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General Practice Test
1.
(16 p.)
The sequence of complex numbers \( z_0,z_1,z_2,\dots \) is defined by \( z_0=1+i/211 \) and \( z_{n+1}=\frac{z_n+i}{z_ni} \). If \( z_{2111}=\frac ab+\frac cdi \) for positive integers \( a,b,c,d \) with \( \gcd(a,b)=\gcd(c,d)=1 \), find \( a+b+c+d \).
2.
(7 p.)
Two students Alice and Bob participated in a twoday math contest. At the end both had attempted questions worth 500 points. Alice scored 160 out of 300 attempted on the first day and 140 out of 200 attempted on the second day, so her twoday success ratio was 300/500 = 3/5. Bob’s scores are different from Alice’s (but with the same twoday total). Bob had a positive integer score on each day. However, for each day Bob’s success ratio was less than Alice’s. Assume that \( p/q \) (\( p \) and \( q \) are relatively prime integers) is the largest possible twoday success ratio that Bob could have achieved. Calculate \( p+q \).
3.
(1 p.)
The square \( \begin{array}{ccc} \hline x&20&151 \\\hline 38 & & \\ \hline & & \\ \hline\end{array} \) is magic, i.e. in each cell there is a number so that the sums of each row and column and of the two main diagonals are all equal. Find \( x \).
4.
(56 p.)
Let \( \triangle ABC \) be a triangle with \( AB=13 \), \( BC=14 \), and \( CA=15 \). Let \( O \) denote its circumcenter and \( H \) its orthocenter. The circumcircle of \( \triangle AOH \) intersects \( AB \) and \( AC \) at \( D \) and \( E \) respectively. Suppose \( \tfrac{AD}{AE}=\tfrac mn \) where \( m \) and \( n \) are positive relatively prime integers. Find \( mn \).
5.
(18 p.)
A frog is jumping in the coordinate plane according to the following rules: (i) From any lattice point \( (a,b) \), the frog can jump to \( (a+1,b) \), \( (a,b+1) \), or \( (a+1,b+1) \). (ii) There are no right angle turns in the frog’s path. How many different paths can the frog take from \( (0,0) \) to \( (5,5) \)?
20052017
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