A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number of pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is $1709$. If $n$ is the last page number, what is the largest prime factor of $n$?
Problem 2 of 30
In a quadrilateral $ABCD$, it is given that $AB = AD = 13$, $BC = CD = 20$, $BD = 24$. If $r$ is the radius of the circle inscribable in the quadrilateral, then what is the integer closest to $r$?
Problem 3 of 30
Consider all $6$-digit numbers of the form $abccba$, where $b$ is odd. Determine the number of all such $6$-digit numbers that are divisible by $7$.
Problem 4 of 30
The equation $166 \times 56 = 8590$ is valid in some base $b \ge 10$ (that is $1,\ 6,\ 5,\ 8,\ 9,\ 0$ are digits in base $b$ in the above equation).
Find the sum of all possible values of $b \ge 10$ satisfying the equation.
Problem 5 of 30
Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$.
Problem 6 of 30
Integers $a,\ b,\ c$ satisfy the $a + b - c = 1$ and $a^2 + b^2 -c^2 = 1$
What is the sum of all possible values of $a^2 + b^2 + c^2$?
Problem 7 of 30
A point $P$ in the interior of a regular hexagon is at a distance $8,\ 8,\ 16$ units from three consecutive vertices of the hexagon, respectively. If $r$ is the radius of the circumscribed circle of the hexagon, what is the integer closest to $r$?
Problem 8 of 30
Let $AB$ be a chord of circle with centre $O$. Let $C$ be a point on the circle such that $\angle ABC = 30^\circ$ and $O$ lies inside the $\triangle ABC$. Let $D$ be a point on $AB$ such that $\angle DCO = \angle OCB = 20^\circ$. Find the measure of $\angle CDO$ in degrees.
Problem 9 of 30
Suppose $a,\ b$ are integers $a + b$ is a root of the equation $x^2 + ax + b = 0$. What is the maximum possible value of $b^2$?
Problem 10 of 30
In a triangle $ABC$, the median from $B$ to $CA$ is perpendicular to the median from $C$ to $AB$. If the median from $A$ to $BC$ is $30$, determine $(BC^2 + CA^2 + AB^2)/100$
Problem 11 of 30
There are several tea cups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly $1200$. What is the maximum possible number of cups in the kitchen?
Problem 12 of 30
Determine the numbers of $8$-tuples $(\epsilon_1,\ \epsilon_2,\ ...\ \epsilon_8)$ such that $\epsilon_1,\ \epsilon_2,\ ...\ \epsilon_8 \ \Large{\epsilon} \normalsize \ [1,\ -1]$
and
$\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 + ... + 8\epsilon_8$ is divisible by $3$.
Problem 13 of 30
In a $\triangle ABC$, right-angled at $A$, the altitude through $A$ and the internal bisector of $\angle A$ have lengths $3$ and $4$ respectively. Find the length of the median through $A$.
Problem 14 of 30
If $x = cos1^\circ cos2^\circ cos3^\circ\ ...\ cos89^\circ$ and
Triangle $ABC$ and $DEF$ are such that $\angle A = \angle D$, $AB = DE = 17$, $BC = EF = 10$ and $AC - DF = 12$. What is $AC + DF$?
Problem 18 of 30
If $a,\ b,\ c \ge 4$ are integers, not all equal, and $4abc = (a + 3)(b+3)(c+3)$, then what is the value of $a + b + c$?
Problem 19 of 30
Let $N = 6 + 66 + 666 +\ ...\ + 666...66$, where there are hundred $6$'s in the last term in the sum. How many times does the digit $7$ occur in the number $N$?
Problem 20 of 30
Determine the sum of all possible positive integers $n$, the product of whose digits equals $n^2 - 15n - 27$.
Problem 21 of 30
Let $ABC$ be an acute-angled triangle and let $H$ be its orthocentre. Let $G_1$, $G_2$ and $G_3$ be the centroids of the triangles $HBC$, $HCA$ and $HAB$ respectively. If the area of triangle $G_1G_2G_3$ is $7$ units, what is the area of triangle $ABC$?
Problem 22 of 30
A positive integer $k$ is said to be good if there exists a partition of $\{1,\ 2,\ 3,\ ...\ 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many good numbers are there?
Problem 23 of 30
What is the largest positive integer $n$ such that
If $N$ be the number of triangles of different shapes (i.e. not similar) whose angles are all integers (in degrees), what is $\dfrac{N}{100}$.
Problem 25 of 30
Let $T$ be the smallest positive integer which, when divided by $11$, $13$, $15$ leaves remainders in the sets $\{7,\ 8,\ 9\}$, $\{1,\ 2,\ 3\}$, $\{4,\ 5,\ 6\}$ respectively. What is the sum of the squares of the digits of $T$?
Problem 26 of 30
What is the number of ways in which one can choose $60$ unit squares from a $11 \times 11$ chessboard, such that no two chosen squares have a side in common?
Problem 27 of 30
What is the number of ways in which one can colour the squares of a $4 \times 4$ chessboard with colours red and blue such that each row as well as each column has exactly two red squares and two blue squares?
Problem 28 of 30
Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N$.
Problem 29 of 30
Let $D$ be an interior point of the side $BC$ of a triangle $ABC$. Let $I_1$ and $I_2$ be the incentres of triangles $ABD$ and $ACD$ respectively. Let $AI_1$ and $AI_2$ meet $BC$ in $E$ and $F$ respectively.
If $\angle BI_1E = 60^\circ$, what is the measure of $\angle CI_2F$ in degrees?
Problem 30 of 30
Let $P(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$ be a polynomial in which $a_i$ is non-negative integer for each $i\ \epsilon\ \{0,\ 1,\ 2,\ 3,\ ...,\ n\}$.
If $P(1) = 4$ and $P(5) = 136$, what is the value of $P(3)$