MathDB

2003 Putnam

Part of Putnam

Subcontests

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Putnam 2003 A5

A Dyck nn-path is a lattice path of nn upsteps (1,1)(1, 1) and nn downsteps (1,−1)(1, -1) that starts at the origin OO and never dips below the xx-axis. A return is a maximal sequence of contiguous downsteps that terminates on the xx-axis. For example, the Dyck 55-path illustrated has two returns, of length 33 and 11 respectively. Show that there is a one-to-one correspondence between the Dyck nn-paths with no return of even length and the Dyck (n−1)(n - 1) paths.
\begin{picture}(165,70) \put(-5,0){O} \put(0,10){\line(1,0){150}} \put(0,10){\line(1,1){30}} \put(30,40){\line(1,-1){15}} \put(45,25){\line(1,1){30}} \put(75,55){\line(1,-1){45}} \put(120,10){\line(1,1){15}} \put(135,25){\line(1,-1){15}} \put(0,10){\circle{1}}\put(0,10){\circle{2}}\put(0,10){\circle{3}}\put(0,10){\circle{4}} \put(15,25){\circle{1}}\put(15,25){\circle{2}}\put(15,25){\circle{3}}\put(15,25){\circle{4}} \put(30,40){\circle{1}}\put(30,40){\circle{2}}\put(30,40){\circle{3}}\put(30,40){\circle{4}} \put(45,25){\circle{1}}\put(45,25){\circle{2}}\put(45,25){\circle{3}}\put(45,25){\circle{4}} \put(60,40){\circle{1}}\put(60,40){\circle{2}}\put(60,40){\circle{3}}\put(60,40){\circle{4}} \put(75,55){\circle{1}}\put(75,55){\circle{2}}\put(75,55){\circle{3}}\put(75,55){\circle{4}} \put(90,40){\circle{1}}\put(90,40){\circle{2}}\put(90,40){\circle{3}}\put(90,40){\circle{4}} \put(105,25){\circle{1}}\put(105,25){\circle{2}}\put(105,25){\circle{3}}\put(105,25){\circle{4}} \put(120,10){\circle{1}}\put(120,10){\circle{2}}\put(120,10){\circle{3}}\put(120,10){\circle{4}} \put(135,25){\circle{1}}\put(135,25){\circle{2}}\put(135,25){\circle{3}}\put(135,25){\circle{4}} \put(150,10){\circle{1}}\put(150,10){\circle{2}}\put(150,10){\circle{3}}\put(150,10){\circle{4}} \end{picture}

Putnam 2003 B5

Let AA, BB and CC be equidistant points on the circumference of a circle of unit radius centered at OO, and let PP be any point in the circle's interior. Let aa, bb, cc be the distances from PP to AA, BB, CC respectively. Show that there is a triangle with side lengths aa, bb, cc, and that the area of this triangle depends only on the distance from PP to OO.
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Putnam 2003 A4

Suppose that a,b,c,A,B,Ca, b, c, A, B, C are real numbers, a≠0a \not= 0 and A≠0A \not= 0, such that ∣ax2+bx+c∣≤∣Ax2+Bx+C∣|ax^2+ bx + c| \le |Ax^2+ Bx + C| for all real numbers xx. Show that ∣b2−4ac∣≤∣B2−4AC∣|b^2- 4ac| \le |B^2- 4AC|

Putnam 2003 B4

Let f(z)=az4+bz3+cz2+dz+e=a(z−r1)(z−r2)(z−r3)(z−r4)f(z) = az^4+ bz^3+ cz^2+ dz + e = a(z -r_1)(z -r_2)(z -r_3)(z -r_4) where a,b,c,d,ea, b, c, d, e are integers, a≠0a \not= 0. Show that if r1+r2r_1 + r_2 is a rational number, and if r1+r2≠r3+r4r_1 + r_2 \neq r_3 + r_4, then r1r2r_1r_2 is a rational number.
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Putnam 2003 A2

Let a1,a2,⋯ ,ana_1, a_2, \cdots , a_n and b1,b2,⋯ ,bnb_1, b_2,\cdots, b_n be nonnegative real numbers. Show that (a1a2⋯an)1/n+(b1b2⋯bn)1/n≤((a1+b1)(a2+b2)⋯(an+bn))1/n(a_1a_2 \cdots a_n)^{1/n}+ (b_1b_2 \cdots b_n)^{1/n} \le ((a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n))^{1/n}

Putnam 2003 B2

Let nn be a positive integer. Starting with the sequence 1,12,13,⋯ ,1n1,\frac{1}{2}, \frac{1}{3} , \cdots , \frac{1}{n}, form a new sequence of n−1n -1 entries 34,512,⋯ ,2n−12n(n−1)\frac{3}{4}, \frac{5}{12},\cdots ,\frac{2n -1}{2n(n -1)}, by taking the averages of two consecutive entries in the first sequence. Repeat the averaging of neighbors on the second sequence to obtain a third sequence of n−2n -2 entries and continue until the final sequence consists of a single number xnx_n. Show that xn<2nx_n < \frac{2}{n}.
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