The sun god has a herd of cows, which are composed of four colors: white, black, flower 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 flowers and cows

It is1/5+1/6 of the total number of 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: De Mezirik's code problem

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 grassland and cattle problem

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

A&# 39; A cow will eat the grass on plot b&# 39 in just a few days.

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

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

Question 04: Bijuk's seven seven questions.

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

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

* * * * * *

* * * * * 7 *

* * * * * * *

* 7 * * * *

* 7 * * * *

* * * * * * *

* * * * 7 * *

* * * * * *

* * * * * *

The number marked with an asterisk was accidentally deleted. What are the missing figures?

Question 05: Female students in Kirkman.

There are fifteen girls in the boarding school. They often walk in groups of three every day and ask how to arrange it so that every student can have a good time.

A girl walks a line with other girls, just once a week?

Question 06 Bernoulli-Euler wrote the wrong letter

To find the arrangement of n elements, it is required that no element is in a proper position in the arrangement.

Question 07 Euler's division of polygons Euler & # 39; S problem of polygon segmentation

How many ways can an N-sided polygon (planar convex polygon) be divided into triangles with diagonal lines?

Question 08: Lucas' spouse and husband and wife problems & # 39; Problems of married couples

N couples sit around the round table. A man sits between two women, but there is no man and himself.

The wife sat side by side and asked how many ways to sit.

Question 09: Kayam Omar Khayyam's binomial expansion S binomial expansion

When n is an arbitrary positive integer, find the n power of binomial a+b expressed by the powers of A and B.

Problem 10 Cauchy mean value theorem Cauchy & # 39; mean value theorem

Verify that the geometric mean of n positive numbers is not greater than the arithmetic mean of these numbers.

Bernoulli power sum 1 1 problem Bernoulli & # 39; S-power sum problem

When the exponent p is a positive integer, the sum of the p powers of the first n natural numbers is determined as S= 1p+2p+3p+…+ mouth.

Problem 12 Euler number Euler number

Find the limit values of functions φ(x)=( 1+ 1/x)x and φ (x) = (1+/kloc-0) x+1when x increases infinitely.

Problem 13 Newton exponential series Newton & # 39; S exponential series

Convert the exponential function ex into a series whose term is the power of X.

Question 14: Nicola Mercator of keitel logarithmic series & # 39; S logarithmic series

Calculate the logarithm of a given number without using a logarithm table.

15 Newton sine and cosine series Newton & # 39; Sine and cosine series

Calculate sine and cosine trigonometric functions with known angles without looking up the table.

Problem 16 Derivation of Andre 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, ..., cn. Deriving the series of secant and tangent by the method of inflectional arrangement.

Question 17 The arctangent sequence of Gregory Gregory & # 39; S arc tangent series

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

18 topic buffon's needle buffon & # 39; 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 actually 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: Gauss's basic algebraic theorem Gauss; fundamental theorem of algebra

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

Question 24: The number of roots of Sturm; The number of roots

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

Question 25 Abel's Impossible Theorem Abel & # 39; impossibility theorem

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

Question 26: Hermite-Lin Deman Transcendence Theorem

The coefficient A is not equal to zero, and the exponent α is the expression A1eα1+A2eα 2+A3eα 3+… No.

May be equal to zero.

Question 27 Euler Straight Euler & # 39; S 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

The three midpoints of three sides, the vertical legs of three heights and the three midpoints of the line segment from the intersection of three heights to each vertex in the triangle are on a circle.

Question 29: Castillon, Castillon & # 39; The problem of

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

Question 30: Malfati's question Malfati & # 39; The problem of

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 Monch, gaspard monge & # 39; The problem of

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: Masoney compass Masoney & # 39; Compass problem

It is proved that any diagram that can be made with compass and ruler can only be made with compass.

Question 34 Steiner's Ruler Question Steiner & # 39; S-straight edge problem

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

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 Archimedes π value determination method Archimedes; Determination of number Pi

Let the perimeters of the circumscribed and inscribed 2vn polygons of a circle be mouth and bv, respectively, and then the Archimedes series of polygon perimeters can be obtained in turn: a0, b0, a 1, b 1, a2, b2, … where mouth+1 is the harmonic term of mouth and bv, and bv+ 1 is. If the first two terms are known, you can use this rule to calculate all terms of a series. This method is called Archimedes algorithm.

Question 39: Make a fuss & # 39; Chord tangent quadrilateral problem

Find 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 Alhazen ' The marble problem & # 39; Billiards problem

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

Question 42: Use * * * to make an ellipse from the radius of the yoke.

Given the size and position of two yoke radii, draw 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 & # 39; S trajectory problem

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: Carden spinning wheel problem Carden & # 39; Spur gear problem

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 Newton & # 39; S 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.