docs: 添加离散数学理论基础 2024 春夏第四次小测试题 (#182)

docs/major_basic/discrete_math/Discrete_Mathematics_Quiz_4_2024.typ

@@ -0,0 +1,117 @@
+#import "@preview/numblex:0.1.1": numblex
+#import "@preview/cetz:0.2.2"
+#import "@preview/wrap-it:0.1.0": wrap-content
+#import "@preview/fletcher:0.5.0": diagram, node, edge
+#import cetz.draw: *
+
+#let graph = x => cetz.canvas(x)
+#set page(margin: 4em)
+#set text(font: "STSong", size: 11pt)
+#set par(leading: 0.6em, justify: true)
+#set enum(numbering: numblex(numberings: ("1.", "a)")), full: true)
+
+#show heading.where(level: 1): it => [
+  #set align(center)
+  #set text(size: 20pt, font: "Playfair Display")
+  #it.body
+]
+
+#show heading.where(level: 2): it => [
+  #set align(center)
+  #set text(size: 13pt, font: "FZXiaoBiaoSong-B05S")
+  #it.body
+]
+
+#show heading.where(level: 3): it => [
+  #set align(center)
+  #set text(size: 15pt, font: "Princess Sofia")
+  #it.body
+]
+
+#let answer = [#box(width: 1fr,repeat("_"))]
+
+= Discrete Mathematics Quiz 4
+== 2023-2024 春夏学期
+=== shrike505
+
+#linebreak()
+
++ Fill in the blanks. (40%, 5% each)
+  + Write a proposition equivalent to $(p and not q)$ that uses only $p$, $q$, $not$, and the connective $or$.
+  + 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.
+  + If $G$ is a planar connected graph with 20 vertices, each of degree 3, then $G$ has #answer regions.
+  + Given a rectangular coordinate system in three-dimensional space, how many points are there whose three coordinate values are all rational numbers? #answer
+  + #let snode = (coord, name, location) => {
+        circle(coord, name: name, radius: 0.06, fill: black)
+        content(name, name, padding:.144, anchor: location)
+        // Credit to Xecades
+    }
+    #let f1 = figure(graph({
+        snode((0, 3), "a","south")
+        snode((1.5, 3), "b","south")
+        snode((3, 3), "c","south")
+        snode((4.5, 3), "d","south")
+        snode((0, 1.5), "e", "south-east")
+        snode((1.5, 1.5), "f","south-east")
+        snode((3, 1.5), "g","south-east")
+        snode((4.5, 1.5), "h","south-east")
+        snode((0, 0), "i","north")
+        snode((1.5, 0), "j","north")
+        snode((3, 0), "k","north")
+        snode((4.5, 0), "l","north")
+        let thinline = (name1,name2) => {
+          line(name1, name2, stroke:(thickness: 0.7pt))
+        }
+        thinline("a", "b")
+        thinline("b","c")
+        thinline("c","d")
+        thinline("a","e")
+        thinline("b","f")
+        thinline("c","g")
+        thinline("d","h")
+        thinline("e","f")
+        thinline("f","g")
+        thinline("g","h")
+        thinline("e","i")
+        thinline("f","j")
+        thinline("g","k")
+        thinline("h","l")
+        thinline("i","j")
+        thinline("j","k")
+        thinline("k","l")
+        thinline("i","l")
+    }))
+    #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.]
+    #v(2em)
+  + 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
+  + 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?
+    #let bignode = (coord, name) => {
+      circle(coord, name: name, radius: 0.3)
+      content(name, name, padding:.1)
+    }
+    #let f2 = figure(diagram(
+      let (a,b,c,d) = ((-2,0), (0.6,0), (-2,1.7), (0.6,1.7)),
+      node(a, [A], stroke: 1pt),
+      node(b, [B], stroke: 1pt),
+      node(c, [C], stroke: 1pt),
+      node(d, [D], stroke: 1pt),
+      edge(b, a, bend: -10deg,"-|>"),
+      edge(a, b, bend: -10deg,"-|>"),
+      edge(a, c, "-|>"),
+      edge(b, d, "-|>"),
+      edge(d, a, "-|>"),
+      edge(c, d, bend: -10deg,"-|>"),
+      edge(d, c, bend: -10deg,"-|>"),
+    ))
+  + #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]
+    #v(2em)
++ 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%)
++ (30%, 5% each) A standard deck of 52 cards consists of 4 suits, with each suit containing 13 cards corresponding to 13 values.
+  + 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)
+  + 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)
+  + How many playing cards do you need to take at random to ensure that there is a pair among them? #v(4em)
+  + 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)
+  + 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)
+  + 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)
++ 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)
++ 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%)

docs/major_basic/discrete_math/index.md

@@ -50,6 +50,7 @@
 - [（22 级）2023 春夏第二次小测](Discrete_Mathematics_Quiz_2_2023.pdf)
 - [（23 级）2024 春夏第一次小测](Discrete_Mathematics_Quiz_1_2024.pdf)
 - [（23 级）2024 春夏第三次小测](Discrete_Mathematics_Quiz_3_2024.pdf)
+- [（23 级）2024 春夏第四次小测](Discrete_Mathematics_Quiz_4_2024.pdf)
 
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