Find R in the figure below : 

A man goes to work in the morning and goes home in the afternoon every day. He uses the bus once, uses the taxi once each day. After x days, he used the bus to go to work 8 times, used the bus to go home 15 times, used the taxi 9 times. Find x

Given \(a,b\in Z^+\) such that their sum, difference, product and quotient are 4 distinct positive integers. Find the minimum value of a + b

Let a, b, \(\sqrt{a^2+b^2}\) be the length of the sides of a triangle. Find the ratio of the area of the subscribe circle to the area of the incircle of the triangle.

Given \(\Delta ABC\left(AB< AC\right)\). Draw D on ray AC such that AD = AB. Draw M on ray BA such that BM = CD. MC cuts BD at I. Prove that IM = IC

Write the integers from 1 to 30 and cross out some numbers such that there's no number which is double another number in the remaining numbers. How many numbers do we cross out at least ?

Given \(\Delta ABC\) isosceles at A. Point M lies between B, C such that MB < MC. Point O lies between A, O. Prove that OB < OC

Given \(\Delta ABC\) and \(\Delta MNP\) such that AB = MN ; AC = MP

Prove that : \(\widehat{A}< \widehat{M}\Leftrightarrow BC< NP\)

Ace have a drawer that contains 4 colors of poms. The numbers of gold, green, blue and red poms arre 90, 70, 50, 40 respectively. She randomly pulls out poms, one at a time, without looking at the colors. Prove that she must remove at least 23 poms to be certain that she has at least 10 pairs of matching poms.

Prove that there are 30 ways to arrange 2 As, 2 Bs, 2 Cs in a row so that no 2 adjacent letters are the same

Let O be a point in the interior of the parallelogram ABCD. Given S_ABCD = 10 cm² ; S_AOD = 2 cm². Calculate S_BOC 

Prove that \(\dfrac{1}{\dfrac{1}{a}+\dfrac{1}{b}}+\dfrac{1}{\dfrac{1}{b}+\dfrac{1}{c}}+\dfrac{1}{\dfrac{1}{a}+\dfrac{1}{c}}\le\dfrac{a+b+c}{2}\left(a,b,c>0\right)\)

Given \(\Delta ABC\) right at A with \(BC=a;AB-AC=b\left(a>0;b\ge0\right)\)

Calculate AB,AC in terms of a,b

Initially, there is a 0 in each square of a 3 x 3 board. In each move, we add 1 to each number in any of the four 2 x 2 sub-boards. After a number of moves, someone erases the numbers at the four corner squares and the central square of the 3 x 3 board. The remaining four numbers are 9, 10, 12 and 13. Find the value of the number at the central square 

P/S : Although you can find the source of these questions, you shouldn't copy the solution. Instead of doing that, you should solve the problem yourselves.

A music store has six display racks each 100 centimeters long. It has 150 CD sets, some of which are singles each 3 centimeters thick, and the others are albums each 6 centimeters thick. All 150 CDs can be placed on the racks with nothing sticking out. Prove that the maximum number of albums among the 150 CD sets is 48

A wire 12 centimeters long is to be cut into a number of pieces, which are bent and welded to form the 12 edges of a cube 1 centimeter on a side. Prove that the minimum number of pieces required is 4

There are 15 pebbles in a single pile. In each move, we divide a pile with at least two pebbles into two piles, and write down the product of the numbers of pebbles in the two newly created piles. After 14 moves, the pebbles are in 15 separate piles. Find the sum of the 14 numbers that has been written down.

On each day of the week except Sunday, 8 students are on patrol duty. In each day, there are exactly 3 students who are on duty only on that day. Prove that 33 is the maximum number of students who are on duty during the week.

RULES :

The members MUSTN'T :

- Ask the questions which are Vietnamese or not related to math or nonsense such as 1 + 1 ; ....

- Cheat to get more points such as : telling other members to give you points ; copying the answer of another member ; making more accounts to tick the answers of your main account

Mathulike can subtract the points or lock the account of the doer forever if he/she violated the rules many times. 

Given \(\Delta ABC\) with point M inside so that S_AMB + S_BMC = S_MAC

Prove that M moves on the midsegment of \(\Delta ABC\) which is parallel to AC