(1) The center of the circle is inside this convex quadrilateral;
(2) The maximum side length is a and the minimum side length is √(4-a2). Make four tangents LA, LB, LC and LD passing through points a, b, c and d into a circle γ in turn. It is known that LA and LB, LB and LC, LC and LD, LD and LA intersect at four points: A', B', C' and D' respectively. Find the maximum and minimum values of the area ratio SA'B'C'D'/SABCD.
Second, let X={ 1, 2,3, … 200 1} find the smallest positive integer m, which is suitable for the requirement that for any m-ary subset w of x, there are u and v (u and v are allowed to be the same), so u+v is the power of 2.
3. Every vertex of a regular N-polygon has a magpie. I was so frightened that all the magpies flew away. After some time, they all returned to these vertices, still one on each vertex, but not all of them returned to their original vertices. Find all positive integers n, so there must be three magpies, and the triangles formed by their front and rear vertices are all acute triangles, right triangles or obtuse triangles.
4. Let A, B, C, A+B-C, A+C-B, B+C-A and A+B+C be seven pairwise different prime numbers, and the sum of two numbers in A, B and C is 800. Let d be the difference between the largest number and the smallest number among the seven prime numbers. Find the maximum possible value of d.
5. Divide the circumference of 24 into 24 equal parts. Select 8 points from 24 points so that the arc length between any two points is not equal to 3 and 8. Q: How many different ways can you get 8 points that meet the requirements? Explain why.
6.a=200 1。 Let a be a set of positive integer pairs (m, n), which is suitable for the following conditions:
( 1)m & lt； 2a； (2)2n |(2am-m2+N2)； (3)n2-m2+2mn ≤2a(n-m). Let f = (2am-m2-mn)/n, and find min(m, n) ∈ Af, max(m, n) ∈ Af.
2003 Winter Camp of Mathematics Olympics for Middle School Students in China
1. Let point I and point H be the center and vertical center of acute triangle respectively, and point B 1 and point C 1 are the midpoint of AC side and AD side respectively. It is known that ray B 1I intersects with AB at point B2(B2≠B), the extension of ray C 1I intersects with AC at C2, B2C2 intersects with BC at k, and A 1 is the outer center of △BHC. Test: Necessary and sufficient conditions for the area of a, i, A 1 three-point * * * line △BKB2 and △CKC2 to be equal.
2. Find the maximum number of elements in the set S that meet the following conditions at the same time:
Every element in (1)S is a positive integer not exceeding 100;
(2) For any two different elements A and B in S, there is an element C in S, so that the greatest common divisor of A and C is equal to 1, and the greatest common divisor of B and C is also equal to1;
(3) For any two different elements A and B in S, one element D is different from A and B in S, so that the greatest common divisor of A and D is greater than 1, and that of B and D is also greater than 1.
3. Given a positive integer n, find the smallest positive number λ, so that for any θi∈(0, π/2), (i= 1, 2, 3, ... n).
As long as tan θ 1 tan θ 2...tan θ n = 2n/2, there are cosθ 1+ cosθ2+...+ cosθn not greater than λ.
4. Find all ternary positive integer groups (a, m, n) satisfying a≥2 and m≥2, so that an+2003 is a multiple of am+ 1.
A company needs to hire a secretary. * * * Number of applicants 10. The company manager decided to interview one by one in the order of application, and the first three people must not be hired after the interview. Starting with the fourth person, compare him with the people who have been interviewed before, and if his ability exceeds everyone who has been interviewed before, hire him; Otherwise, don't hire, continue to interview the next one. If the first nine people are not hired, then hire the last interviewer.
Assuming that this 10 person has different abilities, it can be sorted from strong to weak as 1, 2, …, 10. Obviously, which person the company hired has something to do with the order in which this 10 person signed up. As we all know, this arrangement * * * has 10! Kindness We use Ak to indicate the number of different registered orders that people with ability K can be employed, and use Ak/ 10! Indicate the possibility of his being employed.
Proof: under the guidance of the company manager, there are
( 1)a 1 & gt； A2 & gt…& gt； A8 = A9 = a 10；
(2) The company has more than 70% possibility to hire one of the three people with the strongest ability, and only 10% possibility to hire one of the three people with the weakest ability.
6. Let A, B, C and D be positive real numbers and satisfy AB+CD =1; Points Pi(xi, yi) (i= 1, 2, 3, 4) are four points on the unit circle with the origin as the center. Verification:
(ay 1+by2+cy3+dy4)2+(ax4+bx3+cx2+dx 1)2≤2((a2+B2)/a b+(C2+D2)/CD)
2004 China Middle School Students' Mathematical Olympic Winter Camp
1. Vertices E, F, G and H of convex quadrilateral E, F, G and H are on the sides AB, BC, CD and DA of convex quadrilateral A, B, C and D, respectively, satisfying (AE/EB)(BF/FC)(CG/GD)(DH/HA) and point A, D. On f 1g 1, g 1h 1, h 1e 1 f1‖ ef, f/.
3. Let m be a set of n points on a plane, which satisfies:
There are 7 points in (1)M, which are 7 vertices of a convex heptagon;
(2) Any five points in m, if these five points are five vertices of a convex Pentagon, then the interior of the convex Pentagon contains at least one point in m. 。
Find the minimum value of n.
6. It is proved that all other positive integers n can be expressed as the sum of 2004 positive integers n = a 1+ a2+...+a2004.
And satisfy 1 ≤ A 1 ≤ A2 ≤...≤ An, AI | AI+ 1 (I = 1, 2, ..., 2003).
2005 China Mathematical Olympics National Middle School Students' Mathematics Winter Camp
2. The three sides of the circle intersect △ABC at d 1 and d2, respectively; E 1，E2； F 1, F2. In addition, line D 1E 1 intersects with line D2F2 at point L, line E 1F 1 intersects with E2D2 at point M, and line F 1D 1.
3, as shown in the figure (the figure consists of two concentric circles, n line segments, one endpoint in the center and one endpoint on the great circle. Note: The circular pool is divided into 2n(n≥5) "grids". We call it a "grid" with adjacent partitions (edges or arcs), so that each grid has three adjacent grids. A frog jumped into the pool at 4n+ 1, and it was difficult for the frog to be quiet. As long as there are no fewer than three frogs in a "grid", three frogs will jump into three different adjacent grids sooner or later. Prove: after a period of time, frogs will be roughly evenly distributed in the pool. The so-called roughly uniform distribution means taking any one of the "grids", or there are frogs in it, or there are frogs in three adjacent grids.
4. The known sequence {an} satisfies the condition A1= 21/6, 2an-3an- 1=3/2n+ 1 (where n >: 1).
Let m be a positive integer, m> 1, m≥n, and prove:1/m * [m-(2/3) n (m-1)/m]
5. There are five points in the rectangular ABCD with an area of 1 (including the boundary), and any three of them are not * * * lines. Among all triangles whose vertices are these five points, find the minimum number of triangles whose area does not exceed 1/4.
6. Find all non-negative integer solutions of equation 2 x * 3 y-5 z * 7 w = 1.