Mathematical intelligence problem

Mom's name is red cake. It takes 2 minutes to bake one cake and 1 minute to bake another. The pancake pan can hold up to 4 cakes at a time. How does Xiaohong bake 6 cakes in 5 minutes?

The answer is

First put four cakes on one side that needs to be baked for 2 minutes, then take out two sides, put down the remaining two sides that need to be baked for 2 minutes, and turn the two sides that have been in the pot over. After one minute, take out the two baked sides and put down the two cakes with one baked side. In another minute, four cakes will be baked. Then burn the last two sides for another minute. 2+2+ 1=5 (minutes) so that you can bake 6 cakes in 5 minutes.

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: Germany? Weight problem 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 * *

* * * * * *

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

Question 05: The problem of female students in the problem of female students in Kirkman, Kirkman

There are fifteen girls in the boarding school. They often walk in groups of three every day. They asked how to arrange for each girl to walk on the same line with other girls 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 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's question about married couples

N couples sit around the round table, a man sits between two women, and no man sits with his wife. How many sitting postures are there?

Question 09: 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.

Exponential series of problem 13 Newton

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

Question 14 Nicola Mercator Logarithmic Series Max keitel Logarithmic Series

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

Problem 15 Newton's sine and cosine series

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

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, ..., cn.

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+…+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'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: 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. 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.

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 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 63 The 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 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 conic curve with five known elements (points and tangents), and find their intersection points.

Question 66: Conic curve and a point 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 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 a plane that are not all on the same straight line can be regarded as an oblique mapping of the angles of a tetrahedron similar to a known tetrahedron.

Question 74: Basic Theorem of Gauss Axonometry Basic Theory of Gauss Axonometry

Gauss basic theorem of orthographic projection method: If in the orthographic projection of a three-sided angle, the image plane is regarded as a complex plane, the projection of the vertex of the three-sided angle is regarded as a zero point, and the projection of each endpoint of an edge is regarded as a complex number of the plane, then the sum of squares of these numbers is equal to zero.

Question 75: stereographic projection on the polar plane of hipparchus sphere

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: Double Altitude Problem of Gauss

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

Question 80: Three Altitudinal Problems of Gauss

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

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 curve described by the projection of the straight pole at a certain point in the course of a day is determined.

Question 85 solar and lunar eclipses

If the right ascension, declination and radius of the sun and the moon are known at two moments near the time of the eclipse, the beginning and end of the eclipse and the maximum value of the hidden part of the sun surface are determined.

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 Lambert's comet 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?

Question 90: Fanano's height base point. About the height base point

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

Question 9 1 Fermat problem for Torricelli's problem for Torricelli's problem.

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 seal a regular hexagonal prism with a top cover made of three congruent diamonds, so that the obtained solid has a predetermined volume and the minimum surface area.

Question 94: The biggest problem of Reggio Montanus: The biggest problem of Reggio Omotanus.

In which part of the earth's surface does a vertical boom present the longest? (that is, where is the largest 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) area?

Question 99 Steiner circle problem

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