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Questions ( 1366 )
  • Find the last three digit-numbers of: \(E=2^{2017}\)

  • Find the last two digit-numbers of: \(D=1978^{1986^8}\)

  • Find the last two digit-numbers of: \(C=29^{9^{2012}}\)

  • Find the last two digit-numbers of: \(B=7^{9^{7^9}}\)

  • Find the last two digit-numbers of: \(A=2016^{2017}\)

     

  • Given \(A=n^{n-1}+n^{n-2}+...+n^2+n+1\). Prove that: \(A⋮n-1\) with n\(\in N\)

  • Prove that: \(\left(2^{2^{4n+1}}+7\right)⋮11\) with \(n\in\) N*

  • Prove that: \(\left(2^{2^{2n}}+5\right)⋮7\) with n \(\in\) N

  • Prove that: \(1924^{2003^{2004^n}}+120⋮124\) with n \(\in\) N*.

  • The mean score of the students who took a mathematics test was 6. Exactly 60% of the students passed the test. The mean score of the students who passed the test was 8. What was the mean score of the students who failed the test?

  • In a group of kangaroos, the two lightest kangaroos weigh 25% of the total weight of the group.
    The three heaviest kangaroos weigh 60% of the total weight. How many kangaroos are there in the group?

  • Circles of radius $ r_1, r_2$ and $r_3$ touch each other externally, and they touch a common tangent at points $A, B$ and $C$ respectively, where $B$ lies between $A$ and $C$. Prove that $16(r_1+r_2+r_3) \geq 9(AB+BC+CA).$

  • Let $\Sigma, m, r, \sigma$ denote the mean, median, range and standard deviation of a set, respectively. Let $f(x)=\frac{r(\Sigma-m)}{\sigma}$. Describe a set $x$ that maximises $f(x)$.

  • Given $p\in\mathbb{R}$ and $p^2=1+p$, prove that $p^3=1+2p$.
    Does anyone have an elegant proof for this? My proof was super bashy and horrible.
    Then, generalize this to $p^i$ such that $i\in\mathbb{N}$ and $i\geq 3$, express this as $a+bp$ such that $a,b\in\mathbb{N}$.

  • Triangle ABC has side lengths AB = 9, BC = 10, and AC = 13. If D is the midpoint of BC, what is the length of AD?

  • Suppose $f$ and $g$ are polynomials, and that $h(x)=f(g(x))+g(x)$. Find the degree of $g(x)$ given that the degree of $h(x)$ is $8$ and the degree of $f(x)$ is $4$.

  • Suppose a function $f(x)$ has domain $(-\infty,\infty)$ and range $[-11,3]$. If we define a new function $g(x)$ by$$g(x) = f(6x)+1,$$then what is the range of $g(x)$? Express your answer in interval notation.

  • Yu has $12$ coins, consisting of $5$ pennies, $4$ nickels and $3$ dimes. He tosses them all in the air. What is the probability that the total value of the coins that land heads-up is exactly $30$ cents?

  • How many ordered quintuples $(a, b, c, d, e)$ have coordinates of value $-1, 0$ or $1$ and satisfy $a+b^2+c^3+d^4+e^5=2$

    Could you somehow approach this using stars and bars or something?

  • What is the value of
    tan$(90 - 2a)$ given tan$(a) = \frac{1}{2}$

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