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question:7.3. In a school, there are 1000 students and 35 classes. On the forehead of each student, the number of students in their class is written. What can the sum of the reciprocals of these numbers be? List all the options and prove that there are no others. Recall that the reciprocal of a number a is the number 1 / a.
answer:Answer: 35.Solution: Let there be a people in the class, then the sum of the fractions corresponding to the numbers from this class is 1 (a fractions, each equal to 1/a). There are 35 classes in total. Therefore, the total sum is 35.Criteria:Answer - 0 points.Answer with examples - 0 points.The idea of partitioning into classes and calculating sums separately for each class without further progress - 5 points.
question:Jiaqi is ready to solve the problem: Solve the quadratic equation x^{2}-6x+square =0. If "square" represents a letter, and the quadratic equation x^{2}-6x+square =0 has real roots, then the maximum value of "square" is ______, and the solutions of the equation are ______.
answer:To solve the given problem, we start by substituting the value in "square" with m, turning the equation into x^{2}-6x+m=0. Given that the equation must have real roots, we apply the discriminant condition for real roots, which is Delta geqslant 0. The discriminant Delta of a quadratic equation ax^{2}+bx+c=0 is given by Delta = b^{2}-4ac. For our equation, a=1, b=-6, and c=m, so we have:[Delta = (-6)^{2} - 4(1)(m) geqslant 0]Simplifying the inequality:[36 - 4m geqslant 0]Solving for m:[36 geqslant 4m][9 geqslant m]This means that the maximum value of m (and thus "square") for the equation to have real roots is 9. Substituting m=9 back into the equation, we get:[x^{2} - 6x + 9 = 0]Factoring the quadratic equation:[(x-3)^{2} = 0]This implies that the equation has a repeated root:[x_{1} = x_{2} = 3]Therefore, the maximum value of "square" is boxed{9}, and the solutions of the equation are boxed{x_{1} = x_{2} = 3}.
question:The domestically produced mobile phone chip Kirin 980 is the world's first 7 nanometer process chip. It is known that 1 nanometer =0.000 000 001 meters. Express 7 nanometers in scientific notation as ____ meters.
answer:To express 7 nanometers in scientific notation, we start by understanding that 1 nanometer is equal to 0.000 000 001 meters, which can be written as 1times 10^{-9} meters in scientific notation. Therefore, 7 nanometers can be expressed as:[7 text{ nanometers} = 7 times 0.000 000 001 text{ meters}]Since 0.000 000 001 meters is 1times 10^{-9} meters, we can rewrite the expression as:[7 times 1times 10^{-9} text{ meters}]Simplifying this, we get:[7 times 10^{-9} text{ meters}]Thus, 7 nanometers expressed in scientific notation as meters is boxed{7times 10^{-9}} meters.
question:left[begin{array}{l}text { Symmetry helps solve the problem } text { [The inscribed angle is half the central angle]}end{array}right]A B is the diameter of the circle; C, D, E are points on the same semicircle A C D E B. On the diameter A B, points F and G are taken such that angle C F A = angle D F B and angle D G A = angle E G B. Find angle F D G, if the arc A C is 60^{circ} and the arc B E is 20^{circ}.
answer:Let C_{1} and E_{1} be the points of intersection of the rays D F and D G with the given circle. Then cup A C_{1}=cup A C and cup B E_{1}=cup B E.## SolutionLet C_{1} and E_{1} be the points of intersection of the rays D F and D G with the given circle. Since the circle is symmetric with respect to the diameter A B,the point C_{1} is symmetric to the point C with respect to A B. Therefore, cup A C_{1}=cup A C. Similarly, cup B E_{1}=cup B E. Consequently,angle C_{1} D E_{1}=frac{1}{2} cup C_{1} E_{1}=frac{1}{2}left(180^{circ}-60^{circ}-20^{circ}right)=50^{circ}## Answer50^{circ}.
question:36th Swedish 1996 Problem 1 A triangle area T is divided into six regions by lines drawn through a point inside the triangle parallel to the sides. The three triangular regions have areas T 1 , T 2 , T 3 . Show that √T =√T 1 + √T 2 + √T 3 .
answer:The triangles are all similar, so if √T = kBC, then √T 1 = kUP = kBR, √T 2 = kRS, √T 3 = kPV = kSC. But BC = BR + RS + SC, so √T = √T 1 + √T 2 + √T 3 . 36th Swedish 1996 © John Scholes [email protected] 10 March 2004 Last corrected/updated 10 Mar 04
question:3. The number of triples of positive integers (x, y, z) that satisfy x y z=3^{2010} and x leqslant y leqslant z<x+y is qquad.
answer:3. 336 .Let x=3^{a}, y=3^{b}, z=3^{c}. Then0 leqslant a leqslant b leqslant c, a+b+c=2010 text {. }If c geqslant b+1, thenx+y=3^{a}+3^{b}<3^{b+1} leqslant 3^{c}=z text {, }which contradicts z<x+y. Hence, c=b.Thus, a+2 b=2010.Therefore, 670 leqslant b leqslant 1005.Hence, the number of such positive integer triples, which is the number of b, is 1005-670+1=336.