A. Carrot Cakes time limit per test:1 second memory limit per test:256 megabytes input:standard input output:standard output In some game by Playrix it takes t minutes for an oven to bake k carrot cakes, all cakes are ready at the same moment t minut…
D. Field expansion time limit per test 1 second memory limit per test 256 megabytes input standard input output standard output In one of the games Arkady is fond of the game process happens on a rectangular field. In the game process Arkady can buy…
我从来没想过自己可以被支配的这么惨,大神讲这个场不容易掉分的啊 A. Carrot Cakes time limit per test 1 second memory limit per test 256 megabytes input standard input output standard output In some game by Playrix it takes t minutes for an oven to bake k carrot cakes, all cakes are…
E - Aquarium decoration 枚举两个人都喜欢的个数,就能得到单个喜欢的个数,然后用平衡树维护前k大的和. #include<bits/stdc++.h> #define LL long long #define fi first #define se second #define mk make_pair #define PII pair<int, int> #define PLI pair<LL, int> #define ull unsigned…
题目传送门:http://codeforces.com/contest/799/problem/C C. Fountains time limit per test 2 seconds memory limit per test 256 megabytes input standard input output standard output Arkady plays Gardenscapes a lot. Arkady wants to build two new fountains. T…
C. Fountains time limit per test 2 seconds memory limit per test 256 megabytes input standard input output standard output Arkady plays Gardenscapes a lot. Arkady wants to build two new fountains. There are n available fountains, for each fountain it…
2073. Log Files Time limit: 1.0 secondMemory limit: 64 MB Nikolay has decided to become the best programmer in the world! Now he regularly takes part in various programming contests, attentively listens to problems analysis and upsolves problems. But…
Description Nikolay has decided to become the best programmer in the world! Now he regularly takes part in various programming contests, attentively listens to problems analysis and upsolves problems. But the point is that he had participated in such…
I find it may cost me so much time in doing such solutions to exercises and problems....I am sorry that I could not be persistent in doing it...Wish I could just recover it later on. [Bhatia.Matrix Analysis.Solutions to Exercises and Problems]PrI.6.1…
Mean Requests time limit per test 4 seconds memory limit per test 256 megabytes input standard input output standard output In this problem you will have to deal with a real algorithm that is used in the VK social network. As in any other company tha…
Let $x,y,z$ be linearly independent vectors in $\scrH$. Find a necessary and sufficient condition that a vector $w$ mush satisfy in order that the bilinear functional $$\bex F(u,v)=\sef{x,u}\sef{y,v}+\sef{z,u}\sef{w,v} \eex$$ is elementary. Solution.…
For every matrix $A$, the matrix $$\bex \sex{\ba{cc} I&A\\ 0&I \ea} \eex$$ is invertible and its inverse is $$\bex \sex{\ba{cc} I&-A\\ 0&I \ea}. \eex$$ Use this to show that if $A,B$ are any two $n\times n$ matrices, then $$\bex \sex{\ba{c…
Let $A=A_1\oplus A_2$. Show that (1). $W(A)$ is the convex hull of $W(A_1)$ and $W(A_2)$; i.e., the smallest convex set containing $W(A_1)\cup W(A_2)$. (2). $$\beex \bea \sen{A}&=\max\sed{\sen{A_1},\sen{A_2}},\\ \spr(A)&=\max\sed{\spr(A_1),\spr(A_…
B. Problems for Round 题目连接: http://www.codeforces.com/contest/673/problem/B Description There are n problems prepared for the next Codeforces round. They are arranged in ascending order by their difficulty, and no two problems have the same difficult…
A. Jumping Ball time limit per test 2 seconds memory limit per test 256 megabytes input standard input output standard output In a new version of the famous Pinball game, one of the most important parts of the game field is a sequence of n bumpers. T…
Name Quest time limit per test 2 seconds memory limit per test 256 megabytes input standard input output standard output A Martian boy is named s — he has got this name quite recently from his parents for his coming of age birthday. Now he enjoys loo…
Given a basis $U=(u_1,\cdots,u_n)$ not necessarily orthonormal, in $\scrH$, how would you compute the biorthogonal basis $\sex{v_1,\cdots,v_n}$? Find a formula that expresses $\sef{v_j,x}$ for each $x\in\scrH$ and $j=1,\cdots,k$ in terms of Gram matr…
Every $k\times k$ positive matrix $A=(a_{ij})$ can be realised as a Gram matrix, i.e., vectors $x_j$, $1\leq j\leq k$, can be found so that $a_{ij}=\sef{x_i,x_j}$ for all $i,j$. Solution. By Exercise I.2.2, $A=B^*B$ for some $B$. Let $$\bex B=(x_1,\c…
(Schur's Theorem) If $A$ is positive, then $$\bex \per(A)\geq \det A. \eex$$ Solution. By Exercise I.2.2, $A=T^*T$ for some upper triangular $T$ with non-negative diagonals. Thus $$\beex \bea \det A&=\det T^*\cdot \det T\\ &=\per T^*\cdot \per T\\…
Prove that for any matrices $A,B$ we have $$\bex |\per (AB)|^2\leq \per (AA^*)\cdot \per (B^*B). \eex$$ (The corresponding relation for determinants is an easy equality.) Solution. Let $$\bex A=\sex{\ba{cc} \al_1\\ \vdots\\ \al_n \ea},\quad B=\sex{\b…
Prove that for any vectors $$\bex u_1,\cdots,u_k,\quad v_1,\cdots,v_k, \eex$$ we have $$\bex |\det(\sef{u_i,v_j})|^2 \leq \det\sex{\sef{u_i,u_j}}\cdot \det \sex{\sef{v_i,v_j}}, \eex$$ $$\bex |\per(\sef{u_i,v_j})|^2 \leq \per\sex{\sef{u_i,u_j}}\cdot \…
Let $A$ be a nilpotent operator. Show how to obtain, from aJordan basis for $A$, aJordan basis of $\wedge^2A$. Solution. Since $A$ is nilpotent, each eigenvalue of $A$ is zero, and thus there exists an basis $e_1,\cdot,e_n$ of $\scrH$ such that $$\be…
Show that the inner product $$\bex \sef{x_1\vee \cdots \vee x_k,y_1\vee \cdots\vee y_k} \eex$$ is equal to the permanent of the $k\times k$ matrix $\sex{\sef{x_i,y_j}}$. Solution. $$\beex \bea &\quad \sef{x_1\vee \cdots \vee x_k,y_1\vee \cdots \vee y…
If $\dim \scrH=3$, then $\dim \otimes^3\scrH =27$, $\dim \wedge^3\scrH =1$ and $\dim \vee^3\scrH =10$. In terms of an orthonormal basis of $\scrH$, write an element of $(\wedge^3\scrH )\oplus \vee^3\scrH)^\perp$. Solution. Let $e_1,e_2,e_3$ be an ort…
