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+ Write a proposition equivalent to $(p and not q)$ that uses only $p$, $q$, $not$, and the connective $or$.
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+ Express the negations of the statement $exists x exists y P(x, y) and forall x forall y Q(x, y)$ so that all negation symbols immediately precede predicates.
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+ If $G$ is a planar connected graph with 20 vertices, each of degree 3, then $G$ has #answer regions.
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+ Given a rectangular coordinate system in three-dimensional space, how many points are there whose three coordinate values are all rational numbers? #answer
#wrap-content(f1, align: bottom + right, columns: (50%, 50%),column-gutter:-0.5em)[A dominating set of vertices in a simple graph is a set of vertices suth that every other vertex is adjacent to at least one vertex of this set. A dominating set with the least number of vertices is called a minimum dominating set. Find the number of vertices of minimum dominating set for the given graph.]
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#v(2em)
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+ Does the graph above have a Hamilton circuit? If it does, find such a circuit. If it does not, give an argument to show why no such circuit exists. #answer
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+ What is the worst-case time complexity of Dijkstra#text(['],font: "Garamond")s algorithm for computing the shortest path between two points in a weighted graph with n vertices?
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#letbignode= (coord, name) => {
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circle(coord, name: name, radius: 0.3)
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content(name, name, padding:.1)
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}
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#letf2=figure(diagram(
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let (a,b,c,d) = ((-2,0), (0.6,0), (-2,1.7), (0.6,1.7)),
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node(a, [A], stroke: 1pt),
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node(b, [B], stroke: 1pt),
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node(c, [C], stroke: 1pt),
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node(d, [D], stroke: 1pt),
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edge(b, a, bend: -10deg,"-|>"),
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edge(a, b, bend: -10deg,"-|>"),
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edge(a, c, "-|>"),
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edge(b, d, "-|>"),
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edge(d, a, "-|>"),
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edge(c, d, bend: -10deg,"-|>"),
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edge(d, c, bend: -10deg,"-|>"),
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))
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+ #wrap-content(f2, columns:(50%, 44%), align: right+bottom)[How many distinct paths of length are there in the given graph? (Note: Cycles are allowed.) #answer]
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#v(2em)
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+ There are three consecutive positive integers that are divisible by 5, 7, and 11 respectively （依次能被 5, 7, 11 整除）. Find all the solutions. (6%)
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+ (30%, 5% each) A standard deck of 52 cards consists of 4 suits, with each suit containing 13 cards corresponding to 13 values.
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+ How many different poker hands of 5 cards that containing four cards of the same value? （xxxxy 牌型，问有多少种可能取法）#v(4em)
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+ How many different poker hands of 5 cards that containing 2 pairs （两个对子，对子数字相同但花色可以不同）but not including 3 cards of the same value? （xxyyz 牌型，问有多少种可能取法） #v(4em)
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+ How many playing cards do you need to take at random to ensure that there is a pair among them? #v(4em)
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+ How many playing cards do you need to take at random to ensure that there is a straight（顺子，五张数字连续的牌）among them?（注：A-2-3-4-5 和 10-J-Q-K-A 都算顺子，花色可以不同） #v(4em)
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+ Ignoring the differences in suits, how many different ways are there for taking out 3 cards? （把数字相同但花色不同的牌视为不可分辨） #v(4em)
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+ Put these 52 cards into a grid with 4 rows and 13 columns, and prove that by selecting one card from each column, you can always get all the 13 values. （提示：可看作一个完全匹配问题） #v(8em)
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+ Use generating functions to solve the recurrence relation $a_k= 5 a_(k-1) - 6 a_(k-2)$ with initial conditions $a_0 = 6$ and $a_1 = 30$. (12%) #v(8em)
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+ There are n red points and n green points on the plane, any three of them are not collinear. Please use induction to prove: These $2 n$ points can be connected in pairs to form n non-intersecting line segments. Each line segment connects a red point and a green point. (12%)
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