MathDB

day 1

Part of 2013 Indonesia Juniors

Problems(1)

Indonesia Juniors 2013 day 1 OSN SMP

Source:

11/4/2021
p1. It is known that ff is a function such that f(x)+2f(1x)=3xf(x)+2f\left(\frac{1}{x}\right)=3x for every x≠0x\ne 0. Find the value of xx that satisfies f(x)=f(−x)f(x) = f(-x).
p2. It is known that ABC is an acute triangle whose vertices lie at circle centered at point OO. Point PP lies on side BCBC so that APAP is the altitude of triangle ABC. If ∠ABC+30o≤∠ACB\angle ABC + 30^o \le \angle ACB, prove that ∠COP+∠CAB<90o\angle COP + \angle CAB < 90^o.
p3. Find all natural numbers a,ba, b, and cc that are greater than 11 and different, and fulfills the property that abcabc divides evenly bc+ac+ab+2bc + ac + ab + 2.
p4. Let A,BA, B, and P P be the nails planted on the board ABPABP . The length of AP=aAP = a units and BP=bBP = b units. The board ABPABP is placed on the paths x1x2x_1x_2 and y1y2y_1y_2 so that AA only moves freely along path x1x2x_1x_2 and only moves freely along the path y1y2y_1y_2 as in following image. Let xx be the distance from point PP to the path y1y2y_1y_2 and y is with respect to the path x1x2x_1x_2 . Show that the equation for the path of the point PP is x2b2+y2a2=1\frac{x^2}{b^2}+\frac{y^2}{a^2}=1. https://cdn.artofproblemsolving.com/attachments/4/6/d88c337370e8c3bc5a1833bc9588d3fb047bd0.png
p5. There are three boxes A,BA, B, and CC each containing 33 colored white balls and 22 red balls. Next, take three ball with the following rules: 1. Step 1 Take one ball from box AA. 2. Step 2 ∙\bullet If the ball drawn from box AA in step 1 is white, then the ball is put into box BB. Next from box BB one ball is drawn, if it is a white ball, then the ball is put into box CC, whereas if the one drawn is red ball, then the ball is put in box AA. ∙\bullet If the ball drawn from box AA in step 1 is red, then the ball is put into box CC. Next from box CC one ball is taken. If what is drawn is a white ball then the ball is put into box AA, whereas if the ball drawn is red, the ball is placed in box BB. 3. Step 3 Take one ball each from squares A,BA, B, and CC. What is the probability that all the balls drawn in step 3 are colored red?
algebrageometrycombinatoricsnumber theoryindonesia juniors