Wednesday, March 31, 2021

1979 IMO Problems And Solutions

Problem 01

If $p$ and $q$ are natural numbers so that $\frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319},$ Prove that $p$ is divisible with $1979$ .

Solution

1963 IMO Problems And Solutions

Problem 01

Find all real roots of the equation

$\sqrt{x^2-p}+2\sqrt{x^2-1}=x$ ,

where $p$ is a real parameter.

Solution

Problem 01

Real numbers $a_1, a_2, \dots , a_n$ are given. For each $i$ ( $1\le i\le n$ ) define

$d_i=\max\{a_j:1\le j\le i\}-\min\{a_j:i\le j\le n\}$

and let

$d=\max\{d_i:1\le i\le n\}$ .

(a) Prove that, for any real numbers $x_1\le x_2\le \cdots\le x_n$ ,

$\max\{|x_i-a_i|:1\le i\le n\}\ge \dfrac{d}{2} (*)$

(b) Show that there are real numbers $x_1\le x_2\le x_n$ such that equality holds in (*)

Solution

Problem 01

Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$ . Let $n_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i +a_j$ divides $s_A$ . Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$

Solution

Problem 01

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that, for all integers $a$ and $b$ , $f(2a) + 2f(b) = f(f(a + b)).$

Solutions

Solution 1

Let us substitute $0$ in for $a$ to get $f(0) + 2f(b) = f(f(b)).$

Now, since the domain and range of $f$ are the same, we can let $x = f(b)$ and $f(0)$ equal some constant $c$ to get $c + 2x = f(x).$

Problem 01

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$ .

Solutions

Solution 1

Let $N = 100a + 10b+c$ for some digits $a,b,$ and $c$ . Then $100a + 10b+c = 11m$ for some $m$ . We also have $m=a^2+b^2+c^2$ . Substituting this into the first equation and simplification, we get

Problem 01 (1959 IMO)

Prove that the fraction $\frac{21n+4}{14n+3}$ is irreducible for every natural number.

Solution 01 (Euclidean Algorithm)

Cambodian Olympiad Math

Cambodian Olympiad Math is written by Lim Sovanvichet . This book is shared on social media by many students which is very important document for all Khmer students. In this book, there are many International Math Problems which he translated from other foreign books and wrote them in to Khmer language.

Download This Book Here:

Math Book Cambodia

Pages

Wednesday, March 31, 2021

1979 IMO Problems And Solutions

Problem 01

Solution

Monday, March 29, 2021

1963 IMO Problems And Solutions

Problem 01

Solution

2007 IMO Problems And Solutions

Problem 01

Solution

2011 IMO Problems And Solutions

Problem 01

Solution

2019 IMO Problems And Solutions

Problem 01

Solutions

Solution 1

1960 IMO Problems And Solutions

Problem 01

Solutions

Solution 1

1959 Romania IMO | Problem 01

Problem 01 (1959 IMO)

Solution 01 (Euclidean Algorithm)

Thursday, March 25, 2021

Cambodian Olympiad Math

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