# Recent questions without answers

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1
If a rectangle $R:=\left \{ \left ( x,y \right ) \in \mathbb{R}^{2}\mid A\leq x\leq B,C\leq y\leq D\right \}$ can be covered (allowing overlaps ) by $25$ discs of radius $1$ then it can also be covered by $101$ dics of radius $\frac{1}{2}.$
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2
Given any integer $n\geq 2$, we can always finds an integer $m$ such that each of the $n-1$ consecutive integers $m+2,m+3,\dots,m+n$are composite.
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3
The $10 \times 10$ matrix $\begin{pmatrix} v_{1}w_{1} & \cdots&v_{1}w_{10} \\ v_{2}w_{2}& \cdots & v_{2}w_{10}\\ v_{10}w_{1}&\cdots & v_{10}w_{10} \end{pmatrix}$has rank $2$, where $v_{i},w_{i}\in \mathbb{C}.$
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4
If every continuous function on $X\subset \mathbb{R}^{2}$ is bounded, then $X$ is compact.
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5
The graph of $xy=1$ is $\mathbb{C}^{2}$ is connected.
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6
Consider the differential equations (with $y$ is a function of $x$) $\begin{matrix} \frac{dy}{dx} & = & y\\ y\left ( 0 \right ) & = & 0 \end{matrix}$ $\begin{matrix} \frac{dy}{dx} & = & \left | y \right |^{\frac{1}{3}}\\ y\left ( 0 \right ) & = & 0. \end{matrix}$ Then $(1)$ has infinitely many solutions but $(2)$ has finite number of solutions.
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7
Let $f : \mathbb{R}^{2}\rightarrow \mathbb{R}$ be a continuous function. Then the derivative $\frac{\partial ^{2}f}{\partial x\partial y}$ can exist without $\frac{\partial f}{\partial x}$ existing.
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8
If $f$ is continuous on $\left [ 0,1 \right ]$ and if $\int_{0}^{1}f\left ( x \right )x^{n}dx=0$ for $n=1,2,3,\cdots .$ .Then $\int_{0}^{1}f^{2}\left ( x \right )dx=0.$
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9
Suppose that $f \in \mathfrak{L}^{2} \left ( \mathbb{R} \right )$. Then $f \in \mathfrak{L}^{1} \left ( \mathbb{R} \right )$.
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10
The Integral $\int_{-\infty }^{+\infty }\frac{e^{-x}}{1+x^{2}}\:dx$ is convergent.
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11
If $A\subset \mathbb{R}$ and open then the interior of the closure $\overset{-0}{A}$is $A$.
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12
If $f \in C^{\infty }$ and $f^{\left ( k \right )}\left ( 0 \right )=0$ for all integer $k\geq 0$, then $f\equiv 0$.
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13
Let $f:\left [ 0,1 \right ]\rightarrow \left [ 0,1 \right ]$be continuous then $f$ assumes the value $\int_{0}^{1}f^{2}\left ( t \right )dt$ somewhere in $\left [ 0,1 \right ]$.
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14
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function such that $\underset{h\rightarrow 0}{lim }\:\frac{f\left ( x+h \right )-f\left ( x-h \right )}{h}$ exists for all $x \in \mathbb{R}$. Then $f$ is differentiable in $\mathbb{R}.$
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15
The functions $f\left ( x \right )=x\left | x \right |$ and $x\left | sin\:x \right |$ are not differentiable at $x=0$.
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16
The composition of two uniformly continuous functions need not always be uniformly continuous.
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17
$A\in M_{2}\left ( \mathbb{C} \right )$and $A$ is nilpotent then $A^{2}=0$.
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18
Let $P$ be an $n \times n$ matrix whose row sums equal $1$. Then for any positive integer $m$ the row sums of the matrix $p^{m}$ equal $1$.
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19
There is a non trivial group homomorphism from $C$ to $R$.
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20
If the equation $xyz=1$ holds in a group $G$, does it follow that $yxz=1$.
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21
Any $3 \times3$ and $5\times5$ skew-symmetric matrices have always zero determinants.
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22
The rank of the matrix $\begin{bmatrix} 11 &12 &13 &14 \\ 21& 22 &23 & 24\\ 31& 32 &33 &34 \\ 41&42 & 43 & 44 \end{bmatrix}$ is $2$.
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23
The number $2$ is a prime in $\mathbb{Z}\left [ i \right ]$
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24
If $H_{1}$ & $H_{2}$ are subgroups of a group $G$ then $H_{1} .H_{2}=\left \{ h_{1} h_{2}\in G \mid h_{1}\in H_{1},h_{2}\in H_{2}\right \}$ is a subgroup of $G$.
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25
There exist polynomials $f\left ( x \right )$ and $g\left ( x \right )$, with complex coefficients, such that $\left ( \frac{f\left ( x \right )}{g\left ( x \right )} \right )^{2}=x$.
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26
Let $f$ be real valued, differentiable on $\left ( a,b \right )$ and ${f}'\left ( x \right )\neq 0$ for all $x \in \left ( a,b \right )$. Then $f$ is $1-1$.
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27
The inequality $\sum _{n=0}^{\infty }\frac{\left ( log \: log2 \right )^{n}}{n!}> \frac{3}{5}$ holds.
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28
Every subgroup of order $74$ in a group of order $148$ is normal.
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29
Let $u_{1},u_{2},u_{3},u_{4}$ be vector in $\mathbb{R}^{2}$ and $u=\sum_{j=1}^{4}t_{i}u_{j}\:\:\: ; \:\:\:t_{j}> 0 \:\: and \sum_{j=1}^{4}t_{j}=1.$ Then three vectors $v_{1},v_{2},v_{3}\in \mathbb{R}^{2}$may be chosen from $\left \{ u_{1},u_{2},u_{3},u_{4} \right \}$ such that $u=\sum_{j=1}^{3}s_{i}v_{j}\:\:\: ; \:\:\:s_{j}\geq 0 ,\:\: \sum_{j=1}^{3}s_{j}=1.$
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30
The inequality $\sqrt{1+x}< 1+x/2$ for $x\in \left ( -1, 10 \right )$is true