Olympiad problems

2022 IOQM India, 10

Suppose that $P$ is the polynomial of least degree with integer coefficients such that $$P(\sqrt{7} + \sqrt{5}) = 2(\sqrt{7} - \sqrt{5})$$Find $P(2)$.

2022 IOQM India, 7

Tags: IOQM

Find the number of maps $f: \{1,2,3\} \rightarrow \{1,2,3,4,5\}$ such that $f(i) \le f(j)$ whenever $i < j$.

In a paralleogram $ABCD$ , a point $P$ on the segment $AB$ is taken such that $\frac{AP}{AB}=\frac{61}{2022}$\\ and a point $Q$ on the segment $AD$ is taken such that $\frac{AQ}{AD}=\frac{61}{2065}$.If $PQ$ intersects $AC$ at $T$, find $\frac{AC}{AT}$ to the nearest integer

2022-23 IOQM India, 1

Tags: geometry , IOQM

A triangle $ABC$ with $AC=20$ is inscribed in a circle $\omega$. A tangent $t$ to $\omega$ is drawn through $B$. The distance $t$ from $A$ is $25$ and that from $C$ is $16$.If $S$ denotes the area of the triangle $ABC$, find the largest integer not exceeding $\frac{S}{20}$

2022-23 IOQM India, 6

Tags: algbera , IOQM

Let $a,b$ be positive integers satisfying $a^3-b^3-ab=25$. Find the largest possible value of $a^2+b^3$.

2023-24 IOQM India, 20

Tags: IOQM , combinatorics

For any finite non empty set $X$ of integers, let $\max (X)$ denote the largest element of $X$ and $|X|$ denote the number of elements in $X$. If $N$ is the number of ordered pairs $(A, B)$ of finite non-empty sets of positive integers, such that $$ \begin{aligned} & \max (A) \times|B|=12 ; \text { and } \\ & |A| \times \max (B)=11 \end{aligned} $$ and $N$ can be written as $100 a+b$ where $a, b$ are positive integers less than 100 , find $a+b$.

2024-25 IOQM India, 1

Tags: IOQM , factorization , number theory

The smallest positive integer that does not divide $1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9$ is:

2020-21 KVS IOQM India, 1

Tags: IOQM , KV

If $a,b,c$ are real numbers and $(a+b-5)^2+(b+2c+3)^2+(c+3a-10)^2=0$ find the integer nearest to $a^3+b^3+c^3$.

2022-23 IOQM India, 22

Tags: combinatorics , IOQM

A binary sequence is a sequence in which each term is equal to $0$ or $1$. A binary sequence is called $\text{friendly}$ if each term is adjacent to at least on term that is equal to $1$. For example , the sequence $0,1,1,0,0,1,1,1$ is $\text{friendly}$. Let $F_{n}$ denote the number of $\text{friendly}$ binary sequences with $n$ terms. Find the smallest positive integer $n\ge 2$ such that $F_{n}>100$

2022 IOQM India, 8

Tags: floor function , IOQM

For any real number $t$, let $\lfloor t \rfloor$ denote the largest integer $\le t$. Suppose that $N$ is the greatest integer such that $$\left \lfloor \sqrt{\left \lfloor \sqrt{\left \lfloor \sqrt{N} \right \rfloor}\right \rfloor}\right \rfloor = 4$$Find the sum of digits of $N$.

2023-24 IOQM India, 26

Tags: combinatorics , IOQM

In the land of Binary , the unit of currency is called Ben and currency notes are available in denominations $1,2,2^2,2^3,..$ Bens. The rules of the Government of Binary stipulate that one can not use more than two notes of any one denomination in any transaction. For example, one can give change for $2$ Bens in two ways : $2$ one Ben notes or $1$ two Ben note. For $5$ Ben one can given $1$ one Ben and $1$ four Ben note or $1$ Ben note and $2$ two Ben notes. Using $5$ one Ben notes or $3$ one Ben notes and $1$ two Ben notes for a $5$ Ben transaction is prohibited. Find the number of ways in which one can give a change $100$ Bens following the rules of the Government.

2022 IOQM India, 9

Tags: geometry , IOQM

Let $P_0 = (3,1)$ and define $P_{n+1} = (x_n, y_n)$ for $n \ge 0$ by $$x_{n+1} = - \frac{3x_n - y_n}{2}, y_{n+1} = - \frac{x_n + y_n}{2}$$Find the area of the quadrilateral formed by the points $P_{96}, P_{97}, P_{98}, P_{99}$.

2023-24 IOQM India, 14

Tags: geometry , coordinate geometry , Medians , Centroid , IOQM

Let $A B C$ be a triangle in the $x y$ plane, where $B$ is at the origin $(0,0)$. Let $B C$ be produced to $D$ such that $B C: C D=1: 1, C A$ be produced to $E$ such that $C A: A E=1: 2$ and $A B$ be produced to $F$ such that $A B: B F=1: 3$. Let $G(32,24)$ be the centroid of the triangle $A B C$ and $K$ be the centroid of the triangle $D E F$. Find the length $G K$.