source stringclasses 1
value | problem stringlengths 86 947 | solution stringlengths 1.24k 3.15k | messages listlengths 2 2 |
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amc_aime | Suppose $q_1,q_2,...$ is an infinite sequence of integers satisfying the following two conditions:
(a) $m - n$ divides $q_m - q_n$ for $m>n \geq 0$
(b) There is a polynomial $P$ such that $|q_n|<P(n)$ for all $n$.
Prove that there is a polynomial $Q$ such that $q_n = Q(n)$ for each $n$. |
#### Step 1: Constructing the Polynomial $Q$
Given that $P$ is a polynomial of degree $d$, we construct a polynomial $Q$ of degree at most $d$ such that $Q(n) = q_n$ for $0 \leq n \leq d$. Since the $q_n$ are integers, $Q$ can be determined uniquely by interpolation and will have rational coefficients. Let $k$ be the ... | [
{
"content": "Suppose $q_1,q_2,...$ is an infinite sequence of integers satisfying the following two conditions:\n(a) $m - n$ divides $q_m - q_n$ for $m>n \\geq 0$\n(b) There is a polynomial $P$ such that $|q_n|<P(n)$ for all $n$.\nProve that there is a polynomial $Q$ such that $q_n = Q(n)$ for each $n$.",
... |
amc_aime | Let $\, a =(m^{m+1} + n^{n+1})/(m^m + n^n), \,$ where $\,m\,$ and $\,n\,$ are positive integers. Prove that $\,a^m + a^n \geq m^m + n^n$.
[You may wish to analyze the ratio $\,(a^N - N^N)/(a-N),$ for real $\, a \geq 0 \,$ and integer $\, N \geq 1$.] | 1. **Assumption and Definitions**: Assume without loss of generality that $m \geq n$. Define $a = \frac{m^{m+1} + n^{n+1}}{m^m + n^n}$.
2. **Expressing $m-a$ and $a-n$**:
- For $m-a$, we have:
\[
m-a = \frac{m(m^m + n^n) - (m^{m+1} + n^{n+1})}{m^m + n^n} = \frac{m^m n^n - n^{n+1}}{m^m + n^n} = n^n \frac{m... | [
{
"content": "Let $\\, a =(m^{m+1} + n^{n+1})/(m^m + n^n), \\,$ where $\\,m\\,$ and $\\,n\\,$ are positive integers. Prove that $\\,a^m + a^n \\geq m^m + n^n$.\n[You may wish to analyze the ratio $\\,(a^N - N^N)/(a-N),$ for real $\\, a \\geq 0 \\,$ and integer $\\, N \\geq 1$.]",
"role": "user"
},
{
... |
amc_aime | Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$
and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and
$CE$ meet at $I$. Determine whether or not it is possible for
segments $AB, AC, BI, ID, CI, IE$ to all have integer len... | 1. **Given Information and Assumptions**:
- Triangle $ABC$ is a right triangle with $\angle A = 90^\circ$.
- Points $D$ and $E$ lie on sides $AC$ and $AB$ respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$.
- Segments $BD$ and $CE$ meet at point $I$.
- We need to determine if it... | [
{
"content": "Let $ABC$ be a triangle with $\\angle A = 90^{\\circ}$. Points $D$\nand $E$ lie on sides $AC$ and $AB$, respectively, such that $\\angle ABD = \\angle DBC$ and $\\angle ACE = \\angle ECB$. Segments $BD$ and\n$CE$ meet at $I$. Determine whether or not it is possible for\nsegments $AB, AC, BI, ID, ... |
amc_aime | (Titu Andreescu)
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd. |
We will prove the statement using induction on $n$.
#### Base Case:
For $n=1$, the number $5$ is a one-digit number divisible by $5^1 = 5$, and all its digits (in this case, just one digit) are odd. Thus, the base case holds.
#### Inductive Step:
Assume that for some positive integer $k$, there exists a $k$-digit nu... | [
{
"content": "(Titu Andreescu)\nProve that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.",
"role": "user"
},
{
"content": "\nWe will prove the statement using induction on $n$.\n\n#### Base Case:\nFor $n=1$, the number $5$ is a one-di... |
amc_aime | Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers? | We are given the problem of finding integers $a$ and $b$ such that both $a^5b + 3$ and $ab^5 + 3$ are perfect cubes of integers. We will prove by contradiction that no such integers exist.
**Step 1: Assume the contrary.**
Assume there exist integers $a$ and $b$ such that $a^5b + 3 = x^3$ and $ab^5 + 3 = y^3$ for some ... | [
{
"content": "Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?",
"role": "user"
},
{
"content": "We are given the problem of finding integers $a$ and $b$ such that both $a^5b + 3$ and $ab^5 + 3$ are perfect cubes of integers. We will prove by contrad... |
amc_aime | (Titu Andreescu) Prove that for every [nonnegative](https://artofproblemsolving.com/wiki/index.php/Nonnegative) [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $n$, the number $7^{7^n}+1$ is the [product](https://artofproblemsolving.com/wiki/index.php/Product) of at least $2n+3$ (not necessarily disti... | We will prove the statement using mathematical induction.
#### Base Case:
For $n = 0$, we have:
\[ 7^{7^0} + 1 = 7^1 + 1 = 8 = 2^3. \]
Here, $8$ is the product of $3$ primes (not necessarily distinct, as they are all $2$). This satisfies the condition of having at least $2 \times 0 + 3 = 3$ prime factors.
#### Induct... | [
{
"content": "(Titu Andreescu) Prove that for every [nonnegative](https://artofproblemsolving.com/wiki/index.php/Nonnegative) [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $n$, the number $7^{7^n}+1$ is the [product](https://artofproblemsolving.com/wiki/index.php/Product) of at least $2n+3$ ... |
amc_aime | Let $b\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$. Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer. | 1. **Define the function and terms**: Let $S(n) = n + s_b(n)$, where $s_b(n)$ is the sum of the digits of $n$ in base $b$. We call a number *unrepresentable* if it cannot be written as $S(n)$ for any positive integer $n$.
2. **Claim**: We claim that in every interval $(b^p, b^{p+1}]$, where $p$ is a positive integer, ... | [
{
"content": "Let $b\\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$. Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer.",
"role": "user"
},
{
"content":... |
amc_aime | Let $a_n$ be the number of permutations $(x_1, x_2, \dots, x_n)$ of the numbers $(1,2,\dots, n)$ such that the $n$ ratios $\frac{x_k}{k}$ for $1\le k\le n$ are all distinct. Prove that $a_n$ is odd for all $n\ge 1.$ | To prove that $a_n$ is odd for all $n \geq 1$, where $a_n$ is the number of permutations $(x_1, x_2, \dots, x_n)$ of the numbers $(1, 2, \dots, n)$ such that the $n$ ratios $\frac{x_k}{k}$ for $1 \leq k \leq n$ are all distinct, we will use an involution argument.
#### Step 1: Define the involution
Consider the permut... | [
{
"content": "Let $a_n$ be the number of permutations $(x_1, x_2, \\dots, x_n)$ of the numbers $(1,2,\\dots, n)$ such that the $n$ ratios $\\frac{x_k}{k}$ for $1\\le k\\le n$ are all distinct. Prove that $a_n$ is odd for all $n\\ge 1.$",
"role": "user"
},
{
"content": "To prove that $a_n$ is odd for... |
amc_aime | Let $A_1,A_2,...,A_{n+1}$ be distinct subsets of $[n]$ with $|A_1|=|A_2|=\cdots =|A_{n+1}|=3$. Prove that $|A_i\cap A_j|=1$ for some pair $\{i,j\}$. | 1. **Assumption and Contradiction Setup**: Assume for contradiction that for all pairs $\{i, j\}$, $|A_i \cap A_j| \neq 1$. We know that $n \geq 4$ because there are not enough elements in $[n]$ for $n+1$ subsets of size 3 to all have pairwise intersections not equal to 1 when $n < 4$.
2. **Pigeonhole Principle Applic... | [
{
"content": "Let $A_1,A_2,...,A_{n+1}$ be distinct subsets of $[n]$ with $|A_1|=|A_2|=\\cdots =|A_{n+1}|=3$. Prove that $|A_i\\cap A_j|=1$ for some pair $\\{i,j\\}$.",
"role": "user"
},
{
"content": "1. **Assumption and Contradiction Setup**: Assume for contradiction that for all pairs $\\{i, j\\}... |
amc_aime | The isosceles triangle $\triangle ABC$, with $AB=AC$, is inscribed in the circle $\omega$. Let $P$ be a variable point on the arc $\stackrel{\frown}{BC}$ that does not contain $A$, and let $I_B$ and $I_C$ denote the incenters of triangles $\triangle ABP$ and $\triangle ACP$, respectively.
Prove that as $P$ varies, the ... | 1. **Claim**: The circumcircle of triangle $\triangle PI_BI_C$ passes through the fixed point $M$, where $M$ is the midpoint of arc $BC$ that does not contain $A$.
2. **Strategy**: We will show that $M$, $P$, $I_B$, and $I_C$ are cyclic, meaning they lie on the same circle.
3. **Extension of Lines**: Extend $PI_B$ to... | [
{
"content": "The isosceles triangle $\\triangle ABC$, with $AB=AC$, is inscribed in the circle $\\omega$. Let $P$ be a variable point on the arc $\\stackrel{\\frown}{BC}$ that does not contain $A$, and let $I_B$ and $I_C$ denote the incenters of triangles $\\triangle ABP$ and $\\triangle ACP$, respectively.\nP... |
amc_aime | Let $A_1A_2A_3$ be a [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) and let $\omega_1$ be a [circle](https://artofproblemsolving.com/wiki/index.php/Circle) in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\omega_2, \omega_3, \dots, \omega_7$ such that for $k = 2, 3, \dots,... |
#### Step-by-step Proof:
1. **Setup and Notation:**
Let $\triangle A_1A_2A_3$ be a triangle and $\omega_1$ be a circle passing through $A_1$ and $A_2$. Define $\omega_k$ for $k = 2, 3, \dots, 7$ such that each $\omega_k$ is externally tangent to $\omega_{k-1}$ and passes through $A_k$ and $A_{k+1}$, with $A_{n+3} ... | [
{
"content": "Let $A_1A_2A_3$ be a [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) and let $\\omega_1$ be a [circle](https://artofproblemsolving.com/wiki/index.php/Circle) in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\\omega_2, \\omega_3, \\dots, \\omega_7$ such... |
amc_aime | Prove that for each $n\geq 2$, there is a set $S$ of $n$ integers such that $(a-b)^2$ divides $ab$ for every distinct $a,b\in S$. | We will prove the statement by induction on $n$.
#### Base Case: $n = 2$
Consider the set $S = \{1, 2\}$. For any distinct $a, b \in S$, we have two cases:
- If $a = 1$ and $b = 2$, then $(a-b)^2 = (1-2)^2 = 1^2 = 1$ and $ab = 1 \cdot 2 = 2$. Clearly, $1$ divides $2$.
- If $a = 2$ and $b = 1$, then $(a-b)^2 = (2-1)^2 ... | [
{
"content": "Prove that for each $n\\geq 2$, there is a set $S$ of $n$ integers such that $(a-b)^2$ divides $ab$ for every distinct $a,b\\in S$.",
"role": "user"
},
{
"content": "We will prove the statement by induction on $n$.\n\n#### Base Case: $n = 2$\nConsider the set $S = \\{1, 2\\}$. For any ... |
amc_aime | Let $s_1, s_2, s_3, \ldots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \cdots.$ Suppose that $t_1, t_2, t_3, \ldots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ ... | 1. **Representation of Sequences**: Assume $s_i = \frac{a_i}{b_i}$ and $t_i = \frac{c_i}{d_i}$ for all $i$, where $a_i, b_i, c_i, d_i$ are integers and $b_i, d_i \neq 0$.
2. **Initial Condition**: Consider the first two terms $s_1, s_2$ and $t_1, t_2$. By the problem's condition, $(s_2 - s_1)(t_2 - t_1)$ is an integer... | [
{
"content": "Let $s_1, s_2, s_3, \\ldots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \\cdots.$ Suppose that $t_1, t_2, t_3, \\ldots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ i... |
amc_aime | (Kiran Kedlaya) Let $p$ be a prime number and let $s$ be an integer with $0 < s < p$. Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and
$\left\{ \frac{sm}{p} \right\} < \left\{ \frac{sn}{p} \right\} < \frac{s}{p}$
if and only if $s$ is not a divisor of $p-1$.
Note: For $x$ a real number, let $\lf... | We will prove the statement by considering both directions of the biconditional statement.
#### Forward Direction: Assume $s$ is not a divisor of $p-1$.
Suppose $s$ does not divide $p-1$. Then there exist integers $j$ and $r$ such that $p-1 = sj + r$ with $0 < r < s$. We need to find integers $m$ and $n$ such that $0 ... | [
{
"content": "(Kiran Kedlaya) Let $p$ be a prime number and let $s$ be an integer with $0 < s < p$. Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and\n\n\n$\\left\\{ \\frac{sm}{p} \\right\\} < \\left\\{ \\frac{sn}{p} \\right\\} < \\frac{s}{p}$\n\n\nif and only if $s$ is not a divisor of $p-1$... |
amc_aime | Let $\, a_1, a_2, a_3, \ldots \,$ be a sequence of positive real numbers satisfying $\, \sum_{j = 1}^n a_j \geq \sqrt {n} \,$ for all $\, n \geq 1$. Prove that, for all $\, n \geq 1, \,$
\[\sum_{j = 1}^n a_j^2 > \frac {1}{4} \left( 1 + \frac {1}{2} + \cdots + \frac {1}{n} \right).\] | To prove the given inequality, we start by analyzing the given condition and the desired inequality.
**Given Condition:**
\[
\sum_{j = 1}^n a_j \geq \sqrt{n} \quad \text{for all } n \geq 1.
\]
**Desired Inequality:**
\[
\sum_{j = 1}^n a_j^2 > \frac{1}{4} \left(1 + \frac{1}{2} + \cdots + \frac{1}{n}\right).
\]
We wil... | [
{
"content": "Let $\\, a_1, a_2, a_3, \\ldots \\,$ be a sequence of positive real numbers satisfying $\\, \\sum_{j = 1}^n a_j \\geq \\sqrt {n} \\,$ for all $\\, n \\geq 1$. Prove that, for all $\\, n \\geq 1, \\,$\n\\[\\sum_{j = 1}^n a_j^2 > \\frac {1}{4} \\left( 1 + \\frac {1}{2} + \\cdots + \\frac {1}{n} \\ri... |
amc_aime | $ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country. Also, give a Eucli... | To prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country, we follow these steps:
1. **Uniqueness of Point $O$:**
- Assume there are two points, $P$ and $Q$, on the small map such that both $P$ a... | [
{
"content": "$ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the count... |
amc_aime | (Zoran Sunik) A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n+2^{m_n+1}$ where $2^{m_n}$ is the largest power of 2 that is a factor of $n$. Show that if $k\ge 2$ is a positive integ... |
We need to show that the minimum number of jumps needed to reach $2^i k$ (where $k \geq 2$) is greater than the minimum number of jumps needed to reach $2^i$. We will use the properties of powers of 2 and the divisibility rules for the jumps.
#### Step 1: Understanding the Jump Rule
The frog can jump from $n$ to $n+1... | [
{
"content": "(Zoran Sunik) A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n+2^{m_n+1}$ where $2^{m_n}$ is the largest power of 2 that is a factor of $n$. Show that if $k\\ge... |
amc_aime | $N$ is the north pole. $A$ and $B$ are points on a great circle through $N$ equidistant from $N$. $C$ is a point on the equator. Show that the great circle through $C$ and $N$ bisects the angle $ACB$ in the spherical triangle $ABC$ (a spherical triangle has great circle arcs as sides). | 1. **Coordinate Setup**:
Define the Earth as a sphere of radius one centered at $O = (0,0,0)$. Let $N = (0,0,1)$ be the north pole. Define the $xz$-plane to contain the great circle through $N$, $A$, and $B$. Assume without loss of generality that $A$ and $B$ are symmetric about the $z$-axis, so $A = (\cos \phi, 0,... | [
{
"content": "$N$ is the north pole. $A$ and $B$ are points on a great circle through $N$ equidistant from $N$. $C$ is a point on the equator. Show that the great circle through $C$ and $N$ bisects the angle $ACB$ in the spherical triangle $ABC$ (a spherical triangle has great circle arcs as sides).",
"role... |
amc_aime | An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding 2m to the opposite vertex, which is not adjacent to either of the first two vertice... | 1. **Label the vertices and define variables:**
Let the vertices of the pentagon be labeled as $0, 1, 2, 3, 4$ in a clockwise manner. Let $n_i$ be the integer at vertex $i$. Define $s = \sum_{i=0}^4 n_i = 2011$.
2. **Define the invariant $r_i$:**
For each vertex $i$, define $r_i$ as follows:
\[
r_i \equiv ... | [
{
"content": "An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding 2m to the opposite vertex, which is not adjacent to either of... |
amc_aime | (Zuming Feng) Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on side $BC$. Construct point $C_1$ in such a way that convex quadrilateral $APBC_1$ is cyclic, $QC_1 \parallel CA$, and $C_1$ and $Q$ lie on opposite sides of line $AB$. Construct point $B_1$ in such a way that convex quadrilateral ... |
#### Step 1: Establishing the cyclic nature of quadrilaterals
Given that $APBC_1$ and $APCB_1$ are cyclic, we have:
- $\angle APC_1 = \angle ABC_1$ (since $APBC_1$ is cyclic)
- $\angle APB_1 = \angle ACB_1$ (since $APCB_1$ is cyclic)
#### Step 2: Using parallelism to find relationships
Since $QC_1 \parallel CA$ and $... | [
{
"content": "(Zuming Feng) Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on side $BC$. Construct point $C_1$ in such a way that convex quadrilateral $APBC_1$ is cyclic, $QC_1 \\parallel CA$, and $C_1$ and $Q$ lie on opposite sides of line $AB$. Construct point $B_1$ in such a way tha... |
amc_aime | Two points $P$ and $Q$ lie in the interior of a regular tetrahedron $ABCD$. Prove that angle $PAQ < 60^\circ$. |
1. **Setup and Definitions**: Let $ABCD$ be a regular tetrahedron with side length $a$. Consider two points $P$ and $Q$ inside the tetrahedron. Extend lines $AP$ and $AQ$ to meet the plane containing triangle $BCD$ at points $E$ and $F$, respectively.
2. **Properties of $E$ and $F$**: Since $P$ and $Q$ are inside the... | [
{
"content": "Two points $P$ and $Q$ lie in the interior of a regular tetrahedron $ABCD$. Prove that angle $PAQ < 60^\\circ$.",
"role": "user"
},
{
"content": "\n1. **Setup and Definitions**: Let $ABCD$ be a regular tetrahedron with side length $a$. Consider two points $P$ and $Q$ inside the tetrahe... |
amc_aime | For a positive integer $n\geq 3$ plot $n$ equally spaced points around a circle. Label one of them $A$, and place a marker at $A$. One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of $2n$ distinct moves available; two from each point. ... | 1. **Labeling and Initial Setup**:
We label the points in clockwise order as $1, 2, 3, \ldots, n$, where point $A$ is the same as point $1$. We start and end at point $1$, and we must cross over it, either by visiting it again, or else by making the move from point $n$ to point $2$.
2. **Interpretation as Tiling**... | [
{
"content": "For a positive integer $n\\geq 3$ plot $n$ equally spaced points around a circle. Label one of them $A$, and place a marker at $A$. One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of $2n$ distinct moves available... |
amc_aime | A convex hexagon $ABCDEF$ is inscribed in a circle such that $AB=CD=EF$ and diagonals $AD,BE$, and $CF$ are concurrent. Let $P$ be the intersection of $AD$ and $CE$. Prove that $\frac{CP}{PE}=(\frac{AC}{CE})^2$. | 1. **Identify the Similar Triangles:**
- First, we establish the similarity between triangles $\triangle AEC$ and $\triangle QED$.
- Since $A$, $C$, $D$, and $E$ lie on the circle, by the Inscribed Angle Theorem, $\angle ACE = \angle ADE$.
- Since $AB = CD$, and all points lie on the circle, $\angle AEB =... | [
{
"content": "A convex hexagon $ABCDEF$ is inscribed in a circle such that $AB=CD=EF$ and diagonals $AD,BE$, and $CF$ are concurrent. Let $P$ be the intersection of $AD$ and $CE$. Prove that $\\frac{CP}{PE}=(\\frac{AC}{CE})^2$.",
"role": "user"
},
{
"content": "1. **Identify the Similar Triangles:**... |
amc_aime | If $x, y, z$ are reals such that $0\le x, y, z \le 1$, show that $\frac{x}{y + z + 1} + \frac{y}{z + x + 1} + \frac{z}{x + y + 1} \le 1 - (1 - x)(1 - y)(1 - z)$ | To prove the given inequality $\frac{x}{y + z + 1} + \frac{y}{z + x + 1} + \frac{z}{x + y + 1} \le 1 - (1 - x)(1 - y)(1 - z)$, we will follow the steps outlined in the initial solution and provide more detailed calculations and reasoning.
1. **Rewriting the inequality and defining $f(x, y, z)$**:
Define the functio... | [
{
"content": "If $x, y, z$ are reals such that $0\\le x, y, z \\le 1$, show that $\\frac{x}{y + z + 1} + \\frac{y}{z + x + 1} + \\frac{z}{x + y + 1} \\le 1 - (1 - x)(1 - y)(1 - z)$",
"role": "user"
},
{
"content": "To prove the given inequality $\\frac{x}{y + z + 1} + \\frac{y}{z + x + 1} + \\frac{... |
amc_aime | Prove that there exists a positive integer $k$ such that $k\cdot2^n+1$ is composite for every integer $n$. | To prove that there exists a positive integer $k$ such that $k\cdot2^n+1$ is composite for every integer $n$, we will demonstrate that a specific $k$ satisfies this condition and show why it works for all $n$.
1. **Choice of $k$:** We choose $k = 2935363331541925531$.
2. **Congruence Relations for $k$:** We observe ... | [
{
"content": "Prove that there exists a positive integer $k$ such that $k\\cdot2^n+1$ is composite for every integer $n$.",
"role": "user"
},
{
"content": "To prove that there exists a positive integer $k$ such that $k\\cdot2^n+1$ is composite for every integer $n$, we will demonstrate that a specif... |
amc_aime | Let $a$, $b$, and $c$ denote three distinct integers, and let $P$ denote a polynomial having all integral coefficients. Show that it is impossible that $P(a)=b$, $P(b)=c$, and $P(c)=a$. | To prove that it is impossible for $P(a) = b$, $P(b) = c$, and $P(c) = a$ with $a$, $b$, and $c$ being distinct integers and $P$ having integral coefficients, we proceed as follows:
1. **Divisibility Argument**:
Since $P(x)$ is a polynomial with integer coefficients, for any integers $x$ and $y$, $x - y$ divides $P... | [
{
"content": "Let $a$, $b$, and $c$ denote three distinct integers, and let $P$ denote a polynomial having all integral coefficients. Show that it is impossible that $P(a)=b$, $P(b)=c$, and $P(c)=a$.",
"role": "user"
},
{
"content": "To prove that it is impossible for $P(a) = b$, $P(b) = c$, and $P... |
amc_aime | Karl starts with $n$ cards labeled $1,2,3,\dots,n$ lined up in a random order on his desk. He calls a pair $(a,b)$ of these cards swapped if $a>b$ and the card labeled $a$ is to the left of the card labeled $b$. For instance, in the sequence of cards $3,1,4,2$, there are three swapped pairs of cards, $(3,1)$, $(3,2)$, ... | To prove that the number of swapped pairs (inversions) remains the same after Karl's process, we need to analyze how the inversions change when a card is picked up and reinserted in the opposite position.
#### Step 1: Understanding the effect of moving one card
When Karl picks up a card labeled $i$ and reinserts it in... | [
{
"content": "Karl starts with $n$ cards labeled $1,2,3,\\dots,n$ lined up in a random order on his desk. He calls a pair $(a,b)$ of these cards swapped if $a>b$ and the card labeled $a$ is to the left of the card labeled $b$. For instance, in the sequence of cards $3,1,4,2$, there are three swapped pairs of ca... |
amc_aime | (Titu Andreescu) Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that
\[\frac {1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|.\]
When does equality hold? | 1. **Understanding the Problem Statement**: We are given a quadrilateral $ABCD$ circumscribed about a circle, with all interior and exterior angles at least $60^\circ$. We need to prove the inequality:
\[
\frac{1}{3} |AB^3 - AD^3| \leq |BC^3 - CD^3| \leq 3 |AB^3 - AD^3|.
\]
We will focus on proving the righ... | [
{
"content": "(Titu Andreescu) Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that\n\\[\\frac {1}{3}|AB^3 - AD^3| \\le |BC^3 - CD^3| \\le 3|AB^3 - AD^3|.\\]\nWhen does equality hold?",
"role": "user"
},
{
"content": "1. *... |
amc_aime | Show that, for any fixed integer $\,n \geq 1,\,$ the sequence
\[2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots \pmod{n}\]
is eventually constant.
[The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \pmod{n}$ means the remainder which results from dividing $\,a_i\,$ by $\,n$.] | 1. **Assumption of Contradiction**: Assume that the statement is false for some integer $n \geq 1$. This means there exists a smallest integer $b$ for which the sequence
\[2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots \pmod{b}\]
does not eventually become constant.
2. **Exclusion of $b=1$**: Since all integers... | [
{
"content": "Show that, for any fixed integer $\\,n \\geq 1,\\,$ the sequence\n\\[2, \\; 2^2, \\; 2^{2^2}, \\; 2^{2^{2^2}}, \\ldots \\pmod{n}\\]\nis eventually constant. \n[The tower of exponents is defined by $a_1 = 2, \\; a_{i+1} = 2^{a_i}$. Also $a_i \\pmod{n}$ means the remainder which results from dividi... |
amc_aime | For integer $n \ge 2$, let $x_1$, $x_2$, $\dots$, $x_n$ be real numbers satisfying
\[x_1 + x_2 + \dots + x_n = 0, \quad \text{and} \quad x_1^2 + x_2^2 + \dots + x_n^2 = 1.\]
For each subset $A \subseteq \{1, 2, \dots, n\}$, define
\[S_A = \sum_{i \in A} x_i.\]
(If $A$ is the empty set, then $S_A = 0$.)
Prove that for a... | To solve this problem, we first need to understand the constraints and then use probabilistic methods to estimate the number of subsets $A$ such that $S_A \geq \lambda$.
#### Step 1: Understanding the Constraints
Given are $n$ real numbers $x_1, x_2, \dots, x_n$ such that:
1. $x_1 + x_2 + \dots + x_n = 0$
2. $x_1^2 + ... | [
{
"content": "For integer $n \\ge 2$, let $x_1$, $x_2$, $\\dots$, $x_n$ be real numbers satisfying\n\\[x_1 + x_2 + \\dots + x_n = 0, \\quad \\text{and} \\quad x_1^2 + x_2^2 + \\dots + x_n^2 = 1.\\]\nFor each subset $A \\subseteq \\{1, 2, \\dots, n\\}$, define\n\\[S_A = \\sum_{i \\in A} x_i.\\]\n(If $A$ is the e... |
amc_aime | Two rational numbers $\frac{m}{n}$ and $\frac{n}{m}$ are written on a blackboard, where $m$ and $n$ are relatively prime positive integers. At any point, Evan may pick two of the numbers $x$ and $y$ written on the board and write either their arithmetic mean $\frac{x+y}{2}$ or their harmonic mean $\frac{2xy}{x+y}$ on t... | We need to find all pairs $(m,n)$ such that Evan can write the number $1$ on the board in finitely many steps, starting with the numbers $\frac{m}{n}$ and $\frac{n}{m}$, where $m$ and $n$ are relatively prime positive integers.
#### Step 1: Prove that if $m+n = 2^k$ for some integer $k$, then $1$ can be written on the... | [
{
"content": "Two rational numbers $\\frac{m}{n}$ and $\\frac{n}{m}$ are written on a blackboard, where $m$ and $n$ are relatively prime positive integers. At any point, Evan may pick two of the numbers $x$ and $y$ written on the board and write either their arithmetic mean $\\frac{x+y}{2}$ or their harmonic me... |
amc_aime | Chords $AA'$, $BB'$, and $CC'$ of a sphere meet at an interior point $P$ but are not contained in the same plane. The sphere through $A$, $B$, $C$, and $P$ is tangent to the sphere through $A'$, $B'$, $C'$, and $P$. Prove that $AA'=BB'=CC'$. |
#### Step-by-step Proof:
1. **Understanding the Geometry**:
- We are given that chords $AA'$, $BB'$, and $CC'$ of a sphere meet at an interior point $P$ and are not contained in the same plane.
- Two spheres are described: one through points $A$, $B$, $C$, and $P$, and another through points $A'$, $B'$, $C'$, a... | [
{
"content": "Chords $AA'$, $BB'$, and $CC'$ of a sphere meet at an interior point $P$ but are not contained in the same plane. The sphere through $A$, $B$, $C$, and $P$ is tangent to the sphere through $A'$, $B'$, $C'$, and $P$. Prove that $AA'=BB'=CC'$.",
"role": "user"
},
{
"content": "\n#### S... |
amc_aime | Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$. | 1. **Claim**: The ordered pair $(2^n - 1, 2^n + 1)$ satisfies the criteria for all $n \geq 2$.
2. **Relatively Prime Check**: First, we need to verify that $2^n - 1$ and $2^n + 1$ are relatively prime for $n \geq 2$. Note that the difference between $2^n + 1$ and $2^n - 1$ is $2$. The only divisors of $2$ are $1$ and ... | [
{
"content": "Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.",
"role": "user"
},
{
"content": "1. **Claim**: The ordered pair $(2^n - 1, 2^n + 1)$ satisfies the criteria for all $n \\geq 2$.... |
amc_aime | Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \le 4$. Prove that
\[\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.\] |
#### Step 1: Simplify the given condition
Given the condition:
\[ a^2 + b^2 + c^2 + (a + b + c)^2 \le 4, \]
we can expand and rearrange it as:
\[ a^2 + b^2 + c^2 + a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \le 4, \]
which simplifies to:
\[ 2(a^2 + b^2 + c^2 + ab + bc + ca) \le 4, \]
or:
\[ a^2 + b^2 + c^2 + ab + bc + ca \le 2... | [
{
"content": "Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \\le 4$. Prove that\n\\[\\frac{ab + 1}{(a + b)^2} + \\frac{bc + 1}{(b + c)^2} + \\frac{ca + 1}{(c + a)^2} \\ge 3.\\]",
"role": "user"
},
{
"content": "\n#### Step 1: Simplify the given condition\nGiv... |
amc_aime | Prove that there exists an infinite set of points \[\ldots,\,\,\,\,P_{-3},\,\,\,\,P_{-2},\,\,\,\,P_{-1},\,\,\,\,P_0,\,\,\,\,P_1,\,\,\,\,P_2,\,\,\,\,P_3,\,\,\,\,\ldots\] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $... | 1. **Choice of Elliptic Curve and Generator**: We start by considering an elliptic curve and selecting a generator point $g$ on this curve. The generator $g$ is chosen such that it is not the identity element (often represented as the point at infinity in the context of elliptic curves). This ensures that multiples of ... | [
{
"content": "Prove that there exists an infinite set of points \\[\\ldots,\\,\\,\\,\\,P_{-3},\\,\\,\\,\\,P_{-2},\\,\\,\\,\\,P_{-1},\\,\\,\\,\\,P_0,\\,\\,\\,\\,P_1,\\,\\,\\,\\,P_2,\\,\\,\\,\\,P_3,\\,\\,\\,\\,\\ldots\\] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, poi... |
amc_aime | Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \le N \le 2^{2002}$. Prove that it is possible to color every subset of $S$ either blue or red so that the following conditions hold:
(a) the union of any two red subsets is red;
(b) the union of any two blue subsets is blue;
(c) there are exactly ... |
We will prove that it is possible to color every subset of $S$ either blue or red such that the conditions (a), (b), and (c) are satisfied using two different approaches.
#### Solution 1: Inductive Approach
1. **Base Case ($n = 0$):**
- When $n = 0$, the set $S$ is empty. There is only one subset of $S$, which is... | [
{
"content": "Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \\le N \\le 2^{2002}$. Prove that it is possible to color every subset of $S$ either blue or red so that the following conditions hold:\n(a) the union of any two red subsets is red;\n(b) the union of any two blue subsets is bl... |
amc_aime | Let $u$ and $v$ be real numbers such that
\[(u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8.\]
Determine, with proof, which of the two numbers, $u$ or $v$, is larger. | 1. **Define the functions $U(x)$ and $V(x)$:**
\[
U(x) = (x + x^2 + \cdots + x^8) + 10x^9 = \frac{x(x^9 - 1)}{x - 1} + 10x^9
\]
\[
V(x) = (x + x^2 + \cdots + x^{10}) + 10x^{11} = \frac{x(x^{11} - 1)}{x - 1} + 10x^{11}
\]
Here, we used the formula for the sum of a geometric series, $\sum_{k=1}^n x^k... | [
{
"content": "Let $u$ and $v$ be real numbers such that\n\\[(u + u^2 + u^3 + \\cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \\cdots + v^{10}) + 10v^{11} = 8.\\]\nDetermine, with proof, which of the two numbers, $u$ or $v$, is larger.",
"role": "user"
},
{
"content": "1. **Define the functions $U(x)$ and ... |
amc_aime | If $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ are all [polynomials](https://artofproblemsolving.com/wiki/index.php/Polynomial) such that
\[P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x +1) S(x),\]
prove that $x-1$ is a factor of $P(x)$. |
We are given the polynomial equation:
\[ P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x +1) S(x). \]
We need to prove that $x-1$ is a factor of $P(x)$.
#### Step 1: Evaluate the equation at $x=1$
Substituting $x = 1$ into the equation, we get:
\[ P(1^5) + 1 \cdot Q(1^5) + 1^2 \cdot R(1^5) = (1^4 + 1^3 + 1^2 + ... | [
{
"content": "If $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ are all [polynomials](https://artofproblemsolving.com/wiki/index.php/Polynomial) such that\n\\[P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x +1) S(x),\\]\nprove that $x-1$ is a factor of $P(x)$.",
"role": "user"
},
{
"content": "\nWe are giv... |
amc_aime | Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[\min{\left (\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc\] | 1. **Establishing the inequality for each variable**: We start by considering the inequality $(a-1)^5 \geq 0$, which holds because $a \geq 1$. Expanding this inequality, we have:
\[
a^5 - 5a^4 + 10a^3 - 10a^2 + 5a - 1 \geq 0.
\]
Rearranging terms, we obtain:
\[
10a^2 - 5a + 1 \leq a^5 - 5a^4 + 10a^3 =... | [
{
"content": "Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \\[\\min{\\left (\\frac{10a^2-5a+1}{b^2-5b+10},\\frac{10b^2-5b+1}{c^2-5c+10},\\frac{10c^2-5c+1}{a^2-5a+10}\\right )}\\leq abc\\]",
"role": "user"
},
{
"content": "1. **Establishing the inequality for each variab... |
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