This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 26

2019 India PRMO, 7
Tags: algebra , PRMO , number theory , decimal representation
Let $s(n)$ denote the sum of digits of a positive integer $n$ in base $10$. If $s(m)=20$ and $s(33m)=120$, what is the value of $s(3m)$?

2019 India PRMO, 11
Tags: algebra , PRMO , SFFT
Find the largest value of $a^b$ such that the positive integers $a,b>1$ satisfy $$a^bb^a+a^b+b^a=5329$$

2019 India PRMO, 27
Tags: geometry , PRMO , cone
A conical glass is in the form of a right circular cone. The slant height is $21$ and the radius of the top rim of the glass is $14$. An ant at the mid point of a slant line on the outside wall of the glass sees a honey drop diametrically opposite to it on the inside wall of the glass. If $d$ the shortest distance it should crawl to reach the honey drop, what is the integer part of $d$ ? [center][img]https://i.imgur.com/T1Y3zwR.png[/img][/center]

2019 India PRMO, 8
Tags: algebra , number theory , PRMO , Divisibility
Let $F_k(a,b)=(a+b)^k-a^k-b^k$ and let $S={1,2,3,4,5,6,7,8,9,10}$. For how many ordered pairs $(a,b)$ with $a,b\in S$ and $a\leq b$ is $\frac{F_5(a,b)}{F_3(a,b)}$ an integer?

2019 India PRMO, 20
Tags: combinatorics , PRMO
How many $4-$digit numbers $\overline{abcd}$ are there such that $a<b<c<d$ and $b-a<c-b<d-c$ ?

2019 India PRMO, 9
Tags: geometry , PRMO , circumcircle
The centre of the circle passing through the midpoints of the sides of am isosceles triangle $ABC$ lies on the circumcircle of triangle $ABC$. If the larger angle of triangle $ABC$ is $\alpha^{\circ}$ and the smaller one $\beta^{\circ}$ then what is the value of $\alpha-\beta$?

2019 India PRMO, 4
Tags: number theory , PRMO
Let $a_1=24$ and form the sequence $a_n$, $n\geq 2$ by $a_n=100a_{n-1}+134$. The first few terms are $$24,2534,253534,25353534,\ldots$$ What is the least value of $n$ for which $a_n$ is divisible by $99$?

2019 India PRMO, 19
Tags: geometry , PRMO
If $15$ and $9$ are lengths of two medians of a triangle, what is the maximum possible area of the triangle to the nearest integer ?

2018 India PRMO, 27
Tags: combinatorics , PRMO
What is the number of ways in which one can color the squares of a $4\times 4$ chessboard with colors red and blue such that each row as well as each column has exactly two red squares and two blue squares?

2019 India PRMO, 2
Tags: algebra , PRMO , polynomial
If $x=\sqrt2+\sqrt3+\sqrt6$ is a root of $x^4+ax^3+bx^2+cx+d=0$ where $a,b,c,d$ are integers, what is the value of $|a+b+c+d|$?

2017 India PRMO, 28
Tags: number theory , prime numbers , Fermat Pseudo-Primes , PRMO , Euler s Theorem , carmichael s number , Hi
Let $p,q$ be prime numbers such that $n^{3pq}-n$ is a multiple of $3pq$ for [b]all[/b] positive integers $n$. Find the least possible value of $p+q$.

2019 India PRMO, 26
Tags: geometry , PRMO
A friction-less board has the shape of an equilateral triangle of side length $1$ meter with bouncing walls along the sides. A tiny super bouncy ball is fired from vertex $A$ towards the side $BC$. The ball bounces off the walls of the board nine times before it hits a vertex for the first time. The bounces are such that the angle of incidence equals the angle of reflection. The distance travelled by the ball in meters is of the form $\sqrt{N}$, where $N$ is an integer. What is the value of $N$ ?

2019 India PRMO, 14
Tags: algebra , PRMO
Let $\mathcal{R}$ denote the circular region in the $xy-$plane bounded by the circle $x^2+y^2=36$. The lines $x=4$ and $y=3$ divide $\mathcal{R}$ into four regions $\mathcal{R}_i ~ , ~i=1,2,3,4$. If $\mid \mathcal{R}_i \mid$ denotes the area of the region $\mathcal{R}_i$ and if $\mid \mathcal{R}_1 \mid >$ $\mid \mathcal{R}_2 \mid >$ $\mid \mathcal{R}_3 \mid > $ $\mid \mathcal{R}_4 \mid $, determine $\mid \mathcal{R}_1 \mid $ $-$ $\mid \mathcal{R}_2 \mid $ $-$ $\mid \mathcal{R}_3 \mid $ $+$ $\mid \mathcal{R}_4 \mid $.

2019 India PRMO, 13
Tags: OEIS , PRMO , number theory
Consider the sequence $$1,7,8,49,50,56,57,343\ldots$$ which consists of sums of distinct powers of$7$, that is, $7^0$, $7^1$, $7^0+7^1$, $7^2$,$\ldots$ in increasing order. At what position will $16856$ occur in this sequence?

2019 India PRMO, 1
Tags: PRMO OLYMPIAD , PRMO , PRMO 2019 Q1
Form a square with sides of length $5$, triangular pieces from the four coreners are removed to form a regular octagonn. Find the area [b]removed[/b] to the nearest integer.

2019 India PRMO, 24
Tags: number theory , PRMO
For $n \geq 1$, let $a_n$ be the number beginning with $n$ $9$'s followed by $744$; eg., $a_4=9999744$. Define $$f(n)=\text{max}\{m\in \mathbb{N} \mid2^m ~ \text{divides} ~ a_n \}$$, for $n\geq 1$. Find $f(1)+f(2)+f(3)+ \cdots + f(10)$.

2019 India PRMO, 25
Tags: geometry , PRMO
Let $ABC$ be an isosceles triangle with $AB=BC$. A trisector of $\angle B$ meets $AC$ at $D$. If $AB,AC$ and $BD$ are integers and $AB-BD$ $=$ $3$, find $AC$.

2019 India PRMO, 1
Tags: number theory , PRMO
Consider the sequence of numbers $\left[n+\sqrt{2n}+\frac12\right]$, where $[x]$ denotes the greatest integer not exceeding $x$. If the missing integers in the sequence are $n_1<n_2<n_3<\ldots$ find $n_{12}$

2019 India PRMO, 10
Tags: algebra , Clock , time , PRMO
One day I went for a walk in the morning at $x$ minutes past $5'O$ clock, where $x$ is a 2 digit number. When I returned, it was $y$ minutes past $6'O$ clock, and I noticed that (i) I walked for exactly $x$ minutes and (ii) $y$ was a 2 digit number obtained by reversing the digits of $x$. How many minutes did I walk?

2019 India PRMO, 30
Tags: number theory , floor function , PRMO
For any real number $x$, let $\lfloor x \rfloor$ denote the integer part of $x$; $\{ x \}$ be the fractional part of $x$ ($\{x\}$ $=$ $x-$ $\lfloor x \rfloor$). Let $A$ denote the set of all real numbers $x$ satisfying $$\{x\} =\frac{x+\lfloor x \rfloor +\lfloor x + (1/2) \rfloor }{20}$$ If $S$ is the sume of all numbers in $A$, find $\lfloor S \rfloor$

2019 India PRMO, 29
Tags: geometry , PRMO
Let $ABC$ be an acute angled triangle with $AB=15$ and $BC=8$. Let $D$ be a point on $AB$ such that $BD=BC$. Consider points $E$ on $AC$ such that $\angle DEB=\angle BEC$. If $\alpha$ denotes the product of all possible values of $AE$, find $\lfloor \alpha \rfloor$ the integer part of $\alpha$.

2019 India PRMO, 28
Tags: geometry , PRMO
In a triangle $ABC$, it is known that $\angle A=100^{\circ}$ and $AB=AC$. The internal angle bisector $BD$ has length $20$ units. Find the length of $BC$ to the nearest integer, given that $\sin 10^{\circ} \approx 0.174$

2019 India PRMO, 22
Tags: trigonometry , geometry , PRMO
In parallelogram $ABCD$, $AC=10$ and $BD=28$. The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. What is the maximum walue of $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$ ?

2019 India PRMO, 3
Tags: PRMO , number theory , algebra , difference of squares , special factorizations
Find the number of positive integers less than 101 that [i]can not [/i] be written as the difference of two squares of integers.

2019 India PRMO, 12
Tags: combinatorics , counting , Enumeration , Enumerative Combinatorics , PRMO
Let $N$ be the number of ways of choosing a subset of $5$ distinct numbers from the set $${10a+b:1\leq a\leq 5, 1\leq b\leq 5}$$ where $a,b$ are integers, such that no two of the selected numbers have the same units digits and no two have the same tens digit. What is the remainder when $N$ is divided by $73$?