MathDB

1

Part of 2013 Dutch IMO TST

Problems(2)

4x4 system, ab + c + d = 3, bc + d + a = 5, cd + a + b = 2, da + b + c = 6

Source: Dutch IMO TST 2013 day 1 p1

8/29/2019
Determine all 4-tuples (a,b,c,da, b,c, d) of real numbers satisfying the following four equations: {ab+c+d=3bc+d+a=5cd+a+b=2da+b+c=6\begin{cases} ab + c + d = 3 \\ bc + d + a = 5 \\ cd + a + b = 2 \\ da + b + c = 6 \end{cases}
algebrasystem of equations
Perfect square regarding factorials

Source: Netherlands IMO team Selection Test 2013

10/21/2014
Show that ∑n=020134026!(n!(2013−n)!)2\sum_{n=0}^{2013}\frac{4026!}{(n!(2013-n)!)^2} is a perfect square.
factorialnumber theory unsolvednumber theory