source stringclasses 1
value | problem stringlengths 12 3.55k | solution stringlengths 671 7.11k | messages listlengths 2 2 |
|---|---|---|---|
olympiads | Let \( C \) and \( D \) be two distinct points on a semicircle of diameter \([A B]\). The lines \((A C)\) and \((B D)\) intersect at \( F \), and the lines \((A D)\) and \((B C)\) intersect at \( E \). Show that the midpoints of segments \([A B]\), \([C D]\), and \([E F]\) are collinear. | 1. **Observation**: Consider the semicircle with diameter $[AB]$. Points $C$ and $D$ are distinct points on the semicircle.
2. **Intersection points**:
- Let line $(AC)$ and line $(BD)$ intersect at $F$.
- Let line $(AD)$ and line $(BC)$ intersect at $E$.
3. **Key angles**:
- Since $C$ and $D$ are on the se... | [
{
"content": "Let \\( C \\) and \\( D \\) be two distinct points on a semicircle of diameter \\([A B]\\). The lines \\((A C)\\) and \\((B D)\\) intersect at \\( F \\), and the lines \\((A D)\\) and \\((B C)\\) intersect at \\( E \\). Show that the midpoints of segments \\([A B]\\), \\([C D]\\), and \\([E F]\\) ... |
olympiads | Let \( S_{n} \) be the number of sequences \((a_{1}, a_{2}, \ldots, a_{n})\), where \( a_{i} \in \{0,1\} \), in which no six consecutive blocks are equal. Prove that \( S_{m} \rightarrow \infty \) as \( n \rightarrow \infty \). |
Let \( S_n \) denote the number of sequences \( (a_1, a_2, \ldots, a_n) \) where \( a_i \in \{0, 1\} \) and no six consecutive blocks are equal. To prove that \( S_m \to \infty \) as \( n \to \infty \), we will proceed with the following steps.
1. **Introduction of Sequence Set \( B_n \) and Induction Basis**:
Def... | [
{
"content": "Let \\( S_{n} \\) be the number of sequences \\((a_{1}, a_{2}, \\ldots, a_{n})\\), where \\( a_{i} \\in \\{0,1\\} \\), in which no six consecutive blocks are equal. Prove that \\( S_{m} \\rightarrow \\infty \\) as \\( n \\rightarrow \\infty \\).",
"role": "user"
},
{
"content": "\nLet ... |
olympiads | In $\triangle ABC$, point $D$ lies on segment $BC$ and does not coincide with $B$ or $C$. The circumcircle of $\triangle ABD$ intersects segment $AC$ at another point $E$, and the circumcircle of $\triangle ACD$ intersects segment $AB$ at another point $F$. Let $A'$ be the reflection of point $A$ across line $BC$. Line... |
1. **Let \( \sigma \) represent the reflection (or inversion) about the line \( BC \):**
- Point \( A' \) is the image of \( A \) under \( \sigma \).
2. **Given that points \( A, E, D, F \) are concyclic (\( A, E, D, F \) lie on a common circle):**
\[
\angle BDF = \angle BAC = \angle CDE
\]
- This fo... | [
{
"content": "In $\\triangle ABC$, point $D$ lies on segment $BC$ and does not coincide with $B$ or $C$. The circumcircle of $\\triangle ABD$ intersects segment $AC$ at another point $E$, and the circumcircle of $\\triangle ACD$ intersects segment $AB$ at another point $F$. Let $A'$ be the reflection of point $... |
olympiads | There are 50 boys and 50 girls lined up in some order. In this row, there exists exactly one group of 30 consecutive children where the number of boys and girls is equal. Prove that there exists a group of 70 consecutive children where the number of boys and girls is also equal. |
1. **Transform the problem into a circular arrangement:**
Consider the 100 children (50 boys and 50 girls) arranged in a circle. This transformation will allow us to think of any subsequence of consecutive children in a simpler, more mathematically tractable way.
2. **Identify all possible groups of 30 consecutive... | [
{
"content": "There are 50 boys and 50 girls lined up in some order. In this row, there exists exactly one group of 30 consecutive children where the number of boys and girls is equal. Prove that there exists a group of 70 consecutive children where the number of boys and girls is also equal.",
"role": "use... |
olympiads | Let \( f(m, n) \) satisfy the condition
$$
f(m, n) \leq f(m, n-1) + f(m-1, n),
$$
where \( m \) and \( n \) are positive integers, \( m, n \geq 2 \),
and
$$
f(1, n) = f(m, 1) = 1.
$$
Prove that \( f(m, n) \leq C_{m+n} \). | We are given the following conditions for the function \( f(m, n) \):
\[ f(m, n) \leqslant f(m, n-1) + f(m-1, n), \]
where \( m \) and \( n \) are positive integers with \( m, n \geq 2 \).
Additionally, we have the initial conditions:
\[ f(1, n) = f(m, 1) = 1 \text{ for all positive integers } n \text{ and } m. \]
W... | [
{
"content": "Let \\( f(m, n) \\) satisfy the condition\n$$\nf(m, n) \\leq f(m, n-1) + f(m-1, n),\n$$\nwhere \\( m \\) and \\( n \\) are positive integers, \\( m, n \\geq 2 \\), \nand \n$$\nf(1, n) = f(m, 1) = 1.\n$$\nProve that \\( f(m, n) \\leq C_{m+n} \\).",
"role": "user"
},
{
"content": "We are... |
olympiads | Prove that each odd square ends in the digit 1 in the octal system (base 8); moreover, if this digit is removed, the remaining part will represent a certain triangular number. |
1. **Represent the odd square:**
Consider an odd integer, which can be expressed in the form \(2n + 1\) for some integer \(n\).
The square of this odd integer is:
\[
(2n + 1)^2
\]
2. **Expand the square:**
Expanding the square, we get:
\[
(2n + 1)^2 = 4n^2 + 4n + 1
\]
3. **Re... | [
{
"content": "Prove that each odd square ends in the digit 1 in the octal system (base 8); moreover, if this digit is removed, the remaining part will represent a certain triangular number.",
"role": "user"
},
{
"content": "\n1. **Represent the odd square:**\n Consider an odd integer, which can b... |
olympiads | $a$ and $b$ are distinct elements of the set $\{0,1,2, \cdots, p-1\}$. Then $f(a)$ and $f(b)$ are not congruent modulo $p$. |
Given that $a$ and $b$ are distinct elements in the set $\{0, 1, 2, \ldots, p-1\}$, we aim to show that $f(a)$ and $f(b)$ are distinct modulo $p$.
1. **Case 1: $a \equiv 1 \pmod{p}$**
- When $n \equiv 1 \pmod{p}$, we have
\[
f(n) \equiv 1 + 2 + \cdots + (p-1) \pmod{p}
\]
- The sum of the first $... | [
{
"content": "$a$ and $b$ are distinct elements of the set $\\{0,1,2, \\cdots, p-1\\}$. Then $f(a)$ and $f(b)$ are not congruent modulo $p$.",
"role": "user"
},
{
"content": "\nGiven that $a$ and $b$ are distinct elements in the set $\\{0, 1, 2, \\ldots, p-1\\}$, we aim to show that $f(a)$ and $f(b)... |
olympiads | Let \(\xi_{1}, \ldots, \xi_{m}\) and \(\eta_{1}, \ldots, \eta_{n}\) be random variables. Form the random vectors \(X=\left(\xi_{1}, \ldots, \xi_{m}\right)\) and \(Y=\left(\eta_{1}, \ldots, \eta_{n}\right)\). Suppose the following conditions are satisfied:
1) The random variables \(\xi_{1}, \ldots, \xi_{m}\) are indepen... |
1. **Defining the Problem:**
We are given two sets of random variables:
- $\xi_{1}, \ldots, \xi_{m}$ (random variables)
- $\eta_{1}, \ldots, \eta_{n}$ (random variables)
From these sets, we can form vectors:
- $X=(\xi_{1}, \ldots, \xi_{m}) \in \mathbb{R}^{m}$
- $Y=(\eta_{1}, \ldots, \eta_{n}) ... | [
{
"content": "Let \\(\\xi_{1}, \\ldots, \\xi_{m}\\) and \\(\\eta_{1}, \\ldots, \\eta_{n}\\) be random variables. Form the random vectors \\(X=\\left(\\xi_{1}, \\ldots, \\xi_{m}\\right)\\) and \\(Y=\\left(\\eta_{1}, \\ldots, \\eta_{n}\\right)\\). Suppose the following conditions are satisfied:\n1) The random var... |
olympiads | As shown in the figure, \( AC = BC \). The tangents to the circumcircle of \(\triangle ABC\) at points \(A\) and \(C\) intersect at point \(D\). Line \(BD\) intersects the circle again at point \(E\). Ray \(AE\) intersects \(CD\) at point \(F\). Prove that \(F\) is the midpoint of \(CD\). |
1. We begin by noting from the problem that $AC = BC$ and $ \triangle ABC$ has an external circle, and line $AD$ intersects at point $D$ and line $BD$ intersects the circle at point $E$. The ray $AE$ intersects $CD$ at point $F$.
2. Using the information that $AC = BC$, we know $\triangle ABC$ is isosceles at $C$. Th... | [
{
"content": "As shown in the figure, \\( AC = BC \\). The tangents to the circumcircle of \\(\\triangle ABC\\) at points \\(A\\) and \\(C\\) intersect at point \\(D\\). Line \\(BD\\) intersects the circle again at point \\(E\\). Ray \\(AE\\) intersects \\(CD\\) at point \\(F\\). Prove that \\(F\\) is the midpo... |
olympiads | The coefficients of the quadratic equation \( a x^{2} + b x + c = 0 \) satisfy the condition \( 2a + 3b + 6c = 0 \). Prove that this equation has a root in the interval \( (0, 1) \). | To prove that the quadratic equation \( a x^{2} + b x + c = 0 \) has a root in the interval \((0, 1)\) given that the coefficients satisfy \(2a + 3b + 6c = 0\), we can proceed as follows:
1. **Case when \( c = 0 \):**
- If \( c = 0 \), the quadratic equation reduces to:
\[
ax^2 + bx = 0
\]
... | [
{
"content": "The coefficients of the quadratic equation \\( a x^{2} + b x + c = 0 \\) satisfy the condition \\( 2a + 3b + 6c = 0 \\). Prove that this equation has a root in the interval \\( (0, 1) \\).",
"role": "user"
},
{
"content": "To prove that the quadratic equation \\( a x^{2} + b x + c = 0 ... |
olympiads | Find the value of the following expression:
\(\cos \left(\frac{3}{2} \pi + 4 \alpha \right) + \sin (3 \pi - 8 \alpha) - \sin (4 \pi - 12 \alpha) = 4 \cos 2 \alpha \cos 4 \alpha \sin 6 \alpha\). |
Consider the given expression:
\[ \cos \left(\frac{3}{2}\pi + 4\alpha\right) + \sin (3\pi - 8\alpha) - \sin (4\pi - 12\alpha) \]
1. **Simplify each trigonometric term:**
- Using the trigonometric identity for cosine involving the addition and subtraction of angles:
\[
\cos \left(\frac{3}{2}\pi + 4\alpha\... | [
{
"content": "Find the value of the following expression: \n\\(\\cos \\left(\\frac{3}{2} \\pi + 4 \\alpha \\right) + \\sin (3 \\pi - 8 \\alpha) - \\sin (4 \\pi - 12 \\alpha) = 4 \\cos 2 \\alpha \\cos 4 \\alpha \\sin 6 \\alpha\\).",
"role": "user"
},
{
"content": "\nConsider the given expression:\n\... |
olympiads | Inside a square with side length 2, there are seven polygons each with an area of at least 1.
Prove that there exist two polygons such that the area of their intersection is at least \( \frac{1}{7} \). |
1. **Restate the problem and notation**:
We are given a square with side length of 2 units, thus the area of the square is $4$ square units. Inside this square, there are seven polygons, each with an area of at least $1$ square unit. We need to prove that there exist two polygons with an intersection area of at lea... | [
{
"content": "Inside a square with side length 2, there are seven polygons each with an area of at least 1.\n\nProve that there exist two polygons such that the area of their intersection is at least \\( \\frac{1}{7} \\).",
"role": "user"
},
{
"content": "\n1. **Restate the problem and notation**:\n... |
olympiads | Show that a number is divisible by 3 if and only if the sum of its digits in its base 10 decomposition is divisible by 3. Then, show the same result for 9. |
1. Firstly, we can express any non-negative integer \( n \) in terms of its decimal (base-10) digits. Let these digits be \( a_0, a_1, \ldots, a_k \). Specifically, we write:
\[ n = a_0 \cdot 10^0 + a_1 \cdot 10^1 + a_2 \cdot 10^2 + \ldots + a_k \cdot 10^k \]
2. Observe that the value of each power of 10 modulo 3 and... | [
{
"content": "Show that a number is divisible by 3 if and only if the sum of its digits in its base 10 decomposition is divisible by 3. Then, show the same result for 9.",
"role": "user"
},
{
"content": "\n1. Firstly, we can express any non-negative integer \\( n \\) in terms of its decimal (base-10... |
olympiads | In the diagram, a newly built city's roads form a neat grid. There are $n+1$ east-west roads and $m+1$ north-south roads. A person lives at point $A$ in the southwest corner and rides a bicycle to work at point $B$ in the northeast corner every day. To sightsee along the way, this person often changes their route, with... |
To solve the problem, we need to show that the number of different routes from $A$ to $B$, denoted by $f(m, n)$, satisfies the inequality \( f(m, n) \leqslant 2^{mn} \).
1. **Identify Grid Divisions**:
The road system comprises $n+1$ east-west streets and $m+1$ north-south streets, which divide the city into \(... | [
{
"content": "In the diagram, a newly built city's roads form a neat grid. There are $n+1$ east-west roads and $m+1$ north-south roads. A person lives at point $A$ in the southwest corner and rides a bicycle to work at point $B$ in the northeast corner every day. To sightsee along the way, this person often cha... |
olympiads | As a result of the first successful strike, the number of heads increases by one (event $N$, probability $\frac{3}{8}$). In this case, event $D$ will occur only if the number of heads decreases by 2 - first by one, and then by another one. Therefore, the conditional probability of event $D$ in this case is $x^{2}$.
Us... |
First, we recall the given information:
- Event \( N \) (the number of heads increases by one) occurs with a probability \( \frac{3}{8} \).
- Event \( D \) (the number of heads decreases by one first, and then by one more) happens conditionally based on \( N \). Its conditional probability is given as \( \mathrm{P}(D ... | [
{
"content": "As a result of the first successful strike, the number of heads increases by one (event $N$, probability $\\frac{3}{8}$). In this case, event $D$ will occur only if the number of heads decreases by 2 - first by one, and then by another one. Therefore, the conditional probability of event $D$ in th... |
olympiads | Let $G$ be a graph. Show that it contains an independent set of vertices with a cardinality greater than or equal to
$$
\sum_{v \in V} \frac{1}{1+d(v)}
$$ |
1. **Notation and Initial Claim**: Define \( f(G) = \sum_{v \in G} \frac{1}{1 + d(v)} \), where \( d(v) \) is the degree of vertex \( v \).
2. **Base Case**: When the graph \( G \) has \( n = 1 \) vertices, it is clear that the graph itself is an independent set. Hence, the claim holds true.
3. **Inductive Step**... | [
{
"content": "Let $G$ be a graph. Show that it contains an independent set of vertices with a cardinality greater than or equal to\n\n$$\n\\sum_{v \\in V} \\frac{1}{1+d(v)}\n$$",
"role": "user"
},
{
"content": "\n1. **Notation and Initial Claim**: Define \\( f(G) = \\sum_{v \\in G} \\frac{1}{1 + d(v... |
olympiads | As shown in the figure, \(AD\) is the angle bisector of \(\triangle ABC\) with \(AB < AC\). On \(AC\), point \(E\) is chosen such that \(CE = AB\). Let \(M\) and \(N\) be the midpoints of \(BC\) and \(AE\), respectively. Prove that \(MN \parallel AD\). |
Given: In $\triangle ABC$, $AD$ is the angle bisector of $\angle BAC$ with $AB < AC$. On $AC$, point $E$ is taken such that $CE = AB$. Points $M$ and $N$ are the midpoints of $BC$ and $AE$ respectively. We are to prove that $MN \parallel AD$.
1. **Join point $B$ to point $E$ and find midpoints**:
- Let $F$ be the ... | [
{
"content": "As shown in the figure, \\(AD\\) is the angle bisector of \\(\\triangle ABC\\) with \\(AB < AC\\). On \\(AC\\), point \\(E\\) is chosen such that \\(CE = AB\\). Let \\(M\\) and \\(N\\) be the midpoints of \\(BC\\) and \\(AE\\), respectively. Prove that \\(MN \\parallel AD\\).",
"role": "user"
... |
olympiads | Given \( a, b, c > 0 \), show that
\[
\frac{3}{a^{3} + b^{3} + c^{3}} \leq \frac{1}{a^{3} + b^{3} + abc} + \frac{1}{b^{3} + c^{3} + abc} + \frac{1}{c^{3} + a^{3} + abc} \leq \frac{1}{abc}
\] | **Step 1: Prove the right inequality**
We'll start by showing that:
$$ \frac{1}{a^{3}+b^{3}+abc} \leqslant \frac{c}{abc(a + b + c)} $$
From the Reversal Inequality (Reorganization Inequality), we know that:
$$ a^{3} + b^{3} \geq a^{2}b + b^{2}a $$
Thus:
$$ a^{3} + b^{3} + abc \geq a^{2}b + b^{2}a + abc $$
Taking re... | [
{
"content": "Given \\( a, b, c > 0 \\), show that\n\n\\[ \n\\frac{3}{a^{3} + b^{3} + c^{3}} \\leq \\frac{1}{a^{3} + b^{3} + abc} + \\frac{1}{b^{3} + c^{3} + abc} + \\frac{1}{c^{3} + a^{3} + abc} \\leq \\frac{1}{abc} \n\\]",
"role": "user"
},
{
"content": "**Step 1: Prove the right inequality**\n\nW... |
olympiads | Prove that if the square of a $k$-digit number starts with the same $k$ digits, then this number is a power of 10. |
**Solution I**
1. Define $N$ as a $k$-digit number whose square has the same first $k$ digits as $N$ itself.
2. Consider the division $N^2 / N$. Since $N^2$ starts with the digits of $N$, the operation results in a quotient of $N$ and a remainder of $0$. Thus:
\[
N^2 / N = N
\]
3. Divide $N^2$ by $N$, taking... | [
{
"content": "Prove that if the square of a $k$-digit number starts with the same $k$ digits, then this number is a power of 10.",
"role": "user"
},
{
"content": "\n**Solution I**\n\n1. Define $N$ as a $k$-digit number whose square has the same first $k$ digits as $N$ itself.\n2. Consider the divisi... |
olympiads | Given the number $800 \ldots 008$ (with 80 zeros), replace any two zeros with non-zero digits such that the resulting number is divisible by 198. How many ways can this be done? | Given the number \(800 \ldots 008\) (80 zeros), we must replace some two zeros with non-zero digits so that the resulting number is divisible by 198. Let's start by breaking down the factors of 198.
1. **Factorize 198:**
\[
198 = 2 \times 9 \times 11
\]
This means our number needs to be divisible by 2, 9... | [
{
"content": "Given the number $800 \\ldots 008$ (with 80 zeros), replace any two zeros with non-zero digits such that the resulting number is divisible by 198. How many ways can this be done?",
"role": "user"
},
{
"content": "Given the number \\(800 \\ldots 008\\) (80 zeros), we must replace some t... |
olympiads | Given an acute-angled triangle \(ABC\) and a point \(P\) that does not coincide with the orthocenter of the triangle, prove that the circles passing through the midpoints of the sides of triangles \(PAB\), \(PAC\), \(PBC\), and \(ABC\), as well as the circle passing through the projections of point \(P\) onto the sides... |
1. **Problem description**: We are given an acute triangle $ABC$ and a point $P$ that does not coincide with the orthocenter of triangle $ABC$. We need to prove that the circumcircles passing through the midpoints of the sides of triangles $PAB$, $PAC$, $PBC$, and $ABC$, as well as the circumcircle passing through the... | [
{
"content": "Given an acute-angled triangle \\(ABC\\) and a point \\(P\\) that does not coincide with the orthocenter of the triangle, prove that the circles passing through the midpoints of the sides of triangles \\(PAB\\), \\(PAC\\), \\(PBC\\), and \\(ABC\\), as well as the circle passing through the project... |
olympiads | Prove: If \( x^{2}+y^{2} \leqslant \frac{\pi}{2} \), then
$$
\cos x + \cos y \leqslant 1 + \cos(xy).
$$ |
To prove: If \( x^2 + y^2 \leqslant \frac{\pi}{2} \), then
$$
\cos x + \cos y \leqslant 1 + \cos(xy).
$$
1. **Bounding \( xy \)**:
Given \( x^2 + y^2 \leqslant \frac{\pi}{2} \):
\[
\frac{x^2 + y^2}{2} \leqslant \frac{\pi}{4}.
\]
Since the arithmetic mean is greater than or equal to the geometric mean:
\[
xy \leqslant... | [
{
"content": "Prove: If \\( x^{2}+y^{2} \\leqslant \\frac{\\pi}{2} \\), then\n$$\n\\cos x + \\cos y \\leqslant 1 + \\cos(xy).\n$$",
"role": "user"
},
{
"content": "\nTo prove: If \\( x^2 + y^2 \\leqslant \\frac{\\pi}{2} \\), then\n$$\n\\cos x + \\cos y \\leqslant 1 + \\cos(xy).\n$$\n\n1. **Bounding ... |
olympiads | Find all positive integers \( n \) for which it is possible to color some cells of an infinite grid of unit squares red, such that each rectangle consisting of exactly \( n \) cells (and whose edges lie along the lines of the grid) contains an odd number of red cells. | ```markdown
1. **Statement of Claim**:
We claim that it is possible for all positive integers $n$ to color some cells of an infinite grid of unit squares red, such that each rectangle consisting of exactly $n$ cells (and whose edges lie along the lines of the grid) contains an odd number of red cells. We will call ... | [
{
"content": "Find all positive integers \\( n \\) for which it is possible to color some cells of an infinite grid of unit squares red, such that each rectangle consisting of exactly \\( n \\) cells (and whose edges lie along the lines of the grid) contains an odd number of red cells.",
"role": "user"
},... |
olympiads | Prove that a circle and a tangent line to it, upon inversion, transform into either two tangent circles, a circle and a tangent line, or a pair of parallel lines.
Definition: Let two circles intersect at point \( A \). The angle between the circles is defined as the angle between the tangents to the circles at point \... |
The problem requires proving that circles and lines tangent to each other remain tangent under inversion, or they transform into a pair of tangent circles or a couple of parallel lines. Here’s a more detailed step-by-step solution:
1. **Understanding Inversion**:
- Inversion in geometry refers to transforming poin... | [
{
"content": "Prove that a circle and a tangent line to it, upon inversion, transform into either two tangent circles, a circle and a tangent line, or a pair of parallel lines.\n\nDefinition: Let two circles intersect at point \\( A \\). The angle between the circles is defined as the angle between the tangents... |
olympiads | Let $t$ be a fixed integer. Show that for every odd prime number $p$, there exists a positive integer $n$ such that
$$
(3-7 t) 2^{n}+(18 t-9) 3^{n}+(6-10 t) 4^{n}
$$
is divisible by $p$. |
1. **Definition and Setup:**
Let’s define the expression \( S(n) \) as:
\[
S(n) = (3-7t) \cdot 2^n + (18t-9) \cdot 3^n + (6-10t) \cdot 4^n
\]
2. **Fermat's Little Theorem Application:**
We will utilize Fermat's Little Theorem, which states that for any integer \( a \) such that gcd(\(a, p\)) = 1 (whe... | [
{
"content": "Let $t$ be a fixed integer. Show that for every odd prime number $p$, there exists a positive integer $n$ such that\n\n$$\n(3-7 t) 2^{n}+(18 t-9) 3^{n}+(6-10 t) 4^{n}\n$$\n\nis divisible by $p$.",
"role": "user"
},
{
"content": "\n1. **Definition and Setup:**\n\n Let’s define the exp... |
olympiads | In triangle \(ABC\), squares \(CADE\) and \(CBFG\) are constructed outwardly on sides \(CA\) and \(CB\) with centers at \(O_1\) and \(O_2\). Show that the midpoints of segments \(AB\) and \(EG\), along with the points \(O_1\) and \(O_2\), form the vertices of a square. |
1. **Understanding the Problem**:
We are given a triangle \(ABC\) with squares constructed outwardly on its sides \(CA\) and \(CB\). Specifically:
- The square \(CADE\) is constructed on \(CA\) with center \(O_1\).
- The square \(CBFG\) is constructed on \(CB\) with center \(O_2\).
We need to sho... | [
{
"content": "In triangle \\(ABC\\), squares \\(CADE\\) and \\(CBFG\\) are constructed outwardly on sides \\(CA\\) and \\(CB\\) with centers at \\(O_1\\) and \\(O_2\\). Show that the midpoints of segments \\(AB\\) and \\(EG\\), along with the points \\(O_1\\) and \\(O_2\\), form the vertices of a square.",
... |
olympiads | Let \( M = \{1, 2, \cdots, 20\} \). For any 9-element subset \( S \) of \( M \), the function \( f(S) \) takes an integer value from 1 to 20. Prove that no matter how the function \( f \) is defined, there exists a 10-element subset \( T \) of \( M \) such that for all \( k \in T \), \( f(T - \{k\}) \neq k \) (where \(... | ### Solution:
Let us consider a set \(M\) defined as \(M = \{1, 2, \cdots, 20\}\). We will analyze functions \(f(S)\) which map any 9-element subset \(S\) of \(M\) to an integer value in the range from 1 to 20. We aim to demonstrate the existence of a 10-element subset \(T\) of \(M\) such that for any \(k \in T\), \(... | [
{
"content": "Let \\( M = \\{1, 2, \\cdots, 20\\} \\). For any 9-element subset \\( S \\) of \\( M \\), the function \\( f(S) \\) takes an integer value from 1 to 20. Prove that no matter how the function \\( f \\) is defined, there exists a 10-element subset \\( T \\) of \\( M \\) such that for all \\( k \\in ... |
olympiads | Prove that a polynomial \( P(z) \) is an even function of \( z \in \mathbb{C} \) if and only if there exists a polynomial \( Q(z) \) such that \( P(z) \equiv Q(z) Q(-z) \) for \( z \in \mathbb{C} \). |
1. **Sufficiency:**
Let's start by proving the sufficiency. If the polynomial \( P(z) \) can be expressed as \( P(z) \equiv Q(z) \cdot Q(-z) \), then:
\[
P(-z) \equiv Q(-z) \cdot Q(z) \equiv Q(z) \cdot Q(-z) \equiv P(z)
\]
This shows that \( P(z) \) is an even function since \( P(-z) \equiv P(z) \).
... | [
{
"content": "Prove that a polynomial \\( P(z) \\) is an even function of \\( z \\in \\mathbb{C} \\) if and only if there exists a polynomial \\( Q(z) \\) such that \\( P(z) \\equiv Q(z) Q(-z) \\) for \\( z \\in \\mathbb{C} \\).",
"role": "user"
},
{
"content": "\n1. **Sufficiency:**\n\n Let's sta... |
olympiads | Given that \(a\) and \(b\) are odd integers greater than 1 such that \(a + b = 2^\alpha\) with \(\alpha \geq 1\), show that there are no integers \(k > 1\) such that \(k^2\) divides \(a^k + b^k\). |
1. **Assume an integer \(k > 1\) exists such that \(k^2\) divides \(a^k + b^k\):**
Let us assume, for the sake of contradiction, that there exists an integer \(k > 1\) such that \(k^2\) divides \(a^k + b^k\).
2. **Determine \(k\) modulo 4:**
Since \(a\) and \(b\) are both odd (given \(a, b > 1\) and both are ... | [
{
"content": "Given that \\(a\\) and \\(b\\) are odd integers greater than 1 such that \\(a + b = 2^\\alpha\\) with \\(\\alpha \\geq 1\\), show that there are no integers \\(k > 1\\) such that \\(k^2\\) divides \\(a^k + b^k\\).",
"role": "user"
},
{
"content": "\n1. **Assume an integer \\(k > 1\\) e... |
olympiads | Show that the fraction $\frac{39n + 4}{26n + 3}$ is always in its simplest form (irreducible). |
1. **Supposition Initiale**: Supposons qu'il existe un entier $k \geq 2$ qui divise à la fois le numérateur $39n + 4$ et le dénominateur $26n + 3$ de la fraction $\frac{39n + 4}{26n + 3}$.
2. **Expression Linéaire pour les Diviseurs**:
Si un entier $k$ divise $39n + 4$ et $26n + 3$, alors on peut écrire les deu... | [
{
"content": "Show that the fraction $\\frac{39n + 4}{26n + 3}$ is always in its simplest form (irreducible).",
"role": "user"
},
{
"content": "\n1. **Supposition Initiale**: Supposons qu'il existe un entier $k \\geq 2$ qui divise à la fois le numérateur $39n + 4$ et le dénominateur $26n + 3$ de la ... |
olympiads | In any arithmetic progression \( a, a+d, a+2d, \ldots, a+nd, \ldots \) consisting of natural numbers, there are infinitely many terms whose prime factorization includes exactly the same prime numbers. Prove this. |
Given an arithmetic progression \( a, a+d, a+2d, \ldots, a+nd, \ldots \) composed of natural numbers, we need to prove that there are infinitely many terms in this progression, each having the exact same prime factors in its decomposition.
1. **Consider the GCD of \( a \) and \( d \)**:
- Observe that if the firs... | [
{
"content": "In any arithmetic progression \\( a, a+d, a+2d, \\ldots, a+nd, \\ldots \\) consisting of natural numbers, there are infinitely many terms whose prime factorization includes exactly the same prime numbers. Prove this.",
"role": "user"
},
{
"content": "\nGiven an arithmetic progression \... |
olympiads | Let $A_{1}, B_{1}$ and $C_{1}$ be the projections of the Lemoine point $K$ onto the sides of triangle $ABC$. Prove that $K$ is the intersection point of the medians of triangle $A_{1} B_{1} C_{1}$. | ```markdown
1. Let $M$ be the point of intersection of the medians of triangle $ABC$ and let the distances from points $K$ and $M$ to the sides of the triangle be $a_{1}, b_{1}, c_{1}$ and $a_{2}, b_{2}, c_{2}$ respectively.
2. Since the points $K$ and $M$ are isogonal conjugates, by definition, the following relatio... | [
{
"content": "Let $A_{1}, B_{1}$ and $C_{1}$ be the projections of the Lemoine point $K$ onto the sides of triangle $ABC$. Prove that $K$ is the intersection point of the medians of triangle $A_{1} B_{1} C_{1}$.",
"role": "user"
},
{
"content": "```markdown\n\n1. Let $M$ be the point of intersection... |
olympiads | A maze is a 10x10 grid square where some pairs of adjacent nodes are connected by a "wall" segment, such that it is possible to visit all grid cells by moving from one cell to an adjacent cell through the sides and without passing through walls. The boundary of the square is also considered to be surrounded by walls. I... |
To ensure that the robot can visit all cells of the grid regardless of the labyrinth configuration and its starting position, we can follow these steps:
1. **Enumerate Initial States:**
Each initial state of the robot is defined by the specific labyrinth and the initial position within that labyrinth. Since the gr... | [
{
"content": "A maze is a 10x10 grid square where some pairs of adjacent nodes are connected by a \"wall\" segment, such that it is possible to visit all grid cells by moving from one cell to an adjacent cell through the sides and without passing through walls. The boundary of the square is also considered to b... |
olympiads | In a plane, a square $ABCD$, a line $l$, and a point $M$ are given. Using only a ruler:
a) Draw a line through the point $M$ parallel to $l$;
b) Drop a perpendicular from the point $M$ to the line $l$. | ### Problem:
60. Given a square \(ABCD\), a line \(l\), and a point \(M\) in the plane, using only a ruler:
a) Construct a line through point \(M\) parallel to \(l\);
b) Drop a perpendicular from point \(M\) to line \(l\).
#### Part a)
1. Let \(O\) be the center of the square \(ABCD\).
2. Let points \(K\) and \(L\... | [
{
"content": "In a plane, a square $ABCD$, a line $l$, and a point $M$ are given. Using only a ruler:\n\na) Draw a line through the point $M$ parallel to $l$;\n\nb) Drop a perpendicular from the point $M$ to the line $l$.",
"role": "user"
},
{
"content": "### Problem:\n60. Given a square \\(ABCD\\),... |
olympiads | Let $A, B, C,$ and $D$ be four concyclic points. Let $A'$ and $C'$ be the orthogonal projections of $A$ and $C$ onto $(BD)$, respectively, and let $B'$ and $D'$ be the orthogonal projections of $B$ and $D$ onto $(AC)$. Show that $A', B', C'$ and $D'$ are also concyclic. |
1. Consider four points $A$, $B$, $C$, and $D$ that are cocyclic, meaning they lie on the same circle.
2. Define $A'$ and $C'$ as the orthogonal projections of $A$ and $C$ onto the line segment $BD$, and $B'$ and $D'$ as the orthogonal projections of $B$ and $D$ onto the line segment $AC$.
3. First, observe that the... | [
{
"content": "Let $A, B, C,$ and $D$ be four concyclic points. Let $A'$ and $C'$ be the orthogonal projections of $A$ and $C$ onto $(BD)$, respectively, and let $B'$ and $D'$ be the orthogonal projections of $B$ and $D$ onto $(AC)$. Show that $A', B', C'$ and $D'$ are also concyclic.",
"role": "user"
},
... |
olympiads | Show that among the participants of the Animath training this week, there are two who know the same number of other participants (assuming the "knowing each other" relationship is symmetric). |
1. **Définissons les termes et notation:**
- Soit \( n \) le nombre de stagiaires.
- Chaque stagiaire connaît un certain nombre d'autres stagiaires. La relation "se connaître" est symétrique, ce qui signifie que si le stagiaire \( A \) connaît le stagiaire \( B \), alors le stagiaire \( B \) connaît aussi le s... | [
{
"content": "Show that among the participants of the Animath training this week, there are two who know the same number of other participants (assuming the \"knowing each other\" relationship is symmetric).",
"role": "user"
},
{
"content": "\n1. **Définissons les termes et notation:**\n - Soit \... |
olympiads |
There are three empty jars on the table. Winnie-the-Pooh, Rabbit, and Piglet take turns placing one nut into one of the jars. Their turn order is determined by drawing lots before the game begins. Winnie can only add a nut to the first or the second jar, Rabbit can only add to the second or third jar, and Piglet can o... |
1. **Understanding the problem and initial positioning:**
- Begin by noting that there are three participants: Винни (W), Кролик (K), and Пятачок (P).
- They sequentially place a nut into one of the three jars as per defined restrictions:
- W can place a nut only in jar 1 or jar 2.
- K can place a nut... | [
{
"content": "\nThere are three empty jars on the table. Winnie-the-Pooh, Rabbit, and Piglet take turns placing one nut into one of the jars. Their turn order is determined by drawing lots before the game begins. Winnie can only add a nut to the first or the second jar, Rabbit can only add to the second or thir... |
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