MathDB

1

Part of 2020 Durer Math Competition Finals

Problems(3)

ATOC and BTOC have equal area, circumcenter, altitude

Source: 2020 Dürer Math Competition Finals E1. 1

11/29/2020
Let ABCABC be an acute triangle where AC>BCAC > BC. Let TT denote the foot of the altitude from vertex CC, denote the circumcentre of the triangle by OO. Show that quadrilaterals ATOCATOC and BTOCBTOC have equal area.
geometryequal areaequal areas
dominoes in a 3x3 square

Source: 2020 Dürer Math Competition Finals Day2 E1 https://artofproblemsolving.com/community/c1622639_2020_

1/7/2022
How many ways are there to tile a 3×33 \times 3 square with 44 dominoes of size 1×21 \times 2 and 11 domino of size 1×11 \times 1?
Tilings that can be obtained from each other by rotating the square are considered different. Dominoes of the same size are completely identical
combinatoricsTilingtiles
Help! USAMO 1991 #3

Source: USAMO 1991 #3

9/2/2004
Show that, for any fixed integer  n≥1, \,n \geq 1,\, the sequence 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) is eventually constant.
[The tower of exponents is defined by a1=2,  ai+1=2aia_1 = 2, \; a_{i+1} = 2^{a_i}. Also a_i \; (\mbox{mod} \; n) means the remainder which results from dividing aia_i by nn.]
inductionmodular arithmeticstrong inductionnumber theoryrelatively primeprime numbersnumber theory unsolved