How may positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?
Problem 2 of 30
Suppose $a,\ b$ are positive real numbers such that $a\sqrt{a} + b\sqrt{b} = 183$, $a\sqrt{b} + b\sqrt{a} = 182$.
Find $\dfrac{9}{5}(a + b)$.
Problem 3 of 30
A contractor has two teams of workers: team $A$ and team $B$. Team $A$ can complete a job in $12$ days and team $B$ can do the same job in $36$ days. Team $A$ starts working on the job and team $B$ joins team $A$ after four days. The team $A$ withdraws after two more days. For how many more days should team $B$ work to complete the job?
Problem 4 of 30
Let $a,\ b$ be integers such that all the roots of the equation $(x^2 + ax + 20)(x^2 + 17x + b) = 0$ are negative integers. What is the smallest possible value of $a + b$?
Problem 5 of 30
Let $u,\ v,\ w$ be real numbers in geometric progression such that $u \gt v \gt w$. Suppose $u^{40} = v^n = w^{60}$, find the value of $n$.
Problem 6 of 30
Let the sum $\displaystyle\sum\limits_{n=1}^{9}\dfrac{1}{n(n+1)(n+2)}$ written in its lowest terms be $\dfrac{p}{q}$. Find the value of $q - p$.
Problem 7 of 30
Find the number of positive integers $n$, such that $\sqrt{n} + \sqrt{n + 1} \lt 11$.
Problem 8 of 30
A pen costs $\unicode{0x20B9}\ 11$ and a notebook costs $\unicode{0x20B9}\ 13$. Find the number of ways in which a person can spend exactly $\unicode{0x20B9}\ 1000$ to buy pens and notebooks.
(Note: For international students $\unicode{0x20B9}$ is the symbol for the currency Indian Rupees)
Problem 9 of 30
There are five cities $A,\ B,\ C,\ D,\ E$ on a certain island. Each city is connected to every other city by road. In how many ways can a person, starting from city $A$ come back to $A$ after visiting some cities without visiting a city more than once and without taking the same road more than once? (The order in which he visits the cities also matters: e.g., the routes $A \longrightarrow B \longrightarrow C \longrightarrow A$ and $A \longrightarrow C \longrightarrow B \longrightarrow A$ are different.)
Problem 10 of 30
There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in the four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. In how many ways can the guests be accommodated?
Problem 11 of 30
Let $f(x) = sin\dfrac{x}{3} + cos\dfrac{3x}{10}$ for real $x$. Find the least natural number $n$ such that $f(n\pi + x) = f(x)$ for all real $x$.
Problem 12 of 30
In a class, the total numbers of boys and girls are in the ratio $4:3$. On one day it was found that $8$ boys and $14$ girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class?
Problem 13 of 30
In a rectangle $ABCD$, $E$ is the midpoint of $AB$; $F$ is a point on $AC$ such that $BF$ is perpendicular to $AC$; and $FE$ perpendicular $BD$. Suppose $BC = 8\sqrt{3}$, find $AB$.
Problem 14 of 30
Suppose $x$ is a positive real number such that $\{x\}$, $\lfloor x\rfloor$ and $x$ are in geometric progression. Find the least positive integer $n$ such that $x^n \gt 100$. (Here $\lfloor x\rfloor$ denotes the integer part of $x$ and $\{x\}$ denotes $x - \lfloor x\rfloor$.)
Problem 15 of 30
Integers $1,\ 2,\ 3,\ ...,\ n$ where $n \gt 2$, are written on a board. Two numbers $m,\ k$ such that $1 \lt m \lt k$, $1 \lt k \lt n$ are removed and the average of the remaining numbers is found to be $17$. What is the maximum sum of the two removed numbers?
Problem 16 of 30
Five distinct two digit numbers are in geometric progression. Find the middle term.
Problem 17 of 30
Suppose the altitudes of a triangle are $10,\ 12$ and $15$. What is the integer nearest to the value of the semi-perimeter ?
Problem 18 of 30
If the real numbers $x,\ y,\ z$ are such that $x^2 + 4y^2 + 16z^2 = 48$ and $xy + 4yz + 2zx = 24$, what is the value of $x^2 + y^2 + z^2$?
Problem 19 of 30
Suppose $1,\ 2,\ 3$ are the roots of the equation $x^4 + ax^2 + bx = c$. Find the value of $c$.
Problem 20 of 30
What is the number of triples $(a,\ b,\ c)$ of positive integers such that,
$(i)\ a \lt b \lt c \lt 10$ and
$(ii)\ a,\ b,\ c,\ 10$ form the sides of a quadrilateral?
Problem 21 of 30
Find the number of ordered triples $(a,\ b,\ c)$ of positive integers such that $abc = 108$.
Problem 22 of 30
Suppose in a plane $10$ pairwise nonparallel lines intersect one another. What is the maximum possible number of non-overlapping polygons (with finite areas) that can be formed?
Problem 23 of 30
Suppose an integer $x$, a natural number $n$ and a prime number $p$ satisfy the equation $7x^2 - 44x + 12 = p^n$. Find the largest value of $p$.
Problem 24 of 30
Let $P$ be an interior point of a triangle $ABC$ whose sidelengths are $39,\ 65,\ 78$. The line through $P$ parallel to $BC$ meets $AB$ in $K$ and $AC$ in $L$. The line through $P$ parallel to $CA$ meets $BC$ in $M$ and $BA$ in $N$. The line through $P$ parallel to $AB$ meets $CA$ in $S$ and $CB$ in $T$. If $KL$, $MN$, $ST$ are of equal lengths, find the integer value closest to this common length.
Problem 25 of 30
Let $ABCD$ be a rectangle and let $E$ and $F$ be points on $CD$ and $BC$ respectively such that $area(ADE) = 16$, $area(CEF)=9$ and $area(ABF) = 25$. What is the area of triangle $AEF$?
Problem 26 of 30
Let $AB$ and $CD$ be two parallel chords in a circle with radius $5$ such that the centre $O$ lies between these chords. Suppose $AB = 6$, $CD = 8$. Suppose further that the area of the part of the circle lying between the chords $AB$ and $CD$ is $(m\pi + n)/k$, where $m,\ n,\ k$ are positive integers with $gcd(m, n, k)=1$. What is the value of $m + n + k$?
Problem 27 of 30
Let $\Omega_1$ be a circle with centre $O$ and let $AB$ be a diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$ different from $O$. Suppose another circle $\Omega_2$ with centre $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$ intersecting $\Omega_1$ again at $A_1$ and $B_1$ respectively such that $A_1$ and $B_1$ are on the opposite sides of $AB$. Given that $A_1B = 5$, $AB_1 = 15$ and $OP = 10$, find the radius of $\Omega_1$.
Problem 28 of 30
Let $p,\ q$ be prime numbers such that $n^{3pq}-n$ is a multiple of $3pq$ for $all$ positive integers $n$. Find the least possible value of $p + q$.
Problem 29 of 30
For each positive integer $n$, consider the highest common factor $h_n$ of the two numbers $n!+1$ and $(n + 1)!$. For $n \lt 100$, find the largest value of $h_n$.
Problem 30 of 30
Consider the areas of the four triangles obtained by drawing the diagonals $AC$ and $BD$ of a trapezium $ABCD$. The product of these areas, taken two at time, are computed. If among the six products so obtained, two products are $1296$ and $576$, determine the square root of the maximum possible area of the trapezium to the nearest integer.