MathDB

2016 Taiwan TST Round 1

Part of TST Round 1

Subcontests

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Geometry - 2016 Taiwan TST Round 1 Quiz 1 Problem 2

Circles O1O_1 and O2O_2 intersects at two points BB and CC, and BCBC is the diameter of circle O1O_1. Construct a tangent line of circle O1O_1 at CC and intersecting circle O2O_2 at another point AA. We join ABAB to intersect O1O_1 at point EE, then join CECE and extend it to intersect circle O2O_2 at point FF. Assume that HH is an arbitrary point on the line segment AFAF. We join HEHE and extend it to intersect circle O1O_1 at point GG, and join BGBG and extend it to intersect the extended line of ACAC at point DD. Prove that AHHF=ACCD\frac{AH}{HF}=\frac{AC}{CD}.

Number Theory

Find all ordered pairs (a,b)(a,b) of positive integers that satisfy a>ba>b and the equation (a−b)ab=abba(a-b)^{ab}=a^bb^a.

Inequality (a+b)(b+c)(c+a) neq 0

Let a,b,ca,b,c be nonnegative real numbers such that (a+b)(b+c)(c+a)≠0(a+b)(b+c)(c+a) \neq0. Find the minimum of (a+b+c)2016(1a2016+b2016+1b2016+c2016+1c2016+a2016)(a+b+c)^{2016}(\frac{1}{a^{2016}+b^{2016}}+\frac{1}{b^{2016}+c^{2016}}+\frac{1}{c^{2016}+a^{2016}}).
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2016 Taiwan TST Round 1 Quiz 1 Problem 1: Function

Suppose function f:[0,∞)→[0,∞)f:[0,\infty)\to[0,\infty) satisfies (1)∀x,y≥0,\forall x,y \geq 0, we have f(x)f(y)≤y2f(x2)+x2f(y2)f(x)f(y)\leq y^2f(\frac{x}{2})+x^2f(\frac{y}{2}); (2)∀0≤x≤1,f(x)≤2016\forall 0 \leq x \leq 1, f(x) \leq 2016. Prove that f(x)≤x2f(x)\leq x^2 for all x≥0x\geq 0.

Cards with black and white

Let nn cards are placed in a circle. Each card has a white side and a black side. On each move, you pick one card with black side up, flip it over, and also flip over the two neighboring cards. Suppose initially, there are only one black-side-up card. (a)If n=2015n=2015 , can you make all cards white-side-up through a finite number of moves? (b)If n=2016n=2016 , can you make all cards white-side-up through a finite number of moves?

Circle and lines

Let ABAB be a chord on a circle OO, MM be the midpoint of the smaller arc ABAB. From a point CC outside the circle OO draws two tangents to the circle OO at the points SS and TT. Suppose MSMS intersects with ABAB at the point EE, MTMT intersects with ABAB at the point FF. From E,FE,F draw a line perpendicular to ABAB that intersects with OS,OTOS,OT at the points X,YX,Y, respectively. Draw another line from CC which intersects with the circle OO at the points PP and QQ. Let RR be the intersection point of MPMP and ABAB. Finally, let ZZ be the circumcenter of triangle PQRPQR. Prove that XX,YY and ZZ are collinear.