Difference between revisions of "1997 PMWC Problems"

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== Problem I2 ==
 
== Problem I2 ==
 +
In the multiplication in the image, each letter and each box represent a single digit. Different letters represent different digits but a box can represent any digit. What does the five-digit number <math>\mathrm{HAPPY}</math> stand for?
 +
 +
<cmath> \begin{array}{c c c c c}& &\Box & 1 &\Box\\ &\times & & 9 &\Box\\ \hline &\Box &\Box & 9 &\Box\\ \Box &\Box &\Box & 7 &\\ \hline H & A & P & P & Y\end{array} </cmath>
  
 
[[1997 PMWC Problems/Problem I2|Solution]]
 
[[1997 PMWC Problems/Problem I2|Solution]]
  
 
== Problem I3 ==
 
== Problem I3 ==
Peter is ill. He has to take medicine A every 8 hours,
+
Peter is ill. He has to take medicine <math>A</math> every <math>8</math> hours, medicine <math>B</math> every <math>5</math> hours and medicine <math>C</math> every <math>10</math> hours. If he took all three medicines at <math>7</math> a.m. on Tuesday, when will he take them altogether again?
medicine B every 5 hours and medicine C every 10 hours.
 
If he took all three medicines at 7 a.m. on Tuesday, when will he take them altogether again?
 
  
 
[[1997 PMWC Problems/Problem I3|Solution]]
 
[[1997 PMWC Problems/Problem I3|Solution]]
  
 
== Problem I4 ==
 
== Problem I4 ==
 +
Each of the three diagrams in the image show a balance of weights using different objects. How many squares will balance a circle?
 +
 +
<asy>
 +
/* File unicodetex not found. */
 +
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
 +
import graph; size(5.12cm);
 +
real labelscalefactor = 0.5; /* changes label-to-point distance */
 +
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
 +
pen dotstyle = black; /* point style */
 +
real xmin = -0.43, xmax = 4.69, ymin = -0.49, ymax = 2.22; /* image dimensions */
 +
draw((0.23,0.23)--(0.59,0.23)--(0.41,0.54)--cycle);
 +
draw((1.27,1.04)--(1.63,1.04)--(1.45,1.35)--cycle);
 +
draw((1.45,0.23)--(1.52,0.3)--(1.59,0.3)--(1.56,0.38)--(1.59,0.46)--(1.52,0.46)--(1.45,0.54)--(1.38,0.46)--(1.31,0.46)--(1.35,0.38)--(1.31,0.3)--(1.38,0.3)--cycle);
 +
draw((1.8,1.04)--(1.87,1.11)--(1.94,1.11)--(1.91,1.19)--(1.94,1.27)--(1.87,1.27)--(1.8,1.35)--(1.73,1.27)--(1.66,1.27)--(1.7,1.19)--(1.66,1.11)--(1.73,1.11)--cycle);
 +
draw((1.27,1.85)--(1.63,1.85)--(1.45,2.16)--cycle);
 +
draw((1.73,0.23)--(1.88,0.23)--(1.88,0.37)--(1.73,0.37)--cycle);
 +
draw((0.52,1.04)--(0.66,1.04)--(0.66,1.18)--(0.52,1.18)--cycle);
 +
draw((1.64,1.85)--(1.79,1.85)--(1.79,1.99)--(1.64,1.99)--cycle);
 +
draw((1.82,1.85)--(1.96,1.85)--(1.96,1.99)--(1.82,1.99)--cycle);
 +
/* draw figures */
 +
draw((0,0)--(2.18,0));
 +
draw((0.41,0)--(0.41,0.23));
 +
draw((1.63,0)--(1.63,0.23));
 +
draw((0.95,-0.15)--(1.09,0));
 +
draw((1.23,-0.15)--(1.09,0));
 +
draw((1.23,-0.15)--(0.95,-0.15));
 +
draw((0.23,0.23)--(0.59,0.23));
 +
draw((0.59,0.23)--(0.41,0.54));
 +
draw((0.41,0.54)--(0.23,0.23));
 +
draw((0.08,0.23)--(0.73,0.23));
 +
draw((1.28,0.23)--(1.98,0.23));
 +
draw((1.31,0.46)--(1.35,0.38));
 +
draw((1.59,0.3)--(1.56,0.38));
 +
draw((1.59,0.46)--(1.56,0.38));
 +
draw((1.31,0.3)--(1.35,0.38));
 +
draw((1.31,0.46)--(1.38,0.46));
 +
draw((1.38,0.46)--(1.45,0.54));
 +
draw((1.52,0.3)--(1.45,0.23));
 +
draw((1.59,0.3)--(1.52,0.3));
 +
draw((1.52,0.46)--(1.45,0.54));
 +
draw((1.59,0.46)--(1.52,0.46));
 +
draw((1.31,0.3)--(1.38,0.3));
 +
draw((1.38,0.3)--(1.45,0.23));
 +
draw((0,0.81)--(2.18,0.81));
 +
draw((0.41,0.81)--(0.41,1.04));
 +
draw((1.63,0.81)--(1.63,1.04));
 +
draw((0.95,0.66)--(1.09,0.81));
 +
draw((1.23,0.66)--(1.09,0.81));
 +
draw((1.23,0.66)--(0.95,0.66));
 +
draw((0.08,1.04)--(0.73,1.04));
 +
draw((1.28,1.04)--(1.98,1.04));
 +
draw((1.72,1.04)--(1.89,1.04));
 +
draw(circle((0.25,1.21), 0.17));
 +
draw((1.45,0.23)--(1.52,0.3));
 +
draw((1.52,0.3)--(1.59,0.3));
 +
draw((1.52,0.46)--(1.45,0.54));
 +
draw((1.38,0.46)--(1.31,0.46));
 +
draw((1.31,0.3)--(1.38,0.3));
 +
draw((1.38,0.3)--(1.45,0.23));
 +
draw((1.8,1.04)--(1.87,1.11));
 +
draw((1.87,1.11)--(1.94,1.11));
 +
draw((1.94,1.11)--(1.91,1.19));
 +
draw((1.91,1.19)--(1.94,1.27));
 +
draw((1.94,1.27)--(1.87,1.27));
 +
draw((1.87,1.27)--(1.8,1.35));
 +
draw((1.8,1.35)--(1.73,1.27));
 +
draw((1.73,1.27)--(1.66,1.27));
 +
draw((1.66,1.27)--(1.7,1.19));
 +
draw((1.7,1.19)--(1.66,1.11));
 +
draw((1.66,1.11)--(1.73,1.11));
 +
draw((1.73,1.11)--(1.8,1.04));
 +
draw((0,1.62)--(2.18,1.62));
 +
draw((0.41,1.62)--(0.41,1.85));
 +
draw((1.63,1.62)--(1.63,1.85));
 +
draw((0.95,1.47)--(1.09,1.62));
 +
draw((1.23,1.47)--(1.09,1.62));
 +
draw((1.23,1.47)--(0.95,1.47));
 +
draw((0.23,1.85)--(0.59,1.85));
 +
draw((0.08,1.85)--(0.73,1.85));
 +
draw((1.28,1.85)--(1.98,1.85));
 +
draw(circle((0.41,2.02), 0.17));
 +
draw((1.73,0.23)--(1.88,0.23));
 +
draw((1.88,0.23)--(1.88,0.37));
 +
draw((1.88,0.37)--(1.73,0.37));
 +
draw((1.73,0.37)--(1.73,0.23));
 +
/* dots and labels */
 +
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 +
/* end of picture */
 +
//Credit to dasobson for the diagram</asy>
  
 
[[1997 PMWC Problems/Problem I4|Solution]]
 
[[1997 PMWC Problems/Problem I4|Solution]]
  
 
== Problem I5 ==
 
== Problem I5 ==
 +
Two squares of different sizes overlap as shown in the given figure. What is the difference between the non-overlapping areas?
 +
 +
<asy>
 +
import patterns;
 +
/* modified Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
 +
import graph; usepackage("amsmath"); size(3.95cm);
 +
real labelscalefactor = 0.5; /* changes label-to-point distance */
 +
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
 +
pen dotstyle = black; /* point style */
 +
real xmin = -6.01, xmax = 15.94, ymin = -3.31, ymax = 13.18; /* image dimensions */
 +
draw((3.94,4.18)--(6,5.09)--(6,6)--(3.14,6)--cycle);
 +
/* draw figures */
 +
draw((0,0)--(0,6));
 +
draw((0,0)--(6,0));
 +
draw((0,6)--(6,6));
 +
draw((6,6)--(6,0));
 +
draw((2.33,7.85)--(3.94,4.18));
 +
draw((2.33,7.85)--(5.99,9.45));
 +
draw((5.99,9.45)--(7.6,5.79));
 +
draw((7.6,5.79)--(3.94,4.18));
 +
label("$ 6\text{cm} $",(-1.45,2.86),fontsize(15));
 +
label("$ 4\text{cm} $",(8.46,7.95),fontsize(15));
 +
add("crosshatch",crosshatch(.7mm));
 +
fill((6,5.09)--(3.94,4.18)--(3.14,6)--(6,6)--cycle, pattern("crosshatch"));
 +
/* dots and labels */
 +
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 +
//Credit to dasobson for the diagram</asy>
  
 
[[1997 PMWC Problems/Problem I5|Solution]]
 
[[1997 PMWC Problems/Problem I5|Solution]]
  
 
== Problem I6 ==
 
== Problem I6 ==
John and Mary went to a book shop and bought some exercise books. They had <dollar/>100 each. John could buy 7 large and 4 small ones. Mary could buy 5 large and 6 small ones and had <dollar/>5 left. How much was a small exercise book?
+
John and Mary went to a book shop and bought some exercise books. They had <math>\textdollar 100</math> each. John could buy <math>7</math> large and <math>4</math> small ones. Mary could buy <math>5</math> large and <math>6</math> small ones and had <math>\textdollar 5</math> left. How much was a small exercise book?
  
 
[[1997 PMWC Problems/Problem I6|Solution]]
 
[[1997 PMWC Problems/Problem I6|Solution]]
  
 
== Problem I7 ==
 
== Problem I7 ==
40% of girls and 50% of boys in a class got an 'A'. If there
+
<math>40\%</math> of girls and <math>50\%</math> of boys in a class got an <math>\mathrm{'A'}</math>. If there are only <math>12</math> students in the class who got <math>\mathrm{'A'}</math>s and the ratio of boys and girls in the class is <math>4:5</math>, how many students are there in the class?
are only 12 students in the class who got 'A's and the ratio of
 
boys and girls in the class is 45, how many students are
 
there in the class?
 
  
 
[[1997 PMWC Problems/Problem I7|Solution]]
 
[[1997 PMWC Problems/Problem I7|Solution]]
  
 
== Problem I8 ==
 
== Problem I8 ==
<math>997-996-995+994+993-992+991-990-989+988+989-986+\cdots+7-6-5+4+3-2+1=?</math>
+
<math>997-996-995+994+993-992+991-990-989+988+987-986+\cdots+7-6-5+4+3-2+1=?</math>
  
 
[[1997 PMWC Problems/Problem I8|Solution]]
 
[[1997 PMWC Problems/Problem I8|Solution]]
  
 
== Problem I9 ==
 
== Problem I9 ==
A chemist mixed an acid of 48% concentration with the
+
A chemist mixed an acid of <math>48\%</math> concentration with the same acid of <math>80\%</math> concentration, and then added <math>2</math> litres of distilled water to the mixed acid. As a result, he got <math>10</math> litres of the acid of <math>40\%</math> concentration. How many millilitre of the acid of <math>48\%</math> concentration that the chemist had used? (<math>1</math> litre = <math>1000</math> millilitres)
same acid of 80% concentration, and then added 2 litres of
 
distilled water to the mixed acid. As a result, he got 10
 
litres of the acid of 40% concentration. How many
 
millilitre of the acid of 48% concentration that the chemist
 
had used? (1 litre = 1000 millilitres)
 
  
 
[[1997 PMWC Problems/Problem I9|Solution]]
 
[[1997 PMWC Problems/Problem I9|Solution]]
  
 
== Problem I10 ==
 
== Problem I10 ==
Mary took 24 chickens to the market. In the morning she
+
Mary took <math>24</math> chickens to the market. In the morning she sold the chickens at <math>\textdollar 7</math> each and she only sold out less than half of them. In the afternoon she discounted the price of each chicken but the price was still an integral number in dollar. In the afternoon she could sell all the chickens, and she got totally <math>\textdollar 132</math> for the whole day. How many chickens were sold in the morning?
sold the chickens at <math>\</math>7 each and she only sold out less than
 
half of them. In the afternoon she discounted the price of
 
each chicken but the price was still an integral number in
 
dollar. In the afternoon she could sell all the chickens, and
 
she got totally <math>\</math>132 for the whole day. How many
 
chickens were sold in the morning?
 
  
 
[[1997 PMWC Problems/Problem I10|Solution]]
 
[[1997 PMWC Problems/Problem I10|Solution]]
  
 
== Problem I11 ==
 
== Problem I11 ==
A rectangle <math>ABCD</math> is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of <math>ABCD</math> if its area is <math>6750 \text{cm}^2</math>.  
+
A rectangle <math>ABCD</math> is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of <math>ABCD</math> if its area is <math>6750\text{ cm}^2</math>.  
[[Image:ABCD.gif]]
+
 
 +
<asy>
 +
/* File unicodetex not found. */
 +
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
 +
import graph; size(3.45cm);
 +
real labelscalefactor = 0.5; /* changes label-to-point distance */
 +
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
 +
pen dotstyle = black; /* point style */
 +
real xmin = -19.75, xmax = 39.09, ymin = -10.43, ymax = 20.84; /* image dimensions */
 +
/* draw figures */
 +
draw((0,3.45)--(0,0));
 +
draw((0,0)--(4.29,0));
 +
draw((0,3.45)--(4.29,3.45));
 +
draw((4.29,3.45)--(4.29,0));
 +
draw((0,1.32)--(4.29,1.32));
 +
draw((2.14,0)--(2.14,1.32));
 +
draw((1.43,1.32)--(1.43,3.45));
 +
draw((2.86,1.32)--(2.86,3.45));
 +
/* dots and labels */
 +
label("$B$", (-0.2,0), SW * labelscalefactor);
 +
label("$A$", (-0.2,3.45), NW * labelscalefactor);
 +
label("$C$", (4.29,0), SE * labelscalefactor);
 +
label("$D$", (4.29,3.45), NE * labelscalefactor);
 +
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 +
/* end of picture */
 +
//Credit to dasobson for the diagram</asy>
  
 
[[1997 PMWC Problems/Problem I11|Solution]]
 
[[1997 PMWC Problems/Problem I11|Solution]]
  
 
== Problem I12 ==
 
== Problem I12 ==
In a die, 1 and 6,2 and 5,3 and 4 appear on opposite faces.
+
In a die, <math>1</math> and <math>6</math>, <math>2</math> and <math>5</math>, <math>3</math> and <math>4</math> appear on opposite faces. When <math>2</math> dice are thrown, product of numbers appearing on the top and bottom faces of the <math>2</math> dice are formed as follows:  
When 2 dice are thrown, product of numbers appearing on
+
 
the top and bottom faces of the 2 dice are formed as follows:
+
*number on top face of 1st die times number on top face of 2nd die
  number on top face of 1st die x number on top face of 2nd die
+
*number on top face of 1st die times number on bottom face of 2nd die
  number on top face of 1st die x number on bottom face of 2nd die
+
*number on bottom face of 1st die times number on top face of 2nd die
  number on bottom face of 1st die x number on top face of 2nd die
+
*number on bottom face of 1st die times number on bottom face of 2nd die
  number on bottom face of 1st die x number on bottom face of 2nd die
+
 
What is the sum of these 4 products ?
+
What is the sum of these <math>4</math> products ?
  
 
[[1997 PMWC Problems/Problem I12|Solution]]
 
[[1997 PMWC Problems/Problem I12|Solution]]
  
 
== Problem I13 ==
 
== Problem I13 ==
A truck moved from A to B at a speed of <math>50</math> km/h and returns from B to A at <math>70</math> km/h. It traveled <math>3</math> rounds within 18 hours. What is the distance between A and B?
+
A truck moved from <math>A</math> to <math>B</math> at a speed of <math>50 \text{ km/h}</math> and returns from <math>B</math> to <math>A</math> at <math>70 \text{ km/h}</math>. It traveled <math>3</math> rounds within <math>18</math> hours. What is the distance between <math>A</math> and <math>B</math>?
 +
 
  
 
[[1997 PMWC Problems/Problem I13|Solution]]
 
[[1997 PMWC Problems/Problem I13|Solution]]
Line 91: Line 220:
  
 
== Problem I15 ==
 
== Problem I15 ==
How many paths from A to B consist of exactly six line
+
How many paths from <math>A</math> to <math>B</math> consist of exactly six line segments (vertical, horizontal or inclined)?
segments (vertical, horizontal or inclined)?
 
[[Image:1997_PMWC-I15.png]]
 
  
 +
<asy>
 +
for(int i = 0; i < 3; ++i){
 +
draw((0,i+1)--(0,i)--(4,i)--(4,i+1));
 +
draw((4/3,i+1)--(4/3,i)--(8/3,i+1)--(8/3,i));
 +
}
 +
draw((0,3)--(4,3));
 +
label("$A$",(0,0),SW);
 +
label("$B$",(4,3),NE);
 +
//Credit to chezbgone2 for the diagram</asy>
 
[[1997 PMWC Problems/Problem I15|Solution]]
 
[[1997 PMWC Problems/Problem I15|Solution]]
  
 
== Problem T1 ==
 
== Problem T1 ==
 +
Let <math>PQR</math> be an equilateral triangle with sides of length three units. <math>U</math>, <math>V</math>, <math>W</math>, <math>X</math>, <math>Y</math>, and <math>Z</math> divide the sides into lengths of one unit. Find the ratio of the area of the shaded quadrilateral <math>UWXY</math> to the area of the triangle <math>PQR</math>.
 +
 +
<asy>
 +
draw((1/2,0)--(-1/2,0)--(0,sqrt(3)/2)--cycle);
 +
dot((1/6,sqrt(3)/3));
 +
dot((-1/6,sqrt(3)/3));
 +
dot((1/3,sqrt(3)/6));
 +
dot((-1/3,sqrt(3)/6));
 +
dot((1/6,0));
 +
dot((1/6,0));
 +
dot((-1/6,0));
 +
filldraw((-1/6,sqrt(3)/3)--(1/3,sqrt(3)/6)--(1/6,0)--(-1/6,0)--cycle);
 +
label("$P$",(0,sqrt(3)/2),N);
 +
label("$Z$",(1/6,sqrt(3)/3),NE);
 +
label("$Y$",(1/3,sqrt(3)/6),NE);
 +
label("$R$",(1/2,0),E);
 +
label("$X$",(1/6,0),S);
 +
label("$W$",(-1/6,0),S);
 +
label("$Q$",(-1/2,0),W);
 +
label("$V$",(-1/3,sqrt(3)/6),NW);
 +
label("$U$",(-1/6,sqrt(3)/3),NW);
 +
//Credit to chezbgone2 for the diagram</asy>
  
 
[[1997 PMWC Problems/Problem T1|Solution]]
 
[[1997 PMWC Problems/Problem T1|Solution]]
Line 111: Line 269:
 
&+&9\left(\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)+11\left(\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\
 
&+&9\left(\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)+11\left(\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\
 
&+&13\left(\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)+15\left(\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\
 
&+&13\left(\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)+15\left(\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\
&+&17\left(\dfrac{1}{9}+\dfrac{1}{10}\right)+19\left(\dfrac{1}{10}\right)</cmath>
+
&+&17\left(\dfrac{1}{9}+\dfrac{1}{10}\right)+19\left(\dfrac{1}{10}\right)\end{eqnarray*}</cmath>
  
 
[[1997 PMWC Problems/Problem T2|Solution]]
 
[[1997 PMWC Problems/Problem T2|Solution]]
  
 
== Problem T3 ==
 
== Problem T3 ==
To type all the integers from <tt>1</tt> to <tt>1997</tt> using a typewriter on a piece of paper, how many does the key '<tt>9</tt>' needed to be pressed?
+
To type all the integers from <tt>1</tt> to <tt>1997</tt> using a typewriter on a piece of paper, how many times is the key '<tt>9</tt>' needed to be pressed?
  
 
[[1997 PMWC Problems/Problem T3|Solution]]
 
[[1997 PMWC Problems/Problem T3|Solution]]
  
 
== Problem T4 ==  
 
== Problem T4 ==  
In one morning, a ferry traveled from Hong Kong to Kowloon and another ferry traveled from Kowloon to Hong Kong at a different speed. They started at the same time and met first time at 8:20. The two ferries then sailed to their destinations, stopped for 15 minutes and returned. The two ferries met again at 9:11. Suppose the two ferries traveled at a uniform speed throughout the whole journey, what time did the two ferries start their journey?
+
In one morning, a ferry traveled from Hong Kong to Kowloon and another ferry traveled from Kowloon to Hong Kong at a different speed. They started at the same time and met first time at <math>8:20</math>. The two ferries then sailed to their destinations, stopped for <math>15</math> minutes and returned. The two ferries met again at <math>9:11</math>. Suppose the two ferries traveled at a uniform speed throughout the whole journey, what time did the two ferries start their journey?
  
 
[[1997 PMWC Problems/Problem T4|Solution]]
 
[[1997 PMWC Problems/Problem T4|Solution]]
Line 143: Line 301:
  
 
== Problem T6 ==
 
== Problem T6 ==
During a rebuilding project by contractors 'A', 'B' and 'C', there was a shortage of tractors. The contractors lent each other tractors as needed. At first, 'A' lent 'B' and 'C' as many tractors as they each already had. A few months later, 'B' lent 'A' and 'C' as many as they each already had. Still later, 'C' lent 'A' and 'B' as many as they each already had. By then each contractor had 24 tractors. How many tractors did each contractor originally have?
+
During a rebuilding project by contractors 'A', 'B' and 'C', there was a shortage of tractors. The contractors lent each other tractors as needed. At first, 'A' lent 'B' and 'C' as many tractors as they each already had. A few months later, 'B' lent 'A' and 'C' as many as they each already had. Still later, 'C' lent 'A' and 'B' as many as they each already had. By then each contractor had <math>24</math> tractors. How many tractors did each contractor originally have?
  
 
[[1997 PMWC Problems/Problem T6|Solution]]
 
[[1997 PMWC Problems/Problem T6|Solution]]
  
 
== Problem T7 ==  
 
== Problem T7 ==  
Color the surfaces of a cube of dimension 5*5*5 red, and then cur the cube into smaller cubes of dimension 1*1*1. Take out all the smaller cubes which have at least one red surface and fix a cuboid, keeping the surfaces of the cuboid red. Now what is the maximum possible volume of the cuboid?
+
Color the surfaces of a cube of dimension <math>5\times 5\times 5</math> red, and then cut the cube into smaller cubes of dimension <math>1\times 1\times 1</math>. Take out all the smaller cubes which have at least one red surface and fix a cuboid, keeping the surfaces of the cuboid red. Now what is the maximum possible volume of the cuboid?
  
 
[[1997 PMWC Problems/Problem T7|Solution]]
 
[[1997 PMWC Problems/Problem T7|Solution]]
  
 
== Problem T8 ==
 
== Problem T8 ==
Among the integers 1, 2, ..., 1997, what is the maximum number of integers that can be selected such that the sum of any two selected numbers is not a multiple of 7?
+
Among the integers <math>1, 2,\dots , 1997</math>, what is the maximum number of integers that can be selected such that the sum of any two selected numbers is not a multiple of <math>7</math>?
  
 
[[1997 PMWC Problems/Problem T8|Solution]]
 
[[1997 PMWC Problems/Problem T8|Solution]]
  
 
== Problem T9 ==
 
== Problem T9 ==
Find the two 10-digit numbers which become nine times as large if the order of the digits is reversed.
+
Find the two <math>10</math>-digit numbers which become nine times as large if the order of the digits is reversed.
  
 
[[1997 PMWC Problems/Problem T9|Solution]]
 
[[1997 PMWC Problems/Problem T9|Solution]]
  
 
== Problem T10 ==
 
== Problem T10 ==
The twelve integers 1, 2, 3,..., 12 are arranged in a circle such that the difference of any two adjacent numbers is either 2, 3 or 4. What is the maximum number of the difference '4' can occur in any such arrangement?
+
The twelve integers <math>1, 2, 3,\dots, 12</math> are arranged in a circle such that the difference of any two adjacent numbers is either <math>2, 3,</math> or <math>4</math>. What is the maximum number of the difference <math>4</math> can occur in any such arrangement?
  
 
[[1997 PMWC Problems/Problem T10|Solution]]
 
[[1997 PMWC Problems/Problem T10|Solution]]

Latest revision as of 19:13, 10 March 2015

Problem I1

Evaluate $29 \dfrac{27}{28} \times 27 \frac{14}{15}$

Solution

Problem I2

In the multiplication in the image, each letter and each box represent a single digit. Different letters represent different digits but a box can represent any digit. What does the five-digit number $\mathrm{HAPPY}$ stand for?

\[\begin{array}{c c c c c}& &\Box & 1 &\Box\\ &\times & & 9 &\Box\\ \hline &\Box &\Box & 9 &\Box\\ \Box &\Box &\Box & 7 &\\ \hline H & A & P & P & Y\end{array}\]

Solution

Problem I3

Peter is ill. He has to take medicine $A$ every $8$ hours, medicine $B$ every $5$ hours and medicine $C$ every $10$ hours. If he took all three medicines at $7$ a.m. on Tuesday, when will he take them altogether again?

Solution

Problem I4

Each of the three diagrams in the image show a balance of weights using different objects. How many squares will balance a circle?

[asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(5.12cm);  real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */  pen dotstyle = black; /* point style */  real xmin = -0.43, xmax = 4.69, ymin = -0.49, ymax = 2.22; /* image dimensions */ draw((0.23,0.23)--(0.59,0.23)--(0.41,0.54)--cycle);  draw((1.27,1.04)--(1.63,1.04)--(1.45,1.35)--cycle);  draw((1.45,0.23)--(1.52,0.3)--(1.59,0.3)--(1.56,0.38)--(1.59,0.46)--(1.52,0.46)--(1.45,0.54)--(1.38,0.46)--(1.31,0.46)--(1.35,0.38)--(1.31,0.3)--(1.38,0.3)--cycle);  draw((1.8,1.04)--(1.87,1.11)--(1.94,1.11)--(1.91,1.19)--(1.94,1.27)--(1.87,1.27)--(1.8,1.35)--(1.73,1.27)--(1.66,1.27)--(1.7,1.19)--(1.66,1.11)--(1.73,1.11)--cycle);  draw((1.27,1.85)--(1.63,1.85)--(1.45,2.16)--cycle);  draw((1.73,0.23)--(1.88,0.23)--(1.88,0.37)--(1.73,0.37)--cycle);  draw((0.52,1.04)--(0.66,1.04)--(0.66,1.18)--(0.52,1.18)--cycle);  draw((1.64,1.85)--(1.79,1.85)--(1.79,1.99)--(1.64,1.99)--cycle);  draw((1.82,1.85)--(1.96,1.85)--(1.96,1.99)--(1.82,1.99)--cycle);  /* draw figures */ draw((0,0)--(2.18,0));  draw((0.41,0)--(0.41,0.23));  draw((1.63,0)--(1.63,0.23));  draw((0.95,-0.15)--(1.09,0));  draw((1.23,-0.15)--(1.09,0));  draw((1.23,-0.15)--(0.95,-0.15));  draw((0.23,0.23)--(0.59,0.23));  draw((0.59,0.23)--(0.41,0.54));  draw((0.41,0.54)--(0.23,0.23));  draw((0.08,0.23)--(0.73,0.23));  draw((1.28,0.23)--(1.98,0.23));  draw((1.31,0.46)--(1.35,0.38));  draw((1.59,0.3)--(1.56,0.38));  draw((1.59,0.46)--(1.56,0.38));  draw((1.31,0.3)--(1.35,0.38));  draw((1.31,0.46)--(1.38,0.46));  draw((1.38,0.46)--(1.45,0.54));  draw((1.52,0.3)--(1.45,0.23));  draw((1.59,0.3)--(1.52,0.3));  draw((1.52,0.46)--(1.45,0.54));  draw((1.59,0.46)--(1.52,0.46));  draw((1.31,0.3)--(1.38,0.3));  draw((1.38,0.3)--(1.45,0.23));  draw((0,0.81)--(2.18,0.81));  draw((0.41,0.81)--(0.41,1.04));  draw((1.63,0.81)--(1.63,1.04));  draw((0.95,0.66)--(1.09,0.81));  draw((1.23,0.66)--(1.09,0.81));  draw((1.23,0.66)--(0.95,0.66));  draw((0.08,1.04)--(0.73,1.04));  draw((1.28,1.04)--(1.98,1.04));  draw((1.72,1.04)--(1.89,1.04));  draw(circle((0.25,1.21), 0.17));  draw((1.45,0.23)--(1.52,0.3));  draw((1.52,0.3)--(1.59,0.3));  draw((1.52,0.46)--(1.45,0.54));  draw((1.38,0.46)--(1.31,0.46));  draw((1.31,0.3)--(1.38,0.3));  draw((1.38,0.3)--(1.45,0.23));  draw((1.8,1.04)--(1.87,1.11));  draw((1.87,1.11)--(1.94,1.11));  draw((1.94,1.11)--(1.91,1.19));  draw((1.91,1.19)--(1.94,1.27));  draw((1.94,1.27)--(1.87,1.27));  draw((1.87,1.27)--(1.8,1.35));  draw((1.8,1.35)--(1.73,1.27));  draw((1.73,1.27)--(1.66,1.27));  draw((1.66,1.27)--(1.7,1.19));  draw((1.7,1.19)--(1.66,1.11));  draw((1.66,1.11)--(1.73,1.11));  draw((1.73,1.11)--(1.8,1.04));  draw((0,1.62)--(2.18,1.62));  draw((0.41,1.62)--(0.41,1.85));  draw((1.63,1.62)--(1.63,1.85));  draw((0.95,1.47)--(1.09,1.62));  draw((1.23,1.47)--(1.09,1.62));  draw((1.23,1.47)--(0.95,1.47));  draw((0.23,1.85)--(0.59,1.85));  draw((0.08,1.85)--(0.73,1.85));  draw((1.28,1.85)--(1.98,1.85));  draw(circle((0.41,2.02), 0.17));  draw((1.73,0.23)--(1.88,0.23));  draw((1.88,0.23)--(1.88,0.37));  draw((1.88,0.37)--(1.73,0.37));  draw((1.73,0.37)--(1.73,0.23));  /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);  /* end of picture */ //Credit to dasobson for the diagram[/asy]

Solution

Problem I5

Two squares of different sizes overlap as shown in the given figure. What is the difference between the non-overlapping areas?

[asy] import patterns; /* modified Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; usepackage("amsmath"); size(3.95cm);  real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */  pen dotstyle = black; /* point style */  real xmin = -6.01, xmax = 15.94, ymin = -3.31, ymax = 13.18; /* image dimensions */ draw((3.94,4.18)--(6,5.09)--(6,6)--(3.14,6)--cycle);  /* draw figures */ draw((0,0)--(0,6));  draw((0,0)--(6,0));  draw((0,6)--(6,6));  draw((6,6)--(6,0));  draw((2.33,7.85)--(3.94,4.18));  draw((2.33,7.85)--(5.99,9.45));  draw((5.99,9.45)--(7.6,5.79));  draw((7.6,5.79)--(3.94,4.18));  label("$ 6\text{cm} $",(-1.45,2.86),fontsize(15));  label("$ 4\text{cm} $",(8.46,7.95),fontsize(15));  add("crosshatch",crosshatch(.7mm)); fill((6,5.09)--(3.94,4.18)--(3.14,6)--(6,6)--cycle, pattern("crosshatch")); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); //Credit to dasobson for the diagram[/asy]

Solution

Problem I6

John and Mary went to a book shop and bought some exercise books. They had $\textdollar 100$ each. John could buy $7$ large and $4$ small ones. Mary could buy $5$ large and $6$ small ones and had $\textdollar 5$ left. How much was a small exercise book?

Solution

Problem I7

$40\%$ of girls and $50\%$ of boys in a class got an $\mathrm{'A'}$. If there are only $12$ students in the class who got $\mathrm{'A'}$s and the ratio of boys and girls in the class is $4:5$, how many students are there in the class?

Solution

Problem I8

$997-996-995+994+993-992+991-990-989+988+987-986+\cdots+7-6-5+4+3-2+1=?$

Solution

Problem I9

A chemist mixed an acid of $48\%$ concentration with the same acid of $80\%$ concentration, and then added $2$ litres of distilled water to the mixed acid. As a result, he got $10$ litres of the acid of $40\%$ concentration. How many millilitre of the acid of $48\%$ concentration that the chemist had used? ($1$ litre = $1000$ millilitres)

Solution

Problem I10

Mary took $24$ chickens to the market. In the morning she sold the chickens at $\textdollar 7$ each and she only sold out less than half of them. In the afternoon she discounted the price of each chicken but the price was still an integral number in dollar. In the afternoon she could sell all the chickens, and she got totally $\textdollar 132$ for the whole day. How many chickens were sold in the morning?

Solution

Problem I11

A rectangle $ABCD$ is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of $ABCD$ if its area is $6750\text{ cm}^2$.

[asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(3.45cm);  real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */  pen dotstyle = black; /* point style */  real xmin = -19.75, xmax = 39.09, ymin = -10.43, ymax = 20.84; /* image dimensions */ /* draw figures */ draw((0,3.45)--(0,0));  draw((0,0)--(4.29,0));  draw((0,3.45)--(4.29,3.45));  draw((4.29,3.45)--(4.29,0));  draw((0,1.32)--(4.29,1.32));  draw((2.14,0)--(2.14,1.32));  draw((1.43,1.32)--(1.43,3.45));  draw((2.86,1.32)--(2.86,3.45));  /* dots and labels */ label("$B$", (-0.2,0), SW * labelscalefactor);  label("$A$", (-0.2,3.45), NW * labelscalefactor);  label("$C$", (4.29,0), SE * labelscalefactor);  label("$D$", (4.29,3.45), NE * labelscalefactor);  clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);  /* end of picture */ //Credit to dasobson for the diagram[/asy]

Solution

Problem I12

In a die, $1$ and $6$, $2$ and $5$, $3$ and $4$ appear on opposite faces. When $2$ dice are thrown, product of numbers appearing on the top and bottom faces of the $2$ dice are formed as follows:

  • number on top face of 1st die times number on top face of 2nd die
  • number on top face of 1st die times number on bottom face of 2nd die
  • number on bottom face of 1st die times number on top face of 2nd die
  • number on bottom face of 1st die times number on bottom face of 2nd die

What is the sum of these $4$ products ?

Solution

Problem I13

A truck moved from $A$ to $B$ at a speed of $50 \text{ km/h}$ and returns from $B$ to $A$ at $70 \text{ km/h}$. It traveled $3$ rounds within $18$ hours. What is the distance between $A$ and $B$?


Solution

Problem I14

If we make five two-digit numbers using the digits $0, 1, 2,...9$ exactly once, and the product of the five numbers is maximized, find the greatest number among them.

Solution

Problem I15

How many paths from $A$ to $B$ consist of exactly six line segments (vertical, horizontal or inclined)?

[asy] for(int i = 0; i < 3; ++i){ draw((0,i+1)--(0,i)--(4,i)--(4,i+1)); draw((4/3,i+1)--(4/3,i)--(8/3,i+1)--(8/3,i)); } draw((0,3)--(4,3)); label("$A$",(0,0),SW); label("$B$",(4,3),NE); //Credit to chezbgone2 for the diagram[/asy] Solution

Problem T1

Let $PQR$ be an equilateral triangle with sides of length three units. $U$, $V$, $W$, $X$, $Y$, and $Z$ divide the sides into lengths of one unit. Find the ratio of the area of the shaded quadrilateral $UWXY$ to the area of the triangle $PQR$.

[asy] draw((1/2,0)--(-1/2,0)--(0,sqrt(3)/2)--cycle); dot((1/6,sqrt(3)/3)); dot((-1/6,sqrt(3)/3)); dot((1/3,sqrt(3)/6)); dot((-1/3,sqrt(3)/6)); dot((1/6,0)); dot((1/6,0)); dot((-1/6,0)); filldraw((-1/6,sqrt(3)/3)--(1/3,sqrt(3)/6)--(1/6,0)--(-1/6,0)--cycle); label("$P$",(0,sqrt(3)/2),N); label("$Z$",(1/6,sqrt(3)/3),NE); label("$Y$",(1/3,sqrt(3)/6),NE); label("$R$",(1/2,0),E); label("$X$",(1/6,0),S); label("$W$",(-1/6,0),S); label("$Q$",(-1/2,0),W); label("$V$",(-1/3,sqrt(3)/6),NW); label("$U$",(-1/6,sqrt(3)/3),NW); //Credit to chezbgone2 for the diagram[/asy]

Solution

Problem T2

Evaluate

\begin{eqnarray*} && 1 \left(\dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right) \\ &+& 3\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\ &+&5\left(\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\ &+&7\left(\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\ &+&9\left(\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)+11\left(\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\ &+&13\left(\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)+15\left(\dfrac{1}{8}+\dfrac{1}{9}+\dfrac{1}{10}\right)\\ &+&17\left(\dfrac{1}{9}+\dfrac{1}{10}\right)+19\left(\dfrac{1}{10}\right)\end{eqnarray*}

Solution

Problem T3

To type all the integers from 1 to 1997 using a typewriter on a piece of paper, how many times is the key '9' needed to be pressed?

Solution

Problem T4

In one morning, a ferry traveled from Hong Kong to Kowloon and another ferry traveled from Kowloon to Hong Kong at a different speed. They started at the same time and met first time at $8:20$. The two ferries then sailed to their destinations, stopped for $15$ minutes and returned. The two ferries met again at $9:11$. Suppose the two ferries traveled at a uniform speed throughout the whole journey, what time did the two ferries start their journey?

Solution

Problem T5

During recess, one of five pupils wrote something nasty on the chalkboard. When questioned by the class teacher, the following ensued:

'A': It was 'B' or 'C'

'B': Neither 'E' nor I did it.

'C': You are both lying.

'D': No, either A or B is telling the truth.

'E': No, 'D', that's not true.

The class teacher knows that three of them never lie while the other two cannot be trusted. Who was the culprit?

Solution

Problem T6

During a rebuilding project by contractors 'A', 'B' and 'C', there was a shortage of tractors. The contractors lent each other tractors as needed. At first, 'A' lent 'B' and 'C' as many tractors as they each already had. A few months later, 'B' lent 'A' and 'C' as many as they each already had. Still later, 'C' lent 'A' and 'B' as many as they each already had. By then each contractor had $24$ tractors. How many tractors did each contractor originally have?

Solution

Problem T7

Color the surfaces of a cube of dimension $5\times 5\times 5$ red, and then cut the cube into smaller cubes of dimension $1\times 1\times 1$. Take out all the smaller cubes which have at least one red surface and fix a cuboid, keeping the surfaces of the cuboid red. Now what is the maximum possible volume of the cuboid?

Solution

Problem T8

Among the integers $1, 2,\dots , 1997$, what is the maximum number of integers that can be selected such that the sum of any two selected numbers is not a multiple of $7$?

Solution

Problem T9

Find the two $10$-digit numbers which become nine times as large if the order of the digits is reversed.

Solution

Problem T10

The twelve integers $1, 2, 3,\dots, 12$ are arranged in a circle such that the difference of any two adjacent numbers is either $2, 3,$ or $4$. What is the maximum number of the difference $4$ can occur in any such arrangement?

Solution

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