Problem 1
The numerator and denominator of Juan's fraction are positive integers whose sum is 
.
.
Show / hide solution
.
,
, from which we get that the biggest value of a is 
,
,Problem 2
Rectangle 
intersects a circle at points 
,
, 
,
as shown in the diagram. If 
,
, and 
,
.
Show / hide solution
,
be the projections of 
.
and 
,
.Problem 3
Find the sum of digits of the number 
.
Show / hide solution
.
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?
Show / hide answerShow / hide solution
.
.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.
Show / hide solution
be all the piles. Then 
,
.
and 
,
,
.
,
,
,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 
,.
Show / hide solution
cm.
,
, 
,
be the perimeters of the rectangles 
cm.Problem 7
Find pairwise distinct digits ,, such that
Show / hide solution
,
, 
.
,
, 
.
,
.
,
,
and 
,
,
and 
.
,
,Problem 8
Find the area of a rectangle with perimeter 
cm and diagonal 
cm.
Show / hide solution
.
,
Problem 9
Andrei took 
equally sized white cubes and used them to form one big cube 
.
.
Show / hide solution
,
.
,
.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?
Show / hide solution
be the number of girls in the math circle, and let 
,
,
,
,
.
,
.
,
,
and 
.
,
,
,
.Problem 11
If you increase the number of this problem by 
,
,
. and ,
problems left (including this one).
Show / hide solution
,
.
to be a cube, 
is 
,Problem 12
Find positive integer such that 
.
Show / hide solution
.
Problem 13
Find the smallest positive integer, which ends with number 
,,.
Show / hide solution
.
for some 
.
,Problem 14
Each pair of consecutive digits of a -digit or 
.
.
Show / hide solution
.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?
Show / hide answerShow / hide solution
of the sum of the digits there is exactly one smallest positive integer with this sum of its digits, i.e. one awesome number. Let's denote it 
.
will be two digit integers, whereas 
will have more than three digits. Also the sums of digits of three-digit integers attain all numbers between 
and 
will indeed have three digits and thus we have Problem 16
Tim has found real numbers 
,
, 
satisfying 
.
?
Show / hide solution
.
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?
Show / hide solution
.
,
, … , 
.
to 
the highest digits and 
,
, 
,
. The result is 
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.
Show / hide solution
and 
,
and 
,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
Show / hide solution
.
.
.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.
Show / hide solution
.
.
works, 
does not.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 ,
Show / hide solution
.
.
.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?
Show / hide solution
m.
meters. For the boat which had travelled 
,
.Problem 23
Vertices of a star form a regular heptagon. What is the magnitude of the marked angle?
Show / hide solution
.
,
.
and 
is supplementary.Problem 24
Find such that 
.
Note: the order of operations is: 
.
Show / hide solution
.
,
.Problem 25
How many triplets of positive integers 
exist, such that
?Show / hide solution
.
and 
to get 
,
we get 
and for 
,
we get 
,Problem 26
A circle 
with radius ,
and diameter 
is given in the plane. Draw a line 
through point such that it is perpendicular to .
on such that is outside . the second intersection of 
and and assume 
.
.
Show / hide solution
.
,
is isosceles. Thus 
and also on its hypotenuse, so it is the midpoint of the hypotenuse. Thus 
and Pythagorean theorem gives 
.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?
Show / hide solution
.
soldiers at the beginning of the battle. Also note that 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.
Show / hide solution
.
and 
(dashed in the diagram) all other triangles have a red edge. Every dashed triangle can be colored in 
ways. Finally, we have 
ways to color the dotted diagonals 
,
, 
.
suitable colorings. 
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.
Show / hide answerShow / hide solution
.
.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 
,
Show / hide solution
Indians drinking tea, which is against humanity. The other solution works and yields Problem 31
Three digits were written to the end of a positive integer . to ..
Show / hide solution
.
,
.
,
and greater than 
for 
.
which satisfies the problem for 
.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.
Show / hide solution
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.
Show / hide solution
.
Problem 34
In triangle 
with 
,
on the side 
,
and 
.
such that 
and points and lie each on opposite side of . and 
have integral values (in degrees), find all possible values of angle .
Show / hide solution
.
and 
,
.
a 
.
,
.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?
Show / hide answerShow / hide solution
.
such sequences.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 
.
Show / hide solution
, 
,
so that the pairs 
.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?
Show / hide solution
changes of direction, which means 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.
Show / hide answerShow / hide solution
.Problem 39
Two tetrominoes made of 
squares touch at points ,, as in the diagram. Find the distance .
Show / hide solution
.
are congruent by 
.
a by Pythagorean theorem also 
.
.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?
Show / hide solution
.
midpoints with integers coordinates so the answer is 
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?
Show / hide answerShow / hide solution
.Problem 42
Real numbers and satisfy 
.
.
Show / hide solution
Problem 43
A sequence is definned as follows: 
,
, and
such that 
.Show / hide solution
.Problem 44
Let be an acute-angled triangle with altitudes 
,
, 
,.
.Show / hide solution
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?
Show / hide answerShow / hide solution
Problem 46
Let 
be the midpoint of side 
of rectangle .
and 
have radii both equal to 
and the inradius of 
is .
Show / hide solution
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 
?
Show / hide solution
Problem 48
Find all real numbers such that
Show / hide solution
.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.
Show / hide solution
.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.
Show / hide solution
Problem 51
Find all integers such that both 
and 
are perfect squares.
Show / hide solution
.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?
Show / hide solution
.Problem 53
Let be a triangle with circumradius 
and inradius .
are inscribed in angles 
,
, 
, which is tangent to all three circles. Find .
Show / hide solution
.Problem 54
Real numbers ,, , satisfy
.Show / hide solution
.
