Find the Minimum value of a/b + b/c + c/d + d/a IMO 1964
Given that (a + c)(b + d) = ac + bd
IMO 1964 problem- Finding Positive Integers \(n\) for Which \(2^n – 1\) is Divisible by \(7\)
Solution for this problem: Step 1: Understanding the Problem We need to find all positive integers \(n\) such that: \(2^n−1≡0\) \((mod 7)\) This means that \(2^n≡1\) \((mod 7)\). Step 2:Finding the Order of 2 Modulo 7 The order of \(2\) modulo \(7\) is the smallest positive integer \(d\) such that: \(2^d≡1\) \((mod7)\) We calculate powers […]
“Unique IMO 1962 Puzzle Solved: Finding Smallest Number Digit Rotation Trick”
Step 1: Define the Number n Let \(n\) be a number that ends in \(6\), meaning we can express it as: \(n=10m+6n\) where \(m\) represents the number formed by the digits of \(n\) excluding the last digit \((6)\). Step 2: Rearranging the Digits When we move the last digit \((6)\)to the front, the new number […]
Solution for the IMO 2017 Problem: Finding All Prime Triples \((p,q,r)\) Satisfying \(p^q+q^p = r\)
📌 Given: We need to find all triples (p,q,r) of prime numbers such that: \(p^q + q^p = r\) 🔍 Step 1: Testing Small Prime Numbers Start by testing small prime values for \(p\) and \(q\): \(2^3 + 3^2 = 8 + 9 = 17\) Since 17 is a prime number, the triple \((2,3,17)\) satisfies […]
Proving That n Must Be a Power of 2 – IMO 2020 Solution
🔥 Step-by-Step Solution with Detailed Explanation Let \(a_1, a_2, a_3, \ldots, a_n\) be distinct positive integers such that: \(S = a_1 + a_2 + \ldots + a_n\) divides \(T = a_1^2 + a_2^2 + \ldots + a_n^2\) We need to prove that nnn must be a power of 2. 💡 Step 1: Establishing the Key […]
USAMO 2019: Discover the Only Two Integer Pairs That Solve This Challenging Equation!
Problem Statement: Find all pairs of integers \((x,y)\) such that: \(x^2 + y^2 + 7 = 3xy\) 🧠 Step 1: Rearrange the Given Equation First, rearrange the given equation: \( x^2 + y^2 + 7 = 3xy\) Bring all terms to one side: \( x^2 + y^2 – 3xy + 7 = 0\) 🔍 Step […]
Find All Integers \(n\) for which \(n^4 + 4^n\) is Prime – USAMO 2019
🔎 Step 1: Understand the Problem We need to find all integer values of n for which the expression: \(n^4 + 4^n\) is a prime number. A prime number is a positive integer greater than \(1\) that has no divisors other than \(1\) and itself. 🔥 Step 2: Try Small Values of n Let’s check […]
🧮 Solution to the IMO 2020 Problem – Finding Integer Pairs (a, b)
💡 Problem Statement: Find all pairs of integers \((a,b)\) such that: \(a^3 + b^3 = (a + b)^3 – 2020\) ✅ Step 1: Expand the Cubic Expression First, expand the right-hand side of the equation: \((a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2\) Now, substitute into the original equation: \(a^3 + b^3 […]
IMO 1964 Math Problem: Finding Three-Digit Numbers Satisfying Two Conditions
✅ Solving the IMO 1964 Math Problem: Finding Three-Digit Numbers Satisfying Two Conditions 📚 Problem Statement: We need to find all three-digit numbers \(N\) that satisfy the following conditions: 🔍 Let’s Define the Number: Let \(N=100a+10b+c,\) where: We need to solve for: \( N = a^2 + b^2 + c^2\) And the divisibility condition: \((a […]
Nowruz: The Persian New Year of Renewal and Joy
🌿 Introduction: The Spirit of Nowruz Nowruz, also spelled Nawroz or Norooz, marks the Persian New Year and the arrival of spring. Celebrated around March 20 or 21, it coincides with the vernal equinox, symbolizing new beginnings, hope, and growth. Millions of people across Iran, Central Asia, the Middle East, and South Asia celebrate this […]