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Difference between revisions of "1998 PMWC Problems"

(Added team test--Problem T10 needs diagram!)
m (→‎Problem T10)
 
(19 intermediate revisions by one other user not shown)
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== Problem I2 ==
 
== 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 <math>11</math>. What is the sum of these four numbers ?
  
 
[[1998 PMWC Problems/Problem I2|Solution]]
 
[[1998 PMWC Problems/Problem I2|Solution]]
  
 
== Problem I3 ==
 
== Problem I3 ==
 +
Suppose <math>OB</math> and <math>OA</math> are diameters of the semicircles and <math>OB = OA=3</math> cm. <math>\angle BOA</math> is a right angle. <math>A</math> and <math>B</math> are two points on the circumference of circle of radius <math>OA</math>. Find the area of the shaded region in <math>\text{cm}^2</math>
 +
 +
<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>
  
 
[[1998 PMWC Problems/Problem I3|Solution]]
 
[[1998 PMWC Problems/Problem I3|Solution]]
  
 
== Problem I4 ==
 
== 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?
+
Suppose in each day on a certain planet, there are only <math>10</math> hours and every hour has <math>100</math> minutes. What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand at <math>6</math> o'clock <math>75</math> minutes?
  
 
[[1998 PMWC Problems/Problem I4|Solution]]
 
[[1998 PMWC Problems/Problem I4|Solution]]
  
 
== Problem I5 ==
 
== 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?  
+
There were many balls which were distributed into <math>1998</math> boxes and all these boxes were arranged in a row.  If the second box from the left-hand contained <math>7</math> balls and any <math>4</math> consecutive boxes always had a total of <math>30</math> balls, how many balls were there in the right-hand box?  
  
 
[[1998 PMWC Problems/Problem I5|Solution]]
 
[[1998 PMWC Problems/Problem I5|Solution]]
  
 
== Problem I6 ==
 
== 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  
+
After a mathematics test, each of the <math>25</math> 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?  
 
this test?  
  
Line 29: Line 78:
  
 
== Problem I7 ==
 
== Problem I7 ==
 +
In the figure, <math>ABCD</math> is a <math>6\times 6</math> square with centre <math>O</math>. <math>EOF</math> is a right-angled triangle with <math>OE=8</math> and <math>OF=6</math>. 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>
  
 
[[1998 PMWC Problems/Problem I7|Solution]]
 
[[1998 PMWC Problems/Problem I7|Solution]]
  
 
== Problem I8 ==
 
== 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?  
+
A boy arranges three kinds of books which are <math>30</math> mm, <math>24</math> mm, and <math>18</math> mm thick, respectively.  He places only books of the same thickness into <math>3</math> stacks of equal height, and wants to make the height as small as possible.  How many books would be used in this arrangement?  
  
 
[[1998 PMWC Problems/Problem I8|Solution]]
 
[[1998 PMWC Problems/Problem I8|Solution]]
Line 39: Line 109:
 
== Problem I9 ==
 
== Problem I9 ==
 
How many triangles are there with side lengths whole numbers and  
 
How many triangles are there with side lengths whole numbers and  
−
with a perimeter of 10 cm ?  
+
with a perimeter of <math>10</math> cm ?  
  
 
[[1998 PMWC Problems/Problem I9|Solution]]
 
[[1998 PMWC Problems/Problem I9|Solution]]
  
 
== Problem I10 ==
 
== Problem I10 ==
−
Find the number of  factors of 960.
+
Find the number of  factors of <math>960</math>.
  
 
[[1998 PMWC Problems/Problem I10|Solution]]
 
[[1998 PMWC Problems/Problem I10|Solution]]
Line 54: Line 124:
  
 
== Problem I12 ==
 
== Problem I12 ==
 +
In the pyramid in the diagram, start from the top square containing the number <math>3</math>. 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>
  
 
[[1998 PMWC Problems/Problem I12|Solution]]
 
[[1998 PMWC Problems/Problem I12|Solution]]
Line 63: Line 149:
  
 
== Problem I14 ==
 
== Problem I14 ==
 +
Arrange all fractions of the form <math>\frac{m}{n}</math> in a row where <math>m</math> and <math>n</math> are counting numbers satisfy the following conditions:
 +
 +
(a) If <math>a\times b<m_2\times n_2</math>, then <math>\frac{m_1}{n_1}</math> must be placed before <math>\frac{m_2}{n_2}</math>.
 +
 +
(b) If <math>m_1\times n_1=m_2\times n_2</math>, and <math>n_1<n_2</math>, then <math>\frac{m_1}{n_1}</math> must also be placed before <math>\frac{m_2}{n_2}</math>.
 +
 +
How many fractions are there between <math>\frac{1998}{1}</math> and <math>\frac{1}{1998}</math>?
  
 
[[1998 PMWC Problems/Problem I14|Solution]]
 
[[1998 PMWC Problems/Problem I14|Solution]]
  
 
== Problem I15 ==
 
== 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 ?  
+
Construct a rectangle by putting together nine squares with sides equal to <math>1, 4, 7, 8, 9, 10, 14, 15,</math> and <math>18</math>.  What is the sum of the areas of the squares on the <math>4</math> corners of the resulting rectangle ?  
  
 
[[1998 PMWC Problems/Problem I15|Solution]]
 
[[1998 PMWC Problems/Problem I15|Solution]]
  
 
== Problem T1 ==
 
== Problem T1 ==
−
What is the 1998th number in the following sequence ?
+
What is the <math>1998^{\text{th}}</math> 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,........
+
 
 +
<math>1, -2, 2, -3, 3, -3, 4, -4, 4, -4, 5, -5, 5, -5, 5, -6, 6, -6, 6, -6, 6,\dots</math>
 +
 
 
[[1998 PMWC Problems/Problem T1|Solution]]
 
[[1998 PMWC Problems/Problem T1|Solution]]
  
 
== Problem T2 ==  
 
== Problem T2 ==  
−
Tom started work on a job alone for 30 days. Jerry continued the job
+
Tom started work on a job alone for <math>30</math> days. Jerry continued the job alone for <math>5</math> days, and finally they worked together for another <math>10</math> days to complete that job. For the same job, if Tom and Jerry work together, they can complete it in <math>20</math> days. Assuming Tom and Jerry each work at a constant rate throughout, how many days will Tom take to complete that job alone?
−
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?
 
 
[[1998 PMWC Problems/Problem T2|Solution]]
 
[[1998 PMWC Problems/Problem T2|Solution]]
  
 
== Problem T3 ==
 
== Problem T3 ==
−
The set L consists of all positive integers which leave a remainder of 1
+
The set <math>L</math> consists of all positive integers which leave a remainder of <math>1</math> when divided by <math>3</math>. A member of <math>L</math> (other than <math>1</math>) is called an <math>L</math>-prime if it is not the product of two members of <math>L</math>, other than itself and <math>1</math>. Which is the <math>8</math>th <math>L</math>-prime?
−
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 8th L-prime?
 
 
[[1998 PMWC Problems/Problem T3|Solution]]
 
[[1998 PMWC Problems/Problem T3|Solution]]
  
 
== Problem T4 ==  
 
== Problem T4 ==  
−
There are many circles on a plane. Each is divided into four parts by
+
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?
−
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?
 
 
[[1998 PMWC Problems/Problem T4|Solution]]
 
[[1998 PMWC Problems/Problem T4|Solution]]
  
 
== Problem T5 ==
 
== Problem T5 ==
−
Find the largest positive integer with the following properties :
+
Find the largest positive integer with the following properties:
 +
 
 
(a) all the digits are different.
 
(a) all the digits are different.
−
(b) each two consecutive digits form a number divisible by either 17
+
 
−
or 23.
+
(b) each two consecutive digits form a number divisible by either <math>17</math> or <math>23</math>.
 +
 
 
[[1998 PMWC Problems/Problem T5|Solution]]
 
[[1998 PMWC Problems/Problem T5|Solution]]
  
 
== Problem T6 ==
 
== Problem T6 ==
−
There were 3 students in an athletics competition of at least two
+
There were <math>3</math> 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 <math>3</math> scores. At the end of the competition, the total scores of the <math>3</math> students were <math>5, 9,</math> and <math>16</math>. Determine the first-place score for each event.
−
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.
 
 
[[1998 PMWC Problems/Problem T6|Solution]]
 
[[1998 PMWC Problems/Problem T6|Solution]]
  
 
== Problem T7 ==  
 
== Problem T7 ==  
−
A leaf is torn from a book of not more than 500 pages. The sum of the
+
A leaf is torn from a book of not more than <math>500</math> pages. The sum of the remaining pages numbers is <math>19905</math>. What is the sum of the two page numbers of the leaf torn out?
−
remaining pages numbers is 19905. What is the sum of the two page
+
 
−
numbers of the leaf torn out ?
 
 
[[1998 PMWC Problems/Problem T7|Solution]]
 
[[1998 PMWC Problems/Problem T7|Solution]]
  
 
== Problem T8 ==
 
== Problem T8 ==
−
A rectangular lawn is surrounded by a path 1 meter in width and
+
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 .
−
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 .
 
 
[[1998 PMWC Problems/Problem T8|Solution]]
 
[[1998 PMWC Problems/Problem T8|Solution]]
  
 
== Problem T9 ==
 
== Problem T9 ==
−
A, B, C, D and E play a game in which each is either a lion or a goat.
+
<math>A, B, C, D,</math> and <math>E</math> 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 lion’s statement is always false and a goat’s statement is always true.
+
 
−
A says B is not a goat.
+
<math>A</math> says <math>B</math> is not a goat.
−
C says D is a lion.
+
 
−
E says A is not a lion.
+
<math>C</math> says <math>D</math> is a lion.
−
B says C is not a goat.
+
 
−
D says that E and A are different kinds of animals.
+
<math>E</math> says <math>A</math> is not a lion.
 +
 
 +
<math>B</math> says <math>C</math> is not a goat.
 +
 
 +
<math>D</math> says that <math>E</math> and <math>A</math> are different kinds of animals.
 +
 
 
Who are the lions?
 
Who are the lions?
 +
 
[[1998 PMWC Problems/Problem T9|Solution]]
 
[[1998 PMWC Problems/Problem T9|Solution]]
  
 
== Problem T10 ==
 
== Problem T10 ==
−
In the following expression, each letter represents a digit. Same digits
+
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 <math>\text{ABCBA}</math>?
−
are represented by the same letter, and different letters stand for
+
 
−
different digits. Any digit can replace any square, find the 5-digit
+
<cmath> \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} </cmath>
−
number ABCBA?
 
  
 
[[1998 PMWC Problems/Problem T10|Solution]]
 
[[1998 PMWC Problems/Problem T10|Solution]]

Latest revision as of 18:25, 10 March 2015

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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