source stringclasses 1
value | problem stringlengths 47 1.66k | solution stringlengths 932 6.13k | messages listlengths 2 2 |
|---|---|---|---|
olympiads | a) Does there exist a sequence of natural numbers \( a_{1}, a_{2}, a_{3}, \ldots \) with the following property: no term of the sequence equals the sum of several other terms and \( a_{n} \leq n^{10} \) for any \( n \)?
b) The same question, if \( a_{n} \leq n \sqrt{n} \) for any \( n \). |
To solve both parts of the problem, we need to analyze the given sequences and their properties. We will tackle each part separately.
#### Part (a):
We need to construct a sequence \( \{a_n\} \) of natural numbers such that:
1. No term in the sequence is equal to the sum of some other terms in the sequence.
2. For al... | [
{
"content": "a) Does there exist a sequence of natural numbers \\( a_{1}, a_{2}, a_{3}, \\ldots \\) with the following property: no term of the sequence equals the sum of several other terms and \\( a_{n} \\leq n^{10} \\) for any \\( n \\)?\n\nb) The same question, if \\( a_{n} \\leq n \\sqrt{n} \\) for any \\... |
olympiads | a) The square $ABCD$ with side length $1 \text{ cm}$ is "flipped" around point $D$ as shown in the following figure. The dotted lines indicate the area occupied by the square during its flipping motion. What is the total area occupied by the square from the beginning to the end of its flipping?
b) Similarly to the squ... |
#### Part a)
1. **Identify regions in the figure**:
The figure can be divided into:
- Two right triangles: $\triangle ABD$ and $\triangle DB'C'$
- One quarter circle sector $DBB'$ with a $90^{\circ}$ central angle.
2. **Calculate the radius of the circular sector using the Pythagorean Theorem**:
\[
R ... | [
{
"content": "a) The square $ABCD$ with side length $1 \\text{ cm}$ is \"flipped\" around point $D$ as shown in the following figure. The dotted lines indicate the area occupied by the square during its flipping motion. What is the total area occupied by the square from the beginning to the end of its flipping?... |
olympiads | a) Prove that of all triangles with a given perimeter, the one with the largest area is the equilateral triangle.
b) Prove that of all quadrilaterals with a given perimeter, the one with the largest area is the square. |
To prove that among all triangles with a given perimeter, the one with the largest area is equilateral, we proceed as follows:
1. **Let \( \triangle ABC \) be any triangle:**
Let \( AB = c \), \( BC = a \) and \( CA = b \). The perimeter is given and equals \( P = a + b + c \).
2. **Consider \( \triangle AB'C... | [
{
"content": "a) Prove that of all triangles with a given perimeter, the one with the largest area is the equilateral triangle.\n\nb) Prove that of all quadrilaterals with a given perimeter, the one with the largest area is the square.",
"role": "user"
},
{
"content": "\nTo prove that among all tria... |
olympiads | a) What is the probability that the square of a randomly chosen integer ends in the digit 1? What is the probability that the cube of a randomly chosen integer ends in the digits 11?
b) What is the probability that the tenth power of a randomly chosen integer ends in the digit 6? What is the probability that the twent... |
#### Part (a):
1. **Square Ending in 1:**
- A square of an integer ends with the digit 1 if and only if the integer itself ends with the digit 1 or 9.
- To verify, consider the squares of numbers ending with each digit:
\[
\begin{aligned}
&0^2 = 0 \\
&1^2 = 1 \\
&2^2 = 4 \\
&3^2 = ... | [
{
"content": "a) What is the probability that the square of a randomly chosen integer ends in the digit 1? What is the probability that the cube of a randomly chosen integer ends in the digits 11?\n\nb) What is the probability that the tenth power of a randomly chosen integer ends in the digit 6? What is the pr... |
olympiads | a) Prove that the circle passing through the midpoints of the sides of a triangle touches its incircle and all three excircles (Theorem of Feuerbach).
b) On the sides $AB$ and $AC$ of triangle $ABC$, points $C_1$ and $B_1$ are selected such that $AC_1 = B_1C_1$ and the incircle $S$ of triangle $ABC$ is the excircle of... |
**(a)**
Let $A_1, B_1,$ and $C_1$ be the midpoints of sides $BC$, $CA$, and $AB$ of triangle $ABC$. We need to prove that the circumcircle of triangle $A_1B_1C_1$ is tangent to the incircle $S$ and one of the excircles $S_a$, specifically the excircle that touches side $BC$.
1. Let points $B'$ and $C'$ be the refle... | [
{
"content": "a) Prove that the circle passing through the midpoints of the sides of a triangle touches its incircle and all three excircles (Theorem of Feuerbach).\n\nb) On the sides $AB$ and $AC$ of triangle $ABC$, points $C_1$ and $B_1$ are selected such that $AC_1 = B_1C_1$ and the incircle $S$ of triangle ... |
olympiads | a) Prove that there exists a number of the form 1989...19890... (1989 repeated several times followed by several zeros), which is divisible by 1988.
b) Prove that there exists a number of the form 1988...1988 (1988 repeated several times), which is divisible by 1989. |
Let's go through the solutions for both (a) and (b) and provide more detailed steps and explanations.
#### Part (a):
1. **Construct the Sequence:**
- Consider the sequence of numbers formed by repeating "1989" several times.
\[
1989, \; 19891989, \; 198919891989, \; \ldots, \; 1989 \cdots 1989 \quad (\te... | [
{
"content": "a) Prove that there exists a number of the form 1989...19890... (1989 repeated several times followed by several zeros), which is divisible by 1988.\n\nb) Prove that there exists a number of the form 1988...1988 (1988 repeated several times), which is divisible by 1989.",
"role": "user"
},
... |
olympiads | a) Prove that the solid angle of a dihedral angle is equal to $2 \alpha$, where $\alpha$ is the measure of the dihedral angle in radians.
b) Prove that the solid angle of a polyhedral angle is equal to $\sigma - (n-2) \pi$, where $\sigma$ is the sum of its dihedral angles. | #### Part (a)
To prove that the solid angle of a dihedral angle is equal to $2\alpha$, where $\alpha$ is the measure of the dihedral angle in radians, follow these steps:
1. **Definition and Basic Idea**:
- A dihedral angle is formed by the intersection of two half-planes, and it measures the angle between the two... | [
{
"content": "a) Prove that the solid angle of a dihedral angle is equal to $2 \\alpha$, where $\\alpha$ is the measure of the dihedral angle in radians.\n\nb) Prove that the solid angle of a polyhedral angle is equal to $\\sigma - (n-2) \\pi$, where $\\sigma$ is the sum of its dihedral angles.",
"role": "u... |
olympiads | a) Given an arbitrary tetrahedron \( ABCD \). Prove that \((\overrightarrow{AB}, \overrightarrow{CD}) + (\overrightarrow{AC}, \overrightarrow{DB}) + (\overrightarrow{AD}, \overrightarrow{BC}) = 0\).
b) Prove that if in a tetrahedron two pairs of opposite edges are perpendicular, then the third pair of opposite edges i... | 1. Consider a tetrahedron \(A B C D\) with vertices labeled accordingly. We define the vectors:
\[
\mathbf{a} = \overrightarrow{A B}, \quad \mathbf{b} = \overrightarrow{B C}, \quad \mathbf{c} = \overrightarrow{C D}
\]
2. We need to prove that:
\[
(\overrightarrow{A B}, \overrightarrow{C D})+(\overr... | [
{
"content": "a) Given an arbitrary tetrahedron \\( ABCD \\). Prove that \\((\\overrightarrow{AB}, \\overrightarrow{CD}) + (\\overrightarrow{AC}, \\overrightarrow{DB}) + (\\overrightarrow{AD}, \\overrightarrow{BC}) = 0\\).\n\nb) Prove that if in a tetrahedron two pairs of opposite edges are perpendicular, then ... |
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 | a) The sum of the edge lengths of any convex polyhedron is greater than three times its diameter. Prove this. (The diameter of a polyhedron is the longest possible segment connecting two vertices of the polyhedron.)
b) For any two vertices $A$ and $B$ of any convex polyhedron, there exist three paths, each traveling a... | ### Problem:
1. Prove that the sum of the lengths of the edges of any convex polyhedron is greater than three times its diameter. (The diameter of a polyhedron is defined as the greatest of the lengths of all possible segments with endpoints at the vertices of the polyhedron.)
2. For any two vertices \( A \) and \( B \... | [
{
"content": "a) The sum of the edge lengths of any convex polyhedron is greater than three times its diameter. Prove this. (The diameter of a polyhedron is the longest possible segment connecting two vertices of the polyhedron.)\n\nb) For any two vertices $A$ and $B$ of any convex polyhedron, there exist three... |
olympiads | a) Prove that from a connected graph, several edges can be removed so that a tree remains.
b) Prove that in a tree with $n$ vertices, there are exactly $n-1$ edges.
c) Prove that in a tree, there are at least two leaf vertices.
d) Prove that in a connected graph with $n$ vertices, there are at least $n-1$ edges.
e)... |
#### Part (a)
1. Consider a connected graph $G$ which is not a tree.
2. Since $G$ is connected and not a tree, it contains at least one simple cycle.
3. Choose any edge $e$ from this cycle. Removing $e$ will not disrupt the connectedness of the graph since there are alternative paths in the cycle.
4. Repeat the proces... | [
{
"content": "a) Prove that from a connected graph, several edges can be removed so that a tree remains.\n\nb) Prove that in a tree with $n$ vertices, there are exactly $n-1$ edges.\n\nc) Prove that in a tree, there are at least two leaf vertices.\n\nd) Prove that in a connected graph with $n$ vertices, there a... |
olympiads | a) Two parallel lines \(\ell_{1}\) and \(\ell_{2}\) are tangent to the circle \(\omega_{1}\) with center \(O_{1}\) at points \(A\) and \(B\) respectively. The circle \(\omega_{2}\) with center \(O_{2}\) is tangent to line \(\ell_{1}\) at point \(D\), intersects line \(\ell_{2}\) at points \(B\) and \(E\), and intersect... |
**Part (a):**
Given:
- Two parallel tangents $\ell_1$ and $\ell_2$ touch the circle $\omega_1$ with center $O_1$ at points $A$ and $B$, respectively.
- Circle $\omega_2$ with center $O_2$ touches the line $\ell_1$ at point $D$, intersects line $\ell_2$ at points $B$ and $E$, and intersects circle $\omega_1$ again at ... | [
{
"content": "a) Two parallel lines \\(\\ell_{1}\\) and \\(\\ell_{2}\\) are tangent to the circle \\(\\omega_{1}\\) with center \\(O_{1}\\) at points \\(A\\) and \\(B\\) respectively. The circle \\(\\omega_{2}\\) with center \\(O_{2}\\) is tangent to line \\(\\ell_{1}\\) at point \\(D\\), intersects line \\(\\e... |
olympiads | a) Find all numbers \( n \) between 1 and 100 such that \( (n-1)! \) is not divisible by \( n \).
b) Find all numbers \( n \) between 1 and 100 such that \( (n-1)! \) is not divisible by \( n^2 \). |
#### Part (a):
1. **Understanding the Problem:**
We need to find all numbers \( n \) between 1 and 100 such that \( (n-1)! \) does not divide \( n \).
2. **Prime Number Case:**
Notice that if \( n \) is a prime number:
- For any prime number \( n \), \((n-1)! \) contains all integers from 1 to \( (n-1) \).
... | [
{
"content": "a) Find all numbers \\( n \\) between 1 and 100 such that \\( (n-1)! \\) is not divisible by \\( n \\).\n\nb) Find all numbers \\( n \\) between 1 and 100 such that \\( (n-1)! \\) is not divisible by \\( n^2 \\).",
"role": "user"
},
{
"content": "\n#### Part (a):\n\n1. **Understanding ... |
olympiads | a) Prove geometrically that $\operatorname{ctg} 30^{\circ}+\operatorname{ctg} 75^{\circ}=2$.
b) Prove that $\operatorname{tg} 15^{\circ}=4 \sin ^{2} 15^{\circ}$. |
Let's address each part of the problem one by one.
#### Part (a)
We aim to prove geometrically that:
\[ \operatorname{ctg} 30^{\circ} + \operatorname{ctg} 75^{\circ} = 2 \]
1. **Consider the triangle**:
Consider an isosceles triangle \( \triangle ABC \) with \( AB = AC \) and the angle \( 30^{\circ} \) at vertex ... | [
{
"content": "a) Prove geometrically that $\\operatorname{ctg} 30^{\\circ}+\\operatorname{ctg} 75^{\\circ}=2$.\n\nb) Prove that $\\operatorname{tg} 15^{\\circ}=4 \\sin ^{2} 15^{\\circ}$.",
"role": "user"
},
{
"content": "\nLet's address each part of the problem one by one.\n\n#### Part (a)\nWe aim t... |
olympiads | a) Prove that the area of a convex quadrilateral $ABCD$ is given by the formula
$$
S^2 = (p - a)(p - b)(p - c)(p - d) - abcd \cos^2\left(\frac{B + D}{2}\right)
$$
where $p$ is the semiperimeter, and $a, b, c, d$ are the lengths of the sides.
b) Prove that if the quadrilateral $ABCD$ is inscribed in a circle, then
... |
**(a)** Let $AB = a$, $BC = b$, $CD = c$, and $AD = d$. We aim to prove that the area $S$ of the convex quadrilateral $ABCD$ can be calculated using the given formula:
\[ S^2 = (p - a)(p - b)(p - c)(p - d) - abcd \cos^2\left(\frac{B + D}{2}\right) \]
1. Define $S$ as the sum of the areas of triangles $\triangle ABC$... | [
{
"content": "a) Prove that the area of a convex quadrilateral $ABCD$ is given by the formula\n\n$$\nS^2 = (p - a)(p - b)(p - c)(p - d) - abcd \\cos^2\\left(\\frac{B + D}{2}\\right)\n$$\n\nwhere $p$ is the semiperimeter, and $a, b, c, d$ are the lengths of the sides.\n\nb) Prove that if the quadrilateral $ABCD$... |
olympiads | a) Prove that the feet of the perpendiculars dropped from any point on the circumcircle to the sides of an inscribed triangle lie on one straight line (figure 35a).
b) On three chords \(MA\), \(MB\), and \(MC\) of a circle that pass through one point, three new circles are constructed with these chords as diameters. P... |
#### Part (a)
1. **Problem Restatement**:
Let \(ABC\) be an arbitrary triangle inscribed in a circle, and let \(P\) be any point on the circle. Let \(X, Y, Z\) be the feet of the perpendiculars dropped from \(P\) to the sides \(BC, CA, AB\) of the triangle \(ABC\). We need to prove that points \(X, Y,\) and \(Z\) ... | [
{
"content": "a) Prove that the feet of the perpendiculars dropped from any point on the circumcircle to the sides of an inscribed triangle lie on one straight line (figure 35a).\n\nb) On three chords \\(MA\\), \\(MB\\), and \\(MC\\) of a circle that pass through one point, three new circles are constructed wit... |
olympiads | a) Prove the equality \(1+C_{n}^{3}+C_{n}^{6}+\ldots=\frac{1}{3}\left(2^{n}+2 \cos \frac{n \pi}{3}\right)\).
b) Calculate the sums \(C_{n}^{1}+C_{n}^{4}+C_{n}^{7}+\ldots\) and \(C_{n}^{2}+C_{n}^{5}+C_{n}^{8}+\ldots\). |
#### Part (a)
1. Define the complex number \(\omega\) as follows:
\[
\omega = \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} = -\frac{1}{2} + i \frac{\sqrt{3}}{2}.
\]
This is a cube root of unity, satisfying \(\omega^3 = 1\).
2. Using properties of \(\omega\):
\[
\omega^2 = \cos \frac{4\pi}{3} + i \si... | [
{
"content": "a) Prove the equality \\(1+C_{n}^{3}+C_{n}^{6}+\\ldots=\\frac{1}{3}\\left(2^{n}+2 \\cos \\frac{n \\pi}{3}\\right)\\).\n\nb) Calculate the sums \\(C_{n}^{1}+C_{n}^{4}+C_{n}^{7}+\\ldots\\) and \\(C_{n}^{2}+C_{n}^{5}+C_{n}^{8}+\\ldots\\).",
"role": "user"
},
{
"content": "\n#### Part (a)\... |
olympiads | a) Prove that the sum of the cosines of the dihedral angles of a regular tetrahedron equals 2.
b) The sum of the plane angles of a trihedral angle is $180^{\circ}$. Find the sum of the cosines of its dihedral angles. |
Let's break the problem into the subparts (a) and (b) and approach each in a structured manner.
#### Part (a)
We are asked to prove that the sum of the cosines of dihedral angles in an equilateral tetrahedron is equal to 2.
1. Consider an equilateral tetrahedron with vertices $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf... | [
{
"content": "a) Prove that the sum of the cosines of the dihedral angles of a regular tetrahedron equals 2.\n\nb) The sum of the plane angles of a trihedral angle is $180^{\\circ}$. Find the sum of the cosines of its dihedral angles.",
"role": "user"
},
{
"content": "\nLet's break the problem into ... |
olympiads | a) Do there exist distinct natural numbers \( a, b, c, \) and \( d \) such that \(\frac{1}{a} + \frac{1}{b} = \frac{1}{c} + \frac{1}{d}\)?
b) Do there exist distinct natural numbers \( a, b, c, d, \) and \( e \) such that \(\frac{1}{a} + \frac{1}{b} = \frac{1}{c} + \frac{1}{d} + \frac{1}{e}\)? Justify your answer. |
#### Part (a)
1. Identify the problem:
We need to find distinct natural numbers \(a, b, c, d\) such that
\[
\frac{1}{a} + \frac{1}{b} = \frac{1}{c} + \frac{1}{d}
\]
2. Example 1:
Assume \(a = 12\), \(b = 3\), \(c = 6\) and \(d = 4\).
Calculate the left-hand side:
\[
\frac{1}{12} ... | [
{
"content": "a) Do there exist distinct natural numbers \\( a, b, c, \\) and \\( d \\) such that \\(\\frac{1}{a} + \\frac{1}{b} = \\frac{1}{c} + \\frac{1}{d}\\)?\n\nb) Do there exist distinct natural numbers \\( a, b, c, d, \\) and \\( e \\) such that \\(\\frac{1}{a} + \\frac{1}{b} = \\frac{1}{c} + \\frac{1}{d... |
olympiads | a) Prove that the feet of the perpendiculars dropped from a point \( P \) on the circumcircle of a triangle to its sides or their extensions lie on a single line (Simson line).
b) The feet of the perpendiculars dropped from a point \( P \) to the sides of the triangle or their extensions lie on a single line. Prove th... |
**Let \(A_{1}, B_{1}\), and \(C_{1}\) be the feet of the perpendiculars dropped from point \(P\) to the sides \(BC\), \(CA\), and \(AB\) of the triangle \(ABC\), respectively.**
#### Part (a)
1. Suppose point \(P\) lies on the arc \(AC\) of the circumcircle of triangle \(ABC\).
2. Recall that any cyclic quadrilater... | [
{
"content": "a) Prove that the feet of the perpendiculars dropped from a point \\( P \\) on the circumcircle of a triangle to its sides or their extensions lie on a single line (Simson line).\n\nb) The feet of the perpendiculars dropped from a point \\( P \\) to the sides of the triangle or their extensions li... |
olympiads | a) Diagonals \( AC \) and \( BE \) of a regular pentagon \( ABCDE \) intersect at point \( K \). Prove that the circumcircle of triangle \( CKE \) is tangent to the line \( BC \).
b) Let \( a \) be the side length of a regular pentagon, and \( d \) be the length of its diagonal. Prove that \( d^2 = a^2 + ad \). |
#### Part (a)
1. Assume that $O$ is the center of the circumscribed circle of the triangle $\triangle CKE$.
2. To show that the circumscribed circle of $\triangle CKE$ is tangent to line $BC$, we need to check the angle relationship:
\[ \angle COK = 2 \angle KCB. \]
3. First, note that in a regular pentagon, $\angle... | [
{
"content": "a) Diagonals \\( AC \\) and \\( BE \\) of a regular pentagon \\( ABCDE \\) intersect at point \\( K \\). Prove that the circumcircle of triangle \\( CKE \\) is tangent to the line \\( BC \\).\n\nb) Let \\( a \\) be the side length of a regular pentagon, and \\( d \\) be the length of its diagonal.... |
olympiads | a) Prove that the altitudes of a triangle intersect at a single point.
b) Let \( H \) be the point where the altitudes of triangle \( ABC \) intersect, and \( R \) be the radius of the circumscribed circle. Prove that \( AH^2 + BC^2 = 4R^2 \) and \( AH = BC |\cot \alpha| \). | #### Part (a): Prove that the altitudes of a triangle intersect at one point
1. **Introduction of the concept**:
Given a triangle \(ABC\), we need to prove that the altitudes of the triangle intersect at one point, which is known as the orthocenter.
2. **Constructing parallel lines**:
Through the vertices \(A, B... | [
{
"content": "a) Prove that the altitudes of a triangle intersect at a single point.\n\nb) Let \\( H \\) be the point where the altitudes of triangle \\( ABC \\) intersect, and \\( R \\) be the radius of the circumscribed circle. Prove that \\( AH^2 + BC^2 = 4R^2 \\) and \\( AH = BC |\\cot \\alpha| \\).",
"... |
olympiads | a) Prove that the surface of a cube can be cut in such a way that, when unfolded, it forms the shape shown in Figure 5a.
b) Solve the same problem for the shape shown in Figure 5b. | ### Problem:
0.15.
a) Prove that the surface of a cube can be cut in such a way that, when unfolded, we obtain the shape depicted in Figure 5.
b) Solve the same problem for the shape depicted in Figure 6.
#### Part (a):
1. **Start with the definition of the cube**:
- A cube is a three-dimensional shape with 6 fa... | [
{
"content": "a) Prove that the surface of a cube can be cut in such a way that, when unfolded, it forms the shape shown in Figure 5a.\n\nb) Solve the same problem for the shape shown in Figure 5b.",
"role": "user"
},
{
"content": "### Problem:\n0.15.\na) Prove that the surface of a cube can be cut ... |
olympiads | a) Suppose there is a set of 7 numbers. Then four of them are not less than the median, that is, the number 10. The other three numbers are not less than one. Then the sum of all the numbers in the set is at least
$$
3 + 4 \cdot 10 = 43
$$
Since the arithmetic mean is 6, the sum of the numbers in the set is equal to... | ```markdown
**(a) Proof that the subset cannot have 7 elements:**
1. First, let's assume there is a set with 7 elements. Given the conditions, the median of this set is 10.
2. Therefore, 4 of these elements must be greater than or equal to the median, which is 10.
3. The remaining 3 elements must each be 1 or more.
... | [
{
"content": "a) Suppose there is a set of 7 numbers. Then four of them are not less than the median, that is, the number 10. The other three numbers are not less than one. Then the sum of all the numbers in the set is at least \n\n$$\n3 + 4 \\cdot 10 = 43\n$$\n\nSince the arithmetic mean is 6, the sum of the n... |
olympiads | a) Show that if each side length of a regular octagon is an integer, then the octagon is centrally symmetric.
b) Is a similar statement true if the number of sides of the polygon is 6 or 12? | ### Problem:
a) Prove that if all sides of a regular octagon have lengths that are integers, then the octagon is centrally symmetric.
b) Determine if the same statement is valid for polygons with 6 or 12 sides.
#### Part (a)
1. Let \(A B C D E F G H = N\) be a regular octagon with equal sides and all interior angles ... | [
{
"content": "a) Show that if each side length of a regular octagon is an integer, then the octagon is centrally symmetric.\n\nb) Is a similar statement true if the number of sides of the polygon is 6 or 12?",
"role": "user"
},
{
"content": "### Problem:\na) Prove that if all sides of a regular octa... |
olympiads | a) Let $a>1$. Prove that if $x>y$, then $a^{x}>a^{y}$.
b) Let $a<1$. Prove that if $x>y$, then $a^{x}<a^{y}$. |
#### Part (a)
We are given $a > 1$ and need to prove that if $x > y$, then $a^x > a^y$.
1. **Choose Rational Numbers**:
Select two rational numbers $p$ and $q$ such that $x > p > q > y$. This can always be done since the rationals are dense in the reals.
2. **Define Sequences**:
Define two sequences of ration... | [
{
"content": "a) Let $a>1$. Prove that if $x>y$, then $a^{x}>a^{y}$.\nb) Let $a<1$. Prove that if $x>y$, then $a^{x}<a^{y}$.",
"role": "user"
},
{
"content": "\n\n#### Part (a)\nWe are given $a > 1$ and need to prove that if $x > y$, then $a^x > a^y$.\n\n1. **Choose Rational Numbers**:\n Select tw... |
olympiads | a) Let \( a, b, c, d \) be real numbers with \(0 \leqslant a, b, c, d \leqslant 1\). Prove that
\[ a b(a-b)+b c(b-c)+c d(c-d)+d a(d-a) \leqslant \frac{8}{27}. \]
b) Find all quadruples \((a, b, c, d)\) of real numbers with \(0 \leqslant a, b, c, d \leqslant 1\) for which equality holds in the above inequality. |
#### Part (a):
Let \( S = a b(a - b) + b c(b - c) + c d(c - d) + d a(d - a) \). We start by expanding and simplifying the expression for \( S \):
\[
\begin{aligned}
S &= a b(a - b) + b c(b - c) + c d(c - d) + d a(d - a) \\
&= a^2 b - a b^2 + b^2 c - b c^2 + c^2 d - c d^2 + d^2 a - d a^2 \\
&= a^2 b - a d^2 + b^2 c -... | [
{
"content": "a) Let \\( a, b, c, d \\) be real numbers with \\(0 \\leqslant a, b, c, d \\leqslant 1\\). Prove that\n\n\\[ a b(a-b)+b c(b-c)+c d(c-d)+d a(d-a) \\leqslant \\frac{8}{27}. \\]\n\nb) Find all quadruples \\((a, b, c, d)\\) of real numbers with \\(0 \\leqslant a, b, c, d \\leqslant 1\\) for which equa... |
olympiads | a) Suppose there are 9 numbers in the set. Then five of them do not exceed the median, which is 2. The other four numbers are not greater than 13. Therefore, the sum of all the numbers in the set is at most:
$$
5 \cdot 2 + 4 \cdot 13 = 62
$$
Since the arithmetic mean is 7, the sum of the numbers in the set is $7 \cdot ... |
#### Part a:
1. Let's assume the set contains 9 numbers.
2. We know that out of these 9 numbers, five of them do not exceed the median, which is the number 2.
3. The remaining four numbers do not exceed the number 13.
4. Therefore, the sum of all the numbers in the set is at most:
\[
5 \cdot 2 + 4 \cdot 13 = 10... | [
{
"content": "a) Suppose there are 9 numbers in the set. Then five of them do not exceed the median, which is 2. The other four numbers are not greater than 13. Therefore, the sum of all the numbers in the set is at most:\n$$\n5 \\cdot 2 + 4 \\cdot 13 = 62\n$$\nSince the arithmetic mean is 7, the sum of the num... |
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