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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, ... |
olympiads | (a) Provide at least two different arguments that could justify the interest of the Central Bank of Russia in the use of the described loyalty program. Justify your assumptions.
(b) Provide at least two different arguments justifying the benefit for banks whose customers' cards participate in such a loyalty program. J... | ### Решение:
#### Часть (а)
Для обоснования заинтересованности Центрального банка Российской Федерации (ЦБ РФ) в использовании описанной программы лояльности для карт «Мир», можно привести следующие аргументы:
1. **Повышение числа безналичных операций**:
- Программа лояльности стимулирует держателей карт «Мир» к ... | [
{
"content": "(a) Provide at least two different arguments that could justify the interest of the Central Bank of Russia in the use of the described loyalty program. Justify your assumptions.\n\n(b) Provide at least two different arguments justifying the benefit for banks whose customers' cards participate in s... |
olympiads | Let \( t_n = 1 + x + x^2 + \cdots + x^n \) and \( s_n = 1 + \frac{1+x}{2} + \left( \frac{1+x}{2} \right)^2 + \cdots + \left( \frac{1+x}{2} \right)^n \), for \( n = 1, 2, \cdots \). Prove that
\[ C_{n+1}^{1} + C_{n+1}^{2} t_1 + C_{n+1}^{3} t_2 + \cdots + C_{n+1}^{n+1} t_n = 2^n s_n. \] | ```markdown
To prove the given equation
\[
C_{n+1}^{1} + C_{n+1}^{2} t_{1} + C_{n+1}^{3} t_{2} + \cdots + C_{n+1}^{n+1} t_{n} = 2^{n} s_{n}
\]
we first need to show that the coefficients of $x^k$ in both sides are equal for $k = 0, 1, \ldots, n$. This means proving:
\[
C_{n+1}^{k+1} + C_{n+1}^{k+2} + \cdots + C_{n+... | [
{
"content": "Let \\( t_n = 1 + x + x^2 + \\cdots + x^n \\) and \\( s_n = 1 + \\frac{1+x}{2} + \\left( \\frac{1+x}{2} \\right)^2 + \\cdots + \\left( \\frac{1+x}{2} \\right)^n \\), for \\( n = 1, 2, \\cdots \\). Prove that\n\n\\[ C_{n+1}^{1} + C_{n+1}^{2} t_1 + C_{n+1}^{3} t_2 + \\cdots + C_{n+1}^{n+1} t_n = 2^n... |
olympiads | For any natural number \( n \), the fraction \(\frac{21n + 4}{14n + 3}\) is irreducible. |
**Step 1: Understanding the Given Expression**
We need to prove that the fraction \(\frac{21n + 4}{14n + 3}\) is in its simplest form for any natural number \(n\).
**Step 2: Identifying the Greatest Common Divisor (GCD)**
First, let us denote:
\[
a = 21n + 4 \quad \text{and} \quad b = 14n + 3
\]
Our task is to sho... | [
{
"content": "For any natural number \\( n \\), the fraction \\(\\frac{21n + 4}{14n + 3}\\) is irreducible.",
"role": "user"
},
{
"content": "\n**Step 1: Understanding the Given Expression**\n\nWe need to prove that the fraction \\(\\frac{21n + 4}{14n + 3}\\) is in its simplest form for any natural ... |
olympiads | Let \( \triangle ABC \) be a triangle and let \( D, E, \) and \( F \) be the midpoints of sides \( \overline{BC}, \overline{CA}, \) and \( \overline{AB} \), respectively. Let \( X \neq A \) be the intersection of \( AD \) with the circumcircle of \( \triangle ABC \). Let \( \Omega \) be the circle passing through \( D ... |
As we proceed through the solution presented, we will include more details for clarity and ensure that foundational concepts and theorems are properly cited and explained. The overall task is to demonstrate that the tangents at \( B \) and \( C \) to the circumcircle of \( \triangle ABC \) and the line \( PG \) are co... | [
{
"content": "Let \\( \\triangle ABC \\) be a triangle and let \\( D, E, \\) and \\( F \\) be the midpoints of sides \\( \\overline{BC}, \\overline{CA}, \\) and \\( \\overline{AB} \\), respectively. Let \\( X \\neq A \\) be the intersection of \\( AD \\) with the circumcircle of \\( \\triangle ABC \\). Let \\( ... |
olympiads | In coordinate space, all planes with the equations $x \pm y \pm z = n$ (for all integers $n$) are considered. These planes divide the space into tetrahedra and octahedra. Let the point ($x_{0}, y_{0}, z_{0}$) with rational coordinates not lie on any of the given planes. Prove that there exists a natural number $k$ such... |
To prove the statement given in the problem, we start by considering the **lemma**
#### Lemma:
- Let $a, b, c$ be rational numbers and $a + b + c$ is non-integral.
- There exists a natural number $k$ such that the numbers $ka$, $kb$, $kc$ are non-integral and:
\[
1 < \{ka\} + \{kb\} + \{kc\} < 2
\]
#... | [
{
"content": "In coordinate space, all planes with the equations $x \\pm y \\pm z = n$ (for all integers $n$) are considered. These planes divide the space into tetrahedra and octahedra. Let the point ($x_{0}, y_{0}, z_{0}$) with rational coordinates not lie on any of the given planes. Prove that there exists a... |
olympiads | Let \( ABC \) be a triangle, \( D \) the point of tangency of the incircle with \( [BC] \), \( J \) the center of the \( A \)-excircle, and \( M \) the midpoint of the altitude from \( A \). Show that \( M, D \), and \( I \) are collinear.
Note 37: Recall that the \( A \)-excircle of triangle \( ABC \) is the circle t... |
1. **Definition and Homothety:**
- Consider the $A$-excircle of triangle $ABC$ which is tangent to $BC$, $AB$, and $AC$ at points.
- Let $Z$ be the point where the $A$-excircle is tangent to $BC$.
- An important property of excircles is that the homothety centered at $A$ that maps the incircle of triangle... | [
{
"content": "Let \\( ABC \\) be a triangle, \\( D \\) the point of tangency of the incircle with \\( [BC] \\), \\( J \\) the center of the \\( A \\)-excircle, and \\( M \\) the midpoint of the altitude from \\( A \\). Show that \\( M, D \\), and \\( I \\) are collinear.\n\nNote 37: Recall that the \\( A \\)-ex... |
olympiads | A, B, C, and D are non-coplanar points. ∠ABC = ∠ADC and ∠BAD = ∠BCD. Show that AB = CD and BC = AD. | Given \( A, B, C \), and \( D \) are non-coplanar points such that \( \angle ABC = \angle ADC \) and \( \angle BAD = \angle BCD \). We aim to prove that \( AB = CD \) and \( BC = AD \).
1. **Apply the cosine law to \( \angle ABC \) and \( \angle ADC \)**:
\[
\cos \angle ABC = \cos \angle ADC
\]
Therefore, ... | [
{
"content": "A, B, C, and D are non-coplanar points. ∠ABC = ∠ADC and ∠BAD = ∠BCD. Show that AB = CD and BC = AD.",
"role": "user"
},
{
"content": "Given \\( A, B, C \\), and \\( D \\) are non-coplanar points such that \\( \\angle ABC = \\angle ADC \\) and \\( \\angle BAD = \\angle BCD \\). We aim t... |
olympiads | Given \( N \) lines on a plane (\( N>1 \)), where no three lines intersect at one point and no two lines are parallel. Prove that it is possible to assign nonzero integers to the regions into which these lines divide the plane, such that the absolute values of these integers do not exceed \( N \), and the sum of the nu... |
1. **Coloring the Plane:**
- We color the plane using two colors: white and black. The coloring is done such that neighboring regions (regions sharing a common boundary segment) have different colors.
2. **Assigning Numbers:**
- At each intersection point of the lines, we cyclically assign the numbers \(1, -1... | [
{
"content": "Given \\( N \\) lines on a plane (\\( N>1 \\)), where no three lines intersect at one point and no two lines are parallel. Prove that it is possible to assign nonzero integers to the regions into which these lines divide the plane, such that the absolute values of these integers do not exceed \\( ... |
olympiads | Given \( n \) different natural numbers, prove that for the following cases:
1. \( n=5 \)
2. \( n=1989 \)
there exists an infinite arithmetic sequence of positive integers such that its first term is not greater than its common difference and the sequence contains exactly 3 or 4 of the given numbers. | We need to prove that there exists an infinite sequence of positive integers in arithmetic progression such that the first term is no larger than the common difference, and that there are exactly 3 or 4 of the given `n` distinct natural numbers among its terms.
**Case (1): \( n = 5 \)**
1. Given the 5 distinct natura... | [
{
"content": "Given \\( n \\) different natural numbers, prove that for the following cases:\n\n1. \\( n=5 \\)\n2. \\( n=1989 \\)\n\nthere exists an infinite arithmetic sequence of positive integers such that its first term is not greater than its common difference and the sequence contains exactly 3 or 4 of th... |
olympiads | 51 points are inside a square with side length 1. Show that three of them can be found inside a single circle with radius $1/7$. |
1. **Dividing the Square:**
We begin by dividing the given square of side length 1 into smaller squares. Specifically, we divide it into 25 smaller squares, each with a side length of \(\frac{1}{5}\).
So, each smaller square has area:
\[
\left(\frac{1}{5}\right)^2 = \frac{1}{25}
\]
2. **Applying the ... | [
{
"content": "51 points are inside a square with side length 1. Show that three of them can be found inside a single circle with radius $1/7$.",
"role": "user"
},
{
"content": "\n1. **Dividing the Square:**\n\n We begin by dividing the given square of side length 1 into smaller squares. Specifical... |
olympiads |
(a) To which class (type) of goods (services) can the described service (complete apartment renovation) be attributed? Explain why. What are the characteristics that it has (as opposed to typical consumer goods and services)? Explain.
(b) The decision between a private repair crew and a construction company was diffi... |
#### Problem (a)
Renovation services are a classic example of **credence goods**. These are goods and services where the utility of consumption is difficult or impossible for the consumer to ascertain. Unlike **experience goods**, where the consumer can determine the utility or quality after consumption, credence good... | [
{
"content": "\n(a) To which class (type) of goods (services) can the described service (complete apartment renovation) be attributed? Explain why. What are the characteristics that it has (as opposed to typical consumer goods and services)? Explain.\n\n(b) The decision between a private repair crew and a const... |
olympiads | There are 100 distinct real numbers arranged in a circle. Verify that there exist four consecutive numbers around the circle such that the sum of the two middle numbers is strictly less than the sum of the other two numbers. |
1. **Identify Arrangement Hypothesis and Definitions:**
- Let $a$ be the smallest number written around the circle.
- Let $b$ and $c$ be the two neighbors of $a$ with $b < c$.
- Let $d$ be the other neighbor of $b$.
2. **Order and Positioning on Circle:**
- The numbers $d, b, a,$ and $c$ are written in se... | [
{
"content": "There are 100 distinct real numbers arranged in a circle. Verify that there exist four consecutive numbers around the circle such that the sum of the two middle numbers is strictly less than the sum of the other two numbers.",
"role": "user"
},
{
"content": "\n1. **Identify Arrangement... |
olympiads |
Let \( P(x, y) \) be a polynomial in variables \( x \) and \( y \), and it satisfies \( P(x, y) = P(y, x) \) for any \( x \) and \( y \) (for example, the polynomial \( x^2 - 2xy + y^2 \) satisfies this condition). Given that \( x - y \) is a factor of \( P(x, y) \), prove that \( (x - y)^2 \) is also a factor of \( P... |
1. We start with the given condition: $x - y$ is a factor of $P(x, y)$. Therefore, we can write \( P(x, y) \) in the form:
\[
P(x, y) = (x - y) \cdot Q(x, y)
\]
where \( Q(x, y) \) is a polynomial in the variables \( x \) and \( y \).
2. Since \( P(x, y) = P(y, x) \) by the symmetry condition, substitutin... | [
{
"content": "\nLet \\( P(x, y) \\) be a polynomial in variables \\( x \\) and \\( y \\), and it satisfies \\( P(x, y) = P(y, x) \\) for any \\( x \\) and \\( y \\) (for example, the polynomial \\( x^2 - 2xy + y^2 \\) satisfies this condition). Given that \\( x - y \\) is a factor of \\( P(x, y) \\), prove that... |
olympiads | 1) Consider an arbitrary line \( y = k - b \) that intersects a parabola at points \( A \) and \( B \). The x-coordinates of the endpoints of chord \( AB \) must satisfy the equation \( kx - b = x^2 \), i.e., \( x^2 - kx + b = 0 \). The x-coordinates are \( x_A = \frac{k - \sqrt{k^2 - 4b}}{2} \) and \( x_B = \frac{k + ... |
#### Problem 1:
Given a parabola and a line, we have to prove certain properties about their intersections.
1. Consider the line \( y = kx - b \) that intersects the parabola \( y = x^2 \) at points \( A \) and \( B \).
2. The intersection points of the line and parabola must satisfy:
\[
kx - b = x^2 \implies ... | [
{
"content": "1) Consider an arbitrary line \\( y = k - b \\) that intersects a parabola at points \\( A \\) and \\( B \\). The x-coordinates of the endpoints of chord \\( AB \\) must satisfy the equation \\( kx - b = x^2 \\), i.e., \\( x^2 - kx + b = 0 \\). The x-coordinates are \\( x_A = \\frac{k - \\sqrt{k^2... |
olympiads |
Prove the equality: \( C_{n}^{0}+C_{n-1}^{1}+C_{n-2}^{2}+\ldots=F_{n+1} \).
(The sum on the left-hand side can be interpreted as the sum of the elements of Pascal's triangle that lie on one diagonal.) | :
We aim to prove the equality
\[
C_{n}^{0} + C_{n-1}^{1} + C_{n-2}^{2} + \ldots = F_{n+1}
\]
using the method of mathematical induction.
1. **Base Case:**
We start by checking the base cases.
- For \( n = 0 \):
\[
S_0 = C_0^0 = 1
\]
We know that \( F_1 = 1 \), thus the equality holds.... | [
{
"content": "\nProve the equality: \\( C_{n}^{0}+C_{n-1}^{1}+C_{n-2}^{2}+\\ldots=F_{n+1} \\).\n\n(The sum on the left-hand side can be interpreted as the sum of the elements of Pascal's triangle that lie on one diagonal.)",
"role": "user"
},
{
"content": ": \n\nWe aim to prove the equality \n\n\\[\... |
olympiads | Prove that for any point \( O \) located inside triangle \( A B C \) with semiperimeter \( p \), the following inequality holds:
\[
O A \cdot \cos \frac{\angle B A C}{2} + O B \cdot \cos \frac{\angle A B C}{2} + O C \cdot \cos \frac{\angle A C B}{2} \geq p
\]
Determine when equality is achieved. | ## Solution:
We need to prove that for any point \( O \) inside the triangle \( \triangle ABC \) with semiperimeter \( p \), the following inequality holds:
$$
OA \cdot \cos \frac{\angle BAC}{2} + OB \cdot \cos \frac{\angle ABC}{2} + OC \cdot \cos \frac{\angle ACB}{2} \geq p
$$
### Step-by-step Proof:
### 1. Define... | [
{
"content": "Prove that for any point \\( O \\) located inside triangle \\( A B C \\) with semiperimeter \\( p \\), the following inequality holds:\n\n\\[ \nO A \\cdot \\cos \\frac{\\angle B A C}{2} + O B \\cdot \\cos \\frac{\\angle A B C}{2} + O C \\cdot \\cos \\frac{\\angle A C B}{2} \\geq p \n\\]\n\nDetermi... |
olympiads | Given $\triangle ABC$ with incircle $\Gamma$ that touches sides $BC$ and $CA$ at points $P$ and $Q$ respectively. Let $I_C$ be the excenter opposite vertex $C$. Denote the circumcircle of $\triangle I_CB P$ and $\triangle I_CA Q$, respectively. Let $T$ be the second intersection of these two circumcircles. Prove that t... | 1. **Given setup and angles:**
Let $\Gamma$ be the incircle of $\triangle ABC$. It touches $BC$ at $P$, $CA$ at $Q$, and $AB$ at point $R$. Denote the incenter opposite to vertex $C$ as $I_C$.
We are given the key angles in $\triangle ABC$:
\[
\angle BAC = \alpha, \quad \angle ABC = \beta, \quad \angle A... | [
{
"content": "Given $\\triangle ABC$ with incircle $\\Gamma$ that touches sides $BC$ and $CA$ at points $P$ and $Q$ respectively. Let $I_C$ be the excenter opposite vertex $C$. Denote the circumcircle of $\\triangle I_CB P$ and $\\triangle I_CA Q$, respectively. Let $T$ be the second intersection of these two c... |
olympiads | 1. Show that for all real numbers $\alpha$ and $\beta$:
$$
\lfloor 2 \alpha\rfloor +\lfloor 2 \beta\rfloor \geqslant \lfloor\alpha\rfloor +\lfloor\beta\rfloor +\lfloor\alpha +\beta\rfloor
$$
2. Deduce that for all integers $n$ and $m$:
$$
\frac{(2 m)!(2 n)!}{m!n!(m+n)!}
$$
is an integer. |
#### Part 1:
1. Given the inequality to prove:
\[
\lfloor 2 \alpha \rfloor + \lfloor 2 \beta \rfloor \geq \lfloor \alpha \rfloor + \lfloor \beta \rfloor + \lfloor \alpha + \beta \rfloor
\]
2. This inequality is invariant under the addition of integers to \(\alpha\) and \(\beta\). Therefore, it suffices to ... | [
{
"content": "1. Show that for all real numbers $\\alpha$ and $\\beta$:\n\n$$\n\\lfloor 2 \\alpha\\rfloor +\\lfloor 2 \\beta\\rfloor \\geqslant \\lfloor\\alpha\\rfloor +\\lfloor\\beta\\rfloor +\\lfloor\\alpha +\\beta\\rfloor\n$$\n\n2. Deduce that for all integers $n$ and $m$:\n\n$$\n\\frac{(2 m)!(2 n)!}{m!n!(m+... |
olympiads | It is known that a certain polynomial at rational points takes rational values. Prove that all of its coefficients are rational. |
To prove that if a polynomial takes rational values at rational points, then all its coefficients are rational, we proceed by induction on the degree of the polynomial.
#### Step 1: Base Case
Consider a polynomial of degree zero, i.e., a constant polynomial \( P(x) = c \). If \( c \) is rational, the statement holds ... | [
{
"content": "It is known that a certain polynomial at rational points takes rational values. Prove that all of its coefficients are rational.",
"role": "user"
},
{
"content": "\nTo prove that if a polynomial takes rational values at rational points, then all its coefficients are rational, we procee... |
olympiads | As shown in the figure, \( AB \) is a chord with a slope of 1 in the ellipse \( m x^{2}+n y^{2}=1 \) (where \( m, n>0 \) and \( m \neq n \)). The perpendicular bisector of \( AB \) intersects the major ellipse at two points \( C \) and \( D \). Let \( E \) and \( F \) be the midpoints of \( AB \) and \( CD \), respecti... | :
1. Let's start by considering the diagonal chord \(AB\) of the ellipse given \(mx^2 + ny^2 = 1\) with slope 1. Thus, the equation of \(AB\) can be written as the line \(y = x + b\).
2. We simultaneously solve the equations \(y = x + b\) and the ellipse equation \(mx^2 + ny^2 = 1\):
\[
\begin{cases}
y = x +... | [
{
"content": "As shown in the figure, \\( AB \\) is a chord with a slope of 1 in the ellipse \\( m x^{2}+n y^{2}=1 \\) (where \\( m, n>0 \\) and \\( m \\neq n \\)). The perpendicular bisector of \\( AB \\) intersects the major ellipse at two points \\( C \\) and \\( D \\). Let \\( E \\) and \\( F \\) be the mid... |
olympiads | Find all values of the parameter \( a \) for which the system of equations
$$
\left\{\begin{array}{l}
\left(\frac{2 y}{3}-\frac{|x-1|}{x-1}-1\right)(y-1)=0, \\
y=a+\frac{|x-1|}{(x-1)(x-y)^{2}}
\end{array}\right.
$$
has exactly four distinct solutions. Determine these solutions for each specified \( a \). |
Let us analyze the system of equations:
\[
\begin{cases}
\left( \frac{2 y}{3} - \frac{|x-1|}{x-1} - 1 \right) (y-1) = 0, \\
y = a + \frac{|x-1|}{(x-1)(x-y)^2}
\end{cases}
\]
#### Case 1: \( x > 1 \)
1. **Simplify the first equation:**
Since \( x > 1 \), we have \( \frac{|x - 1|}{x - 1} = 1 \). Thus,
\[
\... | [
{
"content": "Find all values of the parameter \\( a \\) for which the system of equations\n\n$$\n\\left\\{\\begin{array}{l}\n\\left(\\frac{2 y}{3}-\\frac{|x-1|}{x-1}-1\\right)(y-1)=0, \\\\\ny=a+\\frac{|x-1|}{(x-1)(x-y)^{2}}\n\\end{array}\\right.\n$$\n\nhas exactly four distinct solutions. Determine these solut... |
olympiads | In quadrilateral $BEFD$, $BF$ intersects $DE$ at point $C$. Points $U$ and $V$ are on $BE$ and $FD$ respectively, and $\frac{BU}{UE}=\frac{FV}{VD}$. Line $UV$ intersects $DE$ and $BF$ at points $P$ and $Q$ respectively. Prove that:
1. As points $U$ and $V$ move, the circumcircle of $\triangle CPQ$ passes through a fixe... |
1. **Statement of the problem**: In quadrilateral \( BEFD \), \( BF \) intersects \( DE \) at point \( C \). Points \( U \) and \( V \) lie on \( BE \) and \( FD \) respectively such that \( \frac{BU}{UE} = \frac{FV}{VD} \). The line segment \( UV \) intersects \( DE \) and \( BF \) at points \( P \) and \( Q \) respe... | [
{
"content": "In quadrilateral $BEFD$, $BF$ intersects $DE$ at point $C$. Points $U$ and $V$ are on $BE$ and $FD$ respectively, and $\\frac{BU}{UE}=\\frac{FV}{VD}$. Line $UV$ intersects $DE$ and $BF$ at points $P$ and $Q$ respectively. Prove that:\n1. As points $U$ and $V$ move, the circumcircle of $\\triangle ... |
olympiads | Let \( ABCD \) be a quadrilateral with \( AC = BD \). Diagonals \( AC \) and \( BD \) meet at \( P \). Let \( \omega_{1} \) and \( O_{1} \) denote the circumcircle and circumcenter of triangle \( ABP \). Let \( \omega_{2} \) and \( O_{2} \) denote the circumcircle and circumcenter of triangle \( CDP \). Segment \( BC \... |
1. **Introducing the point \( Q \):**
Let \( Q \) be the second intersection point of \( \omega_1 \) and \( \omega_2 \). Note that \( Q \) is the center of a spiral similarity mapping \( \overline{AC} \) to \( \overline{BD} \). Therefore, \( \triangle QAB \) and \( \triangle QCD \) are similar and isosceles sin... | [
{
"content": "Let \\( ABCD \\) be a quadrilateral with \\( AC = BD \\). Diagonals \\( AC \\) and \\( BD \\) meet at \\( P \\). Let \\( \\omega_{1} \\) and \\( O_{1} \\) denote the circumcircle and circumcenter of triangle \\( ABP \\). Let \\( \\omega_{2} \\) and \\( O_{2} \\) denote the circumcircle and circumc... |
olympiads | One tour guide once complained to a small group of tourists: “When I have a tour group of 50 people, after giving them free time during a long excursion, the tourists almost always arrive on time at the appointed time. But if there are few people in the tour group, I constantly encounter the problem that several people... | 1. **Wealth Levels and Cost-Perception Variance**:
- **Wealth Disparity**: Tourists with different wealth levels will perceive additional expenses differently, even if the risk of incurring these expenses is the same for everyone. A less affluent tourist in a small group will value every monetary unit lost (e.g., p... | [
{
"content": "One tour guide once complained to a small group of tourists: “When I have a tour group of 50 people, after giving them free time during a long excursion, the tourists almost always arrive on time at the appointed time. But if there are few people in the tour group, I constantly encounter the probl... |
olympiads | At a conference, there are $n$ professors $A_{1}, A_{2}, A_{3}, \cdots, A_{n}$ participating. Prove: It is possible to divide these $n$ individuals into two groups such that for each person $A_{i}$, the number of people $d_{i}$ they know in the other group is greater than or equal to the number of people $d^{\prime}{ }... |
The problem asks us to show that we can divide \( n \) professors \( A_{1}, A_{2}, A_{3}, \ldots, A_{n} \) into two groups such that for each professor \( A_{i} \), the number of people they recognize in the other group is greater than or equal to the number of people they recognize in their own group.
Let's represen... | [
{
"content": "At a conference, there are $n$ professors $A_{1}, A_{2}, A_{3}, \\cdots, A_{n}$ participating. Prove: It is possible to divide these $n$ individuals into two groups such that for each person $A_{i}$, the number of people $d_{i}$ they know in the other group is greater than or equal to the number o... |
olympiads | In triangle \(ABC\), the orthocenter is \(M\), and its incircle touches sides \(AC\) and \(BC\) at points \(P\) and \(Q\) respectively, with center \(O\). Prove that if \(M\) lies on the line \(PQ\), then the line \(MO\) passes through the midpoint of side \(AB\). | To prove the statement, we will systematically examine the geometric relationships and properties in triangle $ABC$.
1. **Construction and Definitions**:
Let $A$, $B$, and $C$ be the vertices of triangle $ABC$. The orthocenter (intersection of the altitudes) is denoted by $M$. The triangle's incircle touches side $... | [
{
"content": "In triangle \\(ABC\\), the orthocenter is \\(M\\), and its incircle touches sides \\(AC\\) and \\(BC\\) at points \\(P\\) and \\(Q\\) respectively, with center \\(O\\). Prove that if \\(M\\) lies on the line \\(PQ\\), then the line \\(MO\\) passes through the midpoint of side \\(AB\\).",
"role... |
olympiads | Free the stick. We have a loop of rope threaded through one end of a stick, as shown in the picture, but too short to be looped over the opposite end. The puzzle is to suspend the stick from the buttonhole of a jacket (see the picture) and then release it again. |
1. **Initial Setup:**
- We start with a stick that has a rope looped through one of its ends, as illustrated. The main goal is to hang the stick from the loop of a jacket and then free it.
2. **Pass the Jacket Loop through the Rope Loop:**
- Pass the loop of the jacket along with a part of the jacket’s fabric ... | [
{
"content": "Free the stick. We have a loop of rope threaded through one end of a stick, as shown in the picture, but too short to be looped over the opposite end. The puzzle is to suspend the stick from the buttonhole of a jacket (see the picture) and then release it again.",
"role": "user"
},
{
"... |
olympiads | In the country of Dumulandia, exactly 10 roads originate from each city, and each road connects exactly two cities. The network of roads was connected, meaning it was possible to travel from any city to any other city along the roads, possibly through other cities. However, during a flood, two cities connected by a roa... | 1. **Examine the conditions of the problem**:
- Each city has exactly 10 roads leading out of it.
- Each road connects exactly two cities.
- The road network is initially connected, meaning any city can be reached from any other city, potentially via other cities.
- After flooding, which affects two cit... | [
{
"content": "In the country of Dumulandia, exactly 10 roads originate from each city, and each road connects exactly two cities. The network of roads was connected, meaning it was possible to travel from any city to any other city along the roads, possibly through other cities. However, during a flood, two cit... |
olympiads | On a rectangular table, there are equal cardboard squares of \( k \) different colors with sides parallel to the sides of the table. If any \( k \) squares of different colors are considered, then some two of them can be nailed to the table with a single nail. Prove that all squares of a certain color can be nailed to ... |
To prove the proposition, we will use induction on the number of colors \( k \).
1. **Base Case:** \( n = 2 \)
- Consider the leftmost square, call it \( K \).
- If \( K \) is of the first color, then all squares of the second color must intersect with \( K \) because there would be at least one pair of overlap... | [
{
"content": "On a rectangular table, there are equal cardboard squares of \\( k \\) different colors with sides parallel to the sides of the table. If any \\( k \\) squares of different colors are considered, then some two of them can be nailed to the table with a single nail. Prove that all squares of a certa... |
olympiads | Given that \(\triangle A_{1} A_{2} A_{3}\) is a scalene triangle with sides \(a_{1}, a_{2}, a_{3}\), where \(a_{i}\) is opposite to vertex \(A_{i}\) for \(i = 1, 2, 3\). Let \(M_{i}\) be the midpoint of side \(a_{i}\), and \(T_{i}\) be the point where the incircle of \(\triangle A_{1} A_{2} A_{3}\) touches side \(a_{i}... |
Given:
- \( \triangle A_{1} A_{2} A_{3} \) is a scalene triangle with sides \( a_{1}, a_{2}, a_{3} \).
- \( a_{i} \) is opposite to \( A_{i} \).
- \( M_{i} \) are the midpoints of sides \( a_{i} \).
- \( T_{i} \) are the points where the incircle of \( \triangle A_{1} A_{2} A_{3} \) touches \( a_{i} \).
- \( S_{i} \)... | [
{
"content": "Given that \\(\\triangle A_{1} A_{2} A_{3}\\) is a scalene triangle with sides \\(a_{1}, a_{2}, a_{3}\\), where \\(a_{i}\\) is opposite to vertex \\(A_{i}\\) for \\(i = 1, 2, 3\\). Let \\(M_{i}\\) be the midpoint of side \\(a_{i}\\), and \\(T_{i}\\) be the point where the incircle of \\(\\triangle... |
olympiads | Let \( y = f(x) \) be a function defined on \( \mathbf{R} \). If there exists a point \( A \) such that for any point \( P \) on the graph of \( y = f(x) \), the point \( Q \), which is symmetric to \( P \) with respect to \( A \), is also on the graph of \( y = f(x) \), then the graph of \( y = f(x) \) is said to be s... |
#### Part (1)
1. Let \( P(x_1, y_1) \) be an arbitrary point on the graph of the function \( y = g(x) = (x-1)^3 \), so we have:
\[
y_1 = g(x_1) = (x_1 - 1)^3
\]
2. The point \( Q \) is the reflection of \( P \) about \( A(1, 0) \), thus it has the coordinates:
\[
Q(2 - x_1, -y_1)
\]
3. Calc... | [
{
"content": "Let \\( y = f(x) \\) be a function defined on \\( \\mathbf{R} \\). If there exists a point \\( A \\) such that for any point \\( P \\) on the graph of \\( y = f(x) \\), the point \\( Q \\), which is symmetric to \\( P \\) with respect to \\( A \\), is also on the graph of \\( y = f(x) \\), then th... |
olympiads | Construct a circle such that the tangents drawn from three given points \( A \), \( B \), and \( C \) to the circle have lengths \( a \), \( b \), and \( c \) respectively. |
Given three points \(A\), \(B\), and \(C\) with tangents to a desired circle having lengths \(a\), \(b\), and \(c\) respectively, we can find the required circle by following these steps:
1. **Construct Circles \(S_a\), \(S_b\), and \(S_c\)**:
- These circles are centered at points \(A\), \(B\), and \(C\) with ra... | [
{
"content": "Construct a circle such that the tangents drawn from three given points \\( A \\), \\( B \\), and \\( C \\) to the circle have lengths \\( a \\), \\( b \\), and \\( c \\) respectively.",
"role": "user"
},
{
"content": "\nGiven three points \\(A\\), \\(B\\), and \\(C\\) with tangents to... |
olympiads | The line \( y = x - 2 \) intersects the parabola \( y^2 = 2x \) at points \( A \) and \( B \). Prove that \( OA \perp OB \). |
1. **Set up the system of equations:**
The given line and parabola are:
\[
y = x - 2 \quad \text{and} \quad y^2 = 2x.
\]
2. **Substitute the equation of the line into the equation of the parabola:**
Substitute \( y = x - 2 \) into \( y^2 = 2x \):
\[
(x - 2)^2 = 2x.
\]
3. **Expand and solve f... | [
{
"content": "The line \\( y = x - 2 \\) intersects the parabola \\( y^2 = 2x \\) at points \\( A \\) and \\( B \\). Prove that \\( OA \\perp OB \\).",
"role": "user"
},
{
"content": "\n1. **Set up the system of equations:** \n The given line and parabola are:\n \\[\n y = x - 2 \\quad \\text{a... |
olympiads | Given four points on a plane that are not collinear, prove that at least one of the triangles with vertices at these points is not acute-angled. |
Let there be four points on a plane, namely \( A, B, C, \) and \( D \), where no three points are collinear. We are to prove that at least one of the triangles formed by these points is not acute-angled.
Let's consider the following cases:
#### Case 1: The four points form the vertices of a convex quadrilateral
1. ... | [
{
"content": "Given four points on a plane that are not collinear, prove that at least one of the triangles with vertices at these points is not acute-angled.",
"role": "user"
},
{
"content": "\nLet there be four points on a plane, namely \\( A, B, C, \\) and \\( D \\), where no three points are col... |
olympiads | You are given three lists \( A, B \), and \( C \). List \( A \) contains the numbers of the form \( 10^{k} \) in base 10, with \( k \) being any integer greater than or equal to 1. Lists \( B \) and \( C \) contain the same numbers translated into bases 2 and 5, respectively:
| \( A \) | \( B \) | \( C \) |
| :--- | :... |
Given lists $A$, $B$, and $C$, where list $A$ contains powers of 10 in base 10, list $B$ contains the same numbers translated into base 2, and list $C$ contains the same numbers translated into base 5. We are asked to prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists $B$ or ... | [
{
"content": "You are given three lists \\( A, B \\), and \\( C \\). List \\( A \\) contains the numbers of the form \\( 10^{k} \\) in base 10, with \\( k \\) being any integer greater than or equal to 1. Lists \\( B \\) and \\( C \\) contain the same numbers translated into bases 2 and 5, respectively:\n\n| \\... |
olympiads |
In the Umnovka River live Wise gudgeons, and they have a law: as soon as one's eye color changes from gray to green, they must swim away to the Mozgovka River the same night they realize this transformation. There are no mirrors underwater, and although the gudgeons understand everything, they cannot speak or gesture.... |
1. **Case 1: There is one green-eyed fish**
- The singular green-eyed fish sees only gray-eyed fish around it.
- Upon hearing the statement "At least one of you has green eyes" from the old carp, the fish realizes it must be the one with green eyes.
- *Conclusion*: The single green-eyed fish will leave tha... | [
{
"content": "\nIn the Umnovka River live Wise gudgeons, and they have a law: as soon as one's eye color changes from gray to green, they must swim away to the Mozgovka River the same night they realize this transformation. There are no mirrors underwater, and although the gudgeons understand everything, they c... |
olympiads | Given the quadratic equation $a^{3} x^{2}+b^{3} x+c^{3}=0$ which has two roots, prove that the equation $a^{5} x^{2}+b^{5} x+c^{5}=0$ also has two roots. |
Given a quadratic equation \(a^{3} x^{2} + b^{3} x + c^{3} = 0\) that has two roots, we need to prove that the equation \(a^{5} x^{2} + b^{5} x + c^{5} = 0\) also has two roots.
1. First, recall the condition for a quadratic equation \(Ax^2 + Bx + C = 0\) to have two distinct roots: the discriminant must be positive.... | [
{
"content": "Given the quadratic equation $a^{3} x^{2}+b^{3} x+c^{3}=0$ which has two roots, prove that the equation $a^{5} x^{2}+b^{5} x+c^{5}=0$ also has two roots.",
"role": "user"
},
{
"content": "\nGiven a quadratic equation \\(a^{3} x^{2} + b^{3} x + c^{3} = 0\\) that has two roots, we need t... |
olympiads | Cut the figure into 8 identical parts. | To solve the problem of dividing the figure into 8 identical parts, we can proceed as follows:
1. Identify the shape of the figure and assume it is a cube (as suggested by common geometry problems and the typical folktale context).
2. **Divide using plane cuts:**
**Method 1:**
a. **First Cut:**
Make... | [
{
"content": "Cut the figure into 8 identical parts.",
"role": "user"
},
{
"content": "To solve the problem of dividing the figure into 8 identical parts, we can proceed as follows:\n\n1. Identify the shape of the figure and assume it is a cube (as suggested by common geometry problems and the typic... |
olympiads | Prove that there exist infinitely many pairs of relatively prime positive integers \(a, b > 1\) for which \(a + b\) divides \(a^b + b^a\). |
1. **Introduce Variables and Conditions:**
Let's construct integers \(a\) and \(b\) such that \(a + b\) divides \(a^b + b^a\). We will start by setting up the equation.
2. **Choosing \(d\) such that \(d \equiv 1 \pmod{4}\) and \(d > 1\):**
Let \(d\) be an integer such that:
\[
d \equiv 1 \pmod{4} \... | [
{
"content": "Prove that there exist infinitely many pairs of relatively prime positive integers \\(a, b > 1\\) for which \\(a + b\\) divides \\(a^b + b^a\\).",
"role": "user"
},
{
"content": "\n1. **Introduce Variables and Conditions:**\n Let's construct integers \\(a\\) and \\(b\\) such that \\(... |
olympiads | Prove that \( u = \operatorname{ctg} \frac{R}{4} \) satisfies a quadratic equation, and \( v = \frac{1}{\sin \frac{R}{4}} \) satisfies a quartic equation with integer coefficients, where the coefficient of the highest degree term is equal to 1. (Here, \( R \) represents an angle of \( 90^{\circ} \)). |
1. We start with the function $u = \operatorname{ctg} \frac{R}{4}$ where $R = 90^\circ$. Therefore, $\frac{R}{4} = 22.5^\circ$. We need to show that $u$ satisfies a quadratic equation with integer coefficients where the coefficient of the highest degree term is 1.
2. Recall the trigonometric identity for $\operatorn... | [
{
"content": "Prove that \\( u = \\operatorname{ctg} \\frac{R}{4} \\) satisfies a quadratic equation, and \\( v = \\frac{1}{\\sin \\frac{R}{4}} \\) satisfies a quartic equation with integer coefficients, where the coefficient of the highest degree term is equal to 1. (Here, \\( R \\) represents an angle of \\( ... |
olympiads | Given the expression for \( n \geq 1 \):
$$
H_{n} = \sum_{k=1}^{n} \frac{1}{n}
$$
Show that for all \( n \geq 2, H_{n} \) in its irreducible form has an odd numerator and an even denominator. |
To demonstrate that for all \( n \geq 2 \), the harmonic number \( H_n \) in its irreducible form has an odd numerator and an even denominator, we will proceed with strong induction.
1. **Initialization**:
We need to verify the property \( \mathcal{P}_{n} \) for \( n = 2 \). For \( n = 2 \), the harmonic number \... | [
{
"content": "Given the expression for \\( n \\geq 1 \\):\n\n$$\nH_{n} = \\sum_{k=1}^{n} \\frac{1}{n}\n$$\n\nShow that for all \\( n \\geq 2, H_{n} \\) in its irreducible form has an odd numerator and an even denominator.",
"role": "user"
},
{
"content": "\nTo demonstrate that for all \\( n \\geq 2 ... |
olympiads | Show that the function \( y \) satisfies equation (1).
\[ y = 2 \frac{\sin x}{x} + \cos x \]
\[ x \cdot \sin x \cdot y' + (\sin x - x \cdot \cos x) y = \sin x \cdot \cos x - x \] |
To prove that the function \( y \) satisfies the given differential equation:
1. **Given Function and Differential Equation:**
\[
y = 2 \frac{\sin x}{x} + \cos x
\]
The differential equation is:
\[
x \cdot \sin x \cdot y' + (\sin x - x \cdot \cos x) y = \sin x \cdot \cos x - x
\]
2. **Compute th... | [
{
"content": "Show that the function \\( y \\) satisfies equation (1).\n\n\\[ y = 2 \\frac{\\sin x}{x} + \\cos x \\]\n\n\\[ x \\cdot \\sin x \\cdot y' + (\\sin x - x \\cdot \\cos x) y = \\sin x \\cdot \\cos x - x \\]",
"role": "user"
},
{
"content": "\nTo prove that the function \\( y \\) satisfies ... |
olympiads | Prove that the area of a square lying inside a triangle does not exceed half the area of that triangle. |
#### Step 1: Generalization of the Problem
To prove a more general statement, let's demonstrate that the area of any parallelogram lying inside a triangle does not exceed half the area of that triangle.
#### Step 2: Construction
Consider the triangle \(ABC\) and a parallelogram \(MNPQ\) inside of it. Extend the opp... | [
{
"content": "Prove that the area of a square lying inside a triangle does not exceed half the area of that triangle.",
"role": "user"
},
{
"content": "\n#### Step 1: Generalization of the Problem\n\nTo prove a more general statement, let's demonstrate that the area of any parallelogram lying inside... |
olympiads | Prove that the circumscribed circle of a triangle can pass through the center of the excircle. |
1. **Identifying Key Points:**
Consider triangle \(ABC\) with its sides \(AB\), \(AC\), and \(BC\). Let \(O\) denote the center of the circumcircle of triangle \(ABC\).
2. **Circumcircle and Excircle Tangency:**
We are given that there is an excircle of the triangle that is tangent to the extended sides of \(AB... | [
{
"content": "Prove that the circumscribed circle of a triangle can pass through the center of the excircle.",
"role": "user"
},
{
"content": "\n1. **Identifying Key Points:**\n Consider triangle \\(ABC\\) with its sides \\(AB\\), \\(AC\\), and \\(BC\\). Let \\(O\\) denote the center of the circum... |
olympiads | Let \(\varphi\) denote the Euler phi-function. Prove that for every positive integer \(n\)
\[
2^{n(n+1)} \mid 32 \cdot \varphi\left(2^{2^{n}} - 1\right)
\] |
1. **Base Cases:**
We start by validating the base cases for \( n = 1, 2, 3 \):
- **Base Case for \( n = 1 \):**
\[
2^2 = 4 \mid 32 \cdot \varphi(2^2 - 1) = 32 \cdot \varphi(3) = 32 \cdot 2 = 64
\]
Clearly, \( 4 \mid 64 \).
- **Base Case for \( n = 2 \):**
\[
2^{2 \cdot 3} = 2... | [
{
"content": "Let \\(\\varphi\\) denote the Euler phi-function. Prove that for every positive integer \\(n\\)\n\n\\[ \n2^{n(n+1)} \\mid 32 \\cdot \\varphi\\left(2^{2^{n}} - 1\\right) \n\\]",
"role": "user"
},
{
"content": "\n1. **Base Cases:**\n\n We start by validating the base cases for \\( n = ... |
olympiads | Let $P$ be a non-constant polynomial with integer coefficients. Prove that there exists an integer $n$ such that $P\left(n^{2}+2020\right)$ is not a prime number. |
1. **Define a transformation for the polynomial:**
Let \( Q(x) = P(x^2 + 2020) \). The goal now is to show that there exists an integer \( n \) such that \( Q(n) \) is not a prime number.
2. **Check for non-prime value at \( Q(0) \):**
Consider the value \( Q(0) \).
- If \( Q(0) \) is not a prime number, the... | [
{
"content": "Let $P$ be a non-constant polynomial with integer coefficients. Prove that there exists an integer $n$ such that $P\\left(n^{2}+2020\\right)$ is not a prime number.",
"role": "user"
},
{
"content": "\n1. **Define a transformation for the polynomial:**\n Let \\( Q(x) = P(x^2 + 2020) \... |
olympiads | Let \( AB \) be the shortest side of an acute-angled triangle \( ABC \). Points \( X \) and \( Y \) are chosen on sides \( BC \) and \( AC \) respectively. Prove that the length of the broken line \( AXYB \) is not less than twice the length of side \( AB \). |
1. Given a triangle $ABC$ where $AB$ is the smallest side, and points $X$ and $Y$ are chosen on $BC$ and $AC$ respectively. We need to prove that the length of the broken line $AXYB$ is not less than twice the length of side $AB$.
2. Let $A_1$ be the point symmetric to $A$ relative to line $BC$, and $B_1$ be the poin... | [
{
"content": "Let \\( AB \\) be the shortest side of an acute-angled triangle \\( ABC \\). Points \\( X \\) and \\( Y \\) are chosen on sides \\( BC \\) and \\( AC \\) respectively. Prove that the length of the broken line \\( AXYB \\) is not less than twice the length of side \\( AB \\).",
"role": "user"
... |
olympiads | Let \(ABCD\) be a convex quadrilateral with \(\angle ABC = \angle ADC < 90^\circ\). The internal angle bisectors of \(\angle ABC\) and \(\angle ADC\) meet \(AC\) at \(E\) and \(F\) respectively, and meet each other at point \(P\). Let \(M\) be the midpoint of \(AC\) and let \(\omega\) be the circumcircle of triangle \(... |
**Given**: Let \(ABCD\) be a convex quadrilateral with \(\angle ABC = \angle ADC < 90^\circ\). The internal angle bisectors of \(\angle ABC\) and \(\angle ADC\) meet \(AC\) at \(E\) and \(F\) respectively, and meet each other at point \(P\). Let \(M\) be the midpoint of \(AC\) and let \(\omega\) be the circumcircle of... | [
{
"content": "Let \\(ABCD\\) be a convex quadrilateral with \\(\\angle ABC = \\angle ADC < 90^\\circ\\). The internal angle bisectors of \\(\\angle ABC\\) and \\(\\angle ADC\\) meet \\(AC\\) at \\(E\\) and \\(F\\) respectively, and meet each other at point \\(P\\). Let \\(M\\) be the midpoint of \\(AC\\) and le... |
olympiads | Let $d, r \geq 2$ be integers satisfying $d \geq r^{2r+2}$, and let $G$ be a graph that contains a collection $\mathcal{H}$ of $r^2 - 1$ pairwise disjoint paths and a collection $\mathcal{V} = \{V_1, \ldots, V_d\}$ of $d$ pairwise disjoint paths. Suppose that every path in $\mathcal{V}$ intersects every path in $\mathc... |
1. **Illustrate the problem and initial conditions:**
- Let $d, r \geq 2$ satisfy $d \geq r^{2r+2}$.
- Let $G$ be a graph containing:
- A collection $\mathcal{H}$ of $r^2 - 1$ mutually disjoint paths.
- A collection $\mathcal{V} = \{V_1, \ldots, V_d\}$ of $d$ mutually disjoint paths.
- Each pat... | [
{
"content": "Let $d, r \\geq 2$ be integers satisfying $d \\geq r^{2r+2}$, and let $G$ be a graph that contains a collection $\\mathcal{H}$ of $r^2 - 1$ pairwise disjoint paths and a collection $\\mathcal{V} = \\{V_1, \\ldots, V_d\\}$ of $d$ pairwise disjoint paths. Suppose that every path in $\\mathcal{V}$ in... |
olympiads | Using a compass and ruler, construct an equilateral triangle with one vertex at a given point and the other two vertices on two given circles. |
To construct an equilateral triangle where one vertex is at a given point and the other two vertices lie on two given circles using a compass and straightedge, follow these steps:
1. **Identify the Given Elements:**
- Let point \( A \) be the given point which will be one vertex of the equilateral triangle \( \tri... | [
{
"content": "Using a compass and ruler, construct an equilateral triangle with one vertex at a given point and the other two vertices on two given circles.",
"role": "user"
},
{
"content": "\nTo construct an equilateral triangle where one vertex is at a given point and the other two vertices lie on... |
olympiads | Prove that for any prime number \( p > 2 \), the numerator of the fraction \( \frac{m}{n} = \frac{1}{1} + \frac{1}{2} + \ldots + \frac{1}{p-1} \) is divisible by \( p \). | 1. First, consider the sum of the reciprocals $\frac{1}{k}$ such that $k$ ranges from $1$ to $p-1$, where $p$ is a prime greater than $2$:
\[
S = \sum_{k=1}^{p-1} \frac{1}{k}
\]
2. Note that for each term $\frac{1}{k}$, there exists a corresponding term $\frac{1}{p-k}$ in the sum since $p-k$ is also betwee... | [
{
"content": "Prove that for any prime number \\( p > 2 \\), the numerator of the fraction \\( \\frac{m}{n} = \\frac{1}{1} + \\frac{1}{2} + \\ldots + \\frac{1}{p-1} \\) is divisible by \\( p \\).",
"role": "user"
},
{
"content": "1. First, consider the sum of the reciprocals $\\frac{1}{k}$ such that... |
olympiads | a) Given an arbitrary quadrilateral, prove that it is possible to tessellate the plane without gaps and overlaps with quadrilaterals that are congruent to the given one.
b) Does there exist a pentagon whose copies can tessellate the plane?
c) The same question for a pentagon where no two sides are parallel. |
#### Part (a)
1. Consider a quadrilateral with vertices \( A, B, C, D \). The sum of the internal angles of any quadrilateral is \( 360^\circ \).
2. To tile the plane using this quadrilateral, you can position several identical quadrilaterals around a point, ensuring the angles sum up appropriately, covering the plan... | [
{
"content": "a) Given an arbitrary quadrilateral, prove that it is possible to tessellate the plane without gaps and overlaps with quadrilaterals that are congruent to the given one.\n\nb) Does there exist a pentagon whose copies can tessellate the plane?\n\nc) The same question for a pentagon where no two sid... |
olympiads | Let \( O \) be the circumcenter of triangle \( ABC \). Points \( X \) and \( Y \) are found on side \( BC \) such that \( AX = BX \) and \( AY = CY \). Prove that the circumcircle of triangle \( AXY \) passes through the circumcenters of triangles \( AOB \) and \( AOC \). |
1. **Identify Perpendicular Bisectors and Their Implications:**
Given:
- \(O\) is the circumcenter of triangle \(\Delta ABC\).
- Points \(X\) and \(Y\) lie on side \(BC\) such that \(AX = BX\) and \(AY = CY\).
We know that:
- Since \(AX = BX\), point \(X\) lies on the perpendicular bisector of segment... | [
{
"content": "Let \\( O \\) be the circumcenter of triangle \\( ABC \\). Points \\( X \\) and \\( Y \\) are found on side \\( BC \\) such that \\( AX = BX \\) and \\( AY = CY \\). Prove that the circumcircle of triangle \\( AXY \\) passes through the circumcenters of triangles \\( AOB \\) and \\( AOC \\).",
... |
olympiads | Name the theorems related to the triangle and circle, the proofs of which do not rely on the parallel postulate. |
1. **The Perpendicularity Theorem:** Through any point, you can draw one and only one line perpendicular to a given plane.
- *Proof:* This follows directly from basic geometric constructions. Given a plane \( \Pi \) and a point \( P \) not lying in \( \Pi \), construct a perpendicular line by dropping a perpendicu... | [
{
"content": "Name the theorems related to the triangle and circle, the proofs of which do not rely on the parallel postulate.",
"role": "user"
},
{
"content": "\n1. **The Perpendicularity Theorem:** Through any point, you can draw one and only one line perpendicular to a given plane.\n\n - *Proof... |
olympiads | Let \( A B C \) be any acute-angled triangle. \( E \) and \( F \) are the base points of the heights through \( B \) and \( C \). Let \( G \) be the projection of \( B \) onto the line \( E F \) and \( H \) the corresponding projection of \( C \). Show that
\[ |H E| = |F G| \] |
1. **Notation and Definitions**:
Let us define the following:
- \( \alpha = \angle ABC \)
- \( \gamma = \angle BCA \)
- \( H \) is the projection of \( C \) onto the line \( EF \)
- \( G \) is the projection of \( B \) onto the line \( EF \)
Given are the feet of the heights \( E \) ... | [
{
"content": "Let \\( A B C \\) be any acute-angled triangle. \\( E \\) and \\( F \\) are the base points of the heights through \\( B \\) and \\( C \\). Let \\( G \\) be the projection of \\( B \\) onto the line \\( E F \\) and \\( H \\) the corresponding projection of \\( C \\). Show that\n\n\\[ |H E| = |F G|... |
olympiads | There are five infinite geometric progressions, all members of which are integers. Prove that there exists a natural number that is not contained in any of these progressions. |
1. **Identify the problem:** We need to prove that there exists a natural number that is not contained in any of the given five infinite geometric progressions, all of whose terms are integers.
2. **Define a single geometric progression:** Consider one of the five geometric progressions, denoted by \(\{Aq^n\}\), wher... | [
{
"content": "There are five infinite geometric progressions, all members of which are integers. Prove that there exists a natural number that is not contained in any of these progressions.",
"role": "user"
},
{
"content": "\n1. **Identify the problem:** We need to prove that there exists a natural ... |
olympiads | Given \( n \) is a positive integer, prove that \( 2^n + 1 \) does not have a prime factor that is congruent to 7 modulo 8. | 1. Consider a prime number \( p \) such that \( p \equiv -1 \pmod{8} \).
2. We examine the sequence \( 2, 2 \times 2, 2 \times 3, 2 \times 4, \cdots, 2 \times \frac{p-1}{2} \) modulo \( p \).
3. Denote the numbers not greater than \( \frac{p-1}{2} \) as \( r_1, r_2, \dots, r_h \), and the numbers greater than \( \frac{... | [
{
"content": "Given \\( n \\) is a positive integer, prove that \\( 2^n + 1 \\) does not have a prime factor that is congruent to 7 modulo 8.",
"role": "user"
},
{
"content": "1. Consider a prime number \\( p \\) such that \\( p \\equiv -1 \\pmod{8} \\).\n2. We examine the sequence \\( 2, 2 \\times ... |
olympiads | Given a multi-digit number $\overline{a b c \ldots k x y z}$. Separating from it the three-digit number formed by the last three digits, we obtain two numbers: $\overline{a b c \ldots k}$ and $\overline{x y z}$. Prove that if the difference between the obtained numbers is divisible by 7 (or 11, or 13), then the given n... |
We are given a multi-digit number $\overline{a b c \ldots k x y z}$. When we separate this number into two parts: a number formed by the last three digits $\overline{x y z}$ and another number formed by the rest of the digits $\overline{a b c \ldots k}$, we get two numbers. We are to prove that if the difference betwe... | [
{
"content": "Given a multi-digit number $\\overline{a b c \\ldots k x y z}$. Separating from it the three-digit number formed by the last three digits, we obtain two numbers: $\\overline{a b c \\ldots k}$ and $\\overline{x y z}$. Prove that if the difference between the obtained numbers is divisible by 7 (or 1... |
olympiads | Prove that among 2008 arbitrary integers, there exist numbers whose sum is divisible by 2008. | 1. **Define the sequence of partial sums:**
Let \(a_{1}, a_{2}, \ldots, a_{2008}\) be the given integers. Consider the sequence of partial sums \(s_i\) defined by:
\[
s_i = a_1 + a_2 + \cdots + a_i \quad \text{for} \; 1 \leq i \leq 2008.
\]
2. **Check if any partial sum is divisible by 2008:**
We need... | [
{
"content": "Prove that among 2008 arbitrary integers, there exist numbers whose sum is divisible by 2008.",
"role": "user"
},
{
"content": "1. **Define the sequence of partial sums:**\n\n Let \\(a_{1}, a_{2}, \\ldots, a_{2008}\\) be the given integers. Consider the sequence of partial sums \\(s_... |
olympiads | A convex $2n$-gon $A_{1} A_{2} \ldots A_{2n}$ is inscribed in a circle of radius 1. Prove that
$$
\left|\overrightarrow{A_{1} A_{2}}+\overrightarrow{A_{3} A_{4}}+\ldots+\overrightarrow{A_{2n-1} A_{2n}}\right| \leq 2
$$ |
1. Let \(\mathbf{a} = \overrightarrow{A_{1} A_{2}} + \overrightarrow{A_{3} A_{4}} + \cdots + \overrightarrow{A_{2n-1} A_{2n}}\), where \(\mathbf{a} \neq 0\).
2. Introduce a coordinate system such that the axis \(Ox\) is directed along vector \(\mathbf{a}\).
3. Since the sum of the projections of vectors \(\overright... | [
{
"content": "A convex $2n$-gon $A_{1} A_{2} \\ldots A_{2n}$ is inscribed in a circle of radius 1. Prove that\n\n$$\n\\left|\\overrightarrow{A_{1} A_{2}}+\\overrightarrow{A_{3} A_{4}}+\\ldots+\\overrightarrow{A_{2n-1} A_{2n}}\\right| \\leq 2\n$$",
"role": "user"
},
{
"content": "\n1. Let \\(\\mathbf... |
olympiads | Show that for all positive numbers \(a\) and \(b\), the inequality \(\sqrt{a} + \sqrt{b} > \sqrt{a + b}\) holds. |
1. Consider the inequality $\sqrt{a} + \sqrt{b} > \sqrt{a + b}$ for positive numbers $a$ and $b$.
2. To prove this inequality, we will square both sides of the given inequality to eliminate the square roots. Squaring both sides yields:
\[
(\sqrt{a} + \sqrt{b})^2 > (\sqrt{a + b})^2
\]
3. Simplify both sides... | [
{
"content": "Show that for all positive numbers \\(a\\) and \\(b\\), the inequality \\(\\sqrt{a} + \\sqrt{b} > \\sqrt{a + b}\\) holds.",
"role": "user"
},
{
"content": "\n1. Consider the inequality $\\sqrt{a} + \\sqrt{b} > \\sqrt{a + b}$ for positive numbers $a$ and $b$.\n\n2. To prove this inequal... |
olympiads | Two circles, which are located outside of each other, have one external and one internal tangent drawn to them. Consider two lines, each of which passes through the points of tangency that belong to one of the circles. Prove that the intersection point of these lines lies on the line connecting the centers of the circl... |
1. **Define the Circle Centers and Tangent Points:**
Let the centers of the two circles be \( O_{1} \) and \( O_{2} \). An external tangent touches the first circle at point \( K \) and the second circle at point \( L \). An internal tangent touches the first circle at point \( M \) and the second circle at point \( ... | [
{
"content": "Two circles, which are located outside of each other, have one external and one internal tangent drawn to them. Consider two lines, each of which passes through the points of tangency that belong to one of the circles. Prove that the intersection point of these lines lies on the line connecting th... |
olympiads | The squares of an \( n \times n \) chessboard are numbered in an arbitrary manner from 1 to \( n^2 \) (every square has a different number). Show that there are two squares with a common side whose numbers differ by at least \( n \). |
Let's denote the \( n \times n \) chessboard by a grid where each cell is numbered in an arbitrary manner from \( 1 \) to \( n^2 \). Our goal is to demonstrate that there are always two adjacent squares (i.e., two squares sharing a side) whose numbers differ by at least \( n \).
1. **Introductory Definitions and Part... | [
{
"content": "The squares of an \\( n \\times n \\) chessboard are numbered in an arbitrary manner from 1 to \\( n^2 \\) (every square has a different number). Show that there are two squares with a common side whose numbers differ by at least \\( n \\).",
"role": "user"
},
{
"content": "\nLet's den... |
olympiads | A line passing through the center of a parallelogram intersects its sides at points \( P \) and \( Q \), which are connected to the vertices of the parallelogram. Prove that the points of intersection of the segments \( AP, BP, CQ, DQ \) with the diagonals of the parallelogram are the vertices of a new parallelogram. |
To prove the statement, we will consider the properties of diagonals and midpoints in parallelograms:
1. Establish the input details:
- Let the parallelogram be \(ABCD\) with center \(O\).
- The line passing through \(O\) intersects the sides \(AB\) and \(CD\) at points \(P\) and \(Q\) respectively.
2. Identif... | [
{
"content": "A line passing through the center of a parallelogram intersects its sides at points \\( P \\) and \\( Q \\), which are connected to the vertices of the parallelogram. Prove that the points of intersection of the segments \\( AP, BP, CQ, DQ \\) with the diagonals of the parallelogram are the vertic... |
olympiads | The circles \( S_1 \) and \( S_2 \) touch each other externally at point \( F \). A line \( l \) is tangent to \( S_1 \) and \( S_2 \) at points \( A \) and \( B \) respectively. Another line, parallel to line \( l \), is tangent to \( S_2 \) at point \( C \) and intersects \( S_1 \) at two points. Prove that points \(... | 1. Let the centers of the circles \( S_1 \) and \( S_2 \) be \( O_1 \) and \( O_2 \) respectively, and their radii be \( r_1 \) and \( r_2 \).
2. Since the circles \( S_1 \) and \( S_2 \) touch each other externally at point \( F \), the distance between the centers is \( O_1O_2 = r_1 + r_2 \).
3. The line \( l \) touc... | [
{
"content": "The circles \\( S_1 \\) and \\( S_2 \\) touch each other externally at point \\( F \\). A line \\( l \\) is tangent to \\( S_1 \\) and \\( S_2 \\) at points \\( A \\) and \\( B \\) respectively. Another line, parallel to line \\( l \\), is tangent to \\( S_2 \\) at point \\( C \\) and intersects \... |
olympiads | Three bugs start at the same point on a table and begin crawling. After some time, they are at the three vertices of a triangle whose incircle has a radius of 2. Prove that at least one of the bugs has crawled a distance greater than 3.
| 1. Assume initially, the three insects are at the point \( O \). After some time, they are at the vertices of triangle \( \triangle ABC \).
2. Let \( a = BC \), \( b = CA \), \( c = AB \).
3. Denote the distances from the origin \( O \) to the vertices \( A, B, \) and \( C \) as \( x = O A \), \( y = O B \), \( z = O C... | [
{
"content": "Three bugs start at the same point on a table and begin crawling. After some time, they are at the three vertices of a triangle whose incircle has a radius of 2. Prove that at least one of the bugs has crawled a distance greater than 3.\n",
"role": "user"
},
{
"content": "1. Assume ini... |
olympiads | A piece is placed at the point \((0,0)\), and at each step, we are allowed to replace a piece at position \((i, j)\) with two pieces at positions \((i+1, j)\) and \((i, j+1)\), provided that there is not already a piece there. Show that there will always be a piece with coordinates \((a, b)\) such that \(a + b \leq 3\)... |
1. **Introduction et notation:**
On place initialement une pièce sur le point $(0,0)$. À chaque étape, on peut remplacer une pièce à la position $(i, j)$ par deux pièces aux positions $(i+1, j)$ et $(i, j+1)$, à condition qu'il n'y ait pas déjà de pièce à ces positions. Il s'agit de montrer qu'il y aura toujours u... | [
{
"content": "A piece is placed at the point \\((0,0)\\), and at each step, we are allowed to replace a piece at position \\((i, j)\\) with two pieces at positions \\((i+1, j)\\) and \\((i, j+1)\\), provided that there is not already a piece there. Show that there will always be a piece with coordinates \\((a, ... |
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