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- 01-01 · Map points through three graph transformations
- 01-02 · Absolute input discards the left-hand source features
- 01-03 · Locate the minimum after a rational transformation
- 01-04 · Reflect then reciprocate; translate then fold
- 01-05 · Repeat a square-root and line motif
- 01-06 · Read a derivative and reciprocal from a shifted graph
- 01-07 · Reconstruct signs from two absolute-value sketches
- 01-08 · An ellipse becomes a circle
- 01-09 · A turning point fixes a rational curve
- 01-10 · Keep every absolute value in a logarithmic inequality
- 01-11 · Determine when a horizontal level has a negative input
- 01-12 · A family of lines can miss a rational curve
- 01-13 · Use a circle to test negative roots
- 01-14 · Count roots on either side of a pole
- 01-15 · Compare a rational branch to the upper hyperbola
- 01-16 · Count ellipse intersections without losing the domain
- 01-17 · A squared inequality is a region between two branches
- 01-18 · A discriminant reveals a forbidden output interval
- 01-19 · Respect both square-root domains
- 01-20 · Identify a parametric self-intersection
- 02-01 · Track metal amounts through three alloys
- 02-02 · Undo discounts before finding fruit prices
- 02-03 · A nutrient ratio fixes a serving ratio
- 02-04 · Points and gradients identify coefficients
- 02-05 · Four terms identify a cubic
- 02-06 · Select a card for each currency
- 02-07 · Recover a cubic from its extrema
- 02-08 · Area supplies another coefficient equation
- 02-09 · Use place values to find a number
- 02-10 · Use the remainder theorem
- 02-11 · Use the employee discount’s stated basis
- 02-12 · Count complete promotion bundles
- 02-13 · Distinguish full capacity from capacity ceilings
- 02-14 · Compare profit per output barrel
- 02-15 · Fit an elliptical tunnel from boundary points
- 03-01 · Keep the denominator sign in view
- 03-02 · A positive square denominator still has a hole
- 03-03 · Replace the input by its magnitude
- 03-04 · A range proof needs degenerate cases
- 03-05 · Split at the absolute-value boundary
- 03-06 · Use the double-angle identity before solving
- 03-07 · Strict positivity requires a strict discriminant bound
- 03-08 · A chained inequality means an intersection
- 03-09 · A repeated factor does not disappear from a strict inequality
- 03-10 · Square substitution preserves an excluded zero
- 03-11 · Reciprocals reverse negative interval order
- 03-12 · Fold the rational graph above the axis
- 03-13 · A downward absolute-value graph with parameters
- 03-14 · An exponential reflected at its zero
- 04-01 · When can two functions compose?
- 04-02 · Invert, then repeat periodically
- 04-03 · Restrict the domain to choose an inverse
- 04-04 · A composite can reach an interior minimum
- 04-05 · Identity rules still have domains
- 04-06 · Three compositions return the input
- 04-07 · Absolute values and the correct inverse branch
- 04-08 · Invert a special point separately
- 04-09 · The decreasing sine branch
- 04-10 · Equal identity functions need equal domains
- 04-11 · Self-inverse functions alternate by parity
- 04-12 · A logarithmic involution
- 04-13 · A quarter ellipse becomes a self-inverse circle
- 04-14 · Recover the unknown function by undoing a logarithm
- 04-15 · Compose through a restricted range, then use symmetry for area
- 04-16 · A non-invertible function can have an invertible composite
- 04-17 · Composition conditions and a hidden constant case
- 05-01 · Nested derivatives and increasing intervals
- 05-02 · An inverse-trigonometric identity and its endpoints
- 05-03 · Turn a curve sketch into its gradient graph
- 05-04 · Read stationary points and slope limits
- 05-05 · Recover a rational curve from its asymptotes
- 05-06 · Increasing and concave down from a derivative sketch
- 05-07 · Recover the signs hidden by an absolute value
- 06-01 · A vertical tangent forms a square
- 06-02 · Vertical tangents on a tilted ellipse
- 06-03 · All ellipse normals through a fixed point
- 06-04 · Two tangents to an exponential-logarithmic curve
- 06-05 · Horizontal and vertical parametric tangents
- 06-06 · Decreasing curves and the largest tangent intercept
- 06-07 · A normal intersection and a midpoint locus
- 06-08 · Ellipse tangent intercepts and their midpoint
- 06-09 · A particle switches from a curve to its normal
- 06-10 · Two launch angles for an approaching target
- 06-11 · Minimise material in a thick lidless container
- 06-12 · A cone-roof model with least surface area
- 06-13 · A cylinder inscribed in a cone
- 06-14 · The best seat for the viewing angle
- 06-15 · Stationary points of an implicit cubic
- 06-16 · Water rises through a conical container
- 06-17 · Vertical and horizontal gaps to a parabola
- 06-18 · A capsule changes inside a fixed cone
- 07-01 · Recognise the structure before integrating
- 07-02 · Remove the modulus on the actual interval
- 07-03 · Change both the variable and its differential
- 07-04 · Use the supplied derivative, then split the sign
- 07-05 · Distinguish cos(x²) from cos²x
- 07-06 · Build an inverse-sine integral from an earlier result
- 07-07 · Integrate twice and solve for the repeated integral
- 07-08 · Match a denominator derivative and remove a radical
- 07-09 · Use the crossing before integrating an absolute value
- 07-10 · Choose the trigonometric branch before simplifying roots
- 07-11 · Reuse two basic integrals and a radical substitution
- 07-12 · Turn the radical into a trigonometric identity
- 07-13 · Locate the crossing before taking the absolute area
- 07-14 · Check whether the given equality can hold
- 08-01 · Rotate the region between a bell curve and a line
- 08-02 · Change the lower boundary at the intersection
- 08-03 · Find the region before selecting radii
- 08-04 · Subtract the curved core from a cylinder
- 08-05 · Use shells or split the horizontal washers
- 08-06 · Subtract a triangle from an ellipse quadrant
- 08-07 · Cancel whole periods before integrating
- 08-08 · Keep the parameter branch when eliminating t
- 08-09 · Use y as the monotone coordinate
- 08-10 · Carry negative parameter bounds through the substitution
- 08-11 · Use unsigned area and subtract the rotating core
- 08-12 · Use identities for integration and geometry for the regions
- 08-13 · Take the infinite limit and use the bowl radius
- 08-14 · Compare a left-endpoint sum with the integral
- 08-15 · Sum local area inequalities, then shift the axis
- 09-01 · Normalize before expanding
- 09-02 · Multiply two binomial series
- 09-03 · Use partial fractions for distant coefficients
- 09-04 · Compare coefficients without full expansions
- 09-05 · Recover parameters and a general coefficient
- 09-06 · Read a tangent from the first two coefficients
- 09-07 · Repeated poles and general coefficients
- 09-08 · Check an approximation by its error function
- 09-09 · Two routes to the same square-root series
- 09-10 · A vanishing cubic term does not make an integral exact
- 09-11 · Integration can leave the convergence range
- 09-12 · Derivatives from a nonlinear differential identity
- 09-13 · An even function has no linear term
- 09-14 · Factor a logarithm argument
- 09-15 · Use one series to recover its companion
- 09-16 · Compose sine and logarithm carefully
- 09-17 · Small-angle series in a triangle
- 09-18 · Sine rule and a percentage-error interval
- 09-19 · A sector arc contributes a linear term
- 09-20 · Differentiate only after keeping enough terms
- 09-21 · An inverse tangent generates a reciprocal series
- 09-22 · A shifted cotangent is analytic at zero
- 09-23 · Integrate a chain-rule derivative first
- 10-01 · A removable extension of the differential equation
- 10-02 · A second-order equation needs two constants
- 10-03 · The product substitution absorbs a linear term
- 10-04 · Bacteria growth after a product substitution
- 10-05 · Draining a sphere: height is not volume
- 10-06 · Drag models approach different terminal speeds
- 10-07 · Calibrate exponential decay
- 10-08 · Cooling toward ambient temperature
- 10-09 · Constant input shifts a decay equilibrium
- 10-10 · An exponential height law makes the rod never land
- 10-11 · Keep population units consistent
- 10-12 · A decreasing forest model stops at extinction
- 10-13 · A square-root removal law has an implicit solution
- 10-14 · Product substitution and a logistic epidemic
- 10-15 · Compare unbounded growth with a carrying capacity
- 10-16 · A declining species and an increasing logistic species
- 10-17 · Separate the formal model from inconsistent pump directions
- 10-18 · Logistic proportions and the first whole week
- 11-01 · A perpendicular foot organizes four shapes
- 11-02 · An extended segment and coefficient comparison
- 11-03 · Skew lines and a perpendicular foot
- 11-04 · Projection followed by line reflection
- 11-05 · Equal radii make a midpoint perpendicular
- 11-06 · Area identities and a cross-product degeneracy
- 11-07 · A line that meets the x-axis
- 11-08 · Orthogonality and an angle maximisation
- 11-09 · Affine collinearity and a triangle area
- 11-10 · Plane intersections and orthogonal directions
- 11-11 · A parallel line and a perpendicular plane
- 11-12 · Projection onto a plane and a reflected triangle
- 11-13 · Distinguish no intersection from containment
- 11-14 · Build coordinates from a cuboid
- 11-15 · Equal spacing between parallel planes
- 11-16 · Project a point onto an intersection line
- 11-17 · Planes containing a prescribed line
- 11-18 · Parallel lines, a parallelogram and a midplane
- 11-19 · A wardrobe model from lines and planes
- 11-20 · A reflected laser and a sensor-distance check
- 11-21 · A river as the intersection of two planes
- 11-22 · A regular tetrahedron and the methane angle
- 11-23 · The shortest cable to an inclined roof diagonal
- 12-01 · A variable geometric ratio
- 12-02 · A relative geometric-tail error
- 12-03 · Link an AP with a convergent GP
- 12-04 · Odd-indexed terms and an alternating tail
- 12-05 · Selected GP terms form an AP
- 12-06 · Half the infinite sum and alternate indices
- 12-07 · Eliminate two progression parameters
- 12-08 · Alternating terms, even terms and products
- 12-09 · Odd-sized groups of geometric terms
- 12-10 · A half-tail identity and triangular rows
- 12-11 · A filling machine and an insufficient tail
- 12-12 · Positive ribbon lengths and a bounded GP
- 12-13 · AP lap times against GP lap times
- 12-14 · Remove multiples and sum a bouncing path
- 12-15 · Declining sales prices and holding costs
- 12-16 · Selected terms and a rabbit–fox chase
- 12-17 · Monthly deposits and the first crossing day
- 12-18 · A reducing loan and a final smaller payment
- 12-19 · A declining income against rising costs
- 12-20 · Out-and-back circuits and changing growth rates
- 13-01 · Recover terms from a cumulative sum
- 13-02 · A telescoping increment
- 13-03 · Alternation around a fixed point
- 13-04 · Shift an affine recurrence into a GP
- 13-05 · A rational recurrence with period four
- 13-06 · Sum polynomial and exponential increments
- 13-07 · A logarithmic fixed-point iteration
- 13-08 · A square-root iteration crosses its limit
- 13-09 · A rational map approaches two
- 13-10 · Halve the error at every step
- 13-11 · A conditional limit in a two-cycle map
- 13-12 · An exponential denominator creates a stable limit
- 13-13 · An alternating fifth-root iteration
- 14-01 · An integer upper limit in an inequality
- 14-02 · Sum a block of odd squares
- 14-03 · A polynomial term and a middle block
- 14-04 · Combine an alternating GP with logarithms
- 14-05 · A trigonometric block sum
- 14-06 · Telescoping logs and a rational GP ratio
- 14-07 · An infinite tail from a supplied sum
- 14-08 · A factorial tail after reindexing
- 14-09 · An error bound and a shifted rational sum
- 14-10 · Shift an exponential summation identity
- 14-11 · A radical sum with one extra endpoint
- 14-12 · Apply the absolute root test
- 15-01 · Eliminate with complex coefficients
- 15-02 · Reduce a complex system to a quadratic
- 15-03 · Conjugation and component equations
- 15-04 · Arguments after rotation and reflection
- 15-05 · Square roots followed by a quadratic shift
- 15-06 · A modulus equation imposes a sign
- 15-07 · Root midpoint and a parallelogram
- 15-08 · Conjugate roots and transformed equations
- 15-09 · A cubic discriminates real from complex roots
- 15-10 · Use conjugacy only with real coefficients
- 15-11 · A purely imaginary root unlocks a cubic
- 15-12 · Factor a real quartic by a conjugate pair
- 15-13 · Quartic roots and an Argand triangle
- 15-14 · Five points from one polar vector
- 15-15 · A real conjugate sum has two directions
- 15-16 · Find a square using a quarter-turn
- 15-17 · Arguments determine a triangle
- 15-18 · A complex square gives an exact tangent
- 15-19 · Complex division yields an exact sine