There are three kinds of fruits in the market. How many ways are there to purchase 25 fruits from among them if each kind has at least 25 of its fruit available?

Excursion in Mathematics

PRMO 2016

351

The given problem can be expressed in terms of the following equation

$x_1 + x_2 + x_3 = 25$

where $ x_!, x_2, x_3$ are the number of different fruits brought

The solution of the problem is equivalent to finding the non-negative integer solution to this given equation

Try to relate it to the following idea:

There are 25 balls and 2 sticks arranged in a straight line. We want to find the number of different arrangements possible. To the the different possible distinct arrangement we may apply permutation with repetition

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The five digit number $2 a 9 b 1$ is a perfect square. Find the value of $a^{b-1}+b^{a-1}$.

Challenge and Thrill of Pre College Mathematics

PRMO 2016

50

An odd perfect square is of the form $8k+1$

Hence $8| 2a9b0$

Hence $8 | 2a000 + 9b0 $

$8 | 900 +b0 $

$ 8 | b4$

Therefore the possible values of b are $6,2$

So possible numbers are $2a921,2a961$

Now check for the possible values of $a$ and calculate the square root to check if it is a perfect square

The only valid solution is $a=5$

Hence calculate the required expression.

Math Olympiad Program

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The date index of a date is defined as (12 × month number + day number). Three events each with a frequency of once in 21 days, 32 days and 9 days, respectively, occurred simultaneously for the first time on July 31, 1961 (Ireland joining the European Economic Community). Find the date index of the date when they occur simultaneously for the eleventh time.

Excursion in mathematics

PRMO 2016 Problem 3

115

What is the LCM of 21, 32,9.

It is 2016

Observe that the event will occur in multiples of 2016 days.

Now 2016 is divisible by 12

Hence in a year There are 12 months. Hence they will occur on the same date and time but in a different year

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Each of the points $A, B, C, D, E$, and $F$ in the figure below represents a different digit from 1 to 6 . Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is 47 . What is the digit represented by $B$ ?

• 1
• 2
• 3
• 4
• 5

AMC 8 2020 Problem 16

5

Try to find the sumes alongs lines

\begin{aligned}
&A+B+C \
&A+E+F \
&C+D+E \
&B+D \
&B+F
\end{aligned}

Adding The sums all together, we will have $2 A+3 B+2 C+2 D+2 E+2 F=47$, i.e. $2(A+B+C+D+E+F)+B=47$

So we will have $42+B=47$, now we need solve this equation for B.

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Rectangle $A B C D$ is inscribed in a semicircle with diameter $\overline{F E}$, as shown in the figure. Let $D A=16$, and let $F D=A E=9$. What is the area of $A B C D$ ?

• 240
• 248
• 256
• 264
• 272

AMC 8 2020 Problem 13

240

Try to find the diameter of the semicircle. So the diameter will be,

The diameter of the semicircle is $9+16+9=34$, so $O C=17$. By symmetry, $O$ is the midpoint of AD,So, $AO=OD=\frac{16}{2}=8$.

Now, apply Pythagorean Theorem to find CD,

SO the area of ABCD will be=$AD \times CD$

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Consider the areas of the four triangles obtained by drawing the diagonals $A C$ and $B D$ of a trapezium $A B C D$. The product of these areas, taken two at time, are computed. If among the six products so obtained, two products are 1296 and 576 , determine the square root of the maximum possible area of the trapezium to the nearest integer.

Challenge and Trill of Pre College Mathematics

PRMO 2017 Problem 30

The maximum possible area is 13

Let us denote by $\triangle ABC$ the area of triangle ABC.

Lets the point of intersection of the diagonals of the trapezium be O

Hence $\frac{\triangle ABO}{\triangle CDO}$ = $\frac{{AB}^{2}}{{CD}^{2}}$

$\triangle ADO$=$\triangle BCO$

$\frac{\triangle ABO}{\triangle ADO} $= $\frac{BO}{OD}$=$\frac{AB}{CD}$

If$y$=$\frac{AB}{CD}$ and $\triangle ABO$ = $x$ then

$\triangle AOB$, $\triangle BOC$, $\triangle COD$, $\triangle AOD$ = $x$, $xy$, $x{y}^{2}$, $xy$

Then consider different cases and find out the different values of $x$, $y$ and the corresponding values of the area of the triangle and find the maximum of them.

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Consider all possible integers $n \geq 0$ such that
$$
\left(5 \times 3^{m}\right)+4=n^{2}
$$
holds for some corresponding integer $m \geq 0$. Find the sum of all such $n$.

Elementary Number Theory by David Burton

Excursion in mathematics

PRMO 2016

The required answer is 10

$
\left(5 \times 3^{m}\right)+4=n^{2}
$

$\rightarrow \left(5 \times 3^{m}\right) = n^{2}-4$

$\rightarrow \left(5 \times 3^{m}\right) = (n-2)(n+2)$

Observe that $gcd (n-2,n+2)$ = $1$ or $2$ or $4 $

But $\left(5 \times 3^{m}\right) = (n-2)(n+2)$

Hence both $(n-2)$ , $(n+2)$ are odd.

Hence one possible case:

$(n-2)$ = $5$

$(n+2)$ = $3^{m}$

$\rightarrow$ $4$ = $3^{m}$ - $5$

$\rightarrow$ $m$ = $2$

$\rightarrow$ $n$ = $7$

Similarly find other values of n and add

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Jamal has a drawer containing 6 green socks, 18 purple socks, and 12 orange socks. After adding more purple socks, Jamal noticed that there is now a $60 \%$ chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?

• 6
• 9
• 12
• 18
• 24

AMC 8 2020 Problem 13

9

At first Let Jamal adds $x$ purple socks,

So, He have $18+x$ purple Socks now, in total He has $36+x$

So probability of drawing a purple sock is $\frac{18+x}{36+x}$

Hence,

$\frac{18+x}{36+x}=\frac{3}{5}$

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Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?

• 35
• 37
• 39
• 43
• 49

AMC 8 2020 Problem 4

37

Try to see the three figures carefully

First hexagon has one dot and second hexagon has 7 dots. And the 3rd hexagon has dots.

Now, If we notice carefully , we have We have $h_{1}=1$ and $h_{n+1}=h_{n}+6 n$.where, $h_{n}$ denote the number of dots in the $n$th hexagon.

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