1. Suppose that $a$, $b$ and $x$ are positive real numbers. Prove that $$\log_{ab} x =\dfrac{\log_a x\log_b x}{\log_ax+\log_bx}$$
2. Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.
3. Prove that there is no real number $x$ satisfying both equations \begin{align*}2^x+1&=2\sin x \\ 2^x-1&=2\cos x.\end{align*}
4. Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$.
5. The Fibonacci sequence is dened by $f_1=f_2=1$ and $f_n=f_{n-1}+f_{n-2}$ for $n\ge 3$. A Pythagorean triangle is a right-angled triangle with integer side lengths. Prove that $f_{2k+1}$ is the hypotenuse of a Pythagorean triangle for every positive integer $k$ with $k\ge 2$
6. There are $15$ magazines on a table, and they cover the surface of the table entirely. Prove that one can always take away $7$ magazines in such a way that the remaining ones cover at least $\dfrac{8}{15}$ of the area of the table surface
7. If $(a,~b,~c)$ is a triple of real numbers, define $g(a,~b,~c)=(a+b,~b+c,~a+c)$, and $g^n(a,~b,~c)=g(g^{n-1}(a,~b,~c))$ for $n\ge 2$. Suppose that there exists a positive integer $n$ so that $g^n(a,~b,~c)=(a,~b,~c)$ for some $(a,~b,~c)\neq (0,~0,~0)$. Prove that $$g^6(a,~b,~c)=(a,~b,~c)$$
8. Consider three parallelograms $P_1,~P_2,~ P_3$. Parallelogram $P_3$ is inside parallelogram $P_2$, and the vertices of $P_3$ are on the edges of $P_2$. Parallelogram $P_2$ is inside parallelogram $P_1$, and the vertices of $P_2$ are on the edges of $P_1$. The sides of $P_3$ are parallel to the sides of $P_1$. Prove that one side of $P_3$ has length at least half the length of the parallel side of $P_1$.
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