Subcontests
(24)Make c1 large and c2 small
We define f(x,y,z)=∣xy∣x2+y2+∣yz∣y2+z2+∣zx∣z2+x2.
Find the best constants c1,c2∈R such that c1(x2+y2+z2)3/2≤f(x,y,z)≤c1(x2+y2+z2)3/2 hold for all reals x,y,z satisfying x+y+z=0.Proposed by Untro368. Tremendous numbers
We can conduct the following moves to a real number x: choose a positive integer n, and positives reals a1,a2,⋯,an whose reciprocals sum up to 1. Let x0=x, and xk=xk−1ak for all 1≤k≤n. Finally, let y=xn. We said M>0 is "tremendous" if for any x∈R+, we can always choose n,a1,a2,⋯,an to make the resulting y smaller than M. Find all tremendous numbers.Proposed by ckliao914. Operations on numbers
Given integer n≥3. 1,2,…,n were written on the blackboard. In each move, one could choose two numbers x,y, erase them, and write down x+y,∣x−y∣ in the place of x,y. Find all integers X such that one could turn all numbers into X within a finite number of moves. "I can solve this without seeing it"
In an 2023×2023 grid we fill in numbers 1,2,⋯,20232 without duplicating. Find the largest integer M such that there exists a way to fill the numbers, satisfying that any two adjacent numbers has a difference at least M (two squares (x1,y1),(x2,y2) are adjacent if x1=x2 and y1−y2≡±1(mod2023) or y1=y2 and x1−x2≡±1(mod2023)).Proposed by chengbilly. Ghost leg
A ghost leg is a game with some vertical lines and some horizontal lines. A player starts at the top of the vertical line and go downwards, and always walkthrough a horizontal line if he encounters one. We define a layer is some horizontal line with the same height and has no duplicated endpoints. Find the smallest number of layers needed to grant that you can walk from (1,2,…,n) on the top to any permutation (σ1,σ2,…,σn) on the bottom. Cyclic quadrilateral and a tangent
Triangle ABC has circumcenter O. D is the foot from A to BC, and P is apoint on AD. The feet from P to CA,AB are E,F, respectively, and the foot from D to EF is T. AO meets (ABC) again at A′. A′D meets (ABC) again at R. If Q is a point on AO satisfying ∠ABP=∠QBC, prove that D,P,T,R lie on acircle and DQ is tangent to it. Special hexagon
ABCDEF is a cyclic hexagon with circumcenter O, and AD,BE,CF are concurrent at X. P is a point on the plane. The circumenter of PAB is OAB. Define OBC,OCD, ODE,OEF,OFA similarly. Prove that OABODE,OBCOEF,OCDOFA, OX are concurrent. Configuration with cyclic quadrilateral
ABCD is a cyclic quadrilateral with circumcenter O. The lines AC,BD intersect at E and AD,BC intersect at F. O1 and O2 are the circumcenters of △ABE and △CDE, respectively. Assume that (ABCD) and (OO1O2) intersect at two points P,Q. Prove that P,Q,F are collinear. Three concurrent lines in a triangle
P is a point inside △ABC. AP,BP,CP intersects BC,CA,AB at D,E,F, respectively. AD meets (ABC) again at D1. S is a point on (ABC). Lines AS, EF intersect at T, lines TP,BC intersect at K, and KD1 meets (ABC) again at X. Prove that S,D,X are colinear. Number of solution of a system of equations
Let S(b) be the number of nonuples of positive integers (a1,a2,…,a9) satisfying 3b−1=a1+a2+…+a9 and b2+1=a12+…+a92. Prove that for all ϵ>0, there exists Cϵ>0 such that S(b)≤Cϵb3+ϵ.