Langkah Awal Menuju OSN : International Problems

The following Problems from International Mathematics Competition: Please Download All. ~~Good Luck.....!~~
1.
An operation $'\Delta '$ is defined by $a\Delta b= 1-\frac{a}{b}, b\neq 0.$ What is the value of $\left ( 1\Delta 2 \right )\Delta \left ( 3\Delta 4 \right )?$

Answer
-1

2.
The sequence 9, 18, 27, 36, 45, 54, … consists of successive multiples of 9. This sequence is then altered by multiplying every other term by –1, starting with the first term, to produce the new sequence -9, 18, -27, 36, -45, 54, ... . If the sum of the first n terms of this new sequence is 180,determine n.

Answer
40.

3.
The symbol n! is used to represent the product n(n-1)(n-2)...(3)(2)(1). For example, 4!=4(3)(2)(1). Determine n such that $n!=(2^{15})(3^{6})(5^{3})(7^{2})(11)(13)$ .

Answer
16

4.
The symbol $\left \lfloor x \right \rfloor$ . For example $\left \lfloor 5.7 \right \rfloor=5,\left \lfloor \pi \right \rfloor=3,dan \left \lfloor 4 \right \rfloor=4.$
Calculate the value of the sum
$\left \lfloor \sqrt{1} \right \rfloor+\left \lfloor \sqrt{2} \right \rfloor+\left \lfloor \sqrt{3} \right \rfloor+\left \lfloor \sqrt{4} \right \rfloor+....+\left \lfloor \sqrt{50} \right \rfloor.$ .

Answer
217

5.
How many five-digit positive integers have the property that the product of their digits is 2000?

Answer
30

6.
Solve the equation: $4(16^{\sin ^{2}x})=2^{6\sin x}$ , for $0\leq x\leq 2\pi$ .

Answer
$\frac{\pi }{6}, \frac{5\pi }{6},\frac{\pi }{2}$

7.
The sequence of numbers $.... a_{-3}, a_{-2}, a_{-1}, a_{0}, a_{1}, a_{2},....$ is defined by $a_{n}-(n+1)a_{2-n}=(n+3)^{2}$ , for all integers n. Calculate $a_{0}$ .

Answer
-17

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