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1997 PMWC Problems

Problem I1

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

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?

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

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?

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]$

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} $",(8.46,7.95),fontsize(15)); add("crosshatch",crosshatch(.7mm)); fill((6,5.09)--(3.94,4.18)--(3.14,6)--(6,6)--cycle, pattern("crosshatch")); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); //Credit to dasobson for the diagram[/asy]$

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?

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?

Problem I8

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

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)

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?

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$", (4.29,3.45), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ //Credit to dasobson for the diagram[/asy]$

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 ?

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 $B$ ?

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.

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$",(-1/6,sqrt(3)/3),NW); //Credit to chezbgone2 for the diagram[/asy]$

Problem T2

Evaluate

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

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?

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?

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?

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?

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?

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 $7$ ?

Problem T9

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

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?

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