## anayjoshi.com

Problem #1: [RMO-1991] Prove that $$n^4 + 4^n$$ is composite for all integer values of n greater than 1.

Problem #2: [RMO-1992] Determine the set of integers n for which $$n^2 + 19n + 92$$ is a perfect square

Problem #3: [RMO-1999] Let ABCD be a square and $$M$$ & $$N$$ points on sides $$AB$$ & $$BC$$ respectively such that $$\angle MDN = 45^o$$. If $$R$$ is the mid point of $$MN$$, show that $$RP = RQ$$, where $$P$$ & $$Q$$ are the points of intersection of $$AC$$ with the lines $$MD$$ & $$ND$$

Problem #4: [INMO-1995] In an acute-angled traingle $$ABC$$, $$\angle A = 30^0$$, $$H$$ is the orthocentre, and $$M$$ is the midpoint of $$BC$$. On the line $$HM$$, take a point $$T$$ such that $$HM = MT$$. Show that $$AT = 2BC$$.

Problem #5: [INMO-1996] Define a sequence $$a_n$$ by $$a_1 = 1, a_2 = 2$$, and $$a_{n+2} = 2a_{n+1} - a_n + 2$$ for $$n >= 1$$. Prove that for any $$m$$, $$a_m a_{m+1}$$ is also a term in the sequence.

Problem #6: [Tournament of the Towns] The sequence $$a_n$$ is defined as follows: $$a_0 = 9$$, $$a_{n+1} = 3{a_n}^4 + 4{a_n}^3$$ for $$n > 0$$. Show that $$a_{10}$$ contains more than $$1000$$ nines in decimal notation.

Problem #7: [Problem Solving Strategies]: We strike the first digit of $$7^{1996}$$, and then add it to the remaining number. This is repeated until a number with $$10$$ digits remains. Prove that this number has two equal digits