Olympiad problems

1996 Putnam, 4

Tags: Putnam , linear algebra , matrix , trigonometry , college contests

For any square matrix $\mathcal{A}$ we define $\sin {\mathcal{A}}$ by the usual power series. \[ \sin {\mathcal{A}}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\mathcal{A}^{2n+1} \] Prove or disprove : $\exists 2\times 2$ matrix $A\in \mathcal{M}_2(\mathbb{R})$ such that \[ \sin{A}=\left(\begin{array}{cc}1 & 1996 \\0 & 1 \end{array}\right) \]

MIPT student olimpiad spring 2022, 3

Tags: Liniar algebra , college contests , MIPT , linear algebra

Prove that for any two linear subspaces $V, W \subset R^n$ the same dimension there is an orthogonal transformation $A:R^n\to R^n$, such that $A(V )=W$ and $A(W) = V$

2018 Korea USCM, 8

Tags: inequalities , Sequence , series , college contests

Suppose a sequence of reals $\{a_n\}_{n\geq 0}$ satisfies $a_0 = 0$, $\frac{100}{101} <a_{100}<1$, and $$2a_n - a_{n-1} -a_{n+1} \leq 2 (1-a_n )^3$$ for every $n\geq 1$. (1) Define a sequence $b_n = a_n - \frac{n}{n+1}$. Prove that $b_n\leq b_{n+1}$ for any $n\geq 100$. (2) Determine whether infinite series $\sum_{n=1}^\infty \frac{a_n}{n^2}$ converges or diverges.

1955 Miklós Schweitzer, 6

Tags: college contests , NumberTheory

[b]6.[/b] For a prime factorisation of a positive integer $N$ let us call the exponent of a prime $p$ the integer $k$ for which $p^{k} \mid N$ but $p^{k+1} \nmid N$; let, further, the power $p^{k}$ be called the "contribution" of $p$ to $N$. Show that for any positive integer $n$ and for any primes $p$ and $q$ the contibution of $p$ to $n!$ is greater than the contribution of $q$ if and only if the exponent of $p$ is greater than that of $q$.

1954 Miklós Schweitzer, 4

Tags: function , college contests

[b]4.[/b] Find all functions of two variables defined over the entire plane that satisfy the relations $f(x+u,y+u)=f(x,y)+u$ and $f(xv,yv)= f(x,y) v$ for any real numbers $x,y,u,v$. [b](R.12)[/b]

2018 Miklós Schweitzer, 6

Tags: abstract algebra , college contests

Prove that if $a$ is an integer and $d$ is a positive divisor of the number $a^4+a^3+2a^2-4a+3$, then $d$ is a fourth power modulo $13$.

2019 Korea USCM, 8

Tags: vector space , Matrices , linear algebra , college contests , matrix

$M_n(\mathbb{C})$ is the vector space of all complex $n\times n$ matrices. Given a linear map $T:M_n(\mathbb{C})\to M_n(\mathbb{C})$ s.t. $\det (A)=\det(T(A))$ for every $A\in M_n(\mathbb{C})$. (1) If $T(A)$ is the zero matrix, then show that $A$ is also the zero matrix. (2) Prove that $\text{rank} (A)=\text{rank} (T(A))$ for any $A\in M_n(\mathbb{C})$.

2011 Putnam, A2

Tags: Putnam , ratio , geometric series , college contests , Summation

Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1=b_1=1$ and $b_n=b_{n-1}a_n-2$ for $n=2,3,\dots.$ Assume that the sequence $(b_j)$ is bounded. Prove that \[S=\sum_{n=1}^{\infty}\frac1{a_1\cdots a_n}\] converges, and evaluate $S.$

2005 IMC, 4

Tags: algebra , polynomial , IMC , college contests

4) find all polynom with coeffs a permutation of $[1,...,n]$ and all roots rational

1959 Miklós Schweitzer, 1

Tags: number theory , prime numbers , college contests , real analysis

[b]1.[/b] Let $p_n$ be the $n$th prime number. Prove that $\sum_{n=2}^{\infty} \frac{1}{np_n-(n-1)p_{n-1}}= \infty$ [b](N.17)[/b]

1991 Putnam, A3

Tags: Putnam , algebra , polynomial , college contests

Find all real polynomials $ p(x)$ of degree $ n \ge 2$ for which there exist real numbers $ r_1 < r_2 < ... < r_n$ such that (i) $ p(r_i) \equal{} 0, 1 \le i \le n$, and (ii) $ p' \left( \frac {r_i \plus{} r_{i \plus{} 1}}{2} \right) \equal{} 0, 1 \le i \le n \minus{} 1$. [b]Follow-up:[/b] In terms of $ n$, what is the maximum value of $ k$ for which $ k$ consecutive real roots of a polynomial $ p(x)$ of degree $ n$ can have this property? (By "consecutive" I mean we order the real roots of $ p(x)$ and ignore the complex roots.) In particular, is $ k \equal{} n \minus{} 1$ possible for $ n \ge 3$?

2017 IMC, 6

Tags: college contests , IMC , imc 2017

Let $f:[0;+\infty)\to \mathbb R$ be a continuous function such that $\lim\limits_{x\to +\infty} f(x)=L$ exists (it may be finite or infinite). Prove that $$ \lim\limits_{n\to\infty}\int\limits_0^{1}f(nx)\,\mathrm{d}x=L. $$

1989 Putnam, A1

Tags: Putnam , algebra , difference of squares , special factorizations , college contests

How many base ten integers of the form 1010101...101 are prime?

2013 Putnam, 1

Tags: Putnam , college contests

For positive integers $n,$ let the numbers $c(n)$ be determined by the rules $c(1)=1,c(2n)=c(n),$ and $c(2n+1)=(-1)^nc(n).$ Find the value of \[\sum_{n=1}^{2013}c(n)c(n+2).\]

2018 CMI B.Sc. Entrance Exam, 2

Tags: CMI , Chennai Mathematical Institute , 2018 , algebra , college contests

Answer the following questions : $\textbf{(a)}$ Find all real solutions of the equation $$\Big(x^2-2x\Big)^{x^2+x-6}=1$$ Explain why your solutions are the only solutions. $\textbf{(b)}$ The following expression is a rational number. Find its value. $$\sqrt[3]{6\sqrt{3}+10} -\sqrt[3]{6\sqrt{3}-10}$$

2004 IMC, 6

Tags: linear algebra , matrix , IMC , college contests

For $ n\geq 0$ define the matrices $ A_n$ and $ B_n$ as follows: $ A_0 \equal{} B_0 \equal{} (1)$, and for every $ n>0$ let \[ A_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\ A_{n \minus{} 1} & B_{n \minus{} 1} \\ \end{array} \right) \ \textrm{and} \ B_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\ A_{n \minus{} 1} & 0 \\ \end{array} \right). \] Denote by $ S(M)$ the sum of all the elements of a matrix $ M$. Prove that $ S(A_n^{k \minus{} 1}) \equal{} S(A_k^{n \minus{} 1})$, for all $ n,k\geq 2$.

2016 VJIMC, 3

Tags: geometry , college contests

Let $d \geq 3$ and let $A_1 \dots A_{d + 1}$ be a simplex in $\mathbb{R}^d$. (A simplex is the convex hull of $d + 1$ points not lying in a common hyperplane.) For every $i = 1, \dots , d + 1$ let $O_i$ be the circumcentre of the face $A_1 \dots A_{i - 1}A_{i+1}\dots A_{d+1}$, i.e. $O_i$ lies in the hyperplane $A_1 \dots A_{i - 1}A_{i+1}\dots A_{d+1}$ and it has the same distance from all points $A_1, \dots , A_{i-1}, A_{i+1}, \dots , A_{d+1}$. For each $i$ draw a line through $A_i$ perpendicular to the hyperplane $O_1 \dots O_{i-1}O_{i+1} \dots O_{d+1}$. Prove that either these lines are parallel or they have a common point.

2001 Putnam, 3

Tags: Putnam , algebra , polynomial , quadratics , quadratic formula , college contests

For each integer $m$, consider the polynomial \[ P_m(x)=x^4-(2m+4)x^2+(m-2)^2. \] For what values of $m$ is $P_m(x)$ the product of two non-consant polynomials with integer coefficients?

1999 Putnam, 3

Tags: Putnam , limit , algebra , polynomial , modular arithmetic , college contests , Putnam calculus

Let $A=\{(x,y): 0\le x,y < 1\}.$ For $(x,y)\in A,$ let \[S(x,y)=\sum_{\frac12\le\frac mn\le2}x^my^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\to(1,1),(x,y)\in A}(1-xy^2)(1-x^2y)S(x,y).\]

ICMC 2, 5

Tags: college contests

For continuously differentiable function $f : [0, 1] \to\mathbb{R}$ with $f (1/2) = 0$, show that \[\left(\int_0^1 f(x)\mathrm{d}x\right)^2\leq \frac{1}{4}\int_0^1\left(f'(x)\right)^2\mathrm{d}x\]

1990 Putnam, A6