Q1 (10 points)
Give a combinatorial proof of the identity
for such that . No algebraic manipulations are allowed, i.e. you must interpret
both sides directly in the way that they are written.
Q2 (10 points)
You have three red, five green, and eight blue 20-sided dice that you throw randomly on a table. Prove
that there are always three distinct triples of dice, each containing one red, one green, and one blue die,
which have the same sum. For example, in the following configuration, we have three distinct triples,
each summing to 37.
ℕ = {1, 2, 3, …}
[푛] = {1, 2, 3, … , 푛} 푛 ∈ ℕ
( ) − ( ) = ( )
푛
푘
푛 − 푚
푘 ∑
푖=1
푚 푛 − 푖
푘 − 1
푘, 푚, 푛 ∈ ℕ 푘 + 푚 ≤ 푛
Time left Hide
23:43:31
Precisions.
A 20-sided die is a regular icosahedron (a polyhedron of 20 faces) with the numbers 1 to 20 drawn
its faces.
Two triples are considered distinct if they differ by at least one die. In other words, they can have
zero, one, or two dice in common, but not three.
The sum of a triple is the sum of the numbers on the top faces of its dice. Hence, if is
the number on the top face of the red die of the th triple, where , and similarly for
(green) and (blue), the goal of the question is to show that there are triples such that
.
Q3 (10 points)
For all , let be the number of -strings of length such that the difference between any two
consecutive digits is , , or . For example, if , then is valid, but not
because of .
(a) (2 points) Find , , and .
(b) (2 points) Show that for all .
(c) (3 points) Find an explicit expression for using the method of advancement operators.
(d) (3 points) Express the generating function of (i.e. ) as a quotient of two
polynomials.
Q4 (10 points)
Let , . Use the inclusion-exclusion principle to find the number of derangements of
such that .
푟푖 ∈ [20]
푖 푖 ∈ {1, 2, 3} 푔푖
푏푖
푟1 + 푔1 + 푏1 = 푟2 + 푔2 + 푏2 = 푟3 + 푔3 + 푏3
푛 ∈ ℕ 푎푛 [5] 푛
0 1 −1 푛 = 8 32212345 21345443
13
푎1 푎2 푎3
푎푛+3 = 3푎푛+2 − 2푎푛 푛 ∈ ℕ
푎푛
(푎푛 )∞
푛=1 퐹(푥) = ∑∞
푛=1 푎푛푥푛
푛 ∈ ℕ 푛 ≥ 4 휎 [푛]
휎({1, 2}) ∩ {1, 2} = ∅
Q5 (10 points)
Find the number of -strings of length containing an odd number of s, an even
number of s, and at least one .
Q6 (10 points)
Let be a finite graph and an induced subgraph of . Prove that
.
Q7 (10 points)
Consider the following network flow, where the letters on the vertices specify the ordering for the FordFulkerson algorithm. Use the Ford-Fulkerson algorithm to update the given flow until you get a
maximum flow and a minimum cut.
Precisions. Show all your work. For each run of the algorithm, write the list of all the labelings in the
order in which you performed them. For example, the following list taken from the textbook (in a
different example) shows that was labeled first by , then was labeled second by
, then was labeled third by , and so on.
{퐴, 퐵, 퐶, 퐷} 푛 ∈ ℕ 퐴
퐵 퐶
퐆 = (푉 , 퐸) 퐇 = (푊 , 퐹) 퐆
0 ≤ 휒(퐆) − 휒(퐇) ≤ |푉 | − |푊 |
푆 (∗, +, ∞) 퐸
(푆, +, 28) 퐹 (푆, +, 15)
푆
퐸
퐹
퐵
퐷
퐴
퐶
푇
: (∗, +, ∞)
: (푆, +, 28)
: (푆, +, 15)
: (퐸, +, 19)
: (퐸, +, 12)
: (퐹, +, 12)
: (퐵, +, 10)
: (퐴, +, 12).
You must also give the augmenting path and its .
The final answer should show the maximum flow with its value and the minimum cut with its capacity.
훿
Sample Solution
Dante Alighieri played a critical role in the literature world through his poem Divine Comedy that was written in the 14th century. The poem contains Inferno, Purgatorio, and Paradiso. The Inferno is a description of the nine circles of torment that are found on the earth. It depicts the realms of the people that have gone against the spiritual values and who, instead, have chosen bestial appetite, violence, or fraud and malice. The nine circles of hell are limbo, lust, gluttony, greed and wrath. Others are heresy, violence, fraud, and treachery. The purpose of this paper is to examine the Dante’s Inferno in the perspective of its portrayal of God’s image and the justification of hell.
In this epic poem, God is portrayed as a super being guilty of multiple weaknesses including being egotistic, unjust, and hypocritical. Dante, in this poem, depicts God as being more human than divine by challenging God’s omnipotence. Additionally, the manner in which Dante describes Hell is in full contradiction to the morals of God as written in the Bible. When god arranges Hell to flatter Himself, He commits egotism, a sin that is common among human beings (Cheney, 2016). The weakness is depicted in Limbo and on the Gate of Hell where, for instance, God sends those who do not worship Him to Hell. This implies that failure to worship Him is a sin.
God is also depicted as lacking justice in His actions thus removing the godly image. The injustice is portrayed by the manner in which the sodomites and opportunists are treated. The opportunists are subjected to banner chasing in their lives after death followed by being stung by insects and maggots. They are known to having done neither good nor bad during their lifetimes and, therefore, justice could have demanded that they be granted a neutral punishment having lived a neutral life. The sodomites are also punished unfairly by God when Brunetto Lattini is condemned to hell despite being a good leader (Babor, T. F., McGovern, T., & Robaina, K. (2017). While he commited sodomy, God chooses to ignore all the other good deeds that Brunetto did.
Finally, God is also portrayed as being hypocritical in His actions, a sin that further diminishes His godliness and makes Him more human. A case in point is when God condemns the sin of egotism and goes ahead to commit it repeatedly. Proverbs 29:23 states that “arrogance will bring your downfall, but if you are humble, you will be respected.” When Slattery condemns Dante’s human state as being weak, doubtful, and limited, he is proving God’s hypocrisy because He is also human (Verdicchio, 2015). The actions of God in Hell as portrayed by Dante are inconsistent with the Biblical literature. Both Dante and God are prone to making mistakes, something common among human beings thus making God more human.
To wrap it up, Dante portrays God is more human since He commits the same sins that humans commit: egotism, hypocrisy, and injustice. Hell is justified as being a destination for victims of the mistakes committed by God. The Hell is presented as being a totally different place as compared to what is written about it in the Bible. As a result, reading through the text gives an image of God who is prone to the very mistakes common to humans thus ripping Him off His lofty status of divine and, instead, making Him a mere human. Whether or not Dante did it intentionally is subject to debate but one thing is clear in the poem: the misconstrued notion of God is revealed to future generations.
References
Babor, T. F., McGovern, T., & Robaina, K. (2017). Dante’s inferno: Seven deadly sins in scientific publishing and how to avoid them. Addiction Science: A Guide for the Perplexed, 267.
Cheney, L. D. G. (2016). Illustrations for Dante’s Inferno: A Comparative Study of Sandro Botticelli, Giovanni Stradano, and Federico Zuccaro. Cultural and Religious Studies, 4(8), 487.
Verdicchio, M. (2015). Irony and Desire in Dante’s” Inferno” 27. Italica, 285-297.
