1998 PMWC Problems

Problem I1

Calculate: $\frac{1*2*3+2*4*6+3*6*9+4*8*12+5*10*15}{1*3*5+2*6*10+3*9*15+4*12*20+5*15*25}$

Solution

Problem I2

Triangular numbers and Square numbers can be represented in the following manner:

[asy] int triangle(pair z, int n){ for(int i = 0; i < n; ++i){ for(int j = n-i; j > 0; --j){ dot((z.x+j -1 + i/2 ,z.y + i*sqrt(3)/2)); } } return 0; } triangle((0,0),2); label("3",(0.5,0),2S); triangle((5,0),3); label("6",(6,0),2S); triangle((11,0),4); label("10",(12.5,0),2S); int squ(pair z, int n){ for(int i = 0; i < n; ++i){ for(int j = 0; j < n; ++j){ dot((z.x + i , z.y + j)); } } return 0; } squ((0,-6),2); label("4",(0.5,-6),2S); squ((5,-6),3); label("9",(6,-6),2S); squ((11,-6),4); label("16",(12.5,-6),2S); //Credit to chezbgone2 for the diagram[/asy]

Find a pair of consecutive Triangular Numbers and the difference between a pair of consecutive Square Numbers whose difference are both $11$. What is the sum of these four numbers ?

Solution

Problem I3

Suppose $OB$ and $OA$ are diameters of the semicircles and $OB = OA=3$ cm. $\angle BOA$ is a right angle. $A$ and $B$ are two points on the circumference of circle of radius $OA$. Find the area of the shaded region in $\text{cm}^2$

[asy] draw(arc((0,0),(0,1),(-1,0))--(0,0)--cycle); fill(arc((0,0.5),(0,1),(0,0))--cycle,gray); fill(arc((-0.5,0),(0,0),(-1,0))--cycle,gray); unfill(buildcycle(arc((-0.5,0),(0,0),(-1,0)),arc((0,0.5),(0,1),(0,0)))); draw(arc((0,0.5),(0,1),(0,0))--cycle); draw(arc((-0.5,0),(0,0),(-1,0))--cycle); label("3cm",(0,0)--(0,1),E); label("3cm",(0,0)--(-1,0),S); label("O",(0,0),SE); label("A",(0,1),NE); label("B",(-1,0),SW); //Credit to chezbgone2 for the diagram[/asy]

Solution

Problem I4

Suppose in each day on a certain planet, there are only $10$ hours and every hour has $100$ minutes. What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand at $6$ o'clock $75$ minutes?

Solution

Problem I5

There were many balls which were distributed into $1998$ boxes and all these boxes were arranged in a row. If the second box from the left-hand contained $7$ balls and any $4$ consecutive boxes always had a total of $30$ balls, how many balls were there in the right-hand box?

Solution

Problem I6

After a mathematics test, each of the $25$ students in the class got a quick look at the teacher’s grade sheet. Each student noticed five A’s. No student saw all the grades and no student saw her or his own grade. What is the minimum number of students who scored an A on this test?

Solution

Problem I7

In the figure, $ABCD$ is a $6\times 6$ square with centre $O$. $EOF$ is a right-angled triangle with $OE=8$ and $OF=6$. Find the area of the shaded region.

[asy] import olympiad; fill((8cos(5*pi/6),8sin(5*pi/6))--(6cos(4*pi/3),6sin(4*pi/3))--intersectionpoint((-3,-3)--(3,-3),(6cos(4*pi/3),6sin(4*pi/3))--(0,0))--(-3,-3)--intersectionpoint((8cos(5*pi/6),8sin(5*pi/6))--(0,0),(-3,3)--(-3,-3))--cycle,gray); draw((0,3)--(-3,3)--(-3,-3)--(3,-3)--(3,3)--(0,3)--(0,0)--(3,0)); draw((0,0)--(8cos(5*pi/6),8sin(5*pi/6))--(6cos(4*pi/3),6sin(4*pi/3))--cycle); label("O",(0,0),SE); label("B",(3,3),NE); label("A",(-3,3),N); label("D",(-3,-3),S); label("C",(3,-3),SE); label("E",(8cos(5*pi/6),8sin(5*pi/6)),W); label("F",(6cos(4*pi/3),6sin(4*pi/3)),S); label("6",(0,3),N); label("3",(0,0)--(3,0),N); label("3",(0,0)--(0,3),E); label("6",(0,0)--(6cos(4*pi/3),6sin(4*pi/3)),2S); label("8",(0,0)--(8cos(5*pi/6),8sin(5*pi/6)),2NW); rightanglemark((8cos(5*pi/6),8sin(5*pi/6)),(0,0),(6cos(4*pi/3),6sin(4*pi/3)),10); //Credit to chezbgone2 for the diagram[/asy]

Solution

Problem I8

A boy arranges three kinds of books which are $30$ mm, $24$ mm, and $18$ mm thick, respectively. He places only books of the same thickness into $3$ stacks of equal height, and wants to make the height as small as possible. How many books would be used in this arrangement?

Solution

Problem I9

How many triangles are there with side lengths whole numbers and with a perimeter of $10$ cm ?

Solution

Problem I10

Find the number of factors of $960$.

Solution

Problem I11

What is the units digit of $2^{1998}+3^{1998}$?

Solution

Problem I12

In the pyramid in the diagram, start from the top square containing the number $3$. In each step, go to either of the two squares immediately below. Stop when the bottom row is reached. If the number in the seven squares passed over are different from one another, determine the number in the destination square on the bottom row.

[asy] size(2inch); void box(pair p, int a){ draw(p--(p.x+1,p.y)--(p.x+1,p.y+1)--(p.x,p.y+1)--cycle); label(string(a),(p.x+0.5,p.y+0.5)); return; } int rows[][] = {{3},{6,1},{2,3,4},{5,4,7,2},{3,6,2,6,4},{1,7,5,1,3,6},{7,2,4,6,5,1,7}}; for(int i = 0; i < 7; ++i) { for(int j = 0; j <= i; ++j){ box((-0.5*i + j,-1 * i), rows[i][j]); } } //Credit to chezbgone2 and nsun for making the diagram[/asy]

Solution

Problem I13

Every year there is at least one Friday the thirteenth, but no year has more than three. This year there are exactly three : in February, March and November. When will the next year be that contains exactly three Friday the thirteenths?

Solution

Problem I14

Arrange all fractions of the form $\frac{m}{n}$ in a row where $m$ and $n$ are counting numbers satisfy the following conditions:

(a) If $a\times b<m_2\times n_2$, then $\frac{m_1}{n_1}$ must be placed before $\frac{m_2}{n_2}$.

(b) If $m_1\times n_1=m_2\times n_2$, and $n_1<n_2$, then $\frac{m_1}{n_1}$ must also be placed before $\frac{m_2}{n_2}$.

How many fractions are there between $\frac{1998}{1}$ and $\frac{1}{1998}$?

Solution

Problem I15

Construct a rectangle by putting together nine squares with sides equal to $1, 4, 7, 8, 9, 10, 14, 15,$ and $18$. What is the sum of the areas of the squares on the $4$ corners of the resulting rectangle ?

Solution

Problem T1

What is the $1998^{\text{th}}$ number in the following sequence ?

$1, -2, 2, -3, 3, -3, 4, -4, 4, -4, 5, -5, 5, -5, 5, -6, 6, -6, 6, -6, 6,\dots$

Solution

Problem T2

Tom started work on a job alone for $30$ days. Jerry continued the job alone for $5$ days, and finally they worked together for another $10$ days to complete that job. For the same job, if Tom and Jerry work together, they can complete it in $20$ days. Assuming Tom and Jerry each work at a constant rate throughout, how many days will Tom take to complete that job alone?

Solution

Problem T3

The set $L$ consists of all positive integers which leave a remainder of $1$ when divided by $3$. A member of $L$ (other than $1$) is called an $L$-prime if it is not the product of two members of $L$, other than itself and $1$. Which is the $8$th $L$-prime?

Solution

Problem T4

There are many circles on a plane. Each is divided into four parts by two mutually perpendicular diameters. Each part is painted either red, yellow or blue. No matter how the circles are rotated in the plane, they are different from one another. At most how many circles are painted with all three colors?

Solution

Problem T5

Find the largest positive integer with the following properties:

(a) all the digits are different.

(b) each two consecutive digits form a number divisible by either $17$ or $23$.

Solution

Problem T6

There were $3$ students in an athletics competition of at least two events. Each student participated in all events. In each event, student who finished second got more points than the student who finished third but less than the student who finished first. All scores were positive integers and all the events used the same $3$ scores. At the end of the competition, the total scores of the $3$ students were $5, 9,$ and $16$. Determine the first-place score for each event.

Solution

Problem T7

A leaf is torn from a book of not more than $500$ pages. The sum of the remaining pages numbers is $19905$. What is the sum of the two page numbers of the leaf torn out?

Solution

Problem T8

A rectangular lawn is surrounded by a path 1 meter in width and forming a larger rectangle. The dimensions of the lawn are in whole number of metres and the area of the path equals the area of the lawn. Find the smallest possible area of the path in metres .

Solution

Problem T9

$A, B, C, D,$ and $E$ play a game in which each is either a lion or a goat. A lion’s statement is always false and a goat’s statement is always true.

$A$ says $B$ is not a goat.

$C$ says $D$ is a lion.

$E$ says $A$ is not a lion.

$B$ says $C$ is not a goat.

$D$ says that $E$ and $A$ are different kinds of animals.

Who are the lions?

Solution

Problem T10

In the following expression, each letter represents a digit. Same digits are represented by the same letter, and different letters stand for different digits. Any digit can replace any square, find the 5-digit number $\text{ABCBA}$?

\[\begin{array}{c c c c c}& &\Box &\Box & 1\\ &\times & & 9 &\Box\\ \hline &\Box & 9 &\Box &\Box\\ \Box & 8 &\Box &\Box &\\ \hline A & B & C & B & A\\ \end{array}\]

Solution

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