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Arithmetic Expression Decomposition

291 문제 학년 5 –7

A multi-step numerical computation (sum of fractions, rate × time, area × thickness) is attacked by splitting it into a small number of independent subproblems whose answers are then combined. The key move is identifying the clean split, not the underlying topic.

해당 문제

대표 문제

변환 가능한 매개변수

term_count 범위: 2 to 5

How many independent sub-pieces the expression decomposes into

numeric_scale 범위: single digits to thousands

Order of magnitude of the inputs (controls difficulty of the arithmetic)

surface_context 범위: abstract | rate-over-time | area-times-thickness | fuel-economy

Real-world wrapper (fraction sum, CO2 rate, tape on a roll, paint coverage)

rounding_rule 범위: exact | nearest 10 | nearest 100

How the final answer is rounded (exact, nearest 100, nearest integer)

변형 힌트

  • Keep the structure (rate × time) but swap context from CO2 to bacteria population or tuition inflation
  • Make one of the sub-fractions reducible by 11 or 101 to mirror Problem 2's clean cancellation
  • Adjust numeric_scale so the convenient round-number shortcut (e.g. 1.5 × 50) still applies
변형 시 주의사항 (2)
  • Ensure exactly one of A-E is correct after rounding — adjacent answer choices in rate problems often differ by less than the rounding tolerance
  • Sub-problem answers should be clean enough that mental arithmetic stays viable

Compound Area by Decomposition / Difference

108 문제 학년 3 –7

A composite planar region is computed as a sum or difference of simpler shapes (rectangles, squares, circles, annular rings, triangles). The work is in choosing the decomposition; each piece uses a single elementary area formula.

해당 문제

대표 문제

변환 가능한 매개변수

shape_family 범위: square | circle | right-triangle | mixed

Base shape used (squares, circles/rings, 45-45-90 triangles, rectangles)

layer_count 범위: 2 to 5

Number of nested or overlapping shapes

dimension_set 범위: small integers 1-15 (keep arithmetic clean)

Specific side lengths / radii / heights used

configuration 범위: concentric | corner-nested | overlapping-translates

How the pieces sit (concentric, corner-aligned nested, overlapping at an angle)

변형 힌트

  • Change shape_family from squares to circles to convert area answers into multiples of pi
  • Add a layer (e.g. 5 nested squares) to scale difficulty without changing strategy
  • Shift dimension_set so both the total and difference areas remain in the answer-choice range
변형 시 주의사항 (3)
  • After parameter changes, verify the visible shaded region is still a well-defined union/difference
  • For circle problems, keep pi out of the final answer (cancel) so a non-calculator student can finish
  • Be careful with overlaps that change topology when dimensions shift (e.g. overlap may vanish)

Overlap / Complement Counting

48 문제 학년 4 –7

Two classifications partition a population (red/white × high-top/low-top) or we want 'at least one' adjacent pair. Direct counting tangles the cases; the trick is either a 2×2 contingency table (inclusion-exclusion) or counting the complementary event and subtracting from 1.

해당 문제

대표 문제

변환 가능한 매개변수

population_size 범위: 10 to 30

Total count of items being classified or seated

classification_count 범위: 2

Number of cross-cutting classifications (currently always 2 for AMC 8 level)

target_extremum 범위: min-cell | max-cell | p-at-least-one

Minimize one cell of the table, OR compute P(at least one ...)

surface_context 범위: free-form

Wrapping context (sneakers, dogs of color×breed, plane seating, lottery)

변형 힌트

  • Change the two fractions in problem 19 (3/5 and 2/3) to (4/7 and 3/4) — table arithmetic changes, structure identical
  • For problem 25 style, change rows × seats from 4×3 to 3×4 or 5×2 — the per-row adjacent count formula updates
  • Try 'maximum red-high-top' instead of minimum to flip the optimization
변형 시 주의사항 (3)
  • Verify every cell of the 2×2 table remains a non-negative integer for the chosen extremum
  • For probability variants, confirm the final fraction reduces to one of the offered choices
  • When changing seating layout, redefine 'adjacent' carefully (in-row only? or also vertically?)

Branching State Enumeration

43 문제 학년 2 –4

Starting from an initial state, a sequence of choices (each from a small fixed set of actions) generates a tree of reachable states. The question asks for the count of distinct outcomes or valid sequences. Strategy: build the tree level by level, merging duplicate states or pruning states that violate a constraint.

해당 문제

대표 문제

변환 가능한 매개변수

step_count 범위: 3 to 7

Number of moves / hops / days in the process

action_set 범위: 2 to 4 actions

The set of allowed actions at each step (e.g. {+3, *2} or {U, D})

starting_value 범위: 0 to 20

Initial state of the process

state_constraint 범위: none | non-negative | bounded | endpoint-fixed

Boundary or filter rule on states (never negative, never above K, must end at start)

question_type 범위: distinct-endpoints | valid-sequences

Count distinct end values, valid full sequences, or both

변형 힌트

  • Increase step_count from 3 to 4 — tree size grows from 8 to 16 leaves, still listable
  • Add a third action (e.g. +1) to expand the tree without changing strategy
  • Add a 'never go below 0' constraint to convert a state-counting problem into a Dyck-path-flavored one
변형 시 주의사항 (3)
  • Verify the answer (distinct count) lands among A-E choices — duplicate-state collisions are the hardest thing to anticipate
  • Keep the tree small enough (< 32 leaves) that hand enumeration is realistic
  • When adding constraints, make sure at least one valid sequence still exists

Multiplicative Ratio with Integrality Constraints

42 문제 학년 4

Quantities are linked by ratios (R = G/2, B = 2G, or a:b given). Combined with the requirement that all counts are whole numbers, the ratios force totals to lie in a specific arithmetic progression (multiples of some k). The unique answer is the choice that fits the multiplicative scheme.

해당 문제

대표 문제

변환 가능한 매개변수

ratio_chain 범위: any small integer ratio chain summing to 5..20

The list of ratios connecting the quantities (e.g. R:G:B = 1:2:4)

target_quantity 범위: total | category | difference

What the question asks for (total, single category, difference)

surface_context 범위: free-form

Real-world wrapper (marbles, frogs, classes, candies)

perturbation 범위: none | one-step | two-step

Optional change-of-state move (e.g. 3 green become yellow) that re-balances the ratio

변형 힌트

  • Change ratio_chain from 1:2:4 to 2:3:5 — totals must now be multiples of 10 instead of 7
  • Add a perturbation that changes the ratio from 3:1 to 5:2 (problem 21 style)
  • Convert from 'which is possible' to 'find the smallest total possible'
변형 시 주의사항 (3)
  • Exactly one answer choice should be a valid multiple of the ratio sum
  • If using a perturbation, the resulting equation must have a positive-integer solution that lands in the choice set
  • Watch for distractor answer choices that are multiples of a wrong sum

Casework by Position / Vertex Type

38 문제 학년 3

Positions in a configuration (squares on a grid, vertices of a polyhedron) split naturally into a small number of symmetry classes (corner / edge / center; edge-neighbor / face-diagonal / space-diagonal). Counting attacks each class separately and sums via the multiplication principle.

해당 문제

대표 문제

변환 가능한 매개변수

ambient_shape 범위: 3x3 grid | 4x4 grid | cube | regular tetrahedron

The host configuration (n×n grid, cube, tetrahedron, hexagonal grid)

position_classes 범위: 2 to 4

Number of symmetry types of positions

counting_target 범위: free-form

What configurations are being counted (non-attacking pairs, equilateral triangles, collinear triples)

object_count 범위: 2 to 4

How many marked items are placed

변형 힌트

  • Move from 3x3 chess grid to 4x4 — corner/edge/middle counts shift but the casework strategy is identical
  • Replace kings with knights (different attack pattern) so the per-class counts change
  • On the cube, ask for equilateral triangles NOT containing P, or right triangles containing P
변형 시 주의사항 (3)
  • Check whether the symmetry classes really are distinct under the new piece's attack pattern
  • When the host shape grows, verify the total still fits a 5-choice spread (often the answer balloons)
  • Confirm by double-counting (or by listing one full case) that the per-class arithmetic is right

Coordinate Plane Figure Computation

26 문제 학년 6

A polygon or other figure is given by coordinates; the task is to compute an unknown coordinate, length, or area. The structural move is reading off horizontal/vertical distances directly from coordinate differences, then applying a one-line geometric formula (base × height / 2, etc).

해당 문제

대표 문제

변환 가능한 매개변수

figure_type 범위: triangle | parallelogram | trapezoid | rectangle

Polygon being measured (triangle, parallelogram, trapezoid)

known_coords 범위: any combination leaving exactly 1 unknown component

How many coordinates are pinned and which are unknown

target_quantity 범위: coordinate | length | area

What the problem solves for (missing coordinate, area, side length)

alignment 범위: axis-aligned | one-tilted | general

Whether sides are axis-aligned (easy base/height) or tilted (need Pythagorean or shoelace)

변형 힌트

  • Keep base axis-aligned but rotate the apex location (e.g. unknown x instead of unknown y) — symmetric variant
  • Change figure_type to parallelogram so base × height is still the formula
  • For a harder variant, tilt the base so the height must be derived from the perpendicular distance formula
변형 시 주의사항 (2)
  • Ensure the answer choices include both correct value and the natural off-by-one (forgetting the 1/2 in triangle area)
  • Pin the diagram orientation in the problem text to avoid two valid solutions (above vs below the base)

Lattice / Grid Pattern from Small Cases

20 문제 학년 6

The literal scale of the problem (thousands by thousands grid, etc.) makes brute enumeration impossible, but the underlying rule is local. Solve the smallest cases by hand on graph paper, look for a pattern (often involving gcd, lcm, or a sum of dimensions), and generalize to the stated size.

해당 문제

대표 문제

변환 가능한 매개변수

delta_x 범위: any positive integer; keep gcd small for cleanness

Horizontal extent of the segment / grid region

delta_y 범위: any positive integer; keep gcd small for cleanness

Vertical extent of the segment / grid region

gcd_value 범위: 1 to 100

GCD of (delta_x, delta_y) — controls how many lattice corners the segment hits

scale_factor 범위: 100 to 10000

Multiplier applied to the small base case to reach the contest scale

변형 힌트

  • Change (delta_x, delta_y) from (3000, 5000) to (4000, 6000) — gcd shifts from 1000 to 2000, formula stays the same
  • Apply the same dx + dy - gcd pattern to a 'cubes pierced by a diagonal in 3D' variant for harder problems
  • Phrase as a 'rectangle cut by a diagonal' problem instead of a segment on graph paper
변형 시 주의사항 (3)
  • Pattern must be derivable from at most 4-5 small hand-drawn cases
  • Result must round / equal exactly one A-E choice
  • Avoid (delta_x, delta_y) with very large gcd that obscures the pattern

Modular Cycle Frequency Count

18 문제 학년 4 –6

A sequence (often arithmetic, sometimes a schedule of dates) is distributed across residue bins modulo m, and the problem asks which residue is missing, which bin has the most or fewest, or which histogram matches. The decisive move is recognizing that the period m gives a uniform "base" count over a full cycle and a tail correction over the leftover entries.

해당 문제

대표 문제

변환 가능한 매개변수

modulus 범위: 5 to 12 (kid-tractable cycles)

Divisor m that defines the residue bins

sequence_kind 범위: arithmetic-progression | calendar-dates | term-rule

What's being binned — even integers, dates, sequence terms

total_count 범위: must produce a non-trivial tail (not a multiple of m)

How many entries are being binned

target 범위: missing | max-bin | min-bin | match-histogram

What's asked — missing residue, max-bin count, matching histogram

변형 힌트

  • Swap modulus 7 (days of week) for modulus 5 (school days) or modulus 12 (months) — the cycle bookkeeping is structurally the same
  • Replace 'which histogram matches' with 'which residue never appears' to force the same reasoning without picture options
  • Use a non-uniform tail (e.g. last few entries land in a specific cluster) to make the +1 correction step the load-bearing move
변형 시 주의사항 (3)
  • Choose modulus so the full cycle fits comfortably below the total (otherwise there is no 'base' uniform count and the problem degenerates to plain listing)
  • Tail length should be ≥ 1 and ≤ m-1 — if it's 0, every bin gets the same count (trivial)
  • Answer must be uniquely determined; verify no two residues tie at the extremum being asked about

Pair-Sum Invariant

13 문제 학년 4 –7

Partition a universe into forced complementary pairs (or up/down segment pairs) so that any legal configuration has a fixed total. Key move: re-pair the items so the constraint becomes a closure/balance argument, reducing optimization or computation to invariant arithmetic.

해당 문제

대표 문제

변환 가능한 매개변수

pair_structure 범위: complementary-shift | perimeter-up-down | mirror-symmetry

How items are forced into pairs — complementary-value pairs (a, a+k), up/down perimeter segments, or symmetric mirror pairs

universe_size 범위: 6 to 30

Total number of pair-eligible items in the universe (controls arithmetic load, not strategy)

target_invariant 범위: sum | area | count | length

What stays constant across all legal configurations (sum, area, count, length)

surface_context 범위: abstract-numbers | rectilinear-polygon | colored-tokens

Wrapping context (number sets with distance constraints, rectilinear polygons, balanced selections)

변형 힌트

  • Replace the 1-10 / 11-20 sets with 1-12 / 13-24 (shift by 12) — pair-sum changes but the closure argument is identical
  • Build a new rectilinear-polygon closure problem where one vertical side X is unknown and the up-segs = down-segs equation pins it
  • Pair-with-target-sum partition: choose k items from {1..2n} such that no two sum to 2n+1; total is forced
  • Alternate-color balancing on a small graph (each edge has one black + one white endpoint) → counts of black, white are forced
변형 시 주의사항 (4)
  • Ensure the forced pairs actually exist and partition the universe cleanly — if items are unpaired the invariant collapses
  • Verify the invariant is genuinely closed under every legal move (test 2-3 configurations by hand before publishing)
  • Don't conflate with a simple summation problem; the diagnostic feature is that re-pairing exposes the constraint, not raw addition
  • Answer should be unique and land on a single A-E choice; near-misses by ±k often appear when pair-shift k is misread

Meeting Time with Piecewise Rates

11 문제 학년 5 –7

Two or more travelers move along a shared route with piecewise-constant speeds (or with discrete schedules like stops and waits). Find when or where they meet, board, pass, or are at the same position. The decisive move is identifying the segments where each traveler's rate is constant and using "closing-speed × time = gap" or a side-by-side schedule table rather than a single algebraic equation.

해당 문제

대표 문제

변환 가능한 매개변수

traveler_count 범위: 2

Number of moving entities (typically 2)

rate_segments 범위: 1 to 4 per traveler

How many constant-rate segments each traveler has

schedule_style 범위: continuous | discrete | mixed

Continuous-rate (e.g. mph) or discrete (e.g. one stop every N minutes)

target 범위: meet-time | meet-position | catch-which | time-saved

What is asked: meeting time, meeting location, time gap on arrival, or which bus they catch

변형 힌트

  • Swap one traveler's continuous-rate leg for a 'stops every N min' discrete schedule — the simulation table cleanly handles both
  • Introduce a midpoint speed change (e.g. faster on the return trip) to force two-segment reasoning per traveler
  • For an easier Gr5 variant use integer-only meeting times; reserve fractional/decimal meet points for Gr6+ variants
변형 시 주의사항 (3)
  • Verify the meet point falls within the shared route (no traveler overshoots before they meet)
  • Side-by-side schedule table should fit in ≤ 8 rows for the target grade — longer tables need a pattern shortcut
  • Avoid framings that require solving a continuous inequality across multiple segments (that pushes the grade above 7)

Grid Tiling / Packing with Divisibility

7 문제 학년 4

A geometric region must be covered or partitioned by pieces subject to area / divisibility constraints. The decisive observation is usually arithmetic: total area mod tile-area gives the minimum waste, or the constrained items must fit inside a small bounding rectangle.

해당 문제

대표 문제

변환 가능한 매개변수

grid_dimensions 범위: 3x7 to 10x10

Width × height of the region to be covered or filled

tile_set 범위: any combination of 1x1, 2x2, 1x4, L-tromino, etc.

Which tile shapes (or markers) are allowed

divisor_d 범위: 2 to 6

The divisibility constant that controls the packing arithmetic (e.g. mod 4 for 4-area tiles, mod 3 for problem 16)

optimization_goal 범위: minimize | maximize

Minimize a count (1x1 tiles, contaminated rows+cols) or maximize coverage

변형 힌트

  • Change grid_dimensions to 4x9 and reuse the same tile set — the mod-4 leftover shifts to a different value
  • Replace 1x4 with 1x3 tiles to change divisor_d from 4 to 3
  • For an extremal-configuration variant, ask for the MAXIMUM number of clean rows/cols instead of minimum
변형 시 주의사항 (2)
  • The trivial area lower bound must actually be achievable — exhibit a concrete tiling or packing
  • Beware of parity / coloring obstructions (e.g. checkerboard arguments) that may rule out a numerically possible packing

Units-Digit Tracking

7 문제 학년 4

Determine the last digit of a multi-digit arithmetic expression by tracking only the units digit through additions and subtractions. Exploits that the ones column is closed under mod 10, so the bulk of the digits can be ignored.

해당 문제

대표 문제

변환 가능한 매개변수

digit_count 범위: 3 to 8

Maximum number of digits in any operand (controls how scary the expression looks without changing the strategy)

repeated_digit 범위: 1 to 9

The digit that is repeated to build each operand (e.g. 2 in 222222)

operation_mix 범위: subtraction | addition | mixed

Whether the expression is pure subtraction, pure addition, or mixed

term_count 범위: 3 to 8

Number of operands in the expression

변형 힌트

  • Swap repeated_digit from 2 to 7 — units digit pattern shifts but strategy is identical
  • Mix in one addition among the subtractions to force students to track sign carefully
  • For an easier Grade-3 variant use addition only with 3-digit operands
변형 시 주의사항 (3)
  • Answer must be a single digit 0-9 to fit a clean A-E choice set
  • Ensure none of the wrong choices accidentally becomes a valid units digit under a sub-step misread
  • When mixing operations, verify no intermediate borrow propagates from a higher column into the ones (it cannot for pure addition; for mixed, double-check)

Path Length Comparison & Shortest Path

6 문제 학년 2

Compare or minimize total length over a finite collection of routes. The routes may be drawn on a figure (visual comparison: straight vs curved vs zigzag) or described as a small weighted graph (numerical shortest path). The structural move is the same: identify the candidate routes, compute or compare their lengths.

해당 문제

대표 문제

변환 가능한 매개변수

route_count 범위: 3 to 8

Number of paths to compare or routes through the graph

edge_count 범위: 5 to 15

Number of segments / arrows in the figure or graph

comparison_mode 범위: visual | numerical | mixed

Purely visual (no numbers) vs numerical (weights given)

surface_context 범위: free-form

Wrapping context (ice rink, road map, hiking trails, school halls)

변형 힌트

  • Replace one curved segment with a zigzag of the same endpoints to keep the visual ordering intact while changing the look
  • Add a single edge to the graph that opens a tempting longer route, so students must check more options
  • For a numerical variant, choose edge weights so the optimum is unique and not on an obvious 'all small numbers' path
변형 시 주의사항 (2)
  • For visual variants, ensure the figure is unambiguous — students should not need numerical measurement
  • For graph variants, verify the shortest distance matches exactly one A-E choice

Paper Fold-and-Cut Symmetry

5 문제 학년 3 –5

A square (or other shape) is folded one or more times, a piece is cut out of the folded stack, and the unfolded result must be predicted. The underlying mechanic is that each fold creates a reflection axis, so the cut is reflected back across every fold line on unfold. Solving without physically folding requires mentally composing reflections about each fold axis.

해당 문제

대표 문제

변환 가능한 매개변수

fold_count 범위: 1 to 3

Number of folds applied before cutting (each fold doubles the symmetry)

fold_type 범위: half | quarter | diagonal

Whether folds are along straight edges (half) or diagonals (quarter / triangle)

cut_shape 범위: corner-triangle | edge-notch | interior-shape

Shape of the cut piece relative to the folded stack

cut_location 범위: corner | edge | interior

Where on the folded stack the cut is made

변형 힌트

  • Fold once vs twice — single fold gives mirror image, double fold gives 4-fold symmetry
  • Move a triangular cut from a corner to an edge — answer changes from rhombus to bow-tie
  • Use a fold diagonal to introduce 45° rotation into the reflected shape
변형 시 주의사항 (3)
  • Answer must be one of the 5 distinct unfolded patterns — verify no two A-E choices are congruent
  • Distractors should be near-misses (e.g. correct shape but rotated 90°)
  • Keep fold count ≤ 3 so the mental composition stays tractable for the target grade

Net Folding / Face Adjacency

4 문제 학년 4 –7

Given a 2D net of a polyhedron (typically cube, tetrahedron, or octahedron) with labels or markings on the faces, determine which faces are adjacent (share an edge) or opposite once the net is folded into 3D. The decisive move is tracking which net edges glue together during folding, often by rotating around a single fold axis at a time.

해당 문제

대표 문제

변환 가능한 매개변수

polyhedron 범위: cube | tetrahedron | octahedron

Which polyhedron the net folds into

net_shape 범위: cross | T-shape | staircase | strip

Layout of the unfolded net (which arrangement of faces)

target_relation 범위: adjacent-to-X | opposite-to-X | shares-edge-with-X

What relation between labelled faces the problem asks about

marking_style 범위: free-form

How faces are labelled (numbers, letters, colors, dot patterns)

변형 힌트

  • Swap cube for octahedron — face count doubles, but the edge-gluing argument is structurally identical
  • Move the cut/fold sequence so a different face becomes the 'anchor' that stays still
  • Replace 'which face is adjacent to Q?' with 'which face is opposite to Q?' to flip the target relation
변형 시 주의사항 (3)
  • Answer must be uniquely determined by the net (no ambiguity in which way you fold)
  • Distractors should be faces that are adjacent in 2D but NOT adjacent after folding (and vice versa)
  • Keep the net to a single connected piece — disconnected nets are not standard at AMC 8 level