An organizing committee packs Buuz dumplings in small and large boxes of positive integer capacities. A small box holds less than half a large box. One large and one small together hold 63 dumplings. One large and two small hold two-fifths of all prepared dumplings. Packing all into three large and two small boxes leaves a positive integer $n$ unpacked. Find the sum of all possible values of $n$.
Give a positive integer $x$ such that $x,2x,3x,4x,5x$ are all perfect powers, and explain its construction. A perfect power is $a^i$ for positive integers $a,i$ with $i\gt1$.
$ABC$ is right-angled at $B$. Points $D,E\in BC$ satisfy $\angle BAD=20^\circ$, $\angle DAE=15^\circ$, $\angle EAC=20^\circ$. If $EC=4$ cm, find $BD$ in cm.
The board initially contains $1,2,\ldots,33$. Repeatedly erase two numbers, one a multiple of the other, and write their integer quotient. Stop when no number is a multiple of another. What is the least possible number remaining?
A,B,C,D,E are centres of five circles of radius 1 cm. $\angle ABC=90^\circ$, $\angle BAC=60^\circ$, $BC\parallel DE$. The circle pairs (A,D),(B,D),(D,E),(E,C) are externally tangent; all other pairs do not intersect. The maximum distance between two points chosen on the circles is $a+\sqrt b+\sqrt c$ cm, with positive integers $a,b,c$ and $b\lt c$. Find $a+10b+100c$.
A sidewalk has integer length $1\le N\le1000$. Lay Fibonacci-length tiles 1,1,2,3,5,8,… in increasing order starting at the first 1. When the next tile exceeds the uncovered remainder, restart at the first 1. Each start or restart is a new sequence. Choose the smallest N that requires the greatest number of sequences. How many tiles are used altogether? For example N = 26 uses 11 tiles in three sequences: (1,1,2,3,5,8), (1,1,2), (1,1).
A subset $S\subseteq\{1,2,\ldots,2026\}$ is good if the least common multiple of any three of its elements also belongs to S. For example {3,9,12,36} is good. Find the maximum size of a good subset.
Acute triangle $ABC$ has altitudes $AX,BY,CZ$. Let $P=XY\cap CZ$. The line through P parallel to AX meets AC at T and XZ at Q. If H is the orthocentre, prove $\angle HCB=\angle BCQ$.
