MathDB

3

Part of 2006 India IMO Training Camp

Problems(3)

Union of the triangular regions BAD,CBE,ACF covers ABC

Source: India tst 2006 p6

6/27/2012
Let ABCABC be an equilateral triangle, and let D,ED,E and FF be points on BC,BABC,BA and ABAB respectively. Let ∠BAD=α,∠CBE=β\angle BAD= \alpha, \angle CBE=\beta and ∠ACF=γ\angle ACF =\gamma. Prove that if α+β+γ≥120∘\alpha+\beta+\gamma \geq 120^\circ, then the union of the triangular regions BAD,CBE,ACFBAD,CBE,ACF covers the triangle ABCABC.
geometry unsolvedgeometry
n arithmetic progressions of integers each of k terms

Source: India tst 2006 p9

6/27/2012
Let A1,A2,⋯ ,AnA_1,A_2,\cdots , A_n be arithmetic progressions of integers, each of kk terms, such that any two of these arithmetic progressions have at least two common elements. Suppose bb of these arithmetic progressions have common difference d1d_1 and the remaining arithmetic progressions have common difference d2d_2 where 0<b<n0<b<n. Prove that b≤2(k−d2gcd(d1,d2))−1.b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.
number theory unsolvednumber theory
Set: Prove that intersection of all A_i's is not a null set

Source: India tst 2006 p12

6/27/2012
Let A1,A2,…,AnA_1,A_2,\ldots,A_n be subsets of a finite set SS such that ∣Aj∣=8|A_j|=8 for each jj. For a subset BB of SS let F(B)={j∣1≤j≤n  and Aj⊂B}F(B)=\{j \mid 1\le j\le n \ \ \text{and} \ A_j \subset B\}. Suppose for each subset BB of SS at least one of the following conditions holds
(a) ∣B∣>25|B| > 25,
(b) F(B)={\O},
(c) \bigcap_{j\in F(B)} A_j \neq {\O}.
Prove that A_1\cap A_2 \cap \cdots \cap A_n \neq {\O}.
inductioncombinatorics unsolvedcombinatorics