100 famous elementary mathematics problems

Question 0 1 Archimedes problem Bonum

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; There are more black cows than brown cows.

The extra number is equivalent to1/4+1/5 of the number of flowers; The number of flower cattle is more than that of brown cattle, and the extra number is equivalent to 1/6+ 1 of the number of white cattle.

/7.

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 cattle is

1/5+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: the weight of bachet de meziriac code

A businessman has a 40-pound weight, which broke into four pieces because it fell to the ground. Later, each piece was weighed.

It is the whole pound, and these four pieces can be used to weigh any integer pound from 1 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 cow ate up the grass on plot b in c 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 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: The problem of female students in "The Problem of Schoolgirls" by Kirkman, 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 girl can have a good time.

Girls and other girls walk on the same line, and only once a week?

Question 06 Bernoulli-Euler problem of wrong envelope Bernoulli-Euler problem of Misad

Decorated letters

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

Problem 07 Euler's Polygon Segmentation Problem

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

Question 08 Lucas problem of married couples

N couples sit around the round table, a man sits between two women, and there is no man and wife.

Sit side by side and ask how many ways to sit.

Question 09: The Binomial Expansion of Kayam Omar Khayyam

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

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

Problem 1 1 Bernoulli power sum problem

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

Problem 12 Euler number Euler number

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

Problem 13 Newton exponential series

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

Question 14: Nikolai Mercatos Logarithmic Series of Max keitel Logarithmic Series

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

Problem 15 Newton sine and cosine series

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

Question 16 Andre's derivation of secant and tangent series Andre's derivation of secant and Tang.

Ent series

In the arrangement of n numbers 1, 2, 3, ..., n, c 1, c2, ..., cn, if there are no elements with values in between.

Between two adjacent values ci- 1 and ci+ 1, it is called c 1, c2, …, and cn is the inflected arrangement of 1, 2, 3, …, n. 。

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.

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 will touch one of 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: Gaussian fundamental theorem of Gaussian 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 real roots of algebraic equations with real coefficients in known intervals.

Question 25 Abel impossibility theorem Abel impossibility theorem

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 a1eα1+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.

Energy equals zero.

Question 27 Euler straight line Euler straight line

In all triangles, the center of the circumscribed circle, the intersection point of each middle line and the intersection point of each height are all on a straight line-Euler line.

Moreover, the distance between the three points is: the distance from the intersection of each high line (vertical center) to the intersection of each middle line (center of gravity) is twice that of the circumscribed circle.

The distance from the center of the circle to the intersection of the center lines.

Question 28 Feuerbach circle

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

On a circle.

Question 29: The Castilian question.

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

Question 30: marfa's question and 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 Question and 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: Masoroni compass problem.

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

Question 34 Steiner's straightedge problem

It is proved that any diagram that can be made with compasses and rulers can be made with straight edges if a fixed circle is given on the plane.

Can be manufactured.

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 circumscribed and inscribed regular 2vn polygons be av and bv, respectively, and the Archimedes lines of polygon perimeters can be found in turn.

Order: a0, b0, a 1, b 1, a2, b2, … where av+ 1 is the harmonic term of av and bv, bv+ 1 is bv, av+ 1 and so on.

If the first two terms are known, then all the terms of a series can be calculated by this rule. This method is called Aki.

Meade algorithm.

Fuss problem of chord-tangent quadrilateral

Find the relationship between the radius of bipartite quadrilateral and circumscribed circle and inscribed circle.

A polygon is defined as a quadrilateral inscribed with a circle and circumscribed by another circle.

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 alhasen 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: Fanscuton trajectory problem

Two vertices of a fixed triangle on the plane slide along two sides of an angle on the plane, and the trajectory of the third vertex is

What?

Question 48: Universal rotating wheel problem Universal spur gear problem

When a disk rolls along the inner edge of another disk with a radius twice as large, a point marked on this disk is described.

What is the trajectory?

Question 49 Newton elliptic problem.

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

Dream an impossible dream

Defeat the invincible enemy

Correct an uncorrectable mistake

When my arm is tired

To reach those distant stars.

Alte

Old member

Yuan's pity is too tired.

Essence 0

After 2667

Integer 1254

Reading permission 40

Registration telephone number 2003-12/20

The landlord posted a short message on 2005- 12-2000: 00.

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 the line segment E is continuously intercepted n times on the other side.

F, and write down the end points of the line segment by numbers, starting from the vertex, which are 0, 1, 2, …, n and n, n- 1, …, 2, 1 respectively.

,0.

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 Steiner's pointed hypocycloid

Determines the envelope of the Wallace line of the triangle.

Question 54: The closest circular ellipse of a quadrilateral.

Depicted quadrilateral

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 Desargues's homology theorem (homology triangle theorem) and Dasaga's homology theory (theory

Homologous triangle)

If the straight line connecting the corresponding vertices of two triangles passes through a point, the intersection of the corresponding edges of two triangles is on a straight line.

Online.

On the other hand, if the intersections of the corresponding sides of two triangles are on a straight line, the corresponding vertices of the two triangles are connected.

This line crosses a little.

Question 60 Steiner's binary drawing method Steiner's binary drawing method

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

Question 6 1 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 62: Briante-Hungarian Six Linearity Theorem Briante'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 63: Descartes involution theorem and Dasaga involution theorem.

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.

Four pairs meet. The line between a point and three pairs of vertices of a complete quadrilateral *, and the tangent from the point to the quadrilateral.

The tangent drawn by the conic curve of the shape "A" forms a pair of four rays.

* A complete quadrilateral actually contains four points (lines) 1, 2, 3, 4 and their six connecting lines.

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 64: A conic curve of five elements obtained from five elements

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

Question 65: Conic curves and straight lines

A known straight line intersects a quadratic curve and has five known elements: a point and a tangent. Find them.

The intersection of.

Question 66: Conic curve and a point Conic curve and a point.

Given a point and a conic, there are five known elements: a point and a tangent. Make a cylinder from this point to this point.

Tangent of a curve.

Question 67 Steiner divides space by plane.

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

Question 68 Euler tetrahedron problem

The volume of a tetrahedron is represented by six sides.

Question 69: The shortest distance between oblique straight lines

Calculate the angle and distance between two known oblique lines.

Question 70: Draw a tetrahedron on on the sphere.

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

Question 7 1 Five Normals Five Normals

Divide a ball into congruent spherical regular polygons.

Question 72: Square as a quadrilateral image.

It is proved that every quadrilateral can be regarded as a perspective image of a square.

Question 73: Polk-Siegel Theorem Polk-Schwartz Theorem

Any four points on the plane that are not all on the same straight line can be considered as four points similar to the known tetrahedron.

Tilt mapping of each corner of a cube.

Question 74: Gauss basic theorem of basic theory of axonometric measurement.

Gauss basic theorem of orthographic projection method: in a orthographic projection with three corners, if the mapping plane is taken as a complex plane,

The projection of the vertices of three corners is taken as zero, and the projection of each endpoint of an edge is taken as a complex number of a plane, so the sum of squares of these numbers is equal.

At zero.

Question 75: stereographic projection of hipparchus' polar plane.

Try to give a conformal map projection method to transform the circle on the earth into the circle on the map.

Question 76: Mercator projection

Draw a orthographic geographic map, whose coordinate grid is composed of rectangular grids.

The problem of Loxodrome

Determine the longitude of the diagonal line between two points on the earth's surface.

Question 78: Determine the position of the ship at sea.

The position of the ship at sea is determined by the astronomical meridian extrapolation algorithm.

Question 79 Gauss Two Height Problem

Determine the time and position according to the known heights of two planets.

Question 80 Gauss three-altitude problem

The observation time, the latitude of the observation point and the latitude of the planet are determined from the time interval of the time at the same height obtained from the known three-star sphere.

Height.

Question 8 1: Kepler equation

According to the average perigee angle of the planet, the eccentricity and true perigee angle are calculated.

Question 82. Star setting for falling stars

For a given place and date, calculate the time and azimuth of a known star setting.

Question 83. The question of the sundial

Make a sundial.

Question 84: Shadow curve

When the straight pole is placed at the latitude φ and the declination of the sun on that day has a delta value, the daytime pole is determined.

A curve described by a projection.

Question 85 solar and lunar eclipses

If the right ascension, declination, and the radii of the sun and the moon are known at two moments near the eclipse time.

Determine the beginning and end of the solar eclipse and the maximum value of the hidden part of the sun's surface.

Question 86: Stars and Rendezvous Period

Determine the intersection operation period of two * * * plane rotating rays with the known star operation period.

Question 87: Forward and backward motions of planets and forward and backward motions of planes.

When does the planet change from forward motion to reverse motion (or vice versa)?

Question 88 comet Lambert Prolem

With the help of the focal radius and the chord connecting the arc ends, the time required for a comet to move an arc along a parabolic orbit is shown.

Question 89 Steiner problem about Euler number

If x is a positive variable, what is the value of x, and the x-th root of x is the largest?

On the problem of high base point.

In the known acute triangle, make the inscribed triangle with the smallest circumference.

Question 9 1 Fermat's question to Torricelli.

Try to find a point to minimize the sum of the distances between the three vertices of a known triangle.

Question 92: Change course against the wind

How can a sailboat sail due north at the fastest speed against the north wind?

Question 93: Bee cells (Reaumur's question)

Try to close a regular hexagonal prism with a top cover made of three congruent diamonds, so that the obtained solid is predetermined.

The volume is constant and its surface area is the smallest.

Question 94: Reggio Montanus' biggest problem.

In which part of the earth's surface does a vertical boom present the longest? (that is, where is the maximum viewing angle?

? )

Question 95: The maximum brightness of Venus.

Where is Venus the brightest?

Question 96: Comets in Earth's orbit.

How many days can a comet stay in Earth's orbit at most?

Question 97: The shortest dusk.

Where the latitude is known, which day is the shortest in a year?

Question 98 Steiner's Elliptic Problem

Of all the ellipses that can circumscribe (inscribe) a known triangle, which ellipse has the smallest (largest) surface?

Product?

Question 99 Steiner circle problem

In all plane figures with equal perimeters, the circle has the largest area.

On the contrary, in all plane figures with equal areas, the circumference of a circle is the smallest.

Question 100 Steiner ball problem Steiner ball problem

Among all solids with the same surface area, the ball has the largest volume.

Among all solids of equal volume, the surface area of the ball is the smallest.