MathDB

2019 LIMIT Category C

Part of 2019 LIMIT

Subcontests

(12)
Problem 12
2
Problem 11
2

alternating series

Let x=11⋅2−12⋅3+13⋅4−…x=\frac1{1\cdot2}-\frac1{2\cdot3}+\frac1{3\cdot4}-\ldotsThen ex+1e^{x+1} is

find conditional distribution

Let X1,X2,X3X_1,X_2,X_3 be exp⁡(1)\exp(1). Find the conditional distribution of X1∣X1+X2+X3=kX_1|X_1+X_2+X_3=k. <spanclass=′latex−bold′>(A)</span> Uniform⁡(0,k)<span class='latex-bold'>(A)</span>~\operatorname{Uniform}(0,k) <spanclass=′latex−bold′>(B)</span> Uniform⁡(0,k3)<span class='latex-bold'>(B)</span>~\operatorname{Uniform}\left(0,\frac k3\right) <spanclass=′latex−bold′>(C)</span> Uniform⁡(0,2k3)<span class='latex-bold'>(C)</span>~\operatorname{Uniform}\left(0,\frac{2k}3\right) <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
Problem 10
2
Problem 9
2

divisibility statements

Which of the following are true? <spanclass=′latex−bold′>(A)</span> For every n∈N,n3−n is divisible by 6<span class='latex-bold'>(A)</span>~\text{For every }n\in\mathbb N,n^3-n\text{ is divisible by }6 <spanclass=′latex−bold′>(B)</span> For every n∈N,n7−n is divisible by 42<span class='latex-bold'>(B)</span>~\text{For every }n\in\mathbb N,n^7-n\text{ is divisible by }42 <spanclass=′latex−bold′>(C)</span> Every perfect square is of the form 3m or 3m+1 for some n∈N<span class='latex-bold'>(C)</span>~\text{Every perfect square is of the form }3m\text{ or }3m+1\text{ for some }n\in\mathbb N <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}

idempotent matrices

P∈An(R)={Mn×n∣M2=M}P\in A_n(\mathbb R)=\{M_{n\times n}|M^2=M\}. Which of the following are true? <spanclass=′latex−bold′>(A)</span> PT=P,∀P∈An(R)<span class='latex-bold'>(A)</span>~P^T=P,\forall P\in A_n(\mathbb R) <spanclass=′latex−bold′>(B)</span> ∃P≠0,P∈An(R) with tr⁡(P)=0<span class='latex-bold'>(B)</span>~\exists P\ne0,P\in A_n(\mathbb R)\text{ with }\operatorname{tr}(P)=0 <spanclass=′latex−bold′>(C)</span> ∃Xn×r such that Px=X for r=rank⁡(P)<span class='latex-bold'>(C)</span>~\exists X_{n\times r}\text{ such that }Px=X\text{ for }r=\operatorname{rank}(P)
Problem 8
2

floor of infinite sum

The value of ⌊13!+44!+95!+…⌋\left\lfloor\frac1{3!}+\frac4{4!}+\frac9{5!}+\ldots\right\rfloor

Poisson distro

Let X1,X2,…X_1,X_2,\ldots be a sequence of independent random variables distributed exponentially with mean 11. Suppose N\mathbb N is a random variable independent of XiX_i's that has a Poisson distribution with mean λ>0\lambda>0. What is the expected value of X1+X2+…+XNX_1+X_2+\ldots+X_N? <spanclass=′latex−bold′>(A)</span> N2<span class='latex-bold'>(A)</span>~N^2 <spanclass=′latex−bold′>(B)</span> λ+λ2<span class='latex-bold'>(B)</span>~\lambda+\lambda^2 <spanclass=′latex−bold′>(C)</span> λ2<span class='latex-bold'>(C)</span>~\lambda^2 <spanclass=′latex−bold′>(D)</span> λ<span class='latex-bold'>(D)</span>~\lambda
Problem 7
2
Problem 6
2

connectedness

Which of the following are true? <spanclass=′latex−bold′>(A)</span> GL(n,R) is connected<span class='latex-bold'>(A)</span>~GL(n,\mathbb R)\text{ is connected} <spanclass=′latex−bold′>(B)</span> GL(n,C) is connected<span class='latex-bold'>(B)</span>~GL(n,\mathbb C)\text{ is connected} <spanclass=′latex−bold′>(C)</span> O(n,R) is connected<span class='latex-bold'>(C)</span>~O(n,\mathbb R)\text{ is connected} <spanclass=′latex−bold′>(D)</span> O(n,C) is connected<span class='latex-bold'>(D)</span>~O(n,\mathbb C)\text{ is connected}

normal distro, find variance

Let XX be normally distributed with mean μ\mu and variance σ2>0\sigma^2>0. What is the variance of eXe^X?
Problem 5
2

nontrivial subgroups, finitude or infinitude

Let G=(S1,⋅)G=(S^1,\cdot) be a group. Then its nontrivial subgroups <spanclass=′latex−bold′>(A)</span> are necessarily finite<span class='latex-bold'>(A)</span>~\text{are necessarily finite} <spanclass=′latex−bold′>(B)</span> can be infinite<span class='latex-bold'>(B)</span>~\text{can be infinite} <spanclass=′latex−bold′>(C)</span> can be dense in S1<span class='latex-bold'>(C)</span>~\text{can be dense in }S^1 <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}

Uniform(0,1) and Bernoulli(1/4)

Suppose that X∼Uniform⁡(0,1)X\sim\operatorname{Uniform}(0,1) and Y∼Bernoulli⁡(14)Y\sim\operatorname{Bernoulli}\left(\frac14\right), independently of each other. Let Z=X+YZ=X+Y. Then which of the following is true? <spanclass=′latex−bold′>(A)</span> The distribution of Z is symmetric about 1<span class='latex-bold'>(A)</span>~\text{The distribution of }Z\text{ is symmetric about }1 <spanclass=′latex−bold′>(B)</span> Z has a probability density function<span class='latex-bold'>(B)</span>~Z\text{ has a probability density function} <spanclass=′latex−bold′>(C)</span> E(Z)=54<span class='latex-bold'>(C)</span>~E(Z)=\frac54 <spanclass=′latex−bold′>(D)</span> P(Z≤1)=14<span class='latex-bold'>(D)</span>~P(Z\le1)=\frac14
Problem 4
2

existence of matrices

Which of the following are true? <spanclass=′latex−bold′>(A)</span> ∃A∈M3(R) such that A2=−I3<span class='latex-bold'>(A)</span>~\exists A\in M_3(\mathbb R)\text{ such that }A^2=-I_3 <spanclass=′latex−bold′>(B)</span> ∃A,B∈M3(R) such that AB−BA=I3<span class='latex-bold'>(B)</span>~\exists A,B\in M_3(\mathbb R)\text{ such that }AB-BA=I_3 <spanclass=′latex−bold′>(C)</span> ∀A∈M4,det⁡(I4+A2)≥0<span class='latex-bold'>(C)</span>~\forall A\in M_4,\det\left(I_4+A^2\right)\ge0 <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}

conditional distribution

Let X,YX,Y be i.i.d Geom⁡(p)\operatorname{Geom}(p). What is the conditional distribution of X∣X+Y=kX|X+Y=k? <spanclass=′latex−bold′>(A)</span> Uniform⁡{1,2,…,⌊k2⌋}<span class='latex-bold'>(A)</span>~\operatorname{Uniform}\left\{1,2,\ldots,\left\lfloor\frac k2\right\rfloor\right\} <spanclass=′latex−bold′>(B)</span> Uniform⁡{1,2,…,k}<span class='latex-bold'>(B)</span>~\operatorname{Uniform}\left\{1,2,\ldots,k\right\} <spanclass=′latex−bold′>(C)</span> Uniform⁡{1,2,…,⌊k2⌋+1}<span class='latex-bold'>(C)</span>~\operatorname{Uniform}\left\{1,2,\ldots,\left\lfloor\frac k2\right\rfloor+1\right\} <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
Problem 3
2

statements about subgroup

GG be a group and H≤GH\le G. Then which of the following are true? <spanclass=′latex−bold′>(A)</span> a∈G,aHa−1⊂H⇒aHa−1=H<span class='latex-bold'>(A)</span>~a\in G,aHa^{-1}\subset H\Rightarrow aHa^{-1}=H <spanclass=′latex−bold′>(B)</span> ∃G,H and H≤G with H≅G<span class='latex-bold'>(B)</span>~\exists G,H\text{ and }H\le G\text{ with }H\cong G <spanclass=′latex−bold′>(C)</span> All subgroups are normal, then G is abelian.<span class='latex-bold'>(C)</span>~\text{All subgroups are normal, then }G\text{ is abelian.} <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}

convergence of series

Which of the following series are convergent? <spanclass=′latex−bold′>(A)</span> ∑n=1∞2n2+35n3+1<span class='latex-bold'>(A)</span>~\sum_{n=1}^\infty\sqrt{\frac{2n^2+3}{5n^3+1}} <spanclass=′latex−bold′>(B)</span> ∑n=1∞(n+1)nnn+3/2<span class='latex-bold'>(B)</span>~\sum_{n=1}^\infty\frac{(n+1)^n}{n^{n+3/2}} <spanclass=′latex−bold′>(C)</span> ∑n=1∞n2x(1−x2)n<span class='latex-bold'>(C)</span>~\sum_{n=1}^\infty n^2x\left(1-x^2\right)^n <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
Problem 2
2

existence of surjections

Which of the following are true? <spanclass=′latex−bold′>(A)</span> ∃f:N→Z onto and increasing<span class='latex-bold'>(A)</span>~\exists f:\mathbb N\to\mathbb Z\text{ onto and increasing} <spanclass=′latex−bold′>(B)</span> ∃f:Z→Q onto and increasing<span class='latex-bold'>(B)</span>~\exists f:\mathbb Z\to\mathbb Q\text{ onto and increasing} <spanclass=′latex−bold′>(C)</span> ∃f:Q→Z onto and increasing and bounded<span class='latex-bold'>(C)</span>~\exists f:\mathbb Q\to\mathbb Z\text{ onto and increasing and bounded} <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}

inequalities with absolute value

Let x,y∈[0,∞)x,y\in[0,\infty). Which of the following is true? <spanclass=′latex−bold′>(A)</span> ∣log⁡(1+x2)−log⁡(1+y2)∣≤∣x−y∣<span class='latex-bold'>(A)</span>~\left|\log\left(1+x^2\right)-\log\left(1+y^2\right)\right|\le|x-y| <spanclass=′latex−bold′>(B)</span> ∣sin⁡2x−sin⁡2y∣≤∣x−y∣<span class='latex-bold'>(B)</span>~\left|\sin^2x-\sin^2y\right|\le|x-y| <spanclass=′latex−bold′>(C)</span> ∣tan⁡−1x−tan⁡−1y∣≤∣x−y∣<span class='latex-bold'>(C)</span>~\left|\tan^{-1}x-\tan^{-1}y\right|\le|x-y| <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
Problem 1
2

multi-choice matrix statements

Which of the following are true? <spanclass=′latex−bold′>(A)</span> ∀A∈Mn(R),At=X−1AX for some X∈Mn(R)<span class='latex-bold'>(A)</span>~\forall A\in M_n(\mathbb R),A^t=X^{-1}AX\text{ for some }X\in M_n(\mathbb R) <spanclass=′latex−bold′>(B)</span> ∀A∈Mn(R),I+AAt is invertible<span class='latex-bold'>(B)</span>~\forall A\in M_n(\mathbb R),I+AA^t\text{ is invertible} <spanclass=′latex−bold′>(C)</span> tr⁡(AB)=tr⁡(BA),∀A,B∈Mn(R) but ∃A,B,C such that tr⁡(ABC)≠tr⁡(BAC)<span class='latex-bold'>(C)</span>~\operatorname{tr}(AB)=\operatorname{tr}(BA),\forall A,B\in M_n(\mathbb R)\text{ but }\exists A,B,C\text{ such that }\operatorname{tr}(ABC)\ne\operatorname{tr}(BAC) <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}

differentiability of functions at 0

Which of the following functions are differentiable at x=0x=0? <spanclass=′latex−bold′>(A)</span> f(x)={tan⁡−1(1∣x∣)if x≠0π2if x=0<span class='latex-bold'>(A)</span>~f(x)=\begin{cases}\tan^{-1}\left(\frac1{|x|}\right)&\text{if }x\ne0\\\frac\pi2&\text{if }x=0\end{cases} <spanclass=′latex−bold′>(B)</span> f(x)=∣x∣1/2x<span class='latex-bold'>(B)</span>~f(x)=|x|^{1/2}x <spanclass=′latex−bold′>(C)</span> f(x)={x2∣cos⁡πx∣if x≠00if x=0<span class='latex-bold'>(C)</span>~f(x)=\begin{cases}x^2\left|\cos\frac{\pi}x\right|&\text{if }x\ne0\\0&\text{if }x=0\end{cases} <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}