MathDB

1

Part of 2007 Pre-Preparation Course Examination

Problems(4)

Inequality in Triangle

Source: Iran PPCE 2007

3/23/2007
DD is an arbitrary point inside triangle ABCABC, and EE is inside triangle BDCBDC. Prove that SDBC(PDBC)2≥SEBC(PEBC)2\frac{S_{DBC}}{(P_{DBC})^{2}}\geq\frac{S_{EBC}}{(P_{EBC})^{2}}
inequalitiesgeometryinradiusgeometry proposed
Irrational

Source: Iran PPCE 2007

3/23/2007
Let a≥2a\geq 2 be a natural number. Prove that ∑n=0∞1an2\sum_{n=0}^\infty\frac1{a^{n^{2}}} is irrational.
inequalitieslimitfloor functionabsolute valuenumber theory proposednumber theory
Multiplicative function

Source: Iran PPCE 2007

3/25/2007
a) Find all multiplicative functions f:Zp∗⟶Zp∗f: \mathbb Z_{p}^{*}\longrightarrow\mathbb Z_{p}^{*} (i.e. that ∀x,y∈Zp∗\forall x,y\in\mathbb Z_{p}^{*}, f(xy)=f(x)f(y)f(xy)=f(x)f(y).) b) How many bijective multiplicative does exist on Zp∗\mathbb Z_{p}^{*} c) Let AA be set of all multiplicative functions on Zp∗\mathbb Z_{p}^{*}, and VBVB be set of all bijective multiplicative functions on Zp∗\mathbb Z_{p}^{*}. For each x∈Zp∗x\in \mathbb Z_{p}^{*}, calculate the following sums :∑f∈Af(x),  ∑f∈Bf(x)\sum_{f\in A}f(x),\ \ \sum_{f\in B}f(x)
functionnumber theory proposednumber theory
Infinite sequence

Source: Iran PPCE 2007

4/1/2007
a) There is an infinite sequence of 0,10,1, like …,a−1,a0,a1,…\dots,a_{-1},a_{0},a_{1},\dots (i.e. an element of {0,1}Z\{0,1\}^{\mathbb Z}). At each step we make a new sequence. There is a function ff such that for each ii, \mbox{new }a_{i}=f(a_{i-100},a_{i-99},\dots,a_{i+100}). This operation is mapping F:{0,1}Z⟶{0,1}ZF: \{0,1\}^{\mathbb Z}\longrightarrow\{0,1\}^{\mathbb Z}. Prove that if FF is 1-1, then it is surjective. b) Is the statement correct if we have an fif_{i} for each ii?
functionprobabilitytopologycombinatorics proposedcombinatorics