MathDB

1997 Putnam

Part of Putnam

Subcontests

(6)
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Putnam 1997 A6

For a positive integer nn and any real number cc, define xkx_k recursively by : x0=0,x1=1 and for k≥0,  xk+2=cxk+1−(n−k)xkk+1 x_0=0,x_1=1 \text{ and for }k\ge 0, \;x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1} Fix nn and then take cc to be the largest value for which xn+1=0x_{n+1}=0. Find xkx_k in terms of nn and k,  1≤k≤nk,\; 1\le k\le n.

Putnam 1997 B6

The dissection of the 3−4−53-4-5 triangle shown below has diameter 5/25/2. [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(23cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = 1.42, xmax = 24.42, ymin = 3.8, ymax = 15.54; /* image dimensions */ Label laxis; laxis.p = fontsize(10); xaxis(xmin, xmax,defaultpen+black, Ticks(laxis, Step = 1, Size = 2, NoZero), Arrows(6), above = true); yaxis(ymin, ymax,defaultpen+black, Ticks(laxis, Step = 1, Size = 2, NoZero), Arrows(6), above = true); /* draws axes; NoZero hides '0' label */ /* draw figures */ draw((9.44,8.52)--(12.44,8.52)); draw((9.44,12.52)--(9.44,8.52)); draw((9.44,12.52)--(12.44,8.52)); draw((9.44,10.52)--(10.94,10.52)); draw((10.94,10.52)--(10.94,8.52)); draw((9.44,8.52)--(10.94,10.52)); /* dots and labels */ dot((9.44,8.52),dotstyle); label("AA", (9.52,8.64), NE * labelscalefactor); dot((12.44,8.52),dotstyle); label("BB", (12.52,8.64), NE * labelscalefactor); dot((9.44,12.52),dotstyle); label("CC", (9.52,12.64), NE * labelscalefactor); dot((10.94,8.52),dotstyle); label("DD", (11.02,8.64), NE * labelscalefactor); dot((9.44,10.52),dotstyle); label("EE", (9.52,10.64), NE * labelscalefactor); dot((10.94,10.52),dotstyle); label("FF", (11.02,10.64), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy] Find the least diameter of this triangle into four parts. (The diameter of a dissection is the least upper bound of the distances between pairs of points belonging to the same part.)
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Putnam 1997 A4

Let GG be group with identity ee and ϕ:G→G\phi :G\to G be a function such that : ϕ(g1)⋅ϕ(g2)⋅ϕ(g3)=ϕ(h1)⋅ϕ(h2)⋅ϕ(h3) \phi(g_1)\cdot \phi(g_2)\cdot \phi(g_3)=\phi(h_1)\cdot \phi(h_2)\cdot \phi(h_3) Whenever g1⋅g2⋅g3=e=h1⋅h2⋅h3g_1\cdot g_2\cdot g_3=e=h_1\cdot h_2\cdot h_3 Show there exists a∈Ga\in G such that ψ(x)=aϕ(x)\psi(x)=a\phi(x) is a homomorphism. (that is ψ(x⋅y)=ψ(x)⋅ψ(y)\psi(x\cdot y)=\psi (x)\cdot \psi(y) for all x,y∈Gx,y\in G )

Putnam 1997 B4

Let am,na_{m,n} denote the coefficient of xnx^n in the expansion (1+x+x2)n(1+x+x^2)^n. Prove the inequality for all integers k≥0k\ge 0 : 0≤∑ℓ=0⌊2k3⌋(−1)ℓak−ℓ,ℓ≤1 0\le \sum_{\ell=0}^{\left\lfloor{\frac{2k}{3}}\right\rfloor} (-1)^{\ell} a_{k-\ell,\ell}\le 1
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Putnam 1997 B2

ff be a twice differentiable real valued function satisfying f(x)+f′′(x)=−xg(x)f′(x) f(x)+f^{\prime\prime}(x)=-xg(x)f^{\prime}(x) where g(x)≥0g(x)\ge 0 for all real xx. Show that ∣f(x)∣|f(x)| is bounded.

Putnam 1997 A2

Players 1,2,…n1,2,\ldots n are seated around a table, and each has a single penny. Player 11 passes a penny to Player 22, who then passes two pennies to Player 33, who then passes one penny to player 44, who then passes two pennies to Player 55 and so on, players alternately pass one or two pennies to the next player who still has some pennies. The player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers nn for which some player ends up with all the nn pennies.
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Putnam 1997 A1

A rectangle, HOMFHOMF, has sides HO=11HO=11 and OM=5OM=5. A triangle ΔABC\Delta ABC has HH as orthocentre, OO as circumcentre, MM be the midpoint of BCBC, FF is the feet of altitude from AA. What is the length of BCBC ?
[asy] unitsize(0.3 cm); pair F, H, M, O; F = (0,0); H = (0,5); O = (11,5); M = (11,0); draw(H--O--M--F--cycle); label("FF", F, SW); label("HH", H, NW); label("MM", M, SE); label("OO", O, NE); [/asy]

Putnam 1997 B1

For all reals xx define {x}\{x\} to be the difference between xx and the closest integer to xx. For each positive integer nn evaluate : Sn=∑m=16n−1min⁡({m6n},{m3n}) S_n=\sum_{m=1}^{6n-1}\min \left(\left\{\frac{m}{6n}\right\},\left\{\frac{m}{3n}\right\}\right)