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Problem description:

The teacher asked us to be a historical tabloid. Please help! ! !

Analysis:

Archimedes problem

Sun God has a herd of cows, white, black, flowered and brown.

Among the bulls, the number of white cattle is more than that of brown cattle, and the extra number is equivalent to1/2+1/3 of the number of black cattle; The number of black cattle is more than that of brown cattle, and the extra number is equivalent to1/4+1/5 of the number of flower cattle; The number of flower cattle is more than that of brown cattle, and the extra number is equivalent to 1/6+ 1/7 of the number of white cattle.

Among the cows, the number of white cows is1/3+1/4 of all black cows; The number of black cattle is1/4+1/5 of all flower cattle; The number of flower cattle is1/5+1/6 of all brown cattle; The number of brown cattle is 1/6+ 1/7 of the total number of white cattle.

How is this herd made up?

Question 02: the weight of bachet de meziriac code

A businessman had a 40-pound weight, which was smashed into four pieces because it fell to the ground. Later, each piece was weighed by the whole pound, and these four pieces can be used to weigh any integer pound from/kloc-0 to 40 pounds.

How much do these four weights weigh?

Question 03 Newton's questions about fields and cows.

A cow ate up the grass on plot b in c days;

A' A cow ate up B' s grass on C' day;

A "the cow ate up the grass in B" on day C ";

Find the relationship between 9 quantities from A to C "?

Question 04 Bewick's July 7th question Bewick's July 7th question.

In the following division example, the dividend is divided by the dividend:

* * 7 * * * * * * * ÷ * * * * 7 * = * * 7 * *

* * * * * *

* * * * * 7 *

* * * * * * *

* 7 * * * *

* 7 * * * *

* * * * * * *

* * * * 7 * *

* * * * * *

* * * * * *

Numbers marked with an asterisk (*) were accidentally deleted. What are the missing figures?

Question 05 Pascal's hexagon theorem

It is proved that the intersection of three pairs of opposite sides of a hexagon inscribed on a conic curve is on a straight line.

Question 06: Briante-Hungarian Six Linearity Theorem Brian Xiong's Six-pointed Star Theorem.

It is proved that the tangent is among the six lines of the conic, and the three top lines pass through a point.

Question 07 involution theorem of De Sages

The intersection of a straight line with three pairs of opposite sides of a complete quadrilateral * and the conic curve circumscribed by the quadrilateral form a involutory four-point pair. The connecting line between a point and three pairs of vertices of a complete quadrilateral * and the tangent drawn by the conic curve tangent to the quadrilateral from this point form a involutory four-ray pair.

* A complete quadrilateral actually contains four points (lines) 1, 2, 3, 4 and their six connection points 23, 14, 3 1, 24, 12, 34; Where 23 and 14, 3 1 and 24, 12 and 34 are called opposite edges (opposite vertices).

Question 08 A conic curve of five elements obtained from five elements.

Find a conic curve and know its five elements-point and tangent.

Question 09: Conic curves and straight lines

A known straight line intersects a conic curve with five known elements (points and tangents), and find their intersection points.

Problem 10: conic curve and a point

Given a point and a quadratic curve with five known elements (point and tangent), make a tangent from the point to the curve.

Question 1 1 Steiner divides space with planes.

How many parts can n planes divide the whole space into at most?

Problem 12 Euler tetrahedron problem

The volume of a tetrahedron is represented by six sides.

Question 13: the shortest distance between oblique straight lines.

Calculate the angle and distance between two known oblique lines.

Question 14: A spherical circle describes a tetrahedron.

Determine the radius of the circumscribed sphere of a tetrahedron with all six sides known.

Five normal bodies, five normal bodies.

Divide a ball into congruent spherical regular polygons.

Problem 16 Andre's Derivation of Secant and Tangent Series

In the arrangement of n numbers 1, 2, 3, ..., n, if the value of no element ci is between two adjacent values ci- 1 and ci+ 1, it is called c 1, c2, ...

Deriving the series of secant and tangent by the method of inflectional arrangement.

Question 17 Gregory arc tangent series

Knowing the three sides, you don't need to look up the table to find the angle of the triangle.

Question 18: Buffon's needle problem Buffon's needle problem.

Draw a set of parallel lines with a distance of d on the table, and throw a needle with a length of L (less than D) on the table at will. What is the probability that the needle touches one of the two parallel lines?

Problem 19 Fermat-Euler Prime Theorem

Every prime number that can be expressed as 4n+ 1 can only be expressed as the sum of squares of two numbers.

Question 20 Fermat equation Fermat equation

Find the integer solution of the equation x2-dy2 = 1, where d is a non-quadratic positive integer.

Fermat-Gauss impossibility theorem Fermat-Gauss possibility theorem

It is proved that the sum of two cubes cannot be a cube.

Question 22: Law of Quadratic Reciprocity

(Euler-Legendre-Gauss Theorem) Legendre reciprocity sign of odd prime numbers P and Q depends on the formula.

(p/q)(q/p)=(- 1)[(p- 1)/2][(q- 1)/2]。

Question 23: Basic Algebraic Theorem of Gauss

Every equation of degree n Zn+c1Zn-1+c2zn-2+…+= 0 has n roots.

Question 24: the number of roots of Sturm

The number of real roots of algebraic equations with real coefficients in known intervals.

Question 25 Abel's impossibility theorem Abel's possibility theory

Generally, it is impossible to have algebraic solutions for equations higher than quartic.

Question 26: Hermite-Lin Deman Transcendence Theorem Hermite-Lin Deman Transcendence Theorem

The expression a1e1+a2eα 2+a3eα 3+... where the coefficient a is not equal to zero and the exponent α is an algebraic number that is not equal to each other and cannot be equal to zero.

Question 27 Euler straight line Euler straight line

In all triangles, the center of the circumscribed circle, the intersection point of each midline and the intersection point of each height are all on a straight line-Euler line, and the distance between the three points is twice as long as the distance from the intersection point (vertical center) of each height line to the intersection point (center of gravity) of each midline.

Question 28 Feuerbach circle

Three midpoints of three sides in a triangle, three vertical height feet and three midpoints of a line segment from the intersection of heights to each vertex are all on a circle.

Question 29: Castillon's problem, Castillon's problem.

A triangle with three known points is inscribed in a known circle.

Question 30. marfa's question

Draw three circles in the known triangle, each circle is tangent to the other two circles and the two sides of the triangle.

Question 3 1 gaspard monge Gaspard Monge Question

Draw a circle so that it is orthogonal to three known circles.

Tangency of apollonius in Apolloni.

Draw a circle tangent to three known circles.

Question 33: Maceroni's compass problem.

Prove that any diagram that can be made with compasses and straightedge can only be made with compasses.

Question 34 Steiner's straight edge problem

It is proved that as long as a fixed circle is given on the plane, any diagram that can be made with compasses and rulers can be made with rulers.

Question 35: Deliaii cube doubling of Abe cube in Delhi.

Draw one side of a cube twice the volume of a known cube.

Question 36: The bisection of an angle is divided into three parts.

Divide an angle into three equal angles.

Question 37: Regular heptagon

Draw a regular heptagon.

Question 38 How to measure Archimedes π value Archimedes' determination of pi.

Let the perimeters of the circumscribed and inscribed 2vn polygons of a circle be av and bv, respectively, then the Archimedes series of polygon perimeters can be obtained in turn: a0, b0, a 1, b 1, a2, b2, … where av+ 1 is the harmonic term of av and bv, and bv+ 1 is bv and A.

Fuss problem of chord-tangent quadrilateral

Find out the relationship between the radius of bicentric quadrilateral and circumscribed circle and inscribed circle. (Note: A bicentric or chordal quadrilateral is defined as a quadrilateral inscribed in a circle and tangent to another circle at the same time. )

Question 40: Measurement with survey attachment

Use the direction of known points to determine the location of unknown but reachable points on the earth's surface.

Question 4 1 Billiards in Alhazen

Make an isosceles triangle in a known circle, and its two waists pass through two known points in the circle.

Question 42: Ellipse with conjugate radius

Given the size and position of two conjugate radii, make an ellipse.

Question 43: Make an ellipse in a parallelogram.

Make an inscribed ellipse in the specified parallelogram, which is tangent to the parallelogram at the boundary point.

Question 44: Multiply four tangents by four tangents to make a parabola.

We know the four tangents of a parabola and make it a parabola.

Question 45 is a parabola starting from four points.

Draw a parabola through four known points.

Question 46 is a hyperbola starting from four points.

Given four points on a right-angled (isometric) hyperbola, make this hyperbola.

Question 47: Van Short's Trajectory

Two vertices of a fixed triangle on the plane slide along two sides of an angle on the plane. What is the trajectory of the third vertex?

Question 48: The spur gear problem of cardan.

When a disk rolls along the inner edge of another disk with a radius of twice, what is the trajectory drawn by a point marked on this disk?

Question 49 Newton elliptic problem.

Determine the center trajectories of all ellipses inscribed in a known (convex) quadrilateral.

Question 50: Poncelet-Briante-Hungarian Hyperbolic Problem.

Determine the trajectory of the intersection of the top vertical lines of all triangles inscribed with the right-angled hyperbola.

Question 5 1 parabola as envelope.

Starting from the vertex of the angle, any line segment E is continuously intercepted n times on one side of the angle, and line segment F is continuously intercepted n times on the other side. Endpoints of the line segments are numbered from the vertex, which are 0, 1, 2, …, n and n, n- 1, 0 respectively.

It is proved that the envelope of the line connecting points with the same number is a parabola.

Question 52: the star line of the star line

Two calibration points on a straight line slide along two fixed vertical axes to find the envelope of the straight line.

Question 53: Steiner's three-point hypocycloid has three points.

Determines the envelope of the Wallace line of the triangle.

Question 54: The ellipse closest to the circle draws a quadrilateral circumscribed ellipse.

Of all the circumscribed ellipses of a quadrilateral, which deviates from the circle the least?

Question 55 Curvature of conic section

Determine the curvature of a conic curve.

Question 56 Archimedes' calculation of parabola area Archimedes squared parabola.

Determine the area contained by the parabola.

Question 57: Calculate the area square hyperbola of hyperbola.

Determine the area contained in the hyperbola cutting part.

Question 58: Find the long rectification of a parabola.

Determine the length of the parabolic arc.

Question 59: Gilad Girard Des Argues's homology theorem (homology triangle theorem) Desargues' homology theorem (homology triangle theorem)

If the corresponding vertices of two triangles pass through a point, the corresponding edges of the two triangles intersect on a straight line.

On the other hand, if the intersection of the corresponding sides of two triangles is on a straight line, the corresponding vertices of the two triangles pass through a point.

Question 60: Steiner's two-element structure.

The overlapping projective form given by three pairs of corresponding elements makes it a double element.