Competition problems 2011

### Problem 1

The numerator and denominator of Juan's fraction are positive integers whose sum is . The value of this fraction is less than . Find the greatest such fraction.

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### Problem 2

Rectangle intersects a circle at points , , , as shown in the diagram. If , , and , find .

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### Problem 3

Find the sum of digits of the number .

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### Problem 4

T-shirts were made in three colors, red, gray and blue. The number of red T-shirts is six smaller than the total number blue and gray T-shirts. Also, the number of gray T-shirts is ten smaller than the total number red and blue T-shirts. How many blue T-shirts were made?

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### Problem 5

There are walnuts on the table in at least two piles. Each pile consists of at least walnuts. After moving one walnut from each pile to the first one, all piles will have the same number of walnuts. What was the original number of piles? Find all possibilities.

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### Problem 6

A rectangle is divided by two segments parallel to its sides into four smaller rectangles. Let's label them , , , as in the diagram. Given that the perimeters of rectangles , , are , , and , respectively, find all possible values of the perimeter of rectangle .

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### Problem 7

Find pairwise distinct digits , , such that

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### Problem 8

Find the area of a rectangle with perimeter cm and diagonal cm.

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### Problem 9

Andrei took equally sized white cubes and used them to form one big cube . Then he colored its surface in red. Given that now one tenth of the total surface of the cubes is red, find .

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### Problem 10

What is the least possible number of members of a math circle, where girls form more than but less than of the members?

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### Problem 11

If you increase the number of this problem by , you obtain the number of the most shocking problem. Whereas if you increase it by a two-digit number , you obtain the number of the most playful problem. Moreover, we have . Find and , given that you have problems left (including this one).

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### Problem 12

Find positive integer such that .

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### Problem 13

Find the smallest positive integer, which ends with number , which is divisible by , and whose sum of digits is .

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### Problem 14

Each pair of consecutive digits of a -digit number is a multiple of either or . Its last digit is . Find its first digit.

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### Problem 15

A positive integer is called awesome if any other positive integer with the same sum of digits is greater. How many three-digit awesome numbers exist?

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### Problem 16

Tim has found real numbers , , satisfying . What are the possible values of ?

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### Problem 17

The numbers are arranged in some order to form a nine-digit integer. Consider all triplets of consecutive digits and add the corresponding seven three-digit numbers. What is the largest result that can be obtained?

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### Problem 18

A real number is written in each cell of a square. Emily wrote down all products of two numbers from two distinct cells of the table and noticed that exactly of these products were negative. How many times did number appear in the original square? Find all possibilities.

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### Problem 19

Math kingdom started to produce a new set of coins. On the first day they created coins with value MD (Math Dollar). Every other day they created coins with the smallest value which cannot be paid by at most ten existing coins. Which coins did they create on the th day?

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### Problem 20

Let the number be the solution to this problem. Find the probability that a randomly chosen point inside a unit square is at least units away from all sides.

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### Problem 21

A square is filled with integers such that the sums of the horizontal rows increase by two going downwards and the sums of the vertical columns double from left to right. Given that the sum of the numbers in one of the rows is , find the sum of the numbers in the leftmost column.

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### Problem 22

There are 2 boats, one on each side of a river bank. They both sail towards each other at a constant speed (not necessarily the same). The first time they meet, they are meters from one side of the bank. Once they reach the side of the bank, they turn around and move towards each other again. This time they meet meters from the other side of the bank. How wide is the river?

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### Problem 23

Vertices of a star form a regular heptagon. What is the magnitude of the marked angle?

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### Problem 24

Find such that .

Note: the order of operations is: .

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### Problem 25

How many triplets of positive integers exist, such that

and ?

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### Problem 26

A circle with radius , center and diameter is given in the plane. Draw a line through point such that it is perpendicular to . Choose a point on such that is outside . Denote by the second intersection of and and assume . Find .

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### Problem 27

Two nations and are in a battle with soldiers involved altogether. The armies take turns to attack. In each turn every living soldier from the attacking army shoots a soldier from the enemy's army. The battle ended (not necessarily by elimination of one of the sides) after three turns ( was shooting first, then and finally again). What is the least guaranteed number of survivors?

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### Problem 28

All six sides of a convex hexagon are colored in red. Each of the diagonals is either blue or red. Find the number of such colorings that each triangle () has at least one red side.

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### Problem 29

Malcom told each Michal and Shri a positive integer. Further, he told them, that the numbers they heard were distinct and that their sum is a two-digit number. Then the following conversation took place:

Michal: ”I cannot determine which one of us has the greater number.”

Shri: ”I can't determine it either, but I will tell you that my number is divisible by .”

Michal: ”Wow! Now, I can determine the sum of our numbers.”

Find the value of this sum, given that the logic Michal and Shri used was flawless.

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### Problem 30

There are guests in a café, Turks and Indians. Each of them drinks either tea or coffee. An Indian speaks truth if he drinks tea and lies if he drinks coffee, whereas with the Turks it is the other way round. For questions ”Do you drink coffee?”, ”Are you Turkish?” and ”Is it raining outside?” the numbers of positive answers were , , and , respectively. How many Indians drink tea? Find all possibilities.

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### Problem 31

Three digits were written to the end of a positive integer . The resulting number was equal to the sum of numbers from to . Find all possible values of .

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### Problem 32

Alice, Betty, Claudia, Daniel, and Eli were playing doubles tournament in table-tennis. Each pair played against each other pair exactly once. Alice won games and Betty won games. How many games could Claudia win? Find all possibilities.

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### Problem 33

Two players are playing a game on the given plan consisting of 30 cells. The rules are the following.

• players take turns,
• in one move a player colors one cell,
• in the first move, only a cell neighbouring with the edge can be colored. In any other moves, only a cell which is next to the last colored cell and is not further away from the center, can be colored,
• once a cell is colored, it cannot be colored again.
• the player who can no longer make a move, loses.
How many cells will be colored by the end of the game, in which both players play perfectly and the one who cannot win tries to make the game as long as possible?

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### Problem 34

In triangle with , we find a point on the side , such that and . Further, we find point such that and points and lie each on opposite side of . If the internal angles in triangles and have integral values (in degrees), find all possible values of angle .

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### Problem 35

Ten people in the theater are sitting next to each other in one row. After a break they sit in a new arrangement, so that only two people remained in their original positions and the remaining eight sat next to their former position. In how many ways could they have done that?

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### Problem 36

A positive integer is written on each face of a cube. To each vertex we assign the product of the numbers written on the three faces intersecting at that vertex. The sum of the numbers assigned to the vertices is . What are the possible values of the sum of the numbers written on the faces?

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### Problem 37

Two bicyclists are racing on a straight path at constant speeds. They both start at one end and everytime they hit the end of the path, they turn around and go the opposite direction. Eventually, they meet again at one of the endpoints. Before that, the slower bicyclist had traveled the path times (in one of the directions) and the faster one times. How many times did they meet head on?

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### Problem 38

Find the largest positive integer such that all its digits (other than the first one and the last one) are smaller than the arithmetic mean of the two surrounding digits.

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### Problem 39

Two tetrominoes made of squares touch at points , , as in the diagram. Find the distance .

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### Problem 40

There are points with integral coordinates given in the plane. We connect each pair with a segment. How many of these segments are guaranteed to have a midpoint with integral coordinates?

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### Problem 41

A five-digit integer is called irreducible if it cannot be written as a product of two three-digit integers. What is the maximum possible number of consecutive irreducible integers?

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### Problem 42

Real numbers and satisfy . Find the minimum possible value of .

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### Problem 43

A sequence is definned as follows: , , and

as long as the right-hand is well-defined. Find the least such that .

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### Problem 44

Let be an acute-angled triangle with altitudes , , , intersecting at . Given that

find .

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### Problem 45

Every guest at a party (including Tim) knows exactly seven boys and ten girls there. What is the least possible number of people at the party?

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### Problem 46

Let be the midpoint of side of rectangle . The incircles of triangles and have radii both equal to and the inradius of is . Find the sides of the rectangle.

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### Problem 47

We write out all divisors of a positive integer which are less than from the greatest to the lowest. If is the sum of the second and third divisor, we say that is additive. How many additive numbers are there which are less than ?

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### Problem 48

Find all real numbers such that

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### Problem 49

A position of a minute hand and an hour hand on the clock is called valid, if it might occur during one hour cycle. Find the number of valid positions which remain valid after switching the two hands.

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### Problem 50

Let , , be nonzero real numbers, such that the quadratic equations and have a common root. Find all possible real values of this root.

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### Problem 51

Find all integers such that both and are perfect squares.

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### Problem 52

A regular octahedron with side length is given in space. One circle is inscribed in one of the faces and another circle is circumscribed about an adjacent face. What is the minimal distance between the two circles?

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### Problem 53

Let be a triangle with circumradius and inradius . Three circles with radius are inscribed in angles , , , respectively such that the circles are in the interior of the triangle and there exists another circle with radius which is tangent to all three circles. Find .