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Nguyen Trung Tuan – Lecture Notes in Mathematics, High School Olympiads

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Lecture Notes in Mathematics, High School Olympiads

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Alternative text: a,b
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Alternative text: c
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Alternative text: [a,b]\not= [a+c,b+c].
Image source: /latex.php
Alternative text: N
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Alternative text: a_{1}, a_{2},\cdots, a_{N}
Image source: /latex.php
Alternative text: 2^{N+1}
Image source: /latex.php
Alternative text: n\geq N+1
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Alternative text: a_{n}
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Alternative text: a_{k}
Image source: /latex.php
Alternative text: 2^{n}
Image source: /latex.php
Alternative text: a_{1},\cdots, a_{n-1}
Image source: /latex.php
Alternative text: 2^{n}
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Alternative text: a_{n}=2a_{k}
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Alternative text: k
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Alternative text: M
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Alternative text: a_{n}=a_{M}
Image source: /latex.php
Alternative text: n\geq M
Image source: /latex.php
Alternative text: ABC
Image source: /latex.php
Alternative text: O
Image source: /latex.php
Alternative text: D,E
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Alternative text: F
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Alternative text: A,B
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Alternative text: C
Image source: /latex.php
Alternative text: M
Image source: /latex.php
Alternative text: BC
Image source: /latex.php
Alternative text: AD
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Alternative text: EF
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Alternative text: X
Image source: /latex.php
Alternative text: AO
Image source: /latex.php
Alternative text: BC
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Alternative text: Y
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Alternative text: Z
Image source: /latex.php
Alternative text: XY
Image source: /latex.php
Alternative text: A,Z
Image source: /latex.php
Alternative text: M
Image source: /latex.php
Alternative text: n
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Alternative text: n \geq 3
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Alternative text: n
Image source: /latex.php
Alternative text: n
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Alternative text: n
Image source: /avatar/74a60cd44f00304a7cb25541dc261336
Alternative text: Not defined!
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Alternative text: Not defined!
Image source: /latex.php
Alternative text: n-
Image source: /latex.php
Alternative text: P
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Alternative text: 3
Image source: /latex.php
Alternative text: P
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Alternative text: P
Image source: /latex.php
Alternative text: (1,1)
Image source: /latex.php
Alternative text: Oxy
Image source: /latex.php
Alternative text: A
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Alternative text: B
Image source: /latex.php
Alternative text: AOB
Image source: /latex.php
Alternative text: 1/2.
Image source: /latex.php
Alternative text: (m,n)
Image source: /latex.php
Alternative text: (1,1).
Image source: /latex.php
Alternative text: (m,n)
Image source: /latex.php
Alternative text: (m,n)
Image source: /latex.php
Alternative text: (1,1)
Image source: /latex.php
Alternative text: |m-n|
Image source: /latex.php
Alternative text: n \ge 5
Image source: /latex.php
Alternative text: a_i,b_i (i = 1,2, \cdots ,n)
Image source: /latex.php
Alternative text: (a_i,b_i)
Image source: /latex.php
Alternative text: i = 1,2,\cdots,n
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Alternative text: |a_1b_2-a_2b_1| = |a_2b_3-a_3b_2| = \cdots = |a_nb_1-a_1b_n| = 1
Image source: /latex.php
Alternative text: i,j
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Alternative text: 1
Image source: /latex.php
Alternative text: |a_ib_j-a_jb_i|=1
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Alternative text: P
Image source: /latex.php
Alternative text: P
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Alternative text: P
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Alternative text: Not defined!
Image source: /latex.php
Alternative text: BCF
Image source: /latex.php
Alternative text: B
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Alternative text: A
Image source: /latex.php
Alternative text: CF
Image source: /latex.php
Alternative text: FA=FB
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Alternative text: F
Image source: /latex.php
Alternative text: A
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Alternative text: C
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Alternative text: D
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Alternative text: DA=DC
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Alternative text: AC
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Alternative text: \angle{DAB}
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Alternative text: E
Image source: /latex.php
Alternative text: EA=ED
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Alternative text: AD
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Alternative text: \angle{EAC}
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Alternative text: M
Image source: /latex.php
Alternative text: CF
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Alternative text: X
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Alternative text: AMXE
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Alternative text: BD,FX
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Alternative text: ME
Image source: /latex.php
Alternative text: ABC
Image source: /latex.php
Alternative text: \Gamma
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Alternative text: I
Image source: /latex.php
Alternative text: M
Image source: /latex.php
Alternative text: BC
Image source: /latex.php
Alternative text: D
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Alternative text: E
Image source: /latex.php
Alternative text: F
Image source: /latex.php
Alternative text: BC
Image source: /latex.php
Alternative text: CA
Image source: /latex.php
Alternative text: AB
Image source: /latex.php
Alternative text: ID \perp BC
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Alternative text: IE\perp AI
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Alternative text: IF\perp AI
Image source: /latex.php
Alternative text: (AEF)
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Alternative text: \Gamma
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Alternative text: X
Image source: /latex.php
Alternative text: A
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Alternative text: XD
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Alternative text: AM
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Alternative text: \Gamma
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Alternative text: B = (-1, 0)
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Alternative text: C = (1, 0)
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Alternative text: S
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Alternative text: \text{(i)}
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Alternative text: T
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Alternative text: S
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Alternative text: Q
Image source: /latex.php
Alternative text: S
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Alternative text: TQ
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Alternative text: S
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Alternative text: \text{(ii)}
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Alternative text: P_1P_2P_3
Image source: /latex.php
Alternative text: A
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Alternative text: S
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Alternative text: \sigma
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Alternative text: \{1, 2, 3\}
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Alternative text: ABC
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Alternative text: P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}
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Alternative text: S
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Alternative text: S'
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Alternative text: \{(x, y) : x \geq 0, y \geq 0\}
Image source: /latex.php
Alternative text: A \in S
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Alternative text: A' \in S'
Image source: /latex.php
Alternative text: \text{(ii)}
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Alternative text: BA \cdot BA'
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Alternative text: P_1P_2P_3
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Alternative text: Not defined!
Image source: /latex.php
Alternative text: \displaystyle n
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Alternative text: \displaystyle k
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Alternative text: \displaystyle n>k
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Alternative text: \displaystyle n
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Alternative text: \displaystyle n
Image source: /latex.php
Alternative text: \displaystyle k
Image source: /latex.php
Alternative text: \displaystyle n
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Alternative text: \displaystyle n
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Alternative text: \displaystyle n
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Alternative text: \displaystyle 6
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Alternative text: \displaystyle n-
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Alternative text: \displaystyle n
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Alternative text: \displaystyle n \times n
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Alternative text: \displaystyle I,M
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Alternative text: \displaystyle O
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Alternative text: I
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Alternative text: M
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Alternative text: O
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Alternative text: 3
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Alternative text: I
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Alternative text: M
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Alternative text: O
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Alternative text: \displaystyle n \times n
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Alternative text: \displaystyle 1
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Alternative text: \displaystyle n
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Alternative text: \displaystyle (i,j)
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Alternative text: \displaystyle 1 \le i,j \le n
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Alternative text: \displaystyle n>1
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Alternative text: \displaystyle 4n-2
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Alternative text: \displaystyle (i,j)
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Alternative text: \displaystyle i+j
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Alternative text: \displaystyle (i,j)
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Alternative text: \displaystyle i-j
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Alternative text: Not defined!
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Alternative text: \displaystyle k
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Alternative text: \displaystyle S(k)
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Alternative text: \displaystyle k
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Alternative text: \displaystyle P(x)\in\mathbb{Z}[x]
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Alternative text: \displaystyle n\geq 2016
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Alternative text: \displaystyle P(n)
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Alternative text: \displaystyle S(P(n))=P(S(n)).
Image source: /latex.php
Alternative text: \displaystyle n
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Alternative text: \displaystyle \tau (n)
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Alternative text: \displaystyle n
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Alternative text: \displaystyle \tau_1(n)
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Alternative text: \displaystyle n
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Alternative text: \displaystyle 3
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Alternative text: \displaystyle 1
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Alternative text: \displaystyle \dfrac{\tau (10n)}{\tau_1(10n)}.
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Alternative text: \displaystyle P(n)=n^{2}+n+1
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Alternative text: \displaystyle b
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Alternative text: \displaystyle a
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Alternative text: \displaystyle \left \{ P(a+1);P(a+2);...;P(a+b) \right \}
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Alternative text: \displaystyle n,m,k
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Alternative text: \displaystyle l
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Alternative text: \displaystyle n\not=1
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Alternative text: \displaystyle n^k+m.n^l+1
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Alternative text: \displaystyle n^{k+l}-1
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Alternative text: \displaystyle m=1
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Alternative text: \displaystyle l=2k
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Alternative text: \displaystyle l|k
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Alternative text: \displaystyle m=\dfrac{n^{k-l}-1}{n^l-1}.
Image source: /latex.php
Alternative text: \displaystyle a
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Alternative text: \displaystyle A
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Alternative text: \displaystyle k
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Alternative text: \displaystyle k=\dfrac{x^2-a}{x^2-y^2},\quad (1)
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Alternative text: \displaystyle x
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Alternative text: \displaystyle y
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Alternative text: \displaystyle x>\sqrt{a}
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Alternative text: \displaystyle B
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Alternative text: \displaystyle k
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Alternative text: \displaystyle (1)
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Alternative text: \displaystyle x
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Alternative text: \displaystyle y
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Alternative text: \displaystyle 0\leq x
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Alternative text: \displaystyle A=B
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Alternative text: \displaystyle f:\mathbb{N}^*\to\mathbb{N}^*
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Alternative text: \displaystyle m
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Alternative text: \displaystyle n
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Alternative text: \displaystyle f(m)+f(n)-mn
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Alternative text: \displaystyle 0
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Alternative text: \displaystyle mf(m)+nf(n)
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Alternative text: Not defined!
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Alternative text: \displaystyle a,b,c
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Alternative text: \displaystyle \min\{ab,bc,ca\}\geq 1.
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Alternative text: \displaystyle \sqrt[3]{(a^2+1)(b^2+1)(c^2+1)}\leq \left(\frac{a+b+c}{3}\right)^2+1.
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Alternative text: \displaystyle C
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Alternative text: \displaystyle 5
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Alternative text: \displaystyle a_1
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Alternative text: \displaystyle a_2
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Alternative text: \displaystyle a_3
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Alternative text: \displaystyle a_4
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Alternative text: \displaystyle a_5
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Alternative text: \displaystyle i
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Alternative text: \displaystyle j
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Alternative text: \displaystyle k
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Alternative text: \displaystyle l
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Alternative text: \displaystyle \left|\frac{a_i}{a_j}-\frac{a_k}{a_l}\right|\leq C.
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Alternative text: \displaystyle n>2
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Alternative text: \displaystyle 2n
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Alternative text: \displaystyle a_1
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Alternative text: \displaystyle a_2,\cdots
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Alternative text: \displaystyle a_n
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Alternative text: \displaystyle b_1,
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Alternative text: \displaystyle b_2,\cdots,
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Alternative text: \displaystyle b_n
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Alternative text: \displaystyle |a_k|+|b_k|=1\,\forall k=1,2,\cdots,n
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Alternative text: \displaystyle n
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Alternative text: \displaystyle x_1,x_2,\cdots,x_n\in\{-1;1\}
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Alternative text: \displaystyle \left|\sum_{k=1}^nx_ka_k\right|+\left|\sum_{k=1}^nx_kb_k\right|\leq 1.
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Alternative text: \displaystyle f:(0;+\infty)\to (0;+\infty)
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Alternative text: \displaystyle xf(x^2)f(f(y))+f(yf(x))=f(xy)(f(f(x^2))+f(f(y^2))),\quad \forall x,y\in (0;+\infty).
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Alternative text: \displaystyle n
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Alternative text: \displaystyle a/b
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Alternative text: \displaystyle 0
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Alternative text: \displaystyle \sqrt{n}\leq\dfrac{a}{b}\leq\sqrt{n+1}.
Image source: /latex.php
Alternative text: \displaystyle n
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Alternative text: \displaystyle a/b
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Alternative text: \displaystyle 0
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Alternative text: \displaystyle \sqrt{n}\leq\dfrac{a}{b}\leq\sqrt{n+1}.
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Alternative text: Not defined!
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Alternative text: R,S
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Alternative text: \Omega
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Alternative text: RS
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Alternative text: l
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Alternative text: \Omega
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Alternative text: R
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Alternative text: T
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Alternative text: S
Image source: /latex.php
Alternative text: RT
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Alternative text: J
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Alternative text: RS
Image source: /latex.php
Alternative text: \Omega
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Alternative text: (JST)
Image source: /latex.php
Alternative text: l
Image source: /latex.php
Alternative text: A
Image source: /latex.php
Alternative text: R
Image source: /latex.php
Alternative text: l
Image source: /latex.php
Alternative text: (JST)
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Alternative text: AJ
Image source: /latex.php
Alternative text: \Omega
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Alternative text: K
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Alternative text: KT
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Alternative text: (JST)
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Alternative text: N>1
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Alternative text: N(N+1)
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Alternative text: N(N-1)
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Alternative text: N
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Alternative text: \cdots\cdots\cdots
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Alternative text: Not defined!
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Alternative text: a_0 > 1
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Alternative text: (a_n)_{n\geq 0}
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Alternative text: a_{n+1} =\begin{cases}\sqrt{a_n},\quad \sqrt{a_n}\in\mathbb{Z}\\ a_n + 3,\quad \sqrt{a_n}\not\in\mathbb{Z}\end{cases}\forall n\geq 0.
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Alternative text: a_0
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Alternative text: A
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Alternative text: a_n = A
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Alternative text: n
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Alternative text: f: \mathbb{R} \rightarrow \mathbb{R}
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Alternative text: f(f(x)f(y)) + f(x+y) = f(xy),\quad \forall x,y\in \mathbb{R}.
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Alternative text: Not defined!
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Alternative text: \displaystyle n
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Alternative text: \displaystyle \dfrac{p}{q}\in [0;1]
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Alternative text: \displaystyle n
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Alternative text: \displaystyle q\leq n
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Alternative text: \displaystyle n
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Alternative text: \displaystyle n
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Alternative text: \displaystyle F_n
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Alternative text: \displaystyle F_1:\,\frac{0}{1};\frac{1}{1}
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Alternative text: \displaystyle F_2:\,\frac{0}{1};\frac{1}{2};\frac{1}{1}
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Alternative text: \displaystyle F_3:\,\frac{0}{1};\frac{1}{3};\frac{1}{2};\frac{2}{3};\frac{1}{1}.
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Alternative text: \displaystyle F_4:\,\frac{0}{1};\frac{1}{4};\frac{1}{3};\frac{1}{2};\frac{2}{3};\frac{3}{4};\frac{1}{1}.
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Alternative text: \displaystyle n
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Alternative text: \displaystyle F_n
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Alternative text: \displaystyle 1+\sum_{k=1}^n\varphi (k)
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Alternative text: \displaystyle a,b,c
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Alternative text: \displaystyle d
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Alternative text: \displaystyle 0\leq \frac{a}{b}
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Alternative text: \displaystyle bc-ad=1
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Alternative text: \displaystyle \frac{a}{b},\frac{c}{d}
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Alternative text: \displaystyle F_n
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Alternative text: \displaystyle n
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Alternative text: \displaystyle \max\{b,d\}\leq n\leq b+d-1.
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Alternative text: \displaystyle bc-ad=1
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Alternative text: \displaystyle \frac{a}{b},\frac{c}{d}
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Alternative text: \displaystyle \max\{b,d\}\leq n
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Alternative text: \displaystyle F_n
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Alternative text: \displaystyle F_n
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Alternative text: \displaystyle n
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Alternative text: \displaystyle \dfrac{h}{k}
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Alternative text: \displaystyle \displaystyle \frac{a}{b}
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Alternative text: \displaystyle ck-dh\geq 1
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Alternative text: \displaystyle bh-ak\geq 1
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Alternative text: \displaystyle b+d-1\geq n\geq k=(bc-ad)k=b(ck-dh)+d(bh-ak)\geq b+d,
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Alternative text: \Box
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Alternative text: \displaystyle a,b,c
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Alternative text: \displaystyle d
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Alternative text: \displaystyle 0\leq \frac{a}{b}
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Alternative text: \dfrac{a+c}{b+d}
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Alternative text: \displaystyle \dfrac{a}{b}
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Alternative text: \displaystyle \dfrac{c}{d}
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Alternative text: \displaystyle a,b,c
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Alternative text: \displaystyle d
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Alternative text: \displaystyle 0\leq \frac{a}{b}
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Alternative text: \displaystyle bc-ad=1
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Alternative text: \displaystyle \dfrac{h}{k}
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Alternative text: \displaystyle \dfrac{a}{b}
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Alternative text: \dfrac{c}{d}
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Alternative text: \displaystyle \frac{a}{b}
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Alternative text: \displaystyle bh-ak=1,\quad ck-dh=1.
Image source: /latex.php
Alternative text: \displaystyle n
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Alternative text: \displaystyle F_{n+1}
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Alternative text: \displaystyle F_n
Image source: /latex.php
Alternative text: \displaystyle F_n
Image source: /latex.php
Alternative text: \displaystyle n+1
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Alternative text: \displaystyle \dfrac{a}{b}
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Alternative text: \displaystyle F_n
Image source: /latex.php
Alternative text: \displaystyle bc-ad=1
Image source: /latex.php
Alternative text: \displaystyle n
Image source: /latex.php
Alternative text: \displaystyle n\, (n\geq 2)
Image source: /latex.php
Alternative text: \displaystyle n
Image source: /latex.php
Alternative text: \displaystyle \dfrac{a}{b}
Image source: /latex.php
Alternative text: \displaystyle F_n
Image source: /latex.php
Alternative text: \displaystyle bc-ad=1
Image source: /latex.php
Alternative text: \displaystyle F_n
Image source: /latex.php
Alternative text: \displaystyle n+1
Image source: /latex.php
Alternative text: \displaystyle F'_n
Image source: /latex.php
Alternative text: \displaystyle F_{n+1}
Image source: /latex.php
Alternative text: \displaystyle F_{n+1}
Image source: /latex.php
Alternative text: \displaystyle \dfrac{h}{k}
Image source: /latex.php
Alternative text: \displaystyle F'_n
Image source: /latex.php
Alternative text: \displaystyle \dfrac{a}{b}
Image source: /latex.php
Alternative text: \displaystyle F'_n
Image source: /latex.php
Alternative text: \displaystyle \dfrac{a}{b}
Image source: /latex.php
Alternative text: \displaystyle \dfrac{h}{k}
Image source: /latex.php
Alternative text: \displaystyle F'_n
Image source: /latex.php
Alternative text: \displaystyle F_n
Image source: /latex.php
Alternative text: \displaystyle k>n
Image source: /latex.php
Alternative text: \displaystyle k\leq n+1
Image source: /latex.php
Alternative text: \displaystyle k=n+1
Image source: /latex.php
Alternative text: \displaystyle k=n+1\geq b+d\Rightarrow
Image source: /latex.php
Alternative text: \displaystyle \dfrac{a}{b}
Image source: /latex.php
Alternative text: \displaystyle F_n
Image source: /latex.php
Alternative text: \displaystyle b+d\leq n+1
Image source: /latex.php
Alternative text: \displaystyle F'_n
Image source: /latex.php
Alternative text: \displaystyle \Box
Image source: /latex.php
Alternative text: \displaystyle Oxy
Image source: /latex.php
Alternative text: \displaystyle M(1;0)
Image source: /latex.php
Alternative text: \displaystyle N(1;1)
Image source: /latex.php
Alternative text: \displaystyle \dfrac{h}{k}
Image source: /latex.php
Alternative text: \displaystyle F_n
Image source: /latex.php
Alternative text: \displaystyle (k;h)
Image source: /latex.php
Alternative text: \displaystyle OM
Image source: /latex.php
Alternative text: \displaystyle ON
Image source: /latex.php
Alternative text: \displaystyle F_n
Image source: /latex.php
Alternative text: \displaystyle \dfrac{a}{b}
Image source: /latex.php
Alternative text: \displaystyle F_n
Image source: /latex.php
Alternative text: \displaystyle X(b;a),Y(d;c)
Image source: /latex.php
Alternative text: \displaystyle OXY
Image source: /latex.php
Alternative text: \displaystyle S_{OXY}=\dfrac{1}{2}\Rightarrow bc-ad=1
Image source: /latex.php
Alternative text: \displaystyle \Box
Image source: /avatar/74a60cd44f00304a7cb25541dc261336
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Image source: /wp-includes/images/rss.png
Alternative text: RSS
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Alternative text: Not defined!

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