Question bank xIMC 1999

When $122333444455555666666777777788888888999999999$ is divided by $9$, find the remainder.

Find the sum of the angles $a$, $b$, $c$ and $d$ in the diagram below.

How many of the numbers $1^2,2^2,\ldots,1999^2$ have odd numbers as their tens-digits?

The height of a building is $60$ m. At a certain moment during daytime, it casts a shadow of length $40$ m. If a vertical pole of length $2$ m is erected on the roof of the building, find the length, in m, of the shadow of the pole at the same moment.

Calculate $1999^2-1998^2+1997^2-1996^2+\cdots-2^2+1^2$.

Among all four-digit numbers with $3$ as their thousands-digit, how many have exactly two identical digits?

The diagram below shows an equilateral triangle of side $1$ cm. The three circles touch each other and the sides of the triangle. Find the common radius, in cm, of the circles.

Let $a,b,c$ be positive integers. The sum of $160$ and the square of $a$ is equal to the sum of $5$ and the square of $b$. The sum of $320$ and the square of $a$ is equal to the sum of $5$ and the square of $c$. Find $a$.

Let $x$ be a two-digit number. Denote by $f(x)$ the sum of $x$ and its digits minus the product of its digits. Find the value of $x$ which gives the largest possible value for $f(x)$.

The diagram below shows a right triangle $ABC$. The perpendicular sides $AB$ and $AC$ have lengths $15$ cm and $8$ cm respectively. $D,F$ are points on $AB$, and $E,G$ are points on $AC$. The segments $CD,DE,EF,FG$ divide triangle $ABC$ into five triangles of equal area. The length, in cm, of only one of these segments is integral. What is that length?

How many squares are formed by the grid lines in the diagram below?

There are two committees A and B. Committee A had $13$ members while committee B had $6$ members. Each member is paid $6000$ dollars per day for attending the first $30$ days of meetings, and $9000$ dollars per day thereafter. Committee B met twice as many days as committee A, and the expenditure on attendance were the same for the two committees. If the total expenditure on attendance for these two committees was over $3000000$ dollars, how much was it?

The diagram below shows a cubical wire framework of side $1$ cm. An ant starts from a vertex and crawls along the sides of the framework. If it does not repeat any part of its path and finally returns to the starting vertex, what is the longest possible length, in cm, of the path it has travelled?

In the diagram below, $BC\perp AC$. A point $D$ on $BC$ satisfies $BC=4BD$, and a point $E$ on $AC$ satisfies $AC=8CE$. If $AD=164$ cm and $BE=52$ cm, determine the length, in cm, of $AB$.

When a particular six-digit number is multiplied by $2,3,4,5,6$ respectively, each product is still a six-digit number with the same digits as the original number but in a different order. Find the original number.

(a) Decompose $9^8+7^6+5^4+3^2+1$ into prime factors.

(b) Find two distinct prime factors of $2^{30}+3^{30}$.

The cards in a deck are numbered $1,3,\ldots,2n-1$. In the $k$-th step, $1\le k\le n$, $2k-1$ cards from the top of the deck are transferred to the bottom one at a time. We want the new card on top to be $2k-1$, which is then set aside. After $n$ steps, the whole deck should be set aside in increasing order. How should the deck be stacked for this to happen if (a) $n=10$; (b) $n=30$?

(a) Express $1$ as a sum of reciprocals of distinct integers, one of which is $5$.

(b) Express $1$ as a sum of reciprocals of distinct integers, one of which is $1999$.

(a) Show how to dissect a square into $1999$ squares, which may have different sizes.

(b) Dissect the first two shapes shown below into ten or fewer pieces that can be reassembled to form the third shape.

The diagram below shows a blank $5\times5$ table. Each cell is to be filled with one of $1,2,3,4,5$, so there is exactly one of each number in each row, each column and each of the two long diagonals. The score is the sum of the numbers in the four shaded cells. What is the highest possible score?