MathDB

1

Part of 2010 Dutch IMO TST

Problems(2)

min a_{2010}, a_n < a_{n+1}, a_i + a_l > a_j + a_k

Source: 2010 Dutch IMO TST2 p1

1/10/2020
Consider sequences a1,a2,a3,...a_1, a_2, a_3,... of positive integers. Determine the smallest possible value of a2010a_{2010} if (i) an<an+1a_n < a_{n+1} for all n≥1n\ge 1, (ii) ai+al>aj+aka_i + a_l > a_j + a_k for all quadruples (i,j,k,l) (i, j, k, l) which satisfy 1≤i<j≤k<l1 \le i < j \le k < l.
minimumalgebraSequence
&lt;A=45^o, CD \perp AB, P\perp BC iff |AP| = |BC|.

Source:

8/4/2019
Let ABCABC be an acute triangle such that ∠BAC=45o\angle BAC = 45^o. Let DD a point on ABAB such that CD⊥ABCD \perp AB. Let PP be an internal point of the segment CDCD. Prove that AP⊥BCAP\perp BC if and only if ∣AP∣=∣BC∣|AP| = |BC|.
geometryperpendicularequal segments