## Math Circles at FIU Little Circle

• Fall'11 Problems suggested for work at home

October 3rd

1. Is it possible to arrange 6 round long pencils so that each of them touches all the others?

2. In an arithmetic addition problem, the digits were replaced with letters( equal digits with
the same letters, and different digits - with distinct letters). The result is
LOVES + LIVE = THERE.
How many "loves" are "there"? The actual numbers are those for which the value of THERE is maximal
possible.

3. Doing the same procedure as in the previous problem, one gets
BLASE + LBSA = BASES.
Two giant computers came up with two different answers to this riddle. Is this possible, or does one
of them (the computers) need a repair?

September 19th

1. A teacher drew several circles on a sheet of paper. Then he asked a student "How many circles are there?"
"Seven," was the answer. "Correct! So, how many circles are there?" the teacher asked another student. "Five,"
answered the student. "Absolutely right!" replied the teacher.
How many circles were really drawn on the sheet?

2. THree turtles are crawling along a straight road heading in the same direction.
"Two other turtles are behind me," says the first turtle.
"One turtle is behind me and one is ahead," says the second.
"Two turtles are ahead of me and one other is behind,"says the third turtle.
How can this be possible?

3. Three tablespoons of milk from a glass of milk are poured into a glass of tea, and the liquid is
thoroughly mixed. Then three tablespoons of this mixture are poured back into the glass of milk.
Which is greater now: the amount of milk in the glass of tea or the amount of tea in the glass of milk?

• Fall'10 Problems suggested for work at home

September 20th

1. How many zeros are there in the number 40!:= 1x2x...x39x40? In how many zeros does 100! end?
Is it possible that n! ends in five zeros?

2. Woldemort showed on the board his work on multiplying two two-digit numbers. He changed then
the digits involved to letters: equal digits he replaced with the same letters, and different
digits with different letter. The new expression looks like AB*CD = PPQQ. Show that Woldemort
has made a mistake in his compuitations.

3. Can a 300-digit number with a 100 zeros, 100 ones, and a 100 twos be a complete square?

• Spring'10 Problems suggested for work at home

March 27th
1. Can an ordinary chessboard be covered with 2 X 1 dominoes so that only squares a1 and h8
remain uncovered?

2. Twehty five boys and the same number of girls are seated (randomly) at a round table.
Show that both neighbours of at least one student are boys.

3. Of 101 coins, 50 are counterfeit, and differ from the genuine coins in weight by 1 gram.
Peter has a scale in the form of a balance which shows the difference in weights
between the objects placed in each pan. He choses one coin, and wants to find out in
one weighing whether it is counterfeit. Can he do this?

February 13th
1. In a rectangle of size (199 squares) by (991 squares) on a graph paper, how many squares
does the diagonal of the rectangle cross?

2. Peter says: "On the day before yesterday I was 10 years old, and next year I will be 13."
How can that be?

3. The son of Professor's father is talking to the father of the professor's son without
the Professor taking part in the conversation. How can that be?

January 30th
1. The January of certain year had four Fridays and four Mondays. What day of the week was
January 20th of that year?

2. In a bag there are 24 lb of nails. How can one measure off 9 lb of nails by using simple
scales? (no weights are given for the scales)

3. Cross out 10 digits from the number 12345123451234512345 in such a way that the remaining number be the largest possible.

• Fall'09 Problems suggested for work at home
November 23rd
Alice and Bob 42 cents to share their cookies. after each one of them eats 1/3
of the cookies, Bob says that he and Alice should split the 42 cents evenly, and
Alice thinks that she should get 24 cents, and Bob should get 18 cents.
What is the fair division of the money between Alice and Bob?

2. fly sits on the vertex of a wooden cube. What is the shortest path it can follow
to the opposite vertex?

3. A fly is sitting on the outside surface of a cylindrical drinking glass. It must
crawl to another point, situated on the inside surface of the glass. Find the
shortest path possible (neglecting the thickness of the galss).
October 10th
1. All sides of a pentagon ABCDE have the same length, but the pentagon is
not regular (some of its angle measures are not the same). The
adjacent angles A and B are both right angles. What is the measure of
angle C?

2. All angles of a regular polygon are acute. How many sides does it have?

3. Prove that the sum of the diaginals of a quadrilateral is greater than
the half of its perimeter.

September 26th
1. The area nad the perimeter of a square are numericaly equal. What is the length
of a side of the square?

2. In a square of side length a, there is inscribed a circle of largest possible
radius in which, in turn, a largest possible square is inscribed.
How much is the area of the region outside of the smaller square?

3. Two villages lie on opposite sides of a river whose banks are parallel lines.
A bridge is to be built over the river, perpendicular to the banks. Where
should the bridge be built so that the path from one village to the other
is as short as possible?

• Spring'09 Problems suggested for work at home
February 16th
1. Shores A and B are on the opposite sides of a lake. Motorboat M leaves shore
A for shore B as N leaves shore B for shore A; they move across a lake at
a constant speed. The motorboats meet the first time 500 yards from A.
Each returns from the opposite shore without halting, and they meet 300 yards from B.
How long is the lake, and what is the relation between the two boatsĀ“ speeds?
(Sharp wits can sove the problem with minimum calculations!)

2. My watch is one second fast per hour, and my brothersĀ“ is a second and a half
slow per hour. Right now, the watches show the right time. When will they
show the same time again? When will they show the same CORRECT time again?

• Fall'08 Problems suggested for work at home
November 15th
In the problems below, recall that the leghts a = BC, b = CA, c = AB, of the
sides of a triangle ABC satisfy the inequalities, called "triangle
inequalities",
a + b > c b + c > a c + a > b .

1. In how many ways can one choose three numbers from the set {1,2,3,4,5,6}
so that the three could be the lenghts, say in inches, of the sides of a triangle?
Note that the order of the numbers doesn't matter: the triplet (2,3,4) is considered
the same as (3,4,2), (4,3,2) etc.

2. Prove that the lenght of any side of a triangle is less than the half of
the perimeter of that triangle.

3. In the triangle ABC, side AC = 3.8, and side AB = 0.6. Find the lenght of side
BC if it is an integer.

4. Find all positive integers x for which the numbers 2x + 3, 3x + 8, and 6x + 7
can be lenghts of the sides of a triangle.

October 18th
1. (Four Diesel Ships) Four diesel ships left a port at Noon , January 2nd.
The first ship returns to port every 4 weeks, the second - every 8 weeks, the third - every 12
weeks, and the fourth - every 16 weeks.
When did all four ships meet again in the port?

2. The customer said to the cashier: "I have 2 packages of lard at 9 cents; 2 cakes of soap at
27 cents; and 3 packages of sugar and 6 pastries, but I don't remember the prices of
the sugar and pastries."
"That will be \$2.92."
The customer said:"You have made a mistake."
The cashier checked again and agreed.
How did the customer spot the error?

3. Point O is located inside a convex quadrilateral ABCD (convex means that the quadrilateral
is entirely on one side of any line AB, BC, CD, and DA). It turnes out that the
segments OA, OB, OC, and OD divide the quadrilateral into four congruent triangles (having the
corresponding sides of equal lenghts). What kind of quadrilateral is ABCD?

October 4th
1. Suppose we have a barrel of wine, and a cup of tea. A teaspoon of wine is taken from
the barrel, and is poured into the cup of tea. Then the same teaspoon of the mixture is
taken from the cup, and is poured into the barrel. Now, the barrel contains some tea,
and the cup contains some wine. Which volume is larger - that of the tea in the barrel
or that of the wine in the cup?

2. A rancher split his cattle between his sons as follows. The first son got one cow and 1/7
of the rest of the cows; the second son got two of the undistributed cows and 1/7 of
the remaining after that cows; the third son got three of the undistributed yet cows
and 1/7 of the remaining after that cows; and so on. In the end, following this rule
for distributing the cows, the rancher completely split up the entire herd between all
of his sons.
How many sons did the rancher have, and how large was his herd?

3. (a) A teacher from a day care wants to split 9 apples evenly between 12 kids in such a way
that not a single apple gets split into more than four parts. How can this be done?
(b) Is it possible to split 5 apples evenly between 6 people without cutting a single
apple into more than three parts?

September 22nd
1. (Two Trains) A nonstop train leaves Ogden, Utah for Omaha, Nebraska at 60 mph. Another
nonstop train leaves Omaha for Ogden at 40 mph. How far apart are the trains one hour
before they pass each other?

2. (The stopped clock) My only timepiece is a wall clock. One day, I forgot to wind it and it
stopped. I went to visit a friend whose watch is always correct, stayed a while, and
returned home. There, I made a simple calculation, and set the clock right.
How did I do this when I had no watch on me to tell how long it took me to return from my
friend's house?

3. (Riding the train to a cottage) Two school girls were traveling from the city to a summer
cottage on an electric train.
"I noticed," one of the girls said, "that the cottage trains coming in the opposite direction
pass us every 5 minutes. What do you think - how many cottage trains arrive in the city
in one hour, given equal speeds in both directions?"
"Twelve, of course," the other girl answered, "because 60 divided by 5 equals 12."
The first girl did not agree. What do you think?

4. (How long is the train?) A train moving at 45 mph meets and is passed by a train moving at
36 mph. A passenger in the first train sees the second train take 6 second to pass him.
How long is the second train?

• Summer'08 Problems suggested for work at home
July 28th
Mathematical games.
Below, two mathematical games are described. There are two players, player 1 and player 2,
involved in a game. The problem is to decide which one of the players has a winning strategy.
A winning strategy consists of a list of moves the player should do so that they win no
matter what the other player's moves are.

1. There are 25 matches on a table. During each turn, a player can take any number of matches
between 1 and 4. The player who takes the last match (does the last move) wins. Which
of the players has a winning strategy? The same question for a pile of 24 matches.

2. There are two piles of matches, one pile with 10 matches, and another one with 7 matches.
During each turn, a player can take any number of matches from either one of the two
piles. The player who takes the laqst match wins. Which one of the players has a winning
strategy?

July 14th
1. A square has a corner folded over to create a pentagon. The three shorter sides of
thus formed pentagon are all the same length. Find the area of the pentagon as a
fraction of the area of the original square.

2. In a triangle, the lengths of two of the sides are known: 3.8 cm and 0.6 cm, while the
third one is an integer number of cantimeters. Find the length of the third side.

June 23rd
1. A jar filled with honey weighs 500 grams, while filled with gasoline weighs 350 grams.
Honey weighs twice as much as gasoline. How much does the empty jar weigh?

2. Two trains leave their stations, A and B, at the same time, and move toward each other at
a constant speed each. The first train reaches station B one hour, while the second
reaches station B two hours and 15 minutes after the trains meet. How much faster is
the first train moving than the second one?

• Spring'08 Problems suggested for work at home
June 9th
1. One day, at Dawn, two old ladies started walking along the same road: one from point A
to point B, the other - from B to A. They met each other at Noon, but did not stop
and kept going at the same speed. The first lady got to B at 4 p.m., while the second
got to her destination, A, at 9 p.m.. At what time was the Dawn that day?

2. Town X and town Y are 270 km apart. A car started from town X towards town Y at a uniform
speed of 60 km/h while a motorcycle started from town Y to town X at a uniform speed
of 90 km/h. Both the car and the motorcycle started their journey at 5:15 a.m.
(1) At what time did they pass each-other?
(2) How far away was the car from town Y when it passed the motrocycle?

May 26th
1. Angela bought some cloth at \$2p per yard. Using this cloth, she made 3p dresses and
still had 3p^2 yd of cloth left. (p^2 means "p squared") She used 2p yd of cloth to
make each dress.
(a) How much money did she pay for all the cloth? Give your answer in terms of p.
(b) If p = 2, how much money did she pay for all the cloth?

May 12th
1. At an exhibition, the ratio of the number of adults to the number of children was 8:3.
The ratio of the number of men to the number of women was 3:1. If there were 55 adults
and children at the exhibition, how many more children than women were there?

2. The ratio of the number of Ann's picture cards to the number of Wendy's picture cards was
7:8. After Wendy gave 8 picture cards to Ann, they each had the same number of picture cards.
How many picture cards did they have altogether?

3. A square ABCD of side lenght 42 in. is given. Denote by M, N, P, and Q the midpoints of the
sides AB, BC, CD, and DA respectively. Four circles, with centers at the midpoints, and
radii 21 in. are drawn. The circle with center M has AB as a diameter, the one with center
N has BC as a diameter, etc. All the circles meet at the center of the square. The circles
form four circular regions whose overlaps, within the square, form a four-lief "flower".
Find the area of that "flower".

April 21st
1. George and Alfred had the same number of greeting cards. After George used 6 greeting
cards, and Alfred used 30 greeting cards, the number of George's greeting cards to the number
of Alfred's greeting cards was 7:5. How many greeting cards did they have left?

2. Three boxes A, B, and C, had 260 ping pong balls. Lisa added some ping pong balls in box
A, and the number of the ping pong balls in box A tripled. She took half of the ping pong
balls in box B and added 40 ping pong balls into box C. The ratio of the number of ping pong
balls in boxes A, B, and C became 6:4:5. Find the original ratio of the number of ping pong
balls in boxes A, B, and C.

3. Let an isosceles right triangle ABC be given with AC = BC = 40 cm (the right angle is at the
vertex C). Draw three circles with centers the vertices of the triangle, and radii 20 cm each.
These circles cover three pieces of the triangle (each piece is a sector in the respective circle).
Find the area of the uncovered part of the triangle.

March 24th
1. Find the largest natural number which is divisible by 36, and which is written by
using each of the ten digits exactly once.

2. Write digits in the places of ? in 72?3? so that the resulting number be divisible by 45.

3. Find a two digit number the tens' digit of which is equal to the difference between the
number and its reversed: A = ? if AB - BA = A.

4. Is it true that if we write the digits of a natural number in the reversed order, the
difference between the number and the new one is divisible by 9?

5. In a country called Neverland, the currency exchanges in bank notes of worth 1 nev, 10 nevs,
100 nevs, and 1000 nevs. Is it possible to get 1,000,000 nevs in 500,000 bank notes?

March 10th
1. (Double-mysterious numbers) A five-digit number A is called "double-mysterious" if
the decimal notation of A and 2xA contain each one of the digits from 0 to 9 exactly once.
Find the largest and the smallest double-mysterious numbers.

2. Five students, A, B, C, D, and E competed in solving a math problem. The complete solution
of the problem was awarded 10 points, and a partial solution - an integer between 2 and 9.
Each student scored some number of points so that: A, B, and C collected 15 points together;
and B, C, and D collected 12 points together; the student A had the highest score;
and the student E, who scored 6 points, was placed third.
What was the score of student D?

3. A bar of soap has a form of a rectangular solid of dimensions 4cm X 8cm X 2cm. Equal ammount
of soap are washed out every day. In seven days, the bar has the same form but dimensions
two times as short as initially, i.e. 2cm X 4cm X 1 cm.
In how many days will the soap be washed out completely?

February 25th
1. Two positive integers are written using the digits 1, 4, 6, and 9 only. Is it possible
one of the numbers to be 17 times as large as the other one?

2. The National Bank of a certain country printed new series of banknotes of value 20, 50, and 200
units of the currency of that country. A young businessman decided to withdraw 10,000 new
banknotes of total value 1,000,000 currency units. Is this possible to be done? Explain.

3. From a piece of cheese in the form of a cube of edge 10cm, three slices of width 1 cm each are
to be cut. How should the cutting be done so that the rest of the piece has the greatest possible
volume?

4. Before stealing the keys from the sleeping jailer, the prisoner has calculated that he would need
at worst 21 attempts to find out which key to which cell is. How many cells are there
in the jail?

February 11th
1. Find three numbers between 10 and 30 whose sum is 70 and whose squares taken together
contain the digits from 1 to 9 exactly once.

2. The mean of three test scores is 74. What must a fourth score be to increase the mean to 78?

3. In a trivia game, a player receives three ponts for answering an easy question and seven points
for answering a hard one. What is the largest integer that cannot be a contestant's total score
in the game?

4. Suppose 7 people play in a chess tournament. Chess is a one-on-one game: a game is played by two
people only. Show that no matter how the tournament was done, when it is over, the number of people
who played odd number of games will be even.

February 4th
1. (Lucas' problem) Every day, at noon, a ship starts sailing from port A (La Havre, France)
to port B (New York, US), and at the same time a ship starts from B to A. A trip in either direction
takes exactly 7 days. How many ships will the ship sailing today from A meet on its trip?

2. A census-taker knocks on a door, and asks the woman inside how many children she has and
how old they are.

"I have threee daughters, and the product of the ages is 36," says the mother.

"That's not enough information," responds the census-taker.

"I'd tell you the sum of their ages, but you'd still be stumped."

"I wish you'd tell me something more."

"Okay, my oldest daughter Annie likes dogs."

What are the ages of the three daughters?

3. There were 2007 people on a gathering. Every person on that gathering shook hands with
all their friends, and only with them. Show that the number of the people who have
odd number of friends is even.

• Fall'07 Problems suggested for work at home
November 19th
1. A hunter walked 10 km southward from his camp, turned East and walked another 10 km
straight eastward, shot down a bear, turned North, and after a walk of another 10 km
got back to his camp. What was the color of the bear, and where did all that happen?

2. One day, at Dawn, two old ladies started walking along the same road: one from point A
to point B, the other - from B to A. They met each other at Noon, but did not stop
and kept going at the same speed. The first lady got to B at 4 p.m., while the second
got to her destination, A, at 9 p.m.. At what time was the Dawn that day?

3. At Noon today, there was a high tide at Miami Beach. At what time will it happen (at the
same place) tomorrow?
November 5th
1. Peter's sisters are two more than his brothers. How many more daughters than sons
do Peter's parents have?

2. There is a round lake in South America, where every year on June 1st, in the middle of
the lake, a beautiful flower (Victoria Regia) appears. The stem of that flower
rises up from the bottom of the lake, and its petals lie on the surface of the lake.
Every twenty-four hours the area of the petals doubles until July 1st when the flower covers
the whole water surface, the petals drop off, and the seeds of the flower reach the
bottom of the lake.
When does the area of the flower constitute the half of the lake's area?

3. A newborn infant was given 1 trillion dollars in cash at birth. How much would she need
to spend each day (average) to use it all up in eighty years, provided that none was
invested to earn interest?

October 22nd
1. A book for school costs a whole amount of dollars. Mary was 7 dollars short to
buy that book while Mike was only 1 dollar short. They joined to buy that book
together, but were still short of money! How much does the book cost?

2. A bottle with a cork costs 10 cents. The bottle is 9 cents more expensive than
the cork. How much does the bottle without cork cost?

3. The hypotenuse of a right triangle is 10 inches long. The altitude toward the hypotenuse
is 6 inches long. Find the area of the triangle.
The students form a C-school successfully solved this problem (giving the answer 30
square inches) 5 years in a row. But one day, this problem was given to their fellows
from an (A+)-school, and none of the students there solved it! Why?

October 15th
1. Find the year in which the first U.S. transcontinental railroad was completed if
a) the sum of the digits in the year is 24,
b) the ones and the tens digits are multiple of 3,
c) the hundreds digit is one less than the ones digit.

2. If the price is the same, which is a better buy - a round pizza with a diameter
of ten inches or a nine inch square pizza?

3. If a jar of peanut butter that is 3 inches in diameter and 4 inches high sells for
60 cents, what is a fair price for a jar that is 6 inches in diameter and 6
inches high?

4. A cube has edges of length 10 cm. If a fly lands on a vertex and then walks along
only the edges and never retraces an edge, what is the greatest distance the fly
could walk before coming to a vertex a second time?

5. Break 20 into two integers whose squares differ by 120.

October 1st
1. Demeter Dunce wrote four letters and correctly addressed four envelopes.
He was careless, however, and put some of the letters in the wrong envelopes.
(He put one letter in each envelope.) As it happened, he either got exactly three
of them right, or he got exactly two of them right, or he got exactly one of them
wrong. How many did he get right?

2. In a twelve hour period of time on an electronic digital clock, for how many
minutes is the sum of the digits greater than 15?

3. (Limerick Arithmetic)

A multiple of seven I be,
Not odd, but even you see;
My digits (a pair),
When multiplied there,
Make a cube and a square out of me.

Determine the number.

4. What is the greatest area of a rectangle having a perimeter of 26 cm and sides
of integral measure?

5. A six-foot-tall person walks around the Earth. Haw much farther does the person's
head travel than the person's feet?

September 17th
1. A Goose (with weak computational skills) met a flock of other geese. "Hello, a
hundred geese!" he greeted them. The flock's leader answered, "We are not a hundred
geese! But if one takes our flock, adds again our flock, adds half of our flock,
adds quarter of our flock, and than adds you Goose, then there will be a hundred."
How big was the flock?

2. The units' digit of a three-digit number N is 5, and its hundred and tens digits
are the same. When a one-digit number divides this number the remainder is 8.
What is N?

3. What is the remainder when 1492 x 1776 x 1812 x 1996 is divided by 5?

4. (A Roman Poem) A Roman boy was given this puzzle:

Take one hundred one, and to it affix
The half of a dozen, or if you please six
Put fifty to this, and then you will see
What ev'ry good boy should be.

Can you help that boy solve the puzzle?

Florida International University
Mathematics Department, FIU