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\\ $Find the last three digits of the number $ 7^{9999}

\\ $Let $ a,b,c $ denote three distinct integers, and let $ P(x) $ be a polynomial with integer coefficients.$ \\\\ $Show that there does NOT exist a polynomial, where $ P(a)=b, P(b)=c $ and $ P(c)=a.

N points are given on the circumference of a circle, and all possible chords are constructed. The points are chosen such that no three chords are concurrent. How many triangles are there with all of its vertices lying inside the circle?

\\ $Define $ f_n(x)=f_0(f_{n-1}(x)) $ and let $ f_0(x)=\frac{1}{1-x}. \\\\ $Evaluate $ f_{2012}(2012)

\\ $Find all integer triplets $ (x,y,z) $ that satisfy $ x^3+y^3+z^3=(x+y+z)^3

\\ $Suppose $ n \in \mathbb{N}. $ Find the sum of the digits appearing in the integers$\\\\ 1,2,3,...,10^n-2,10^n-1

The square given below has the unique property, that the products of the rows, columns, and the two diagonals, yield the same value k. A B C D E F G H I So ABC=DEF=GHI=ADG=BEH=CFI=AEI=CEG=k. Prove that if A,B,..,I are all integers, then k must be a perfect cube.

$Determine all pairs of rational numbers $ (x,y) $ such that $ x^3+y^3=x^2+y^2

\\ $Define the sequence $ T_{n+1} = \frac{T_n}{1+nT_n}. \\\\ $Find the value of T_{2012}.$

\\ $A number $ N $ is said to be 'stubborn' if when multiplied by some positive integer $ k $, the end product always contains the digits $ 0,1,2,...,9 $ in any permutation, allowing repetition.$ \\\\ $Prove, or disprove, that the number $ N=526315789473684210 $ is 'stubborn'. If so, do there...

\\ $The Fibbonaci Sequence $ f_1, f_2,..., f_n $ is defined by $ f_1 = f_2 = 1 $ and $ f_n = f_{n-1} + f_{n-2}, n \geq 3. \\\\ $Let $ Q = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}.$ Prove that $ Q^n = \begin{bmatrix} f_{n+1} & f_n \\ f_n & f_{n-1} \end{bmatrix} $ and hence prove that $ f_{3n}...

\\ $Find all positive integers $ n $ for which the quadratic equation$ \\\\ a_{n+1}x^2-2x\sqrt{\sum_{i=1}^{n+1}a_i ^2} + \sum_{i=1}^{n} a_i=0 \\\\ $has real roots $ \forall a_i \in \mathbb{R} $ where $ i \in [1,n+1]

$Find the number of terms in the expansion of $ \prod_{j=1}^{n}\left (\sum_{i=1}^{j} x_i \right )

\\ $Consider the polynomial $ P(x)=x^2+bx+c. $ Show that $ |x_i| < 1 $ if and only if $ |b| < 1+c < 2. \\\\ $ Extend this criterion to a cubic polynomial $ P(x)=x^3+bx^2+cx+d $ by showing that $ |x_i| < 1 $ if and only if $ |bd-c| < 1-d^2 $ and $ |b+d| < |1+c|

\\ $Consider the function $ f(x) = \sqrt{x+a \sqrt x + b} + \sqrt x -c $ for some real constants $ a,b,c. $ For what values of $ a,b,c $ does the function have infinitely many roots?$

\\ $Consider the polynomial $ P(x) = x^3 - 7x^2 + 14x -7 $, which has roots $ x_i $ for $ i=1,2,3. $ Show that $ \max (x_i) = 4 \cos ^2 \left ( \frac{\pi}{14} \right )

\\ $Define $ P(x,y,z) = \frac{x+1}{xy+x+1} + \frac{y+1}{yz+y+1} + \frac{z+1}{xz+z+1} $ where $ x,y,z \geq 0 $ and $ xyz=1. \\\\ $Find $ \max \left ( P(x,y,z) \right ) $ and $ \min \left ( P(x,y,z) \right ).

Here's a fun question I did a couple weeks ago.

Here is a neat result. Prove that for all natural numbers N, \int_{0}^{\frac{\pi}{2}} \cos^{2n}x dx = \frac{1}{2^{2n}} \binom{2n}{n}\frac{\pi}{2}

Consider a straight line and a piece of string with length L. In what shape should the string be made to maximise the area bounded by the straight line and the string, given that the string must touch the line at least once? Give proof.