problem solving

euclidean plane geometry

• an exercise with radical centers: suppose triangle ABC is isosceles with apex A. Let D and E denote the midpoints of [AB] and [AC] respectively. Let F be the unique point on DE so that FAC is a right angle. Let G be the projection of A on FC. Show that the circle passing through B, C and G is tangent to both AB and AC.

• IMO 2016, problem 1 (Shortlisted as G1): Triangle BCF has a right angle at B. Let A be the point on the line CF such that |FA|=|FB| and F lies between A and C. Point D is chosen such that |DA|=|DC| and AC is the bisector of angle DAB. Point E is chosen such that |EA|=|ED| and AD is the bisector of angle EAC. Let M be the midpoint of CF. Let X be the point so that AMXE is a parallelogram. Prove that lines BD, FX, and ME are concurrent. (Hint: Either try to use the radical center of three circles, or Desargues' perspectivity theorem)

• JEMC 2020, problem 1. Suppose triangle ABC is isosceles with apex A. Let D and E denote the midpoints [AB] and [AC] respectively. Let F be the unique point so that D is the midpoint of [EF] and let Gamma be the circumscribed circle of FDB. Let G be the unique point on CD so that the midpoint of [BG] lies on Gamma. Let H be the second intersection of Gamma and FC. Show that BHGC is cyclic. (bonus: suppose T is the intersection of CF and BG. Then show that BH,CD en AT are concurrent).

• Triangle ABC is isosceles with apex A and angle at A is obtuse. Denote its circumscribed circle by Gamma. The points D and E are the midpoints of [AB] and [BC] respectively. Let Omega denote the circumscribed circle of triangle ADE. We denote by F and G the second intersection points of Omega with BC and Gamma respectively, so that B,F,G,C lie in this order on BC. Show that AF, CD and BE are concurrent.

• BxMO 2020, problem 3 (with Pierre Haas and Jeroen Huijben): Let ABC be a triangle. The circle ω_A through A is tangent to line BC at B. The circle ω_C through C is tangent to line AB at B. Let ω_A and ω_C meet again at D. Let M be the midpoint of line segment [BC], and let E be the intersection of lines MD and AC. Show that E lies on ω_A 

• A equilateral triangle of area 1/4 is placed within an equilateral triangle of area 1, in such a way so that the edges are parallel. One connects the endpoints for each pair of parallel edges. Show that the area of the resulting hexagon is 1/3. 

combinatorial geometry

• Consider n circles in the plane, and the regions they form. Show that at least one region is convex.

• Consider a configuration of n lines in the plane so that no two lines are parallel and no three lines are concurrent. Such a configuration divides up the plane in regions. Show that there are at least n intersection points which are neighbouring to exactly one triangular region.

• JEMC 2020, problem 3. A variant on a classical tiling problem. Two types of tiles are given: an F-tile consisting of 6 squares and a Z-tile consisting of 4 squares: Find all positive integers n such that an n × n board consisting of n 2 unit squares can be covered without gaps with these two types of tiles (rotations and reflections are allowed) so that no two tiles overlap and no part of any tile covers an area outside the n × n board. 

• IMOLL 2021 (longlisted as BEL1, with Stijn Cambie) Show that a convex polygon of area 1 is circumscribed by a triangle of area 2.

other

• BxMO 2020, problem 1 (with Pierre Haas and Stijn Cambie): Find all positive integers d with the following property: there exists a polynomial P of degree d with integer coefficients such that |P(m)| = 1 for at least d + 1 different integers m.

• Suppose P(X) is an integer polynomial of degree 2n+1 so that P(0),P(1),...,P(4n+4) are squares of prime numbers. Show that P(X) is irreducible.

• IMOSL 2022, N8. Show that for all integers n>0 the integer 2^n+65 does not divide 5^n-3^n. (Hint: Jacobi reciprocity)

• IMOSL 2023, A4 (with Tijs Buggenhout) determine all functions f:R_{>0}->R_{>0} so that for all x,y in R_{>0} the following inequality holds: y(f(x)+f(y)) \ge (x+f(f(y)))f(x) 

• Show that x²+y²+z²+w²=0 admits no nontrivial solutions over Q(sqrt(-7))

• Let p be a prime. Show that there are exactly p² solutions to the diophantine equation x²+y²+z²=0 modulo p.  hint: expand the sum \sum_(0<a,b,c<p) (1+(abc/p)), where (abc/p) denotes the legendre symbol.

• (by Heegner) Suppose f(X) is a degree 4 polynomial with coefficients in a field K, whose leading coefficient is not a square in K. Assume that the equation y²=f(x) has a solution over a finite field extension L/K of odd degree. Then show that there already exists a solution to y²=f(x) over K. (note: a related result of Artin-Springer deals with descending solutions of quadratic forms along odd degree extensions)

• (by Dedekind) Let S(N) denote the sum of all positive reduced fractions with both enumerator and denominator at most N. Show that S(N)/N^2->9/(2pi²)  as N->infty.

• Show that for every prime p=1 mod 3, there exist invertible p-adic integers x,y,z so that x^p+y^p=z^p. (note: it seems it is an open problem whether there exist infinitely many p=2 mod 3 satisfying this property)

• A few friends go to a bar and during the night one each of them orders a different bottle of wine at a different price. Each of them is sure they have enough money to pay for their bottle. Because in the end they get too mixed up in talking and drinking, they forget how much each of them has to pay and what wine they ordered. Each of the friends only has a limited amount of cash and no credit cards. The bartender suggests that the person with the least cash pays the cheapest wine, and so fort. Prove that the strategy of the bartender works.

• the 'lost boarding pass' problem: A fully booked plane of 100 passengers is ready to take off. One by one, each of the passengers enters the plane. Unfortunately the first passenger has lost their ticket, and can't remember where to sit. They decide to sit somewhere randomly. Subsequently each of the other 99 passengers finds their spot, all knowing their correct seat number. In case their seat is taken they go sit somewhere randomly. show that in the end the last passenger has a chance of exactly 50% of sitting in the correct seat.