MathDB

2022 Taiwan TST Round 2

Part of TST Round 2

Subcontests

(7)
6
1

Electrified

Let N>sN>s be positive integers. Electricity park has a number of buildings; exactly NN of them are power plants, and another one of them is the headquarter. Some pairs of buildings have one-way power cables between them, satisfying: (i) The cables connected to a power plant will only send the power out of the plant. (ii) For each non-headquarter building, there is a unique sequence of cables that can transport the power from that building to the headquarter. A building is ss-electrifed if, after removing any one cable from the park, the building can still receive power from at least ss different power plants. Find the maximum possible number of ss-electrifed buildings.
Note: There seems to be confusion about whether a power plant is 11-electrified. For the sake of simplicity let's say that any power plant is not ss-electrified for any s≥1s\geq 1.
Proposed by usjl
5
1
C
1
2
1
A
1
N
2

Large and small number

For any two coprime positive integers p,qp, q, define f(i)f(i) to be the remainder of p⋅ip\cdot i divided by qq for i=1,2,…,q−1i = 1, 2,\ldots,q -1. The number ii is called a large number (resp. small number) when f(i)f(i) is the maximum (resp. the minimum) among the numbers f(1),f(2),…,f(i)f(1), f(2),\ldots,f(i). Note that 11 is both large and small. Let a,ba, b be two fixed positive integers. Given that there are exactly aa large numbers and bb small numbers among 1,2,…,q−11, 2,\ldots , q - 1, find the least possible number for qq.
Proposed by usjl

Cute Palindromic Problem

A positive integer is said to be palindromic if it remains the same when its digits are reversed. For example, 12211221 or 7484774847 are both palindromic numbers. Let kk be a positive integer that can be expressed as an nn-digit number an−1an−2⋯a0‾\overline{a_{n-1}a_{n-2} \cdots a_0}. Prove that if kk is a palindromic number, then k2k^2 is also a palindromic number if and only if a02+a12+⋯+an−12<10a_0^2 + a^2_1 + \cdots + a^2_{n-1} < 10.
Proposed by Ho-Chien Chen
G
2

Yet Another OI Geometry Problem

Let II, OO, HH, and Ω\Omega be the incenter, circumcenter, orthocenter, and the circumcircle of the triangle ABCABC, respectively. Assume that line AIAI intersects with Ω\Omega again at point M≠AM\neq A, line IHIH and BCBC meets at point DD, and line MDMD intersects with Ω\Omega again at point E≠ME\neq M. Prove that line OIOI is tangent to the circumcircle of triangle IHEIHE.
Proposed by Li4 and Leo Chang.

OH parallel to BC

Let ABCABC be a triangle with circumcenter OO and orthocenter HH such that OHOH is parallel to BCBC. Let AHAH intersects again with the circumcircle of ABCABC at XX, and let XB,XCXB, XC intersect with OHOH at Y,ZY, Z, respectively. If the projections of Y,ZY,Z to AB,ACAB,AC are P,QP,Q, respectively, show that PQPQ bisects BCBC.
Proposed by usjl