source stringclasses 1
value | problem stringlengths 11 3.55k | solution stringlengths 190 7.07k | messages listlengths 2 2 | is_sufficient_information bool 2
classes | explanation stringlengths 39 835 |
|---|---|---|---|---|---|
olympiads | Determine with reasoning whether the number \(\lambda_{n}=\sqrt{3 n^{2}+2 n+2}\) is irrational for all non-negative integers \(n\). |
1. **Supposition: $n$ is even**
- Let \( n = 2k \), where \( k \) is an integer.
- Write \( n \) in the even form:
\[
3n^2 + 2n + 2 = 3(2k)^2 + 2(2k) + 2 = 12k^2 + 4k + 2
\]
- Observe the divisibility properties:
- \( 12k^2 + 4k \) is a multiple of 4.
- Adding 2 to it gives:
... | [
{
"content": "Determine with reasoning whether the number \\(\\lambda_{n}=\\sqrt{3 n^{2}+2 n+2}\\) is irrational for all non-negative integers \\(n\\).",
"role": "user"
},
{
"content": "\n1. **Supposition: $n$ is even**\n - Let \\( n = 2k \\), where \\( k \\) is an integer.\n - Write \\( n \\)... | true | The question provides a mathematical expression for \\lambda_{n} and asks whether it is irrational for all non-negative integers n. An expert can analyze the expression for various values of n, consider the nature of the square root, and apply properties of rationality to determine whether \\lambda_{n} is irrational fo... |
olympiads | Suppose that \( m \) and \( n \) are integers such that \( m, n \geq 2 \). For any set of \( n \) integers \( A = \{a_1, a_2, \dots, a_n\} \), form a sequence by taking the difference \( a_i - a_j \) for each pair \( a_i, a_j \). This sequence, composed of \( \binom{n}{2} \) differences, is called the "derived sequence... |
1. **Notation and Given Data:**
Let \( m, n \geq 2 \) be integers.
We are given two sets:
- A set \( A = \{a_1, a_2, \ldots, a_n\} \) consisting of \( n \) integers.
- A set \( B = \{1, 2, \ldots, n\} \) also consisting of \( n \) integers.
2. **Definition of Generated Sequence:**
For any set \( C... | [
{
"content": "Suppose that \\( m \\) and \\( n \\) are integers such that \\( m, n \\geq 2 \\). For any set of \\( n \\) integers \\( A = \\{a_1, a_2, \\dots, a_n\\} \\), form a sequence by taking the difference \\( a_i - a_j \\) for each pair \\( a_i, a_j \\). This sequence, composed of \\( \\binom{n}{2} \\) d... | true | The problem defines a derived sequence based on differences between pairs of integers in a set, and it involves proving a generalized inequality about the count of differences divisible by an integer m. An expert in the subject matter, particularly in number theory or combinatorics, would have sufficient information to... |
olympiads | Let \( n \) be a natural number greater than 6, and let \( X \) be a set with \( n \) elements. Consider choosing \( m \) distinct 5-element subsets \( A_1, A_2, \ldots, A_m \) of \( X \). Prove that if \( m > \frac{n(n-1)(n-2)(n-3)(4n-15)}{600} \), there exist six of these subsets \( A_{i_1}, A_{i_2}, \ldots, A_{i_6} ... |
1. **Construct a Bipartite Graph $G$**:
Let's construct a bipartite graph $G = (A \cup B, E)$ where:
- $A = \{A_1, A_2, \ldots, A_m\}$, where each $A_i$ is an arbitrary 5-element subset of the set $X$.
- $B$ consists of ${n \choose 4}$ vertices, where each vertex represents a 4-element subset of $X$.
... | [
{
"content": "Let \\( n \\) be a natural number greater than 6, and let \\( X \\) be a set with \\( n \\) elements. Consider choosing \\( m \\) distinct 5-element subsets \\( A_1, A_2, \\ldots, A_m \\) of \\( X \\). Prove that if \\( m > \\frac{n(n-1)(n-2)(n-3)(4n-15)}{600} \\), there exist six of these subsets... | true | The problem provides specific parameters related to the natural number n and the constraints on m as well as the nature of the subsets A_i. An expert in combinatorics would have sufficient information to approach the problem, using methods from extremal set theory to show that the given condition ensures the existence ... |
olympiads | How many two-digit numbers have the tens digit greater than the units digit? |
We need to count the number of two-digit numbers where the tens digit is greater than the units digit. Let's denote the tens digit by \( t \) and the units digit by \( u \).
Given that \( t \) and \( u \) are digits, they must satisfy \( 1 \leq t \leq 9 \) and \( 0 \leq u \leq 9 \). We require that \( t > u \).
Let'... | [
{
"content": "How many two-digit numbers have the tens digit greater than the units digit?",
"role": "user"
},
{
"content": "\nWe need to count the number of two-digit numbers where the tens digit is greater than the units digit. Let's denote the tens digit by \\( t \\) and the units digit by \\( u ... | true | The problem asks for two-digit numbers where the tens digit is greater than the units digit. An expert can analyze the range of digits for the tens place (1 to 9) and the units place (0 to 9) and determine the possible combinations. Since the question is clearly defined and provides the necessary conditions, an expert ... |
olympiads | Isosceles triangles \(ABC (AB = BC)\) and \(A_1B_1C_1 (A_1B_1 = B_1C_1)\) are similar and \(AB : A_1B_1 = 2 : 1\).
Vertices \(A_1, B_1\), and \(C_1\) are located on sides \(CA, AB,\) and \(BC\) respectively, with \(A_1B_1 \perp AC\). Find angle \(B\). | 1. **Define the problem and assumptions:**
- We are given two similar isosceles triangles, \( \triangle ABC \) and \( \triangle A_1B_1C_1 \), with given side ratios \( AB : A_1B_1 = 2 : 1 \).
- It’s known that \( A_1B_1 \perp AC \) and \( A_1B_1 = B_1C_1 = 1 \).
- Let \( \angle A = \alpha \).
- Need to find... | [
{
"content": "Isosceles triangles \\(ABC (AB = BC)\\) and \\(A_1B_1C_1 (A_1B_1 = B_1C_1)\\) are similar and \\(AB : A_1B_1 = 2 : 1\\).\n\nVertices \\(A_1, B_1\\), and \\(C_1\\) are located on sides \\(CA, AB,\\) and \\(BC\\) respectively, with \\(A_1B_1 \\perp AC\\). Find angle \\(B\\).",
"role": "user"
}... | true | The problem describes two similar isosceles triangles and provides the ratio of their sides. The conditions given, including the fact that the triangles are similar and that the length ratio is specified, allow for the application of properties related to similar triangles. Additionally, knowing that one triangle's ver... |
olympiads | Given positive numbers \( x, y, z \). Prove the inequality \(\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1} \leq \frac{x}{y}+\frac{y}{z}+\frac{z}{x}\). |
1. **Identify and Transform Inequality:**
Given inequality:
\[
\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1} \leq \frac{x}{y}+\frac{y}{z}+\frac{z}{x}
\]
Rewrite each term:
\[
\frac{x+1}{y+1} - \frac{x}{y}
\]
Simplify:
\[
\frac{x+1}{y+1} - \frac{x}{y} = \frac{(x+1)y - x(y+1)}{... | [
{
"content": "Given positive numbers \\( x, y, z \\). Prove the inequality \\(\\frac{x+1}{y+1}+\\frac{y+1}{z+1}+\\frac{z+1}{x+1} \\leq \\frac{x}{y}+\\frac{y}{z}+\\frac{z}{x}\\).",
"role": "user"
},
{
"content": "\n1. **Identify and Transform Inequality:**\n Given inequality:\n \\[\n \\frac{x+1... | false | The problem statement requires more context regarding the specific conditions or constraints for the variables x, y, and z to prove the given inequality. As stated, there might be cases where the inequality does not hold, hence an expert would not have enough information to conclude definitively. |
olympiads | Given \( X \) is the closed interval \([0, 1]\), find all functions \( f: X \times X \to X \) such that:
1. \( f(x, 1) = f(1, x) = x \) for all \( x \in X \),
2. \( f(f(x, y), z) = f(x, f(y, z)) \) for all \( x, y, z \in X \),
3. For some fixed positive real \( k \), \( f(xy, xz) = x^k f(y, z) \) for all \( x, y, z \in... | Given the problem requires finding all functions \( f: X \times X \to X \) satisfying three conditions:
1. \( f(x, 1) = f(1, x) = x \) for all \( x \) in \( X \).
2. \( f(f(x, y), z) = f(x, f(y, z)) \) for all \( x, y, z \) in \( X \).
3. \( f(xy, xz) = x^k f(y, z) \) for all \( x, y, z \) in \( X \) and some fixed po... | [
{
"content": "Given \\( X \\) is the closed interval \\([0, 1]\\), find all functions \\( f: X \\times X \\to X \\) such that:\n1. \\( f(x, 1) = f(1, x) = x \\) for all \\( x \\in X \\),\n2. \\( f(f(x, y), z) = f(x, f(y, z)) \\) for all \\( x, y, z \\in X \\),\n3. For some fixed positive real \\( k \\), \\( f(x... | true | The problem presents a function defined on the closed interval
[0, 1] and provides three specific criteria that the function must satisfy. An expert in mathematical analysis or functional equations would have sufficient information to explore the properties of the function based on these criteria and attempt to find a... |
olympiads | In a vacation town where many adults spend their holidays, $28\%$ of the adults read "Le Monde", $25\%$ read "Le Figaro", and $20\%$ read "L'Aurore". Additionally, $11\%$ of the adults read both "Le Monde" and "Le Figaro", $3\%$ read both "Le Monde" and "L'Aurore", and $2\%$ read both "Le Figaro" and "L'Aurore". Furthe... |
1. **Define Variables and Sets**:
Let's define the following:
- Let $x$ be the percentage of vacationers who read all three newspapers.
- Let $V$ be the set of all vacationers.
- Let $M$ be the set of readers of "Mond."
- Let $F$ be the set of readers of "Figaro."
- Let $A$ be the set of readers of "... | [
{
"content": "In a vacation town where many adults spend their holidays, $28\\%$ of the adults read \"Le Monde\", $25\\%$ read \"Le Figaro\", and $20\\%$ read \"L'Aurore\". Additionally, $11\\%$ of the adults read both \"Le Monde\" and \"Le Figaro\", $3\\%$ read both \"Le Monde\" and \"L'Aurore\", and $2\\%$ re... | true | The problem provides relevant data about the percentages of adults reading each newspaper, as well as the overlaps for pairs of newspapers. Additionally, it states that 42% of adults do not read any of the newspapers. This information, particularly the inclusion-exclusion principle and the percentages of overlaps, allo... |
olympiads | Solve the system $\left\{\begin{array}{l}x^{2}+y^{2} \leq 2, \\ x^{4}-8 x^{2} y^{2}+16 y^{4}-20 x^{2}-80 y^{2}+100=0 .\end{array}\right.$ |
1. Given the system of equations:
\[
\begin{cases}
x^2 + y^2 \leq 2, \\
x^4 - 8x^2 y^2 + 16y^4 - 20x^2 - 80y^2 + 100 = 0.
\end{cases}
\]
2. We start with the second equation:
\[
x^4 - 8x^2 y^2 + 16y^4 - 20x^2 - 80y^2 + 100 = 0.
\]
3. We can rearrange the terms as follows:
\[
x^4 + 8... | [
{
"content": "Solve the system $\\left\\{\\begin{array}{l}x^{2}+y^{2} \\leq 2, \\\\ x^{4}-8 x^{2} y^{2}+16 y^{4}-20 x^{2}-80 y^{2}+100=0 .\\end{array}\\right.$",
"role": "user"
},
{
"content": "\n1. Given the system of equations:\n \\[\n \\begin{cases} \n x^2 + y^2 \\leq 2, \\\\\n x^4 - 8x^2... | true | The problem presents a system of inequalities and an equation that define a specific region and relationship between the variables x and y. An expert in mathematics would have sufficient information to analyze the constraints and find the solution. |
olympiads | Prove the following:
(1) \(\sin^2 \alpha + \sin^2 (\alpha + \beta) + \cdots + \sin^2 (\alpha + n \beta)\)
$$
=\frac{n+1}{2}-\frac{\sin (n+1) \beta \cos (2 \alpha + n \beta)}{2 \sin \beta};
$$
(2) \(\sin \alpha \sin (\alpha + \beta) + \sin (\alpha + \beta) \sin (\alpha + 2 \beta) + \cdots + \sin[\alpha + (n-1) \beta] ... |
To prove the given identities, we will use trigonometric identities such as the product-to-sum formulas and leveraging known sum properties.
#### Part (1)
We aim to prove:
$$
\sin^2 \alpha + \sin^2 (\alpha + \beta) + \cdots + \sin^2 (\alpha + n \beta) = \frac{n+1}{2} - \frac{\sin (n+1) \beta \cos (2\alpha + n \beta)}... | [
{
"content": "Prove the following:\n\n(1) \\(\\sin^2 \\alpha + \\sin^2 (\\alpha + \\beta) + \\cdots + \\sin^2 (\\alpha + n \\beta)\\)\n$$\n=\\frac{n+1}{2}-\\frac{\\sin (n+1) \\beta \\cos (2 \\alpha + n \\beta)}{2 \\sin \\beta};\n$$\n\n(2) \\(\\sin \\alpha \\sin (\\alpha + \\beta) + \\sin (\\alpha + \\beta) \\si... | true | The problem presents two mathematical expressions that need to be proven. Although it does not provide diagrams or specific numbers, it lays out clear mathematical identities involving sine functions. An expert with knowledge of trigonometric identities and series would have sufficient information to approach and solve... |
olympiads | In triangle $ABC$, it is known that $AB=3$, $AC=4$, and $\angle BAC=60^\circ$. The extension of the angle bisector $AA_1$ intersects the circumcircle of triangle $ABC$ at point $A_2$. Find the areas of triangles $OA_2C$ and $A_1A_2C$ ($O$ is the center of the circumcircle of triangle $ABC$). | Given the triangle \(ABC\) where \(AB=3\), \(AC=4\), and \(\angle BAC=60^\circ\).
1. **Using the Law of Cosines to find \(BC\):**
The Law of Cosines states: \[ BC^2 = AB^2 + AC^2 - 2 \cdot AB \cdot AC \cdot \cos(\angle BAC) \]
Plugging in the values: \[ BC^2 = 3^2 + 4^2 - 2 \cdot 3 \cdot 4 \cdot \cos(60^\circ)... | [
{
"content": "In triangle $ABC$, it is known that $AB=3$, $AC=4$, and $\\angle BAC=60^\\circ$. The extension of the angle bisector $AA_1$ intersects the circumcircle of triangle $ABC$ at point $A_2$. Find the areas of triangles $OA_2C$ and $A_1A_2C$ ($O$ is the center of the circumcircle of triangle $ABC$).",
... | false | Although the problem provides some dimensions and angles of triangle ABC, it lacks sufficient information about the size and position of the circumcircle, as well as the method to calculate the areas of the specified triangles without additional geometric information. |
olympiads | Given a positive integer \( n \), it is known that using \( k \) weights, each of which has a whole number of grams, and a balance scale, one can measure all items with a mass of \( 1, 2, 3, \cdots, n \) grams.
(1) Find the smallest value \( f(n) \) for \( k \).
(2) For which values of \( n \) is the composition of t... |
#### Part (1):
1. Let the weights of these $k$ blocks be $a_1, a_2, \ldots, a_k$ such that \(1 \leq a_1 \leq a_2 \leq \cdots \leq a_k\) and all $a_i$ are integers.
2. To measure any integer weight from $1$ to $n$, we can place weights on either side of the balance. Hence, we can express the total weight as \(\sum_{i=1... | [
{
"content": "Given a positive integer \\( n \\), it is known that using \\( k \\) weights, each of which has a whole number of grams, and a balance scale, one can measure all items with a mass of \\( 1, 2, 3, \\cdots, n \\) grams.\n\n(1) Find the smallest value \\( f(n) \\) for \\( k \\).\n\n(2) For which valu... | true | The problem provides a clear mathematical scenario involving the number of weights needed to measure integers within a certain range, along with specific questions regarding functions and their properties. An expert with knowledge of number theory and combinatorial mathematics would have sufficient information to deter... |
olympiads | Let \( f(n) \in \mathbb{N} \) be the smallest number such that the sum \( \sum_{k=1}^{f(n)} k \) is divisible by \( n \). Prove that \( f(n) = 2n - 1 \) if and only if \( n = 2^m \) where \( m \in \mathbb{Z}^+ \). |
1. **Step 1: Proving that if \( n = 2^m \), then \( f(n) = 2n - 1 \).**
Consider the sum of the first \( 2n - 1 \) natural numbers:
\[
\sum_{k=1}^{2n-1} k = \frac{(2n-1) \cdot 2n}{2}
\]
Simplifying the right-hand side:
\[
\frac{(2n-1) \cdot 2n}{2} = \frac{(2^m \cdot 2 - 1) \cdot 2^m \cdot ... | [
{
"content": "Let \\( f(n) \\in \\mathbb{N} \\) be the smallest number such that the sum \\( \\sum_{k=1}^{f(n)} k \\) is divisible by \\( n \\). Prove that \\( f(n) = 2n - 1 \\) if and only if \\( n = 2^m \\) where \\( m \\in \\mathbb{Z}^+ \\).",
"role": "user"
},
{
"content": "\n1. **Step 1: Provin... | false | The problem states a property involving the sum of integers and divisibility, but does not provide sufficient context or definitions for how to approach the proof. An expert may recognize that more information about the properties of the function and what exactly needs proving is necessary to evaluate the statement abo... |
olympiads | Find the odd prime numbers $p_1, p_2, p_3, p_4, p_5$ and a natural number $a$ such that the numbers in (1) and (2) respectively form a pair of amicable numbers.
$$
\begin{array}{lll}
p_{1}^{a} \cdot p_{2} \cdot 7 \cdot 71 & \text { and } & p_{1}^{a} \cdot p_{2} \cdot 17 \cdot 31 \\
3^{2} \cdot 5 \cdot 13 \cdot 79 \cdo... |
Given the expressions for the numbers (1) and (2):
$$
\begin{array}{lll}
p_{1}^{a} \cdot p_{2} \cdot 7 \cdot 71 & \text{ and } & p_{1}^{a} \cdot p_{2} \cdot 17 \cdot 31 \\
3^{2} \cdot 5 \cdot 13 \cdot 79 \cdot p_{3} & \text{ and } & 3^{2} \cdot 5 \cdot 13 \cdot p_{4} \cdot p_{5}
\end{array}
$$
**Step-by-Step Solutio... | [
{
"content": "Find the odd prime numbers $p_1, p_2, p_3, p_4, p_5$ and a natural number $a$ such that the numbers in (1) and (2) respectively form a pair of amicable numbers.\n\n$$\n\\begin{array}{lll}\np_{1}^{a} \\cdot p_{2} \\cdot 7 \\cdot 71 & \\text { and } & p_{1}^{a} \\cdot p_{2} \\cdot 17 \\cdot 31 \\\\\... | true | The problem provides equations involving prime numbers and requires finding specific odd primes and a natural number that make the pairs amicable. An expert familiar with number theory would understand the concept of amicable numbers and how to derive the primes and exponent required to satisfy the conditions set out i... |
olympiads | Several consecutive natural numbers are written on the board. Exactly 52% of them are even. How many even numbers are written on the board? |
1. Прежде всего, заметим, что записанные числа являются последовательными натуральными числами. В последовательности натуральных чисел четные и нечетные числа чередуются.
2. По условию задачи нам дано, что 52% из них являются четными числами. Это число больше половины (50%), что указывает на наличие большего числа... | [
{
"content": "Several consecutive natural numbers are written on the board. Exactly 52% of them are even. How many even numbers are written on the board?",
"role": "user"
},
{
"content": "\n1. Прежде всего, заметим, что записанные числа являются последовательными натуральными числами. В последовател... | false | The problem does not provide the total number of consecutive natural numbers written on the board, which is necessary to determine how many are even. Without knowing either the count of the numbers or the specific range of the consecutive natural numbers, an expert cannot calculate the number of even numbers. |
olympiads | What is the largest finite number of roots that the equation
$$
\left|x-a_{1}\right| + \cdots + |x-a_{50}| = \left|x-b_{1}\right| + \cdots + |x-b_{50}|
$$
can have, where \( a_{1}, a_{2}, \dots, a_{50}, b_{1}, b_{2}, \dots, b_{50} \) are distinct numbers? |
1. Define the function:
\[
f(x) = \left| x - a_1 \right| + \left| x - a_2 \right| + \cdots + \left| x - a_{50} \right| - \left| x - b_1 \right| - \left| x - b_2 \right| - \cdots - \left| x - b_{50} \right|
\]
The given equation can then be rewritten as:
\[
f(x) = 0
\]
2. Let \(c_1 < c_2 < \cdots ... | [
{
"content": "What is the largest finite number of roots that the equation \n\n$$\n\\left|x-a_{1}\\right| + \\cdots + |x-a_{50}| = \\left|x-b_{1}\\right| + \\cdots + |x-b_{50}|\n$$\n\ncan have, where \\( a_{1}, a_{2}, \\dots, a_{50}, b_{1}, b_{2}, \\dots, b_{50} \\) are distinct numbers?",
"role": "user"
... | true | The question asks about the largest number of finite roots for a specific equation involving absolute values of distinct numbers. While such problems may require knowledge of properties of absolute value functions and intersections, an expert can analyze and derive insights about the behavior of these functions over in... |
olympiads | Consider the sequences \(\{a_n\}\) and \(\{b_n\}\) defined as follows:
\[
\begin{array}{l}
a_{0}=\frac{\sqrt{2}}{2}, \quad a_{n+1}=\frac{\sqrt{2}}{2} \sqrt{1-\sqrt{1-a_{n}^{2}}}, \quad n=0, 1, 2, \ldots \\
b_{0}=1, \quad b_{n+1}=\frac{\sqrt{1+b_{n}^{2}}-1}{b_{n}}, \quad n=0, 1, 2, \ldots
\end{array}
\]
Prove that for... |
Given two sequences $\{a_n\}$ and $\{b_n\}$ defined by:
$$
\begin{aligned}
a_0 &= \frac{\sqrt{2}}{2}, \quad a_{n+1} = \frac{\sqrt{2}}{2} \sqrt{1 - \sqrt{1 - a_n^2}}, \quad n = 0, 1, 2, \ldots, \\
b_0 &= 1, \quad b_{n+1} = \frac{\sqrt{1 + b_n^2} - 1}{b_n}, \quad n = 0, 1, 2, \ldots,
\end{aligned}
$$
We are required t... | [
{
"content": "Consider the sequences \\(\\{a_n\\}\\) and \\(\\{b_n\\}\\) defined as follows:\n\\[ \n\\begin{array}{l}\na_{0}=\\frac{\\sqrt{2}}{2}, \\quad a_{n+1}=\\frac{\\sqrt{2}}{2} \\sqrt{1-\\sqrt{1-a_{n}^{2}}}, \\quad n=0, 1, 2, \\ldots \\\\\nb_{0}=1, \\quad b_{n+1}=\\frac{\\sqrt{1+b_{n}^{2}}-1}{b_{n}}, \\qu... | true | The problem provides explicit definitions for the sequences \(\{a_n\}\) and \(\{b_n\}\), as well as an inequality that they need to satisfy. An expert in mathematics can analyze the properties of these sequences, calculate their values, and discuss their limits to prove the stated inequality. Thus, there is sufficient ... |
olympiads | An ellipse, with a major axis equal to $2a$ and a minor axis equal to $2b (a > b)$, rotates: 1) around the major axis; 2) around the minor axis. Find the volumes of the resulting ellipsoids of revolution. In a specific case, determine the volume of a sphere. |
Let's consider the problem of finding the volumes of ellipsoids of revolution formed when an ellipse rotates around its major axis and minor axis.
#### Step 1: Define the Ellipse
First, we start with the equation of the ellipse given by:
\[
\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1
\]
where $2a$ is the length o... | [
{
"content": "An ellipse, with a major axis equal to $2a$ and a minor axis equal to $2b (a > b)$, rotates: 1) around the major axis; 2) around the minor axis. Find the volumes of the resulting ellipsoids of revolution. In a specific case, determine the volume of a sphere.",
"role": "user"
},
{
"cont... | true | The problem provides the necessary parameters (major and minor axes of the ellipse) to find the volumes of the ellipsoids when rotated around both axes. An expert in geometry and volume calculations would have sufficient information to derive the formulas for the volumes and also compute the specific case of the sphere... |
olympiads | Let \( p(x) \equiv x^3 + ax^2 + bx - 1 \) and \( q(x) \equiv x^3 + cx^2 + dx + 1 \) be polynomials with integer coefficients. Let \( \alpha \) be a root of \( p(x) = 0 \). The polynomial \( p(x) \) is irreducible over the rationals. Given that \( \alpha + 1 \) is a root of \( q(x) = 0 \), find an expression for another... | 1. Let \( p(x) \equiv x^3 + ax^2 + bx - 1 \) and \( q(x) \equiv x^3 + cx^2 + dx + 1 \) be polynomials with integer coefficients, and let \( \alpha \) be a root of \( p(x) = 0 \). Given that \( p(x) \) is irreducible over the rationals, we know that its roots are not rational.
2. Since \( \alpha + 1 \) is a root of \( q... | [
{
"content": "Let \\( p(x) \\equiv x^3 + ax^2 + bx - 1 \\) and \\( q(x) \\equiv x^3 + cx^2 + dx + 1 \\) be polynomials with integer coefficients. Let \\( \\alpha \\) be a root of \\( p(x) = 0 \\). The polynomial \\( p(x) \\) is irreducible over the rationals. Given that \\( \\alpha + 1 \\) is a root of \\( q(x)... | true | The problem gives two specific polynomial functions and conditions about their roots and irreducibility. An expert in the subject matter (polynomial roots and algebra) would have sufficient information to manipulate the given polynomial equations and relations involving their roots to find another root as requested, us... |
olympiads | Let \( A, B, C, D \) be four collinear points in that order. Consider \(\Gamma_{1}\), the circle with diameter \([A C]\), and \(\Gamma_{2}\), the circle with diameter \([B D]\). These circles intersect at points \( X \) and \( Y \). Let \( P \) be any point on the line \((X Y)\), \( M \) be the intersection point of \(... |
1. **Identification of Radical Axes:**
Let us first consider the radical axes. We have two circles \( \Gamma_1 \) with diameter \([A, C]\) and \( \Gamma_2 \) with diameter \([B, D]\). These two circles intersect at points \( X \) and \( Y \). The radical axis theorem states that the radical axis of two circles is t... | [
{
"content": "Let \\( A, B, C, D \\) be four collinear points in that order. Consider \\(\\Gamma_{1}\\), the circle with diameter \\([A C]\\), and \\(\\Gamma_{2}\\), the circle with diameter \\([B D]\\). These circles intersect at points \\( X \\) and \\( Y \\). Let \\( P \\) be any point on the line \\((X Y)\\... | true | The problem provides a detailed geometric scenario involving circles, their diameters, and specific intersections. An expert in geometry would understand how to handle the elements outlined (collinear points, intersecting circles, and the properties related to concurrency) and would have the necessary information to pr... |
olympiads | Compute the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{\sqrt{n+1}-\sqrt[3]{n^{3}+1}}{\sqrt[4]{n+1}-\sqrt[5]{n^{5}+1}}
\] |
1. **Rewrite the Expressions under the Limits:**
Consider the limit:
\[\lim_{n \to \infty} \frac{\sqrt{n+1} - \sqrt[3]{n^3 + 1}}{\sqrt[4]{n+1} - \sqrt[5]{n^5 + 1}}.\]
We start by factoring out the dominant term, which helps simplify the expressions under the limit.
2. **Simplifying Numerator and Denominat... | [
{
"content": "Compute the limit of the numerical sequence:\n\n\\[\n\\lim _{n \\rightarrow \\infty} \\frac{\\sqrt{n+1}-\\sqrt[3]{n^{3}+1}}{\\sqrt[4]{n+1}-\\sqrt[5]{n^{5}+1}}\n\\]",
"role": "user"
},
{
"content": "\n1. **Rewrite the Expressions under the Limits:**\n\n Consider the limit:\n\n \\[\\... | true | The problem provides a clearly defined limit involving a sequence and includes the necessary mathematical expressions to analyze the behavior as n approaches infinity. An expert in limits would have sufficient information to solve the problem by applying limit properties, L'Hospital's rule, or other limit evaluation te... |
olympiads | Eight identical spheres, each of diameter 20, fit tightly into a cube of side length 40 so that each sphere just touches three of the faces of the cube. The radius of the largest sphere that will fit in the central space, just touching all eight spheres, is closest to
(A) 7.0
(B) 7.3
(C) 7.6
(D) 7.9
(E) 8.2 | 1. **Divide the Large Cube:**
- The large cube has a side length of 40 units.
- Divide this large cube into 8 smaller cubes, each with a side length of \( 20 \) units, by making three cuts through the center using planes parallel to each pair of faces.
2. **Enclose the Spheres in the Smaller Cubes:**
- Eac... | [
{
"content": "Eight identical spheres, each of diameter 20, fit tightly into a cube of side length 40 so that each sphere just touches three of the faces of the cube. The radius of the largest sphere that will fit in the central space, just touching all eight spheres, is closest to\n(A) 7.0\n(B) 7.3\n(C) 7.6\n(... | true | The question provides specific dimensions for the spheres and the cube, as well as the requirement for the central sphere to touch all eight spheres. An expert familiar with geometry and spatial reasoning would have enough information to calculate the radius of the largest sphere that can fit in the given space. |
olympiads | Points \( M \) and \( N \) lie on the sides \( AB \) and \( AC \) of triangle \( ABC \), respectively, such that \( AM = CN \) and \( AN = BM \).
Prove that the area of quadrilateral \( BMNC \) is at least three times the area of triangle \( AMN \). |
1. **Notations and Setup**:
We'll denote the lengths as follows:
\[
AM = NC = x \quad \text{and} \quad AN = BM = y.
\]
Thus, the sides of $\triangle ABC$ can be expressed as:
\[
AB = x + y \quad \text{and} \quad AC = x + y.
\]
2. **Area of $\triangle AMN$**:
Using the formula for t... | [
{
"content": "Points \\( M \\) and \\( N \\) lie on the sides \\( AB \\) and \\( AC \\) of triangle \\( ABC \\), respectively, such that \\( AM = CN \\) and \\( AN = BM \\).\n\nProve that the area of quadrilateral \\( BMNC \\) is at least three times the area of triangle \\( AMN \\).",
"role": "user"
},
... | true | The problem specifies relationships between segments in triangle ABC and asks to prove a geometric relationship involving areas. An expert in geometry would be familiar with the concepts of area ratios and the properties of triangles. Thus, given these relationships, an expert has sufficient information to approach the... |
olympiads | How many people are in the team if the average age of all team members is 25 years, the team leader is 45 years old, and the average age of the team members without the team leader is 23 years? |
1. Define \( x \) as the number of people in the brigade.
2. The average age of all members in the brigade is given as 25 years. Thus, the total age sum of the brigade is \( 25x \).
3. The foreman's age is 45 years and he is included in the brigade.
4. The average age of the members excluding the foreman is 23 year... | [
{
"content": "How many people are in the team if the average age of all team members is 25 years, the team leader is 45 years old, and the average age of the team members without the team leader is 23 years?",
"role": "user"
},
{
"content": "\n1. Define \\( x \\) as the number of people in the briga... | true | The problem provides information about the average age of the team with the team leader and without the team leader, along with the age of the team leader. This allows an expert to set up equations to determine the total number of team members and thus solve for the number of people in the team. |
olympiads | Pirates led by John Silver on Treasure Island found Billy Bones' chest, which contained 40 coins worth 1 ducat each and 40 coins worth 5 ducats each. John Silver has not yet decided how to distribute this money among all the pirates (he does not want to take any for himself). For the maximum number of pirates, any deci... |
We need to determine the maximum number of pirates such that any decision by John Silver regarding the distribution of coins can be executed, ensuring each pirate receives a whole number of ducats.
1. **Define Variables**:
Let \( S_i \) be the number of ducats received by the \( i \)-th pirate. Given that there ar... | [
{
"content": "Pirates led by John Silver on Treasure Island found Billy Bones' chest, which contained 40 coins worth 1 ducat each and 40 coins worth 5 ducats each. John Silver has not yet decided how to distribute this money among all the pirates (he does not want to take any for himself). For the maximum numbe... | true | The problem provides detailed information about the total number of coins and their values, allowing an expert to analyze the situation. They can determine how to distribute the coins among the maximum number of pirates in various ways, considering that the pirates can receive whole coins or none at all. Thus, there is... |
olympiads | If the parallel lines \(EF\), \(MN\) intersect with the intersecting lines \(AB\) and \(CD\) to form the figure as shown in the diagram, then the total number of pairs of consecutive interior angles is:
(A) 4
(B) 8
(C) 12
(D) 16 |
1. **Understand the Problem Setup:**
We need to find the total number of pairs of corresponding interior angles formed by the intersecting lines $EF$, $MN$, $AB$, and $CD$.
2. **Identify the Basic Shape:**
The setup forms multiple basic "three-line eight-angle" shapes. In every such shape, there are two pairs o... | [
{
"content": "If the parallel lines \\(EF\\), \\(MN\\) intersect with the intersecting lines \\(AB\\) and \\(CD\\) to form the figure as shown in the diagram, then the total number of pairs of consecutive interior angles is:\n(A) 4 \n(B) 8\n(C) 12\n(D) 16",
"role": "user"
},
{
"content": "\n1. **Und... | false | The question references a diagram that is essential for understanding the relationships between the lines and angles involved. Without the diagram, an expert would lack the necessary visual information to determine the total number of pairs of consecutive interior angles. |
olympiads | Given 2017 points distributed on a circle, each point being labeled either +1 or -1, if from any point onward, in any direction around the circle to any other point, the sum of the numbers passed through is always positive, then that point is called "good." Prove that if the points labeled -1 do not exceed 672, then th... | ### Problem Statement:
Given 2017 points distributed on the circumference of a circle, each marked with +1 or -1. If starting from any point and moving in any direction around the circle, the sum of the values of the points passed is always a positive number, we call that point a "good" point. Prove that if the number... | [
{
"content": "Given 2017 points distributed on a circle, each point being labeled either +1 or -1, if from any point onward, in any direction around the circle to any other point, the sum of the numbers passed through is always positive, then that point is called \"good.\" Prove that if the points labeled -1 do... | true | The problem provides a clear mathematical scenario with sufficient definitions and conditions to analyze the arrangement of points on a circle labeled +1 or -1. An expert with knowledge in combinatorial or number theory could apply existing theorems related to the distribution of these labels to solve for the existence... |
olympiads | From the 100 natural numbers between 1 and 100, select at least ______ numbers to ensure that among them there are at least two numbers whose product is 105. | :
To solve this problem, we need to ensure that among the selected numbers from 1 to 100, there exist at least two numbers whose product is 105.
Firstly, let's consider the factorization of 105. The factors of 105 are:
\[ 105 = 3 \times 5 \times 7 \]
Next, we list pairs of factors of 105 within the range from 1 to... | [
{
"content": "From the 100 natural numbers between 1 and 100, select at least ______ numbers to ensure that among them there are at least two numbers whose product is 105.",
"role": "user"
},
{
"content": ": \n\nTo solve this problem, we need to ensure that among the selected numbers from 1 to 100, ... | true | The problem requires identifying factors of the product 105. The prime factorization of 105 is 3 \ imes 5 \ imes 7. The task is to find a selection strategy that guarantees selecting at least two of these factors in any chosen group. An expert could use the pigeonhole principle for this combinatorial problem even witho... |
olympiads | In an olympiad, 55 students participated, and all of them submitted their work. For each problem, one of the three assessments was given: "+" - the problem was solved, "-" - the problem was attempted, "0" - the problem was not attempted. After evaluating all the works, it was found that no two works had the same number... | 1. Let's denote the number of problems given in the olympiad as \(a\).
2. Each student can receive one of three possible scores for each problem: \(+\) (solved), \(-\) (attempted), or \(0\) (not attempted).
3. Each student's results can thus be represented as a sequence of length \(a\) with three possible values for ea... | [
{
"content": "In an olympiad, 55 students participated, and all of them submitted their work. For each problem, one of the three assessments was given: \"+\" - the problem was solved, \"-\" - the problem was attempted, \"0\" - the problem was not attempted. After evaluating all the works, it was found that no t... | true | The problem provides the necessary information regarding the number of students, their submissions, and the unique combinations of assessments for each problem. An expert in combinatorial mathematics or competition settings would have sufficient information to deduce relationships and constraints to determine the minim... |
olympiads | There are 10 bags full of coins. All coins look the same and all weigh 10 grams, except the coins from one bag that are fake and all weigh 9 grams. Given a scale, how could you tell which bag has the fake coins in just one measurement? Explain your answer! |
1. **Conceptual Step:**
- We have 10 bags, each containing coins that should weigh 10 grams each. However, one of these bags contains fake coins that weigh 9 grams each.
- We need to identify the bag with the fake coins using only one measurement on the scale.
2. **Setting Up the Experiment:**
- Label the ba... | [
{
"content": "There are 10 bags full of coins. All coins look the same and all weigh 10 grams, except the coins from one bag that are fake and all weigh 9 grams. Given a scale, how could you tell which bag has the fake coins in just one measurement? Explain your answer!",
"role": "user"
},
{
"conten... | true | The problem provides information about the situation (10 bags of coins with all but one being fake) and asks for a method to identify the bag with the fake coins in a specified manner (one measurement). An expert with knowledge of the subject matter would understand how to strategically use the weight differences to de... |
olympiads | For which natural numbers \( n \) can the fraction \(\frac{5}{n}\) be represented as a repeating decimal of the form \(0.1(a_1 a_2 a_3)\) with a period containing at least two different digits? |
1. To determine for which natural numbers \( n \) the fraction \( \frac{5}{n} \) can be represented as a periodic decimal of the form \( 0,\overline{1a_{1}a_{2}a_{3}} \) with a period containing at least two different digits, we first must recall a few concepts related to periodic decimals.
2. A fraction \( \frac{a}{... | [
{
"content": "For which natural numbers \\( n \\) can the fraction \\(\\frac{5}{n}\\) be represented as a repeating decimal of the form \\(0.1(a_1 a_2 a_3)\\) with a period containing at least two different digits?",
"role": "user"
},
{
"content": "\n1. To determine for which natural numbers \\( n \... | true | The problem requires determining the natural numbers n for which the fraction 5/n produces a repeating decimal of a specific form. An expert in mathematics, particularly in number theory and fractions, has sufficient information to analyze the properties of repeating decimals and their periods. They would understand ho... |
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