MathDB

9

Part of 2013 Stanford Mathematics Tournament

Problems(6)

2013 General Problem 9

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2/4/2013
A tree has 10 pounds of apples at dawn. Every afternoon, a bird comes and eats x pounds of apples. Overnight, the amount of food on the tree increases by 10%. What is the maximum value of x such that the bird can sustain itself indefinitely on the tree without the tree running out of food?
SMT 2013 Calculus #9

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2/3/2013
Evaluate ∫0π/2dx(sin⁡x+cos⁡x)4\int_{0}^{\pi/2}\frac{dx}{\left(\sqrt{\sin x}+\sqrt{\cos x}\right)^4}.
calculusintegrationtrigonometry
2013 SMT Geometry #9

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11/2/2014
In tetrahedron ABCDABCD, AB=4AB=4, CD=7CD=7, and AC=AD=BC=BD=5AC=AD=BC=BD=5. Let IAI_A, IBI_B, ICI_C, and IDI_D denote the incenters of the faces opposite vertices AA, BB, CC, and DD, respectively. It is provable that AIAAI_A intersects BIBBI_B at a point XX, and CICCI_C intersects DIDDI_D at a point YY. Compute XYXY.
geometry3D geometrytetrahedronincenter
SMT 2013 Algebra #9

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2/3/2013
Let a=−3+5+7,b=3−5+7,c=3+5−7a=-\sqrt{3}+\sqrt{5}+\sqrt{7}, b=\sqrt{3}-\sqrt{5}+\sqrt{7}, c=\sqrt{3}+\sqrt{5}-\sqrt{7}. Evaluate a4(a−b)(a−c)+b4(b−c)(b−a)+c4(c−a)(c−b).\frac{a^4}{(a-b)(a-c)}+\frac{b^4}{(b-c)(b-a)}+\frac{c^4}{(c-a)(c-b)}.
algebrapolynomialfactoring polynomials
SMT 2013 Team #9

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2/4/2013
Charles is playing a variant of Sudoku. To each lattice point (x,y)(x, y) where 1≤x,y<1001\le x,y <100, he assigns an integer between 11 and 100100 inclusive. These integers satisfy the property that in any row where y=ky=k, the 9999 values are distinct and never equal to kk; similarly for any column where x=kx=k. Now, Charles randomly selects one of his lattice points with probability proportional to the integer value he assigned to it. Compute the expected value of x+yx+y for the chosen point (x,y)(x, y).
probabilityexpected value
SMT 2013 Advanced Topics #9

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12/31/2016
Big candles cost 16 cents and burn for exactly 16 minutes. Small candles cost 7 cents and burn for exactly 7 minutes. The candles burn at possibly varying and unknown rates, so it is impossible to predictably modify the amount of time for which a candle will burn except by burning it down for a known amount of time. Candles may be arbitrarily and instantly put out and relit. Compute the cost in cents of the cheapest set of big and small candles you need to measure exactly 1 minute.
2013Advanced Topics