times 3^1
\]
Here, the number of prime factors is \(2 + 1 = 3\). This is an odd number.
2. **Operations Analysis:**
Each minute, the number on the board is either ... | [
{
"content": "The number 12 is written on the whiteboard. Each minute, the number on the board is either multiplied or divided by one of the numbers 2 or 3 (a division is possible only if the result is an integer). Prove that the number that will be written on the board in exactly one hour will not be equal to ... |
olympiads | Prove that it is possible to write a non-negative number on each edge of an arbitrary tetrahedron such that the sum of the numbers on the edges of each face is equal to the area of that face. |
Let's prove that it is possible to assign a non-negative number to each edge of any given tetrahedron such that the sum of the numbers on the edges of each face equals the area of that face.
1. **Identify the areas of the faces:**
Let the areas of the four faces of the tetrahedron be denoted as follows:
- The ... | [
{
"content": "Prove that it is possible to write a non-negative number on each edge of an arbitrary tetrahedron such that the sum of the numbers on the edges of each face is equal to the area of that face.",
"role": "user"
},
{
"content": "\nLet's prove that it is possible to assign a non-negative n... |
olympiads | Construct the triangle given its area \(\left(q^{2}\right)\) and its three angles. |
1. **Constructing a Triangle with Given Angles:**
- Begin by constructing a triangle \( \triangle ABC \) where the angles are given.
2. **Constructing a Square with the Same Area as the Triangle \( \triangle ABC \):**
- Let's first assume the area of \( \triangle ABC \) is \( A = q^2 \).
- To construct a squ... | [
{
"content": "Construct the triangle given its area \\(\\left(q^{2}\\right)\\) and its three angles.",
"role": "user"
},
{
"content": "\n1. **Constructing a Triangle with Given Angles:**\n - Begin by constructing a triangle \\( \\triangle ABC \\) where the angles are given.\n\n2. **Constructing a ... |
olympiads | The space is partitioned by the set of planes
$$
A=\{x \pm y \pm z=n, n \in \mathbf{Z}\}
$$
into some combination of tetrahedra and octahedra. Suppose a rational point $\left(x_{0}, y_{0}, z_{0}\right)$ does not belong to any plane in $A$. Prove that there exists a positive integer $k$ such that the point $\left(k x_{... |
1. **Define Greatest Integer and Fractional Part Functions**:
- Let $[x]$ denote the greatest integer less than or equal to $x$, and $\{x\} = x - [x]$ the fractional part of $x$.
2. **Stating and Proving the Lemma**:
- Assume there exist rational numbers $a, b, c$ such that none of $a, b, c, a+b+c$ is an intege... | [
{
"content": "The space is partitioned by the set of planes\n$$\nA=\\{x \\pm y \\pm z=n, n \\in \\mathbf{Z}\\}\n$$\n\ninto some combination of tetrahedra and octahedra. Suppose a rational point $\\left(x_{0}, y_{0}, z_{0}\\right)$ does not belong to any plane in $A$. Prove that there exists a positive integer $... |
olympiads | Color each positive integer either red or blue. Prove that there exists an infinite sequence of positive integers \(a_{1} < a_{2} < \cdots < a_{n} < \cdots\) such that the infinite sequence \(a_{1}, \frac{a_{1}+a_{2}}{2}, a_{2}, \frac{a_{2}+a_{3}}{2}, a_{3}, \cdots\) is a sequence of integers all of the same color. |
We are tasked with proving that there exists an infinite sequence of integers \(a_1 < a_2 < \cdots < a_n < \cdots\) such that the infinite sequence \(a_1, \frac{a_1 + a_2}{2}, a_2, \frac{a_2 + a_3}{2}, a_3, \cdots\) consists of integers of the same color (either all red or all blue) when each positive integer is color... | [
{
"content": "Color each positive integer either red or blue. Prove that there exists an infinite sequence of positive integers \\(a_{1} < a_{2} < \\cdots < a_{n} < \\cdots\\) such that the infinite sequence \\(a_{1}, \\frac{a_{1}+a_{2}}{2}, a_{2}, \\frac{a_{2}+a_{3}}{2}, a_{3}, \\cdots\\) is a sequence of inte... |
olympiads | A chord AB of constant length slides around the curved part of a semicircle. M is the midpoint of AB, and C is the foot of the perpendicular from A onto the diameter. Show that angle ACM does not change. |
Given a semicircle with center \( O \) and a chord \( AB \) of constant length sliding around the curved part of the semicircle. We need to show that the angle \( \angle ACM \) does not change, where \( M \) is the midpoint of \( AB \), and \( C \) is the foot of the perpendicular from \( A \) onto the diameter.
1. *... | [
{
"content": "A chord AB of constant length slides around the curved part of a semicircle. M is the midpoint of AB, and C is the foot of the perpendicular from A onto the diameter. Show that angle ACM does not change.",
"role": "user"
},
{
"content": "\nGiven a semicircle with center \\( O \\) and a... |
olympiads | In an acute-angled triangle \(ABC\), the altitudes \(AA_1\), \(BB_1\), and \(CC_1\) intersect at point \(H\). From point \(H\), perpendiculars are drawn to the lines \(B_1C_1\) and \(A_1C_1\), which intersect the rays \(CA\) and \(CB\) at points \(P\) and \(Q\) respectively. Prove that the perpendicular dropped from po... |
Let's denote \(N\) as the projection of point \(C\) onto the line \(A_1 B_1\) (reference the figure for clarity). We need to demonstrate that the perpendicular dropped from point \(C\) onto line \(A_1 B_1\) goes through the midpoint of segment \(PQ\).
1. **Homothety Transformation**:
Consider a homothety cente... | [
{
"content": "In an acute-angled triangle \\(ABC\\), the altitudes \\(AA_1\\), \\(BB_1\\), and \\(CC_1\\) intersect at point \\(H\\). From point \\(H\\), perpendiculars are drawn to the lines \\(B_1C_1\\) and \\(A_1C_1\\), which intersect the rays \\(CA\\) and \\(CB\\) at points \\(P\\) and \\(Q\\) respectively... |
olympiads | Using a compass and a ruler, construct a triangle from the midpoints of its three sides. | 1. **Identify and Label the Given Points:**
Let's designate the given points as \( A, B \), and \( C \). These points are the midpoints of the sides of the triangle we need to construct.
2. **Draw Lines Parallel to Opposite Sides:**
Through each point, draw lines parallel to the sides opposite to the other two p... | [
{
"content": "Using a compass and a ruler, construct a triangle from the midpoints of its three sides.",
"role": "user"
},
{
"content": "1. **Identify and Label the Given Points:**\n Let's designate the given points as \\( A, B \\), and \\( C \\). These points are the midpoints of the sides of the... |
olympiads | S is a parabola with focus F and axis L. Three distinct normals to S pass through P. Show that the sum of the angles which these make with L less the angle which PF makes with L is a multiple of π. |
1. **Understanding the Problem**:
We need to prove that the sum of the angles formed by the normals to a parabola at point \( P \) with the parabola's axis, less the angle between the line connecting \( P \) to the focus \( F \) and the axis, is a multiple of \( \pi \).
2. **Mathematical Set-Up**:
Let's consid... | [
{
"content": "S is a parabola with focus F and axis L. Three distinct normals to S pass through P. Show that the sum of the angles which these make with L less the angle which PF makes with L is a multiple of π.",
"role": "user"
},
{
"content": "\n1. **Understanding the Problem**:\n We need to pro... |
olympiads | João has a sheet of graph paper with 10 squares in length and 7 squares in width. He chose 30 triangles with vertices at the intersections of the lines on this graph paper. Explain why there must necessarily be at least two chosen triangles with common vertices. | ### Solução:
1. **Revisão da configuração do papel quadriculado**:
- O papel possui 10 quadrados de comprimento e 7 quadrados de largura.
- Cada quadrado tem interseções nas linhas; essas interseções são os vértices dos triângulos.
2. **Determinação do total de pontos de interseção**:
- Como há 10 quadrados ... | [
{
"content": "João has a sheet of graph paper with 10 squares in length and 7 squares in width. He chose 30 triangles with vertices at the intersections of the lines on this graph paper. Explain why there must necessarily be at least two chosen triangles with common vertices.",
"role": "user"
},
{
"... |
olympiads | Prove that inside a cube with edge length \(a\), two regular tetrahedra with edge length \(a\) can be placed without sharing any common points. | 1. Consider a cube with edge length \( a \) and denote the vertices of the cube as \( A_1, A_2, A_3, A_4, A_1', A_2', A_3', A_4' \).
2. Introduce a plane passing through the center \( O \) of the cube and perpendicular to the diagonal \( A_1A_3' \). This plane intersects the midpoints of the edges \( A_1'A_4' \), \( A_... | [
{
"content": "Prove that inside a cube with edge length \\(a\\), two regular tetrahedra with edge length \\(a\\) can be placed without sharing any common points.",
"role": "user"
},
{
"content": "1. Consider a cube with edge length \\( a \\) and denote the vertices of the cube as \\( A_1, A_2, A_3, ... |
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