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What is the greatest amount of rooks that can be placed on an $n\times n$ board, such that each rooks beats an even number of rooks? A rook is considered to beat another rook, if they lie on one vertical or one horizontal line and no rooks are between them.
[I]Proposed by D. Karpov[/i] | To determine the greatest number of rooks that can be placed on an $n \times n$ board such that each rook beats an even number of other rooks, we need to carefully analyze the placement and interactions of the rooks.
1. **Initial Placement**:
- Place a rook on every square of the $n \times n$ board. This gives us $... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given real numbers $x,y,z,t\in (0,\pi /2]$ such that
$$\cos^2 (x)+\cos^2 (y) +\cos^2 (z) +\cos^2 (t)=1.$$
What is the minimum possible value of $$\cot (x) +\cot (y) +\cot (z) +\cot (t)?$$ | Given the problem, we need to find the minimum possible value of \(\cot(x) + \cot(y) + \cot(z) + \cot(t)\) under the constraint \(\cos^2(x) + \cos^2(y) + \cos^2(z) + \cos^2(t) = 1\) with \(x, y, z, t \in (0, \pi/2]\).
1. **Expressing the cotangent in terms of cosine and sine:**
\[
\cot(x) = \frac{\cos(x)}{\sin(x... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Given are $n$ distinct natural numbers. For any two of them, the one is obtained from the other by permuting its digits (zero cannot be put in the first place). Find the largest $n$ such that it is possible all these numbers to be divisible by the smallest of them? | 1. Let the smallest number among the \( n \) distinct natural numbers be \( x \).
2. Since all numbers are permutations of \( x \), they must have the same digits in some order. Therefore, they must have the same number of digits.
3. For any two numbers \( a \) and \( b \) in this set, \( a \) is obtained by permuting ... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A few (at least $5$) integers are put on a circle, such that each of them is divisible by the sum of its neighbors. If the sum of all numbers is positive, what is its minimal value? | 1. **Understanding the Problem:**
We need to find the minimal positive sum of integers placed on a circle such that each integer is divisible by the sum of its neighbors. The sum of all numbers must be positive.
2. **Analyzing the Divisibility Condition:**
Let the integers be \(a_1, a_2, \ldots, a_n\) arranged i... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different? | 1. **Understanding the problem**: We need to find the smallest \( n \) such that there exists a convex \( n \)-gon where the sines of all angles are equal and the lengths of all sides are different.
2. **Analyzing the angles**: If the sines of all angles are equal, then each angle must be either \( x^\circ \) or \( (1... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
(F.Nilov) Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \angle A \equal{} 50^{\circ}$. Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \angle KAC \equal{} \angle LAB \equal{} 10^{\circ}$. Determine the ratio $ CK/LB$. | 1. **Given Information and Setup:**
- We have a right triangle \( \triangle ABC \) with \( \angle A = 50^\circ \) and hypotenuse \( AC \).
- Points \( K \) and \( L \) are on the cathetus \( BC \) such that \( \angle KAC = \angle LAB = 10^\circ \).
- We need to determine the ratio \( \frac{CK}{LB} \).
2. **Us... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a fixed triangle in the plane. Let $D$ be an arbitrary point in the plane. The circle with center $D$, passing through $A$, meets $AB$ and $AC$ again at points $A_b$ and $A_c$ respectively. Points $B_a, B_c, C_a$ and $C_b$ are defined similarly. A point $D$ is called good if the points $A_b, A_c,B_a, B_c, ... | 1. **Identify the circumcenter \( O \) as a good point:**
- The circumcenter \( O \) of \( \triangle ABC \) is always a good point because the circle centered at \( O \) passing through \( A \) will also pass through \( B \) and \( C \), making \( A_b, A_c, B_a, B_c, C_a, C_b \) concyclic.
2. **Define the isogonal ... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a right angled triangle with $\angle A = 90^o$and $BC = a$, $AC = b$, $AB = c$. Let $d$ be a line passing trough the incenter of triangle and intersecting the sides $AB$ and $AC$ in $P$ and $Q$, respectively.
(a) Prove that $$b \cdot \left( \frac{PB}{PA}\right)+ c \cdot \left( \frac{QC}{QA}\right) =a$$
(b... | Given:
- $\triangle ABC$ is a right-angled triangle with $\angle A = 90^\circ$.
- $BC = a$, $AC = b$, $AB = c$.
- $d$ is a line passing through the incenter of the triangle and intersecting the sides $AB$ and $AC$ at $P$ and $Q$, respectively.
We need to prove:
(a) \( b \cdot \left( \frac{PB}{PA}\right)+ c \cdot \lef... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$. | 1. **Identify the roots and the polynomial:**
Given the polynomial \( P(x) = x^5 - x^2 + 1 \) with roots \( r_1, r_2, r_3, r_4, r_5 \), we need to find the value of the product \( Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5) \) where \( Q(x) = x^2 + 1 \).
2. **Express the product in terms of the roots:**
We need to evaluate:
... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Points $M$ and $N$ are considered in the interior of triangle $ABC$ such that $\angle MAB = \angle NAC$ and $\angle MBA = \angle NBC$. Prove that $$\frac{AM \cdot AN}{AB \cdot AC}+ \frac{BM\cdot BN}{BA \cdot BC}+ \frac{CM \cdot CN }{CA \cdot CB}=1$$ | To prove the given identity, we will use the concept of areas and the Law of Sines. Let's denote the area of triangle $ABC$ by $\Delta$.
1. **Using the Law of Sines in $\triangle AMN$:**
\[
\frac{AM}{\sin(\angle NAC)} = \frac{AN}{\sin(\angle MAB)} = \frac{MN}{\sin(\angle MAN)}
\]
Since $\angle MAB = \angle... | 1 | Geometry | proof | Yes | Yes | aops_forum | false |
How many natural numbers $n$ satisfy the following conditions:
i) $219<=n<=2019$,
ii) there exist integers $x, y$, so that $1<=x<n<y$, and $y$ is divisible by all natural numbers from $1$ to $n$ with the exception of the numbers $x$ and $x + 1$ with which $y$ is not divisible by. | To solve the problem, we need to find the number of natural numbers \( n \) that satisfy the given conditions. Let's break down the problem step by step.
1. **Range of \( n \)**:
We are given that \( 219 \leq n \leq 2019 \).
2. **Existence of integers \( x \) and \( y \)**:
There must exist integers \( x \) and... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine all primes $p$, for which there exist positive integers $m, n$, such that $p=m^2+n^2$ and $p|m^3+n^3+8mn$. | To determine all primes \( p \) for which there exist positive integers \( m \) and \( n \) such that \( p = m^2 + n^2 \) and \( p \mid m^3 + n^3 + 8mn \), we proceed as follows:
1. **Express the given conditions:**
\[
p = m^2 + n^2
\]
\[
p \mid m^3 + n^3 + 8mn
\]
2. **Use the fact that \( p \) is a... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given is a chessboard 8x8. We have to place $n$ black queens and $n$ white queens, so that no two queens attack. Find the maximal possible $n$.
(Two queens attack each other when they have different colors. The queens of the same color don't attack each other) | To solve this problem, we need to place $n$ black queens and $n$ white queens on an 8x8 chessboard such that no two queens attack each other. Queens attack each other if they are on the same row, column, or diagonal. However, queens of the same color do not attack each other.
1. **Understanding the Constraints:**
-... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest positive integer $a$ for which there exist a prime number $p$ and a positive integer $b \ge 2$ such that $$\frac{a^p -a}{p}=b^2.$$ | To determine the smallest positive integer \( a \) for which there exist a prime number \( p \) and a positive integer \( b \ge 2 \) such that
\[ \frac{a^p - a}{p} = b^2, \]
we start by rewriting the given equation as:
\[ a(a^{p-1} - 1) = pb^2. \]
1. **Initial Example:**
- Consider \( a = 9 \), \( p = 2 \), and \(... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$ . Find the maximum value of the following expression $$P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c$$ | 1. Given the non-negative real numbers \(a, b, c\) satisfying \(a^2 + b^2 + c^2 = 2\), we need to find the maximum value of the expression:
\[
P = \frac{\sqrt{b^2+c^2}}{3-a} + \frac{\sqrt{c^2+a^2}}{3-b} + a + b - 2022c
\]
2. We start by considering the cyclic sum:
\[
P = \sum_{\text{cyc}} \left( \frac{\... | 3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Tatjana imagined a polynomial $P(x)$ with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer $k$ and Tatjana tells her the value of $P(k)$. Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined. | To determine the polynomial \( P(x) \) with nonnegative integer coefficients, Danica can use the following steps:
1. **Step 1: Choose \( k = 2 \) and find \( P(2) \).**
- Let \( P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \).
- When \( k = 2 \), we have \( P(2) = a_n 2^n + a_{n-1} 2^{n-1} + \cdots + ... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$, $y$, $z$ be arbitrary positive numbers such that $xy+yz+zx=x+y+z$.
Prove that $$\frac{1}{x^2+y+1} + \frac{1}{y^2+z+1} + \frac{1}{z^2+x+1} \leq 1$$.
When does equality occur?
[i]Proposed by Marko Radovanovic[/i] | 1. Given the condition \(xy + yz + zx = x + y + z\), we start by noting that this implies \(xy + yz + zx \geq 3\). This follows from the AM-GM inequality, which states that for any non-negative real numbers \(a, b, c\), we have:
\[
\frac{a + b + c}{3} \geq \sqrt[3]{abc}
\]
Applying this to \(x, y, z\), we g... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
We call polynomials $A(x) = a_n x^n +. . .+a_1 x+a_0$ and $B(x) = b_m x^m +. . .+b_1 x+b_0$
($a_n b_m \neq 0$) similar if the following conditions hold:
$(i)$ $n = m$;
$(ii)$ There is a permutation $\pi$ of the set $\{ 0, 1, . . . , n\} $ such that $b_i = a_{\pi (i)}$ for each $i \in {0, 1, . . . , n}$.
Let $P(x)$ and ... | Given that \( P(x) = a_n x^n + \ldots + a_1 x + a_0 \) and \( Q(x) = b_n x^n + \ldots + b_1 x + b_0 \) are similar polynomials, we know:
1. \( n = m \)
2. There exists a permutation \(\pi\) of the set \(\{0, 1, \ldots, n\}\) such that \( b_i = a_{\pi(i)} \) for each \( i \in \{0, 1, \ldots, n\} \).
Given \( P(16) = 3^... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$. | To solve the given system of equations for positive integer solutions \((a, b, c, d)\) where \(p\) is a prime number, we start by analyzing the equations:
1. \(ac + bd = p(a + c)\)
2. \(bc - ad = p(b - d)\)
We will consider different cases based on the divisibility of \(a, b, c,\) and \(d\) by \(p\).
### Case 1: \(p... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer and $d$ be the greatest common divisor of $n^2+1$ and $(n + 1)^2 + 1$. Find all the possible values of $d$. Justify your answer. | 1. Let \( d \) be the greatest common divisor (gcd) of \( n^2 + 1 \) and \( (n+1)^2 + 1 \). We need to find all possible values of \( d \).
2. First, we compute the difference between \( (n+1)^2 + 1 \) and \( n^2 + 1 \):
\[
(n+1)^2 + 1 - (n^2 + 1) = (n^2 + 2n + 1 + 1) - (n^2 + 1) = 2n + 2
\]
Therefore, \( ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $\overline{a_1a_2...a_{2009}}$ is a $2009$-digit integer such that for each $i = 1,2,...,2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$
(Note: $\overline{xyz...}$ denotes an integer whose digits are $x, y,z,...$.) | 1. We are given a 2009-digit integer $\overline{a_1a_2 \ldots a_{2009}}$ such that for each $i = 1, 2, \ldots, 2007$, the 2-digit integer $\overline{a_ia_{i+1}}$ contains 3 distinct prime factors.
2. We need to find the value of $a_{2008}$.
Let's analyze the given condition:
- For each $i$, $\overline{a_ia_{i+1}}$ mus... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all prime numbers which can be presented as a sum of two primes and difference of two primes at the same time. | 1. Let the prime be \( p \). Then we have \( p = a + b \) and \( p = c - d \), where \( a \), \( b \), \( c \), and \( d \) are also primes. Without loss of generality, assume \( a > b \).
2. First, consider the case \( p = 2 \). If \( p = 2 \), then \( 2 = a + b \) and \( 2 = c - d \). Since 2 is the only even prime,... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be a set of numbers chosen from $1,2,..., 2015$ with the property that any two distinct numbers, say $x$ and $y$, in $A$ determine a unique isosceles triangle (which is non equilateral) whose sides are of length $x$ or $y$. What is the largest possible size of $A$? | 1. **Understanding the Problem:**
We need to find the largest possible size of a set \( A \) of numbers chosen from \( \{1, 2, \ldots, 2015\} \) such that any two distinct numbers \( x \) and \( y \) in \( A \) determine a unique isosceles triangle (which is non-equilateral) with sides of length \( x \) or \( y \).
... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Show that any representation of 1 as the sum of distinct reciprocals of numbers drawn from the arithmetic progression $\{2,5,8,11,...\}$ such as given in the following example must have at least eight terms: \[1=\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{20}+\frac{1}{41}+\frac{1}{110}+\frac{1}{1640}\] | 1. **Define the Arithmetic Progression and the Problem:**
We are given an arithmetic progression (AP) of the form \(\{2, 5, 8, 11, \ldots\}\) with the first term \(a = 2\) and common difference \(d = 3\). We need to show that any representation of 1 as the sum of distinct reciprocals of numbers from this AP must hav... | 8 | Number Theory | proof | Yes | Yes | aops_forum | false |
What is the smallest possible value of $\left|12^m-5^n\right|$, where $m$ and $n$ are positive integers? | To find the smallest possible value of $\left|12^m - 5^n\right|$ where $m$ and $n$ are positive integers, we start by evaluating some initial values and then proceed to prove that the smallest possible value is indeed $7$.
1. **Initial Evaluation:**
\[
\left|12^1 - 5^1\right| = \left|12 - 5\right| = 7
\]
T... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all prime numbers $p$ for which the number $p^2+11$ has less than $11$ divisors. | To find all prime numbers \( p \) for which the number \( p^2 + 11 \) has less than 11 divisors, we need to analyze the number of divisors of \( p^2 + 11 \).
1. **Divisor Function Analysis**:
The number of divisors of a number \( n \) is given by the product of one plus each of the exponents in its prime factorizat... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The convex quadrilateral $ABCD$ has area $1$, and $AB$ is produced to $E$, $BC$ to $F$, $CD$ to $G$ and $DA$ to $H$, such that $AB=BE$, $BC=CF$, $CD=DG$ and $DA=AH$. Find the area of the quadrilateral $EFGH$. | 1. **Cheap Method:**
- Assume without loss of generality that $ABCD$ is a square with area $1$.
- Since $AB = BE$, $BC = CF$, $CD = DG$, and $DA = AH$, each of the triangles $\triangle AHE$, $\triangle BEF$, $\triangle CFG$, and $\triangle DGH$ will have the same area as the square $ABCD$.
- Therefore, the are... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$ABCD$ is a square of side 1. $P$ and $Q$ are points on $AB$ and $BC$ such that $\widehat{PDQ} = 45^{\circ}$. Find the perimeter of $\Delta PBQ$. | 1. **Define the problem and variables:**
- Let $ABCD$ be a square with side length 1.
- Points $P$ and $Q$ are on sides $AB$ and $BC$ respectively.
- Let $AP = x$ and $CQ = y$, where $0 \leq x, y \leq 1$.
- Therefore, $BP = 1 - x$ and $BQ = 1 - y$.
- We need to prove that the perimeter of $\triangle PBQ$... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Prove that the sum $\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}} +\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}$
is independent of $a$ for $ a \ge - \frac{3}{4}$ and evaluate it. | 1. Let \( s = \sqrt[3]{\frac{a+1}{2} + \frac{a+3}{6}\sqrt{\frac{4a+3}{3}}} \) and \( t = \sqrt[3]{\frac{a+1}{2} - \frac{a+3}{6}\sqrt{\frac{4a+3}{3}}} \). We need to prove that \( s + t \) is independent of \( a \) for \( a \ge -\frac{3}{4} \).
2. Consider the identity for the sum of cubes:
\[
(s + t)^3 = s^3 + t... | 1 | Algebra | proof | Yes | Yes | aops_forum | false |
The function $g$ is defined about the natural numbers and satisfies the following conditions:
$g(2) = 1$
$g(2n) = g(n)$
$g(2n+1) = g(2n) +1.$
Where $n$ is a natural number such that $1 \leq n \leq 2002$.
Find the maximum value $M$ of $g(n).$ Also, calculate how many values of $n$ satisfy the condition of $g(n) = M.$ | 1. **Understanding the function \( g(n) \)**:
- Given conditions:
\[
g(2) = 1
\]
\[
g(2n) = g(n)
\]
\[
g(2n+1) = g(2n) + 1
\]
- We claim that \( g(n) \) is equal to the number of ones in the binary representation of \( n \).
2. **Base cases**:
- For \( n = 1 \):
... | 0 | Other | math-word-problem | Yes | Yes | aops_forum | false |
How many possible areas are there in a convex hexagon with all of its angles being equal and its sides having lengths $1, 2, 3, 4, 5$ and $6,$ in any order?
| 1. **Identify the properties of the hexagon:**
- The hexagon is convex and has all its internal angles equal.
- The sum of the internal angles of a hexagon is \(720^\circ\). Since all angles are equal, each angle is \(120^\circ\).
2. **Assign side lengths:**
- The side lengths of the hexagon are \(1, 2, 3, 4,... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest real number $a$ such that \[\left\{ \begin{array}{l}
x - 4y = 1 \\
ax + 3y = 1\\
\end{array} \right.
\] has an integer solution. | 1. We start with the given system of linear equations:
\[
\begin{cases}
x - 4y = 1 \\
ax + 3y = 1
\end{cases}
\]
2. To eliminate \( y \), we can manipulate the equations. Multiply the first equation by 3 and the second equation by 4:
\[
3(x - 4y) = 3 \implies 3x - 12y = 3
\]
\[
4(ax + ... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There are 7 lines in the plane. A point is called a [i]good[/i] point if it is contained on at least three of these seven lines. What is the maximum number of [i]good[/i] points? | 1. **Counting Intersections:**
- There are 7 lines in the plane. The maximum number of intersections of these lines can be calculated by choosing 2 lines out of 7 to intersect. This is given by the binomial coefficient:
\[
\binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21
\]
- Therefore, there are ... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest natural number $n$ such that for all real numbers $a, b, c, d$ the following holds:
$$(n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d)$$ | 1. **Understanding the problem**: We need to find the largest natural number \( n \) such that the inequality
\[
(n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d)
\]
holds for all real numbers \( a, b, c, d \).
2. **Using the Cauchy-Schwarz Inequality**: To a... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
An integer $n\ge2$ is called [i]resistant[/i], if it is coprime to the sum of all its divisors (including $1$ and $n$).
Determine the maximum number of consecutive resistant numbers.
For instance:
* $n=5$ has sum of divisors $S=6$ and hence is resistant.
* $n=6$ has sum of divisors $S=12$ and hence is not resistant... | 1. **Define the sum of divisors function**: Let \( d(n) \) denote the sum of the divisors of \( n \). For example, if \( n = 6 \), then the divisors of \( 6 \) are \( 1, 2, 3, 6 \), and thus \( d(6) = 1 + 2 + 3 + 6 = 12 \).
2. **Definition of resistant number**: An integer \( n \ge 2 \) is called resistant if it is co... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $q(n)$ denote the sum of the digits of a natural number $n$. Determine $q(q(q(2000^{2000})))$. | 1. **Simplify the problem using properties of digit sums and modular arithmetic:**
We need to determine \( q(q(q(2000^{2000}))) \). Notice that the sum of the digits of a number \( n \) modulo 9 is congruent to \( n \) modulo 9. Therefore, we can simplify the problem by considering \( 2000^{2000} \mod 9 \).
2. **Ca... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Positive real numbers $x,y,z$ have the sum $1$. Prove that $\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7$.
Can number $7$ on the right hand side be replaced with a smaller constant? | Given positive real numbers \( x, y, z \) such that \( x + y + z = 1 \), we need to prove that:
\[
\sqrt{7x + 3} + \sqrt{7y + 3} + \sqrt{7z + 3} \le 7
\]
We start by using the known inequality for non-negative real numbers \( a, b, c \) such that \( a + b + c = 1 \):
\[
2 + \sqrt{\lambda + 1} \leq \sqrt{\lambda a + 1}... | 7 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
An airport contains 25 terminals which are two on two connected by tunnels. There is exactly 50 main tunnels which can be traversed in the two directions, the others are with single direction. A group of four terminals is called [i]good[/i] if of each terminal of the four we can arrive to the 3 others by using only the... | 1. **Understanding the Problem:**
- We have 25 terminals.
- There are 50 main tunnels, each of which is bidirectional.
- The rest of the tunnels are unidirectional.
- A group of four terminals is called *good* if each terminal in the group can reach the other three using only the tunnels connecting them.
2... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a given positive integer $n,$ find
$$\sum_{k=0}^{n} \left(\frac{\binom{n}{k} \cdot (-1)^k}{(n+1-k)^2} - \frac{(-1)^n}{(k+1)(n+1)}\right).$$ | To solve the given sum, we start by defining two functions \( f(x) \) and \( g(x) \) and then use these functions to simplify the given expression.
1. Define the function \( f(x) \) as follows:
\[
f(x) = \sum_{k=0}^n \binom{n}{k} \frac{x^{k+1}}{(k+1)^2}
\]
2. Note that:
\[
(xf'(x))' = \sum_{k=0}^n \bin... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let trapezoid $ABCD$ inscribed in a circle $O$, $AB||CD$. Tangent at $D$ wrt $O$ intersects line $AC$ at $F$, $DF||BC$. If $CA=5, BC=4$, then find $AF$. | 1. **Define Variables and Given Information:**
- Let $AF = x$.
- Given $CA = 5$ and $BC = 4$.
- $AB \parallel CD$ and $DF \parallel BC$.
- $D$ is a point on the circle $O$ such that the tangent at $D$ intersects $AC$ at $F$.
2. **Power of a Point (PoP) Theorem:**
- By the Power of a Point theorem, the p... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Put $1,2,....,2018$ (2018 numbers) in a row randomly and call this number $A$. Find the remainder of $A$ divided by $3$. | 1. We are given the sequence of numbers \(1, 2, \ldots, 2018\) and we need to find the remainder when the number \(A\) formed by arranging these numbers in a row is divided by 3.
2. To solve this, we first consider the sum of the numbers from 1 to 2018. The sum \(S\) of the first \(n\) natural numbers is given by the f... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Assume $A=\{a_{1},a_{2},...,a_{12}\}$ is a set of positive integers such that for each positive integer $n \leq 2500$ there is a subset $S$ of $A$ whose sum of elements is $n$. If $a_{1}<a_{2}<...<a_{12}$ , what is the smallest possible value of $a_{1}$? | 1. Given that \( A = \{a_1, a_2, \ldots, a_{12}\} \) is a set of positive integers, and for each positive integer \( n \leq 2500 \), there exists a subset \( S \) of \( A \) such that the sum of the elements of \( S \) is \( n \).
2. We need to find the smallest possible value of \( a_1 \) given that \( a_1 < a_2 < \ld... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, let $\omega(n)$ denote the number of positive prime divisors of $n$. Find the smallest positive tinteger $k$ such that $2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}$. | 1. Let \( p_0 = 2 < p_1 < p_2 < \cdots \) be the sequence of all prime numbers. For any positive integer \( n \), there exists an index \( i \) such that
\[
p_0 p_1 p_2 \cdots p_{i-1} \leq n < p_0 p_1 p_2 \cdots p_i.
\]
2. The number of distinct prime divisors of \( n \), denoted by \( \omega(n) \), satisfies... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$. | To find the minimum value of \( p \) such that the quadratic equation \( px^2 - qx + r = 0 \) has two distinct real roots in the open interval \( (0,1) \), we need to ensure that the roots satisfy the following conditions:
1. The roots are real and distinct.
2. The roots lie within the interval \( (0,1) \).
Given the ... | 5 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Starting from 37, adding 5 before each previous term, forms the following sequence:
\[37,537,5537,55537,555537,...\]
How many prime numbers are there in this sequence? | 1. **Understanding the Sequence:**
The sequence starts with 37 and each subsequent term is formed by adding 5 before the previous term. The sequence is:
\[
37, 537, 5537, 55537, 555537, \ldots
\]
We can observe that each term in the sequence can be written as:
\[
a_n = 5^n \cdot 10^k + 37
\]
... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For each lattice point on the Cartesian Coordinate Plane, one puts a positive integer at the lattice such that [b]for any given rectangle with sides parallel to coordinate axes, the sum of the number inside the rectangle is not a prime. [/b] Find the minimal possible value of the maximum of all numbers. | To solve this problem, we need to ensure that for any rectangle with sides parallel to the coordinate axes, the sum of the numbers inside the rectangle is not a prime number. We will consider two interpretations of the problem: one where the boundary is included and one where it is not.
### Case 1: Boundary is not inc... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find positive reals $a, b, c$ which maximizes the value of $abc$ subject to the constraint that $b(a^2 + 2) + c(a + 2) = 12$. | To solve the problem, we need to maximize the value of \(abc\) subject to the constraint \(b(a^2 + 2) + c(a + 2) = 12\). We will use the method of Lagrange multipliers to find the maximum value.
1. **Define the function and constraint:**
Let \(f(a, b, c) = abc\) be the function we want to maximize, and let \(g(a, b... | 3 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a unit square. Points $E, F, G, H$ are chosen outside $ABCD$ so that $\angle AEB =\angle BF C = \angle CGD = \angle DHA = 90^o$ . Let $O_1, O_2, O_3, O_4$, respectively, be the incenters of $\vartriangle ABE, \vartriangle BCF, \vartriangle CDG, \vartriangle DAH$. Show that the area of $O_1O_2O_3O_4$ is at... | 1. **Identify the given conditions and setup:**
- We have a unit square \(ABCD\).
- Points \(E, F, G, H\) are chosen outside \(ABCD\) such that \(\angle AEB = \angle BFC = \angle CGD = \angle DHA = 90^\circ\).
- \(O_1, O_2, O_3, O_4\) are the incenters of \(\triangle ABE, \triangle BCF, \triangle CDG, \triangl... | 1 | Geometry | proof | Yes | Yes | aops_forum | false |
Find the maximum number of colors used in coloring integers $n$ from $49$ to $94$ such that if $a, b$ (not necessarily different) have the same color but $c$ has a different color, then $c$ does not divide $a+b$. | To solve the problem, we need to find the maximum number of colors that can be used to color the integers from 49 to 94 such that if \(a\) and \(b\) (not necessarily different) have the same color but \(c\) has a different color, then \(c\) does not divide \(a + b\).
1. **Define the coloring function:**
Let \(\chi(... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f, g$ be bijections on $\{1, 2, 3, \dots, 2016\}$. Determine the value of $$\sum_{i=1}^{2016}\sum_{j=1}^{2016}[f(i)-g(j)]^{2559}.$$
| 1. We start by considering the given double sum:
\[
S = \sum_{i=1}^{2016} \sum_{j=1}^{2016} [f(i) - g(j)]^{2559}
\]
Since \( f \) and \( g \) are bijections on the set \(\{1, 2, 3, \ldots, 2016\}\), we can reindex the sums using \( k = f(i) \) and \( l = g(j) \). This reindexing does not change the value of... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer and let $P$ be the set of monic polynomials of degree $n$ with complex coefficients. Find the value of
\[ \min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \} \] | 1. **Define the problem and the set \( P \):**
Let \( n \) be a positive integer and let \( P \) be the set of monic polynomials of degree \( n \) with complex coefficients. We need to find the value of
\[
\min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \}.
\]
2. **Consider a specific polynomial \( p... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find all primes that can be written both as a sum of two primes and as a difference of two primes. | 1. Let \( p \) be a prime number that can be written both as a sum of two primes and as a difference of two primes. Therefore, we can write:
\[
p = q + r \quad \text{and} \quad p = s - t
\]
where \( q \) and \( r \) are primes, and \( s \) and \( t \) are primes with \( r < q \) and \( t < s \).
2. Since \... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of primes$ p$, such that $ x^{3} \minus{}5x^{2} \minus{}22x\plus{}56\equiv 0\, \, \left(mod\, p\right)$ has no three distinct integer roots in $ \left[0,\left. p\right)\right.$.
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$ | 1. We start with the polynomial \( x^3 - 5x^2 - 22x + 56 \). We need to find the number of primes \( p \) such that the polynomial has no three distinct integer roots in the range \([0, p)\).
2. First, we factorize the polynomial:
\[
x^3 - 5x^2 - 22x + 56 = (x - 2)(x - 7)(x + 4)
\]
This factorization can b... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Find the minimal value of integer $ n$ that guarantees:
Among $ n$ sets, there exits at least three sets such that any of them does not include any other; or there exits at least three sets such that any two of them includes the other.
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)... | To solve this problem, we need to find the minimal value of \( n \) such that among \( n \) sets, there exist at least three sets such that any of them does not include any other, or there exist at least three sets such that any two of them include the other.
1. **Understanding the Problem:**
- We need to ensure th... | 5 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $ p$ and $ q$ be two consecutive terms of the sequence of odd primes. The number of positive divisor of $ p \plus{} q$, at least
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$ | 1. Let \( p \) and \( q \) be two consecutive terms of the sequence of odd primes. By definition, both \( p \) and \( q \) are odd numbers.
2. The sum of two odd numbers is always even. Therefore, \( p + q \) is an even number.
3. Let \( r = p + q \). Since \( r \) is even, it can be written as \( r = 2k \) for some in... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If two faces of a dice have a common edge, the two faces are called adjacent faces. In how many ways can we construct a dice with six faces such that any two consecutive numbers lie on two adjacent faces?
$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 56 \qquad\textbf{(E)}\ \t... | 1. Let's start by placing the numbers \(1\) and \(2\) on the dice. We can visualize the dice in an unfolded (net) form as follows:
\[
\begin{array}{ccc}
& 1 & \\
& 2 & \\
A & B & C \\
& D & \\
\end{array}
\]
2. We need to ensure that any two consecutive numbers lie on adjacent faces. Let's consider the placemen... | 10 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Find the number of distinct integral solutions of $ x^{4} \plus{}2x^{3} \plus{}3x^{2} \minus{}x\plus{}1\equiv 0\, \, \left(mod\, 30\right)$ where $ 0\le x<30$.
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$ | To find the number of distinct integral solutions of \( x^{4} + 2x^{3} + 3x^{2} - x + 1 \equiv 0 \pmod{30} \) where \( 0 \le x < 30 \), we need to consider the polynomial modulo the prime factors of 30, which are 2, 3, and 5.
1. **Modulo 2:**
\[
x^4 + 2x^3 + 3x^2 - x + 1 \equiv x^4 + 0x^3 + x^2 - x + 1 \pmod{2}
... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
$ ABCD$ is a $ 4\times 4$ square. $ E$ is the midpoint of $ \left[AB\right]$. $ M$ is an arbitrary point on $ \left[AC\right]$. How many different points $ M$ are there such that $ \left|EM\right|\plus{}\left|MB\right|$ is an integer?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\... | 1. **Define the problem geometrically:**
- Let \( ABCD \) be a \( 4 \times 4 \) square.
- \( E \) is the midpoint of \( [AB] \), so \( E \) has coordinates \( (2, 4) \).
- \( M \) is an arbitrary point on \( [AC] \). Since \( A \) is at \( (0, 0) \) and \( C \) is at \( (4, 4) \), the coordinates of \( M \) ca... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If $ a,b,c\in {\rm Z}$ and
\[ \begin{array}{l} {x\equiv a\, \, \, \pmod{14}} \\
{x\equiv b\, \, \, \pmod {15}} \\
{x\equiv c\, \, \, \pmod {16}} \end{array}
\]
, the number of integral solutions of the congruence system on the interval $ 0\le x < 2000$ cannot be
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(... | 1. We start with the system of congruences:
\[
\begin{array}{l}
x \equiv a \pmod{14} \\
x \equiv b \pmod{15} \\
x \equiv c \pmod{16}
\end{array}
\]
where \(a, b, c \in \mathbb{Z}\).
2. According to the Chinese Remainder Theorem (CRT), a solution exists if and only if the moduli are pairwise cop... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If inequality $ \frac {\sin ^{3} x}{\cos x} \plus{} \frac {\cos ^{3} x}{\sin x} \ge k$ is hold for every $ x\in \left(0,\frac {\pi }{2} \right)$, what is the largest possible value of $ k$?
$\textbf{(A)}\ \frac {1}{2} \qquad\textbf{(B)}\ \frac {3}{4} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \frac {3}{2} \qquad\tex... | To solve the given inequality \( \frac{\sin^3 x}{\cos x} + \frac{\cos^3 x}{\sin x} \ge k \) for \( x \in \left(0, \frac{\pi}{2}\right) \), we will use the AM-GM inequality and some trigonometric identities.
1. **Rewrite the expression:**
\[
\frac{\sin^3 x}{\cos x} + \frac{\cos^3 x}{\sin x} = \sin^2 x \cdot \frac... | 1 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
If the polynomial $ P\left(x\right)$ satisfies $ 2P\left(x\right) \equal{} P\left(x \plus{} 3\right) \plus{} P\left(x \minus{} 3\right)$ for every real number $ x$, degree of $ P\left(x\right)$ will be at most
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)... | 1. Given the polynomial \( P(x) \) satisfies the functional equation:
\[
2P(x) = P(x + 3) + P(x - 3)
\]
for every real number \( x \).
2. Assume \( P(x) \) is a polynomial of degree \( n \). We will analyze the implications of the given functional equation on the degree of \( P(x) \).
3. Let \( P(x) = a_n... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $ t_{k} \left(n\right)$ show the sum of $ k^{th}$ power of digits of positive number $ n$. For which $ k$, the condition that $ t_{k} \left(n\right)$ is a multiple of 3 does not imply the condition that $ n$ is a multiple of 3?
$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 15 ... | 1. We need to determine for which \( k \) the condition that \( t_k(n) \) is a multiple of 3 does not imply that \( n \) is a multiple of 3. Here, \( t_k(n) \) represents the sum of the \( k \)-th powers of the digits of \( n \).
2. First, observe the behavior of \( a^k \mod 3 \) for different values of \( k \). We kn... | 6 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many pairs of real numbers $ \left(x,y\right)$ are there such that $ x^{4} \minus{} 2^{ \minus{} y^{2} } x^{2} \minus{} \left\| x^{2} \right\| \plus{} 1 \equal{} 0$, where $ \left\| a\right\|$ denotes the greatest integer not exceeding $ a$.
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad... | 1. **Rewrite the given equation:**
\[
x^4 - 2^{-y^2} x^2 - \left\| x^2 \right\| + 1 = 0
\]
We can rewrite this equation by letting \( z = x^2 \). Thus, the equation becomes:
\[
z^2 - 2^{-y^2} z - \left\| z \right\| + 1 = 0
\]
2. **Analyze the equation:**
Since \( z = x^2 \), \( z \geq 0 \). Als... | 2 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Find the number of functions defined on positive real numbers such that $ f\left(1\right) \equal{} 1$ and for every $ x,y\in \Re$, $ f\left(x^{2} y^{2} \right) \equal{} f\left(x^{4} \plus{} y^{4} \right)$.
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text... | 1. Given the function \( f \) defined on positive real numbers such that \( f(1) = 1 \) and for every \( x, y \in \mathbb{R} \), \( f(x^2 y^2) = f(x^4 + y^4) \).
2. Let's analyze the functional equation \( f(x^2 y^2) = f(x^4 + y^4) \).
3. Set \( x = 1 \) and \( y = 1 \):
\[
f(1^2 \cdot 1^2) = f(1^4 + 1^4) \imp... | 1 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
$ a_{i} \in \left\{0,1,2,3,4\right\}$ for every $ 0\le i\le 9$. If $ 6\sum _{i \equal{} 0}^{9}a_{i} 5^{i} \equiv 1\, \, \left(mod\, 5^{10} \right)$, $ a_{9} \equal{} ?$
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$ | 1. We start with the given equation:
\[
6 \sum_{i=0}^{9} a_i 5^i \equiv 1 \pmod{5^{10}}
\]
where \(a_i \in \{0, 1, 2, 3, 4\}\) for \(0 \leq i \leq 9\).
2. Notice that \(5^i \equiv 0 \pmod{5^{i+1}}\) for any \(i\). This means that each term \(a_i 5^i\) for \(i < 9\) will be divisible by \(5^{i+1}\) and thus... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
For how many primes $ p$, there exits unique integers $ r$ and $ s$ such that for every integer $ x$ $ x^{3} \minus{} x \plus{} 2\equiv \left(x \minus{} r\right)^{2} \left(x \minus{} s\right)\pmod p$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}... | To solve the problem, we need to find the number of primes \( p \) for which there exist unique integers \( r \) and \( s \) such that for every integer \( x \), the congruence
\[ x^3 - x + 2 \equiv (x - r)^2 (x - s) \pmod{p} \]
holds.
1. **Equating the polynomials:**
\[ x^3 - x + 2 \equiv (x - r)^2 (x - s) \pmod{... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Square $ BDEC$ with center $ F$ is constructed to the out of triangle $ ABC$ such that $ \angle A \equal{} 90{}^\circ$, $ \left|AB\right| \equal{} \sqrt {12}$, $ \left|AC\right| \equal{} 2$. If $ \left[AF\right]\bigcap \left[BC\right] \equal{} \left\{G\right\}$ , then $ \left|BG\right|$ will be
$\textbf{(A)}\ 6 \minu... | 1. **Identify the given information and set up the coordinate system:**
- Given: $\angle A = 90^\circ$, $|AB| = \sqrt{12}$, $|AC| = 2$.
- Place $A$ at the origin $(0,0)$, $B$ at $(0, \sqrt{12})$, and $C$ at $(2, 0)$.
2. **Construct the square $BDEC$ with center $F$:**
- Since $BDEC$ is a square, $D$ and $E$ a... | 2 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Some of $A,B,C,D,$ and $E$ are truth tellers, and the others are liars. Truth tellers always tell the truth. Liars always lie. We know $A$ is a truth teller. According to below conversation,
$B: $ I'm a truth teller.
$C: $ $D$ is a truth teller.
$D: $ $B$ and $E$ are not both truth tellers.
$E: $ $A$ and $B$ are tr... | 1. We know that $A$ is a truth teller.
2. Let's analyze the statements made by $B, C, D,$ and $E$.
### Case 1: Assume $B$ is a truth teller.
- Since $B$ is a truth teller, $B$'s statement "I'm a truth teller" is true.
- $E$'s statement "A and B are truth tellers" must also be true because $E$ is a truth teller.
- Sinc... | 3 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Let $f(x)=x^3+7x^2+9x+10$. Which value of $p$ satisfies the statement
\[
f(a) \equiv f(b) \ (\text{mod } p) \Rightarrow a \equiv b \ (\text{mod } p)
\]
for every integer $a,b$?
$ \textbf{(A)}\ 5
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 11
\qquad\textbf{(D)}\ 13
\qquad\textbf{(E)}\ 17
$ | To solve the problem, we need to find a prime number \( p \) such that if \( f(a) \equiv f(b) \pmod{p} \), then \( a \equiv b \pmod{p} \) for the polynomial \( f(x) = x^3 + 7x^2 + 9x + 10 \).
1. **Evaluate \( f(x) \) at different values:**
\[
f(0) = 0^3 + 7 \cdot 0^2 + 9 \cdot 0 + 10 = 10
\]
\[
f(1) = 1... | 11 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
A committee with $20$ members votes for the candidates $A,B,C$ by a different election system. Each member writes his ordered prefer list to the ballot (e.g. if he writes $BAC$, he prefers $B$ to $A$ and $C$, and prefers $A$ to $C$). After the ballots are counted, it is recognized that each of the six different permuta... | 1. Define the number of ballots for each permutation of the candidates \(A, B, C\):
\[
\begin{cases}
p = \#(ABC) \\
q = \#(ACB) \\
r = \#(CAB) \\
s = \#(CBA) \\
t = \#(BCA) \\
u = \#(BAC)
\end{cases}
\]
We know that the total number of ballots is 20:
\[
p + q + r + s + t + u = 20
... | 8 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Each of the football teams Istanbulspor, Yesildirek, Vefa, Karagumruk, and Adalet, played exactly one match against the other four teams. Istanbulspor defeated all teams except Yesildirek; Yesildirek defeated Istanbulspor but lost to all the other teams. Vefa defeated all except Istanbulspor. The winner of the game Kar... | To solve this problem, we need to determine the number of ways to order the five football teams such that each team, except the last, defeated the next team. Let's denote the teams as follows:
- \( I \) for Istanbulspor
- \( Y \) for Yesildirek
- \( V \) for Vefa
- \( K \) for Karagumruk
- \( A \) for Adalet
From the ... | 9 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
How many primes $p$ are there such that $2p^4-7p^2+1$ is equal to square of an integer?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. We need to find the number of prime numbers \( p \) such that \( 2p^4 - 7p^2 + 1 \) is a perfect square. Let's denote this perfect square by \( k^2 \), where \( k \) is an integer. Therefore, we have:
\[
2p^4 - 7p^2 + 1 = k^2
\]
2. First, let's check small prime numbers to see if they satisfy the equation.... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many real solution does the equation $\dfrac{x^{2000}}{2001} + 2\sqrt 3 x^2 - 2\sqrt 5 x + \sqrt 3$ have?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 11
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | To determine how many real solutions the equation
\[
\frac{x^{2000}}{2001} + 2\sqrt{3} x^2 - 2\sqrt{5} x + \sqrt{3} = 0
\]
has, we need to analyze the behavior of the function
\[
f(x) = \frac{x^{2000}}{2001} + 2\sqrt{3} x^2 - 2\sqrt{5} x + \sqrt{3}.
\]
1. **Analyze the term \(\frac{x^{2000}}{2001}\):**
- The t... | 0 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
Which of the followings gives the product of the real roots of the equation $x^4+3x^3+5x^2 + 21x -14=0$?
$
\textbf{(A)}\ -2
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ -14
\qquad\textbf{(D)}\ 21
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. Given the polynomial equation:
\[
x^4 + 3x^3 + 5x^2 + 21x - 14 = 0
\]
We need to find the product of the real roots of this equation.
2. We start by attempting to factorize the polynomial. Notice that we can group terms to facilitate factorization:
\[
x^4 + 3x^3 + 5x^2 + 21x - 14 = (x^4 + 5x^2 - 1... | -2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
For how many integers $n$, does the equation system \[\begin{array}{rcl}
2x+3y &=& 7\\
5x + ny &=& n^2
\end{array}\] have a solution over integers?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | To determine the number of integer values of \( n \) for which the system of equations
\[
\begin{array}{rcl}
2x + 3y &=& 7 \\
5x + ny &=& n^2
\end{array}
\]
has integer solutions, we will analyze the system step-by-step.
1. **Express \( x \) in terms of \( y \) from the first equation:**
\[
2x + 3y = 7 \implies ... | 8 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many different solutions does the congruence $x^3+3x^2+x+3 \equiv 0 \pmod{25}$ have?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 6
$ | To determine the number of different solutions to the congruence \(x^3 + 3x^2 + x + 3 \equiv 0 \pmod{25}\), we start by factoring the polynomial.
1. **Factor the polynomial:**
\[
x^3 + 3x^2 + x + 3 = (x+3)(x^2 + 1)
\]
This factorization can be verified by expanding:
\[
(x+3)(x^2 + 1) = x^3 + x + 3x^2... | 6 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
A convex polygon has at least one side with length $1$. If all diagonals of the polygon have integer lengths, at most how many sides does the polygon have?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 7
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. Let \( A_1A_2\cdots A_n \) be a convex polygon with \( A_1A_2 = 1 \). We need to determine the maximum number of sides \( n \) such that all diagonals have integer lengths.
2. Consider the triangle inequality for \( n \geq 6 \). For any three vertices \( A_1, A_2, A_4 \), the triangle inequality states:
\[
|A... | 5 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Berk tries to guess the two-digit number that Ayca picks. After each guess, Ayca gives a hint indicating the number of digits which match the number picked. If Berk can guarantee to guess Ayca's number in $n$ guesses, what is the smallest possible value of $n$?
$
\textbf{(A)}\ 9
\qquad\textbf{(B)}\ 10
\qquad\textbf{(... | To solve this problem, we need to determine the smallest number of guesses, \( n \), that Berk needs to guarantee finding Ayca's two-digit number.
1. **Understanding the Problem:**
- Ayca picks a two-digit number, which means there are 90 possible numbers (from 10 to 99).
- After each guess, Ayca provides a hin... | 10 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
If decimal representation of $2^n$ starts with $7$, what is the first digit in decimal representation of $5^n$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ 9
$ | 1. We start by noting that the decimal representation of \(2^n\) starts with 7. This implies that there exists some integer \(k\) such that:
\[
7 \times 10^k \leq 2^n < 8 \times 10^k
\]
Taking the logarithm base 10 of all parts of the inequality, we get:
\[
\log_{10}(7 \times 10^k) \leq \log_{10}(2^n)... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $ABCD$ be a isosceles trapezoid such that $AB || CD$ and all of its sides are tangent to a circle. $[AD]$ touches this circle at $N$. $NC$ and $NB$ meet the circle again at $K$ and $L$, respectively. What is $\dfrac {|BN|}{|BL|} + \dfrac {|CN|}{|CK|}$?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ ... | 1. **Understanding the Problem:**
We are given an isosceles trapezoid \(ABCD\) with \(AB \parallel CD\) and all sides tangent to a circle. The tangency points imply that the circle is the incircle of the trapezoid. We need to find the value of \(\dfrac{|BN|}{|BL|} + \dfrac{|CN|}{|CK|}\).
2. **Using the Power of a P... | 10 | Geometry | MCQ | Yes | Yes | aops_forum | false |
What is the $33$-rd number after the decimal point of $(\sqrt {10} + 3)^{2001}$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 8
$ | 1. We start by considering the expression \((\sqrt{10} + 3)^{2001}\). To find the 33rd digit after the decimal point, we need to understand the behavior of this expression.
2. By the Binomial Theorem, we can write:
\[
(\sqrt{10} + 3)^{2001} = \sum_{k=0}^{2001} \binom{2001}{k} (\sqrt{10})^k \cdot 3^{2001-k}
\]... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $f$ be a real-valued function defined over ordered pairs of integers such that \[f(x+3m-2n, y-4m+5n) = f(x,y)\] for every integers $x,y,m,n$. At most how many elements does the range set of $f$ have?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 15
\qquad\textbf{(D)}\ 49
\qquad\textbf{(E)}\ \text{In... | 1. Given the functional equation:
\[
f(x+3m-2n, y-4m+5n) = f(x,y)
\]
for all integers \(x, y, m, n\), we need to determine the maximum number of distinct values that the function \(f\) can take.
2. First, consider the case when \(m = 1\) and \(n = 1\):
\[
f(x + 3 \cdot 1 - 2 \cdot 1, y - 4 \cdot 1 + ... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many ordered pairs $(p,n)$ are there such that $(1+p)^n = 1+pn + n^p$ where $p$ is a prime and $n$ is a positive integer?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 1
\qquad\textbf{(D)}\ 0
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | To solve the problem, we need to find the ordered pairs \((p, n)\) such that \((1 + p)^n = 1 + pn + n^p\), where \(p\) is a prime number and \(n\) is a positive integer.
1. **Consider the parity of the equation:**
- If \(p\) is an odd prime, then \(1 + p\) is even.
- Therefore, \((1 + p)^n\) is even for any posi... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many integers $0 \leq x < 125$ are there such that $x^3 - 2x + 6 \equiv 0 \pmod {125}$?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 3
\qquad\textbf{e)}\ \text{None of above}
$ | 1. We start by solving the congruence \( x^3 - 2x + 6 \equiv 0 \pmod{125} \). To do this, we first solve the congruence modulo 5.
2. Consider the congruence \( x^3 - 2x + 6 \equiv 0 \pmod{5} \). We test all possible values of \( x \) modulo 5:
- For \( x \equiv 0 \pmod{5} \): \( 0^3 - 2 \cdot 0 + 6 \equiv 6 \not\equ... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many positive roots does polynomial $x^{2002} + a_{2001}x^{2001} + a_{2000}x^{2000} + \cdots + a_1x + a_0$ have such that $a_{2001} = 2002$ and $a_k = -k - 1$ for $0\leq k \leq 2000$?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 1001
\qquad\textbf{e)}\ 2002
$ | To determine the number of positive roots of the polynomial \( P(x) = x^{2002} + 2002x^{2001} - 2001x^{2000} - 2000x^{1999} - \cdots - 2x - 1 \), we can use Descartes' Rule of Signs. Descartes' Rule of Signs states that the number of positive real roots of a polynomial is either equal to the number of sign changes betw... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many real roots does the polynomial $x^5 + x^4 - x^3 - x^2 - 2x - 2$ have?
$
\textbf{a)}\ 1
\qquad\textbf{b)}\ 2
\qquad\textbf{c)}\ 3
\qquad\textbf{d)}\ 4
\qquad\textbf{e)}\ \text{None of above}
$ | 1. We start with the polynomial \( P(x) = x^5 + x^4 - x^3 - x^2 - 2x - 2 \).
2. We test for possible rational roots using the Rational Root Theorem, which states that any rational root of the polynomial \( P(x) \) must be a factor of the constant term (-2) divided by a factor of the leading coefficient (1). The possib... | 3 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The lengths of two altitudes of a triangles are $8$ and $12$. Which of the following cannot be the third altitude?
$
\textbf{a)}\ 4
\qquad\textbf{b)}\ 7
\qquad\textbf{c)}\ 8
\qquad\textbf{d)}\ 12
\qquad\textbf{e)}\ 23
$ | 1. Denote the area of the triangle by $\Delta$ and the altitudes from vertices $A, B, C$ to the opposite sides by $h_a, h_b, h_c$, respectively. Given $h_a = 8$ and $h_b = 12$.
2. The area $\Delta$ can be expressed in terms of the altitudes and the corresponding sides:
\[
\Delta = \frac{1}{2} a h_a = \frac{1}{2} ... | 4 | Geometry | MCQ | Yes | Yes | aops_forum | false |
The thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of this number?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 2
\qquad\textbf{c)}\ 4
\qquad\textbf{d)}\ 6
\qquad\textbf{e)}\ 8
$ | 1. **Determine the least common multiple (LCM) of 37 and 173:**
\[
\text{LCM}(37, 173) = 37 \times 173 = 6401
\]
Since 37 and 173 are both prime numbers, their LCM is simply their product.
2. **Identify the five-digit multiples of 6401:**
\[
6401 \times 2 = 12802
\]
\[
6401 \times 3 = 19203
... | 2 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
What is the least number of weighings needed to determine the sum of weights of $13$ watermelons such that exactly two watermelons should be weighed in each weigh?
$
\textbf{a)}\ 7
\qquad\textbf{b)}\ 8
\qquad\textbf{c)}\ 9
\qquad\textbf{d)}\ 10
\qquad\textbf{e)}\ 11
$ | 1. **Understanding the problem**: We need to determine the sum of the weights of 13 watermelons using the least number of weighings, where each weighing involves exactly two watermelons.
2. **Initial analysis**: Let's denote the weights of the 13 watermelons as \( x_1, x_2, \ldots, x_{13} \). We need to find the sum \... | 8 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $ABC$ be triangle such that $|AB| = 5$, $|BC| = 9$ and $|AC| = 8$. The angle bisector of $\widehat{BCA}$ meets $BA$ at $X$ and the angle bisector of $\widehat{CAB}$ meets $BC$ at $Y$. Let $Z$ be the intersection of lines $XY$ and $AC$. What is $|AZ|$?
$
\textbf{a)}\ \sqrt{104}
\qquad\textbf{b)}\ \sqrt{145}
\qquad... | 1. **Apply Menelaus' Theorem**: Menelaus' Theorem states that for a triangle \( \triangle ABC \) with a transversal line intersecting \( BC \), \( CA \), and \( AB \) at points \( D \), \( E \), and \( F \) respectively, the following relation holds:
\[
\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = 1
... | 10 | Geometry | MCQ | Yes | Yes | aops_forum | false |
How many primes $p$ are there such that $39p + 1$ is a perfect square?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 3
\qquad\textbf{e)}\ \text{None of above}
$ | 1. Let \( 39p + 1 = n^2 \) for some integer \( n \). This implies:
\[
39p = n^2 - 1
\]
Since \( n^2 - 1 \) can be factored as \( (n+1)(n-1) \), we have:
\[
39p = (n+1)(n-1)
\]
2. The factors of \( 39p \) are \( 1, 3, 13, 39, p, 3p, 13p, \) and \( 39p \). We need to test each of these factors to se... | 3 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $ABCD$ be a trapezoid and a tangential quadrilateral such that $AD || BC$ and $|AB|=|CD|$. The incircle touches $[CD]$ at $N$. $[AN]$ and $[BN]$ meet the incircle again at $K$ and $L$, respectively. What is $\dfrac {|AN|}{|AK|} + \dfrac {|BN|}{|BL|}$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 9
\qquad\textbf{(C)}\ 1... | 1. **Understanding the Problem:**
We are given a trapezoid \(ABCD\) with \(AD \parallel BC\) and \(|AB| = |CD|\). The quadrilateral is tangential, meaning it has an incircle that touches all four sides. The incircle touches \(CD\) at point \(N\). We need to find the value of \(\dfrac{|AN|}{|AK|} + \dfrac{|BN|}{|BL|}... | 10 | Geometry | MCQ | Yes | Yes | aops_forum | false |
For how many integers $x$ is $|15x^2-32x-28|$ a prime number?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 4
\qquad\textbf{e)}\ \text{None of above}
$ | 1. We start with the given expression \( |15x^2 - 32x - 28| \) and need to determine for how many integer values of \( x \) this expression is a prime number.
2. First, we factorize the quadratic expression inside the absolute value:
\[
15x^2 - 32x - 28 = (3x + 2)(5x - 14)
\]
3. We need \( |(3x + 2)(5x - 14)| ... | 2 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many positive integers $A$ are there such that if we append $3$ digits to the rightmost of decimal representation of $A$, we will get a number equal to $1+2+\cdots + A$?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 2
\qquad\textbf{d)}\ 2002
\qquad\textbf{e)}\ \text{None of above}
$ | 1. Let the three digits appended be \( B = \overline{abc} \), where \( a, b, c \) are digits (possibly all zeros). The new number formed by appending these digits to \( A \) is \( 1000A + B \).
2. According to the problem, this new number equals the sum of the first \( A \) positive integers:
\[
1000A + B = 1 + ... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If $2^n$ divides $5^{256} - 1$, what is the largest possible value of $n$?
$
\textbf{a)}\ 8
\qquad\textbf{b)}\ 10
\qquad\textbf{c)}\ 11
\qquad\textbf{d)}\ 12
\qquad\textbf{e)}\ \text{None of above}
$ | To determine the largest possible value of \( n \) such that \( 2^n \) divides \( 5^{256} - 1 \), we need to analyze the factorization of \( 5^{256} - 1 \) and count the powers of 2 in the factorization.
1. **Factorization using difference of powers:**
\[
5^{256} - 1 = (5 - 1)(5 + 1)(5^2 + 1)(5^4 + 1)(5^8 + 1)(5... | 10 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many positive integers $n$ are there such that the equation $\left \lfloor \sqrt[3] {7n + 2} \right \rfloor = \left \lfloor \sqrt[3] {7n + 3} \right \rfloor $ does not hold?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 7
\qquad\textbf{d)}\ \text{Infinitely many}
\qquad\textbf{e)}\ \text{None of above}... | To determine how many positive integers \( n \) exist such that the equation
\[
\left\lfloor \sqrt[3]{7n + 2} \right\rfloor = \left\lfloor \sqrt[3]{7n + 3} \right\rfloor
\]
does not hold, we need to analyze the conditions under which the floor functions of the cube roots of \( 7n + 2 \) and \( 7n + 3 \) are equal.
1.... | 0 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
How many positive integers $n$ are there such that $3n^2 + 3n + 7$ is a perfect cube?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 1
\qquad\textbf{c)}\ 3
\qquad\textbf{d)}\ 7
\qquad\textbf{e)}\ \text{Infinitely many}
$ | To determine how many positive integers \( n \) exist such that \( 3n^2 + 3n + 7 \) is a perfect cube, we start by setting up the equation:
\[ 3n^2 + 3n + 7 = y^3 \]
We need to analyze this equation modulo 3. First, we consider the expression \( 3n^2 + 3n + 7 \) modulo 3:
\[ 3n^2 + 3n + 7 \equiv 0n^2 + 0n + 7 \equiv... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If $a^5 +5a^4 +10a^3 +3a^2 -9a-6 = 0$ where $a$ is a real number other than $-1$, what is $(a + 1)^3$?
$
\textbf{a)}\ 1
\qquad\textbf{b)}\ 3\sqrt 3
\qquad\textbf{c)}\ 7
\qquad\textbf{d)}\ 8
\qquad\textbf{e)}\ 27
$ | 1. Given the polynomial equation:
\[
a^5 + 5a^4 + 10a^3 + 3a^2 - 9a - 6 = 0
\]
we need to find the value of \((a + 1)^3\).
2. Notice that the polynomial can be rewritten in terms of \((a + 1)\). Let \(b = a + 1\). Then \(a = b - 1\).
3. Substitute \(a = b - 1\) into the polynomial:
\[
(b-1)^5 + 5(b-... | 7 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $P$ be a polynomial such that $(x-4)P(2x) = 4(x-1)P(x)$, for every real $x$. If $P(0) \neq 0$, what is the degree of $P$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. Given the polynomial equation:
\[
(x-4)P(2x) = 4(x-1)P(x)
\]
for every real \( x \), we need to determine the degree of \( P \) given that \( P(0) \neq 0 \).
2. First, substitute \( x = 4 \) into the equation:
\[
(4-4)P(8) = 4(4-1)P(4) \implies 0 = 12P(4)
\]
This implies \( P(4) = 0 \). Ther... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many primes $p$ are there such that $5p(2^{p+1}-1)$ is a perfect square?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. Let \( N = 5p(2^{p+1} - 1) \). For \( N \) to be a perfect square, the product \( 5p(2^{p+1} - 1) \) must be a perfect square.
2. Since \( N \) includes a factor of \( p \), for \( N \) to be a perfect square, one of \( 5 \) and \( 2^{p+1} - 1 \) must be divisible by \( p \).
3. First, consider the case where \( p ... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
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