Recent questions tagged isi2020-pcb-mathematics

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Deduce that if $N, H, K$ are normal subgroups of a group $G$ such that $$ N \bigcap H=N \bigcap K=H \bigcap K=\left\{e_{G}\right\} $$ and $G=H K$, then $N$ is an Abelian ...
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Let $\epsilon>0$. Prove that there exists $n_{0} \in \mathbb{N}$ such that $$ n \geq n_{0} \Rightarrow 2-\epsilon<\frac{2 n+1}{n+2}<2+\epsilon. $$
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Show that the sequence $\left\{x_{n}\right\}, n>0$, defined by $$ x_{n}=\int_{1}^{n} \frac{\cos (t)}{t^{2}} d t $$ is convergent.
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Let $c$ be a positive real number for which the equation $$ x^{4}-x^{3}+x^{2}-(c+1) x-\left(c^{2}+c\right)=0 $$ has a real root $\alpha$. Prove that $c=\alpha^{2}-\alpha$...
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Prove that if $N$ and $K$ are normal subgroups of a group $G$ such that $N \cap K=\left\{e_{G}\right\}$, then $x y=y x, \forall x \in N, \forall y \in K$.
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7
Find all possible integers $n$ for which $n^{2}+20 n+15$ is a perfect square.
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8
Let $A=\{1,2,3, \cdots, 50\}$. In how many ways can three distinct numbers $x<y<z$ be chosen from $A$ such that the product $x y z$ is divisible by $125?$
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Prove that the function defined by $$ f(x)=\sum_{n=0}^{\infty}\left(\frac{x^{n}}{n !}\right)^{2} $$ is continuous on $\mathbb{R}$, for any real number $x$.
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11
Prove that if $T_{1}, T_{2}, \ldots, T_{k}$ are pairwise-intersecting subtrees of a tree $T$, then $T$ has a vertex that belongs to all of $T_{1}, T_{2}, \ldots, T_{k}$.
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