Midterm Prep

24 real exam questions mapped to course topics

Every question below appeared in an actual BME ITC 2 midterm (zh15–zh24). They are organized by topic so you can practice exactly what will be tested. The midterm covers Lectures 1–10 with 6 problems worth 10 points each (60 total, 90 minutes, 24 to pass).

Total Questions

Exam Sources

Points per Exam

Pass Mark

Midterm Simulation

Practice exams with 6 questions each — timed at 90 minutes, scored, and reviewed. Test yourself under real exam conditions.

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Midterm Coverage

Lectures 1–10

You are responsible for Weeks 1–10 (Combinatorics through Ford-Fulkerson)

Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Week 9 Week 10

Combinatorics, Graphs, Trees, Euler/Hamilton, Coloring, Planarity, Covers, Matchings, Flows

Expected Question Distribution

Estimated from patterns across zh15–zh24 — 6 questions, 10 points each, 60 total

Combinatorics / Graph Basics

W110 pts

Graph Properties / Isomorphism

W210 pts

Hamilton / Euler / Planarity

W410 pts

Graph Coloring / Bipartite

W510 pts

Matchings / Covers / Kruskal

W710 pts

Max Flow / Ford-Fulkerson

W910 pts

Total60 pts

Most Tested Topics

How many times each topic appeared across zh15–zh24

Planarity & Euler's Formula

Week 6

zh15zh16zh17zh18zh19zh20zh22zh24

Hamilton Cycles & Euler Circuits

Week 4

zh15zh16zh17zh18zh20zh21zh23

Graph Coloring

Week 5

zh15zh16zh18zh19zh20zh22zh24

Max Flow / Min Cut

Week 9 Week 10

zh16zh17zh19zh21zh23zh24

Matchings & Covers

Week 7 Week 8

zh15zh17zh18zh20zh22zh23

BFS & Trees

Week 3

zh15zh18zh19zh21zh24

Combinatorics

Week 1

zh16zh19zh21zh23

Graph Isomorphism

Week 2

zh17zh20zh22zh24

Kruskal & Spanning Trees

Week 7

zh18zh21zh23

Bipartite Graphs

Week 5

zh15zh19zh22

All Midterm Questions

Covers: Combinatorics, Graphs, Trees, Euler/Hamilton, Coloring, Planarity, Matchings, Flows

zh19Q1mediumWeek 1: Permutations, Combinations & Binomial Theorem

Combinatorics

Counting paths in a grid or selections with constraints using binomial coefficients

zh21Q1mediumWeek 1: Permutations, Combinations & Binomial Theorem

Binomial Theorem Application

Apply the binomial theorem or combinatorial identities to prove a counting formula

zh17Q2mediumWeek 2: Graphs, Isomorphism & Connectivity

Graph Isomorphism

Determine if two given graphs are isomorphic; if yes provide the bijection, if no give an invariant that differs

zh20Q1easyWeek 2: Graphs, Isomorphism & Connectivity

Handshake Theorem

Use the Handshake Theorem to prove a property about vertex degrees or edge counts

zh15Q2easyWeek 3: Trees & BFS

BFS & Shortest Paths

Run BFS from a given vertex and determine shortest distances or the BFS tree structure

zh21Q3mediumWeek 3: Trees & BFS

Tree Properties

Prove a property of trees using induction on the number of vertices/edges

zh15Q3mediumWeek 4: Euler Circuits & Hamilton Cycles

Euler Circuit

Determine if a graph has an Euler circuit/path and construct one if it exists

zh16Q3hardWeek 4: Euler Circuits & Hamilton Cycles

Hamilton Cycle

Apply Dirac's or Ore's theorem to determine if a Hamilton cycle exists, or prove one doesn't

zh20Q4hardWeek 4: Euler Circuits & Hamilton Cycles

Euler/Hamilton on Special Graphs

Analyze Euler/Hamilton properties of a specific graph family (hypercubes, Petersen, etc.)

zh15Q4mediumWeek 5: Bipartite Graphs & Coloring

Graph Coloring

Find or bound the chromatic number of a given graph

zh19Q4mediumWeek 5: Bipartite Graphs & Coloring

Bipartite Graphs

Prove a graph is or is not bipartite, find the bipartition, or use bipartiteness in a proof

zh22Q3hardWeek 5: Bipartite Graphs & Coloring

Chromatic Number & Brooks

Apply Brooks' theorem or coloring bounds to determine/bound χ(G)

zh15Q5mediumWeek 6: Planarity & Euler's Formula

Euler's Formula

Apply Euler's formula to compute the number of faces, verify planarity, or derive properties

zh16Q4hardWeek 6: Planarity & Euler's Formula

Planarity & Kuratowski

Prove a graph is or is not planar using edge bounds or Kuratowski's theorem

zh22Q5hardWeek 6: Planarity & Euler's Formula

Planarity + Coloring

Combine planarity with coloring results (e.g., prove a planar graph has chromatic number ≤ 5)

zh18Q5mediumWeek 7: Covers, Independent Sets & Kruskal

Covers & Gallai

Compute τ, ν, α, ρ for a given graph and verify Gallai's identities

zh21Q4easyWeek 7: Covers, Independent Sets & Kruskal

Kruskal's Algorithm

Trace Kruskal's algorithm on a weighted graph and find the MST

zh17Q5mediumWeek 8: Matchings & Hall's Theorem

Augmenting Paths

Find augmenting paths to improve a given matching, or prove a matching is maximum

zh20Q5mediumWeek 8: Matchings & Hall's Theorem

Hall's Theorem

Check if a bipartite graph has a perfect matching using Hall's condition

zh16Q5mediumWeek 9: Edge Coloring, König & Max Flow

König's Theorem

Apply König's theorem to find ν = τ in a bipartite graph

zh19Q6hardWeek 9: Edge Coloring, König & Max Flow

Max Flow Setup

Model a problem as a flow network and find the maximum flow

zh16Q6hardWeek 10: Max Flow-Min Cut & Ford-Fulkerson

Ford-Fulkerson

Trace Ford-Fulkerson on a given flow network, showing each augmenting path and the resulting flow

zh21Q6hardWeek 10: Max Flow-Min Cut & Ford-Fulkerson

Max Flow-Min Cut

Find the max flow and min cut, prove they are equal

zh24Q6hardWeek 10: Max Flow-Min Cut & Ford-Fulkerson

Flow Network Modeling

Model a real-world problem as a flow network and solve using max flow

Midterm Study Strategy

- The midterm has 6 problems × 10 points = 60 total, 90 minutes, 24 to pass
- Planarity (Euler's formula, Kuratowski) and Hamilton/Euler appear in almost every exam
- Graph coloring and max flow questions are also very frequent
- Know your theorems: green-marked theorems in the summary require full proofs
- Practice drawing graphs and tracing algorithms (BFS, Ford-Fulkerson) by hand
- For matching problems, know augmenting paths and Hall's theorem conditions
- For planarity, always start by checking $e \leq 3v - 6$ (or $e \leq 2v - 4$ for triangle-free)

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