Some of my personal favourites
by navi_09220114, Nov 15, 2023, 5:27 PM
I have been creating proposals for the past 4-5 years now, and I would like to make a small collection of my own favourites here. Hope you (the reader) will like them
Algebra:
1. Find all functions such that for all real numbers
[It was restricted to continuous functions in the actual test to lower its difficulty, but it can be solved in more generality]
2. A sequence of reals satisfies for all ,Prove that for all , the sequence satisfies the equation
3. For each integer , determine all infinite sequences of positive integers , , for which there exists a polynomial of the formwhere , , , are non-negative integers, such that
Combinatorics
1. Let be a strictly increasing sequence on positive integers.
Is it always possible to partition the set of natural numbers into infinitely many subsets with infinite cardinality , so that for every subset , if we denote be the elements of , then for every and for every , it satisfies ?
2. Let be a set of points in a plane, and it is known that the distances of any two different points in are all distinct. Ivan colors the points with colors such that for every point , the closest and the furthest point from in also have the same color as .
What is the maximum possible value of ?
3. Ivan is playing Lego with blocks. First, he places blocks to fit a square as the bottom layer. Then he builds the top layer on top of the bottom layer using the remaining blocks. Note that the blocks in the bottom layer are connected to the blocks above it in the top layer, just like real Lego blocks. He wants the whole two-layered building to be connected and not in seperate pieces.
Prove that if he can do so, then the four blocks connecting the four corners of the bottom layer, must be all placed horizontally or all vertically.
4. Let be a fixed integer. In the town of Ivanland, there are at least citizens standing on a plane such that the distances between any two citizens are distinct. An election is to be held such that every citizen votes the -th closest citizen to be the president. What is the maximal number of votes a citizen can have?
5. Given two positive integers and , find the largest in terms of and such that the following condition holds:
Any tree graph with vertices has two (possibly equal) vertices and such that for any other vertex in , either there is a path of length at most from to , or there is a path of length at most from to .
Geometry:
1. Given an acute triangle , mark points in the interior of the triangle. Let be the projections of to respectively, and define the points similarly for .
a) Suppose that for all , prove that .
b) Prove that this is not neccesarily true, if triangle is allowed to be obtuse.
2. Given a triangle with and circumcenter . Let and be midpoints of and respectively, and let intersect at . Denote to be the circle . Let intersect again at and let intersect again at .
Suppose the line parallel to passing through meets at . Prove that the lines and meet at .
3. Let triangle with has orthocenter , and let the midpoint of be . The internal angle bisector of meet at , and the external angle bisector of meet at . The circles and meet again at .
Prove that .
4. Let be an acute triangle with . Let be the midpoints of the sides , , and respectively, and be the midpoints of minor arc not containing and major arc respectively. Suppose are the incenter, -excenter, -excenter, and -excenter of triangle respectively.
Prove that the circumcircles of the triangles , , meet at a common point.
Number Theory:
1. Given a natural number , call a divisor of to be if . A natural number is if one or more distinct nontrivial divisors of sum up to .
Prove that every natural number has a multiple that is good.
2. Given a four digit string , , prove that there exist a such that contains as a substring when written in base .
3. Find all polynomials with integer coefficients such that for all positive integers , the sequenceis eventually constant modulo .
Algebra:
1. Find all functions such that for all real numbers
[It was restricted to continuous functions in the actual test to lower its difficulty, but it can be solved in more generality]
2. A sequence of reals satisfies for all ,Prove that for all , the sequence satisfies the equation
3. For each integer , determine all infinite sequences of positive integers , , for which there exists a polynomial of the formwhere , , , are non-negative integers, such that
Combinatorics
1. Let be a strictly increasing sequence on positive integers.
Is it always possible to partition the set of natural numbers into infinitely many subsets with infinite cardinality , so that for every subset , if we denote be the elements of , then for every and for every , it satisfies ?
2. Let be a set of points in a plane, and it is known that the distances of any two different points in are all distinct. Ivan colors the points with colors such that for every point , the closest and the furthest point from in also have the same color as .
What is the maximum possible value of ?
3. Ivan is playing Lego with blocks. First, he places blocks to fit a square as the bottom layer. Then he builds the top layer on top of the bottom layer using the remaining blocks. Note that the blocks in the bottom layer are connected to the blocks above it in the top layer, just like real Lego blocks. He wants the whole two-layered building to be connected and not in seperate pieces.
Prove that if he can do so, then the four blocks connecting the four corners of the bottom layer, must be all placed horizontally or all vertically.
4. Let be a fixed integer. In the town of Ivanland, there are at least citizens standing on a plane such that the distances between any two citizens are distinct. An election is to be held such that every citizen votes the -th closest citizen to be the president. What is the maximal number of votes a citizen can have?
5. Given two positive integers and , find the largest in terms of and such that the following condition holds:
Any tree graph with vertices has two (possibly equal) vertices and such that for any other vertex in , either there is a path of length at most from to , or there is a path of length at most from to .
Geometry:
1. Given an acute triangle , mark points in the interior of the triangle. Let be the projections of to respectively, and define the points similarly for .
a) Suppose that for all , prove that .
b) Prove that this is not neccesarily true, if triangle is allowed to be obtuse.
2. Given a triangle with and circumcenter . Let and be midpoints of and respectively, and let intersect at . Denote to be the circle . Let intersect again at and let intersect again at .
Suppose the line parallel to passing through meets at . Prove that the lines and meet at .
3. Let triangle with has orthocenter , and let the midpoint of be . The internal angle bisector of meet at , and the external angle bisector of meet at . The circles and meet again at .
Prove that .
4. Let be an acute triangle with . Let be the midpoints of the sides , , and respectively, and be the midpoints of minor arc not containing and major arc respectively. Suppose are the incenter, -excenter, -excenter, and -excenter of triangle respectively.
Prove that the circumcircles of the triangles , , meet at a common point.
Number Theory:
1. Given a natural number , call a divisor of to be if . A natural number is if one or more distinct nontrivial divisors of sum up to .
Prove that every natural number has a multiple that is good.
2. Given a four digit string , , prove that there exist a such that contains as a substring when written in base .
3. Find all polynomials with integer coefficients such that for all positive integers , the sequenceis eventually constant modulo .
This post has been edited 2 times. Last edited by navi_09220114, Nov 15, 2023, 5:29 PM