MathDB

2

Part of 2003 Junior Balkan Team Selection Tests - Romania

Problems(3)

30 divides n_1^4 + n_2^4+...+n_{31}^4, among 31 primes exist 3 consecutive

Source: 2003 Romania JBMO TST 1.2

6/1/2020
Consider the prime numbers n1<n2<...<n31n_1< n_2 <...< n_{31}. Prove that if 3030 divides n14+n24+...+n314n_1^4 + n_2^4+...+n_{31}^4, then among these numbers one can find three consecutive primes.
primesnumber theoryconsecutivedividesSum of powers
intersecting circles are seen from intersection of tangents under same angle.

Source: 2003 Romania JBMO TST2 p2

5/23/2020
Two circles C1(O1)C_1(O_1) and C2(O2)C_2(O_2) with distinct radii meet at points AA and BB. The tangent from AA to C1C_1 intersects the tangent from BB to C2C_2 at point MM. Show that both circles are seen from MM under the same angle.
geometryequal anglescirclesTangents
a^n has an odd number of digits in the decimal representation for all n &gt;0

Source: 2003 Romania JBMO TST 3.2

6/1/2020
Let aa be a positive integer such that the number ana^n has an odd number of digits in the decimal representation for all n>0n > 0. Prove that the number aa is an even power of 1010.
oddnumber theorydecimal representationPower