Olympiad problems

1976 Miklós Schweitzer, 1

Tags: advanced fields , advanced fields unsolved , Discrete Mathematics , computer science , college contests , Miklos Schweitzer

Assume that $ R$, a recursive, binary relation on $ \mathbb{N}$ (the set of natural numbers), orders $ \mathbb{N}$ into type $ \omega$. Show that if $ f(n)$ is the $ n$th element of this order, then $ f$ is not necessarily recursive. [i]L. Posa[/i]

1994 Putnam, 5

Tags: Putnam , floor function , college contests

For each $\alpha\in \mathbb{R}$ define $f_{\alpha}(x)=\lfloor{\alpha x}\rfloor$. Let $n\in \mathbb{N}$. Show there exists a real $\alpha$ such that for $1\le \ell \le n$ : \[ f_{\alpha}^{\ell}(n^2)=n^2-\ell=f_{\alpha^{\ell}}(n^2).\] Here $f^{\ell}_{\alpha}(x)=(f_{\alpha}\circ f_{\alpha}\circ \cdots \circ f_{\alpha})(x)$ where the composition is carried out $\ell$ times.

1961 Miklós Schweitzer, 8

Tags: college contests , real analysis

[b]8.[/b] Let $f(x)$ be a convex function defined on the interval $[0, \frac {1}{2}]$ with $f(0)=0$ and $f(\frac{1}{2})=1$; Let further $f(x)$ be differentiable in $(0, \frac {1}{2})$, and differentiable at $0$ and $\frac{1}{2}$ from the right and from the left, respectively. Finally, let $f'(0)>1$. Extend $f(x)$ to $[0.1]$ in the following manner: let $f(x)= f(1-x)$ if $x \in (\frac {1} {2}, 1]$. Show that the set of the points $x$ for shich the terms of the sequence $x_{n+1}=f(x_n)$ ($x_0=x; n = 0, 1, 2, \dots $) are not all different is everywhere dense in $[0,1]$; [b](R. 10)[/b]

1979 Putnam, B4

Tags: college contests

(a) Find a solution that is not identically zero, of the homogeneous linear differential equation $$(3x^2-x-1)y''-(9x^2+9x-2)y'+(18x+3)y=0.$$ Intelligent guessing of the form of a solution may be helpful. (b) Let $y=f(x)$ be the solution of the [i]nonhomogeneous[/i] differential equation $$(3x^2-x-1)y''-(9x^2+9x-2)y'+(18x+3)y=6(6x+1)$$ that has $f(0)=1$ and $(f(-1)-2)(f(1)-6)=1.$ Find integers $a,b,c$ such that $(f(-2)-a)(f(2)-b)=c.$

1959 Miklós Schweitzer, 4

Tags: college contests , geometry

[b]4.[/b] Consider $n$ circles of radius $1$ in the planea. Prove that at least one of the circles contains an are of length greater than $\frac{2\pi}{n}$ not intersected by any other of these circles. [b](G. 4)[/b]

2000 Miklós Schweitzer, 2

Tags: college contests , Miklos Schweitzer , point

Let $n$ red and $n$ blue subarcs of a circle be given such that each red subarc intersects each blue subarc. Prove that there is a point which is covered by at least $n$ of the given (red or blue) subarcs.

2014 Paenza, 2

Tags: college contests

There are $n$ cards numbered and stacked in increasing order from up to down (i.e. the card in the top is the number 1, the second is the 2, and so on...). With this deck, the next steps are followed: -the first card (from the top) is put in the bottom of the deck. -the second card (from the top) is taken away of the deck. -the third card (from the top) is put in the bottom of the deck. -the fourth card (from the top) is taken away of the deck. - ... The proccess goes on always the same way: the card in the top is put at the end of the deck and the next is taken away of the deck, until just one card is left. Determine which is that card.

ICMC 6, 5

Tags: geometry , combinatorics , ICMC , college contests , expected value , area of a triangle

A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$? [i]Proposed by Dylan Toh[/i]

2015 IMC, 9

Tags: college contests , linear algebra , Matrices , IMC2015

An $n \times n$ complex matrix $A$ is called \emph{t-normal} if $AA^t = A^t A$ where $A^t$ is the transpose of $A$. For each $n$, determine the maximum dimension of a linear space of complex $n \times n$ matrices consisting of t-normal matrices. Proposed by Shachar Carmeli, Weizmann Institute of Science

MIPT student olimpiad spring 2023, 1

Tags: Liniar algebra , vector , MIPT , college contests

In $R^n$ is given $n-1$ vectors, the coordinates of each are zero-sum integers. Prove that the $(n-1)$-dimensional volume of an $(n-1)$-dimensional parallelepiped $P$ stretched by these vectors, is the product of an integer and $\sqrt(n)$.

ICMC 4, 5

Tags: number theory , Powers of 2 , elementary-number-theory , ICMC , college contests

Find all composite positive integers $m$ such that, whenever the product of two positive integers $a$ and $b$ is $m$, their sum is a power of $2$. [i]Proposed by Harun Khan[/i]

1961 Miklós Schweitzer, 6

Tags: college contests , real analysis

[b]6.[/b] Consider a sequence $\{ a_n \}_{n=1}^{\infty}$ such that, for any convergent subsequence $\{ a_{n_k} \}$ of $\{a_n\}$, the sequence $\{ a_{n_k +1} \}$ also is convergent and has the same limit as $\{ a_{n_k}\}$. Prove that the sequence $\{ a_n \}$ is either convergent of has infinitely many accumulation points the set of which is dense in itself. Give an example for the second case. (A sequence $ x_n \to \infty $ or $-\infty$ is considered to be convergente, too) [b](S. 13)[/b]

1994 Putnam, 2

Tags: Putnam , analytic geometry , graphing lines , slope , college contests

For which real numbers $c$ is there a straight line that intersects the curve \[ y = x^4 + 9x^3 + cx^2 + 9x + 4\] in four distinct points?

2006 IMC, 3

Tags: trigonometry , logarithms , IMC , college contests

Compare $\tan(\sin x)$ with $\sin(\tan x)$, for $x\in \left]0,\frac{\pi}{2}\right[$.

2001 Putnam, 2

Tags: Putnam , USAMTS , algebra , system of equations , college contests

Find all pairs of real numbers $(x,y)$ satisfying the system of equations: \begin{align*}\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2)\\ \frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4)\end{align*}

2019 Korea USCM, 2

Tags: Volume , calculus , college contests , linear algebra

Matrices $A$, $B$ are given as follows. \[A=\begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0 \\ 2 & 4 & 0 \\ 0 & 0 & 12\end{pmatrix}\] Find volume of $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$.

2007 Putnam, 4

Tags: Putnam , algebra , polynomial , logarithms , quadratics , college contests

A [i]repunit[/i] is a positive integer whose digits in base $ 10$ are all ones. Find all polynomials $ f$ with real coefficients such that if $ n$ is a repunit, then so is $ f(n).$

1985 Miklós Schweitzer, 5

Tags: Miklos Schweitzer , college contests , number theory , relatively prime , algebra , polynomial , absolute value

Let $F(x,y)$ and $G(x,y)$ be relatively prime homogeneous polynomials of degree at least one having integer coefficients. Prove that there exists a number $c$ depending only on the degrees and the maximum of the absolute values of the coefficients of $F$ and $G$ such that $F(x,y)\neq G(x,y)$ for any integers $x$ and $y$ that are relatively prime and satisfy $\max \{ |x|,|y|\} > c$. [K. Gyory]

2024 SEEMOUS, P2

Tags: college contests , linear algebra

Let $A,B\in\mathcal{M}_n(\mathbb{R})$ two real, symmetric matrices with nonnegative eigenvalues. Prove that $A^3+B^3=(A+B)^3$ if and only if $AB=O_n$.

1985 Miklós Schweitzer, 4

Tags: college contests

[b]4.[/b] Call a subset $S$ of the set $\{1,\dots,n\}$ [i]exceptional[/i] if any pair of distinct elements of $S$ are coprime. Consider an exceptional set with a maximal sum of elements (among all exceptional sets for a fixed $n$). Prove that if $n$ is sufficiently large, then each element of $S$ has at most two distinct prime divisors. ([b]N.17[/b]) [P. Erdos]

2002 Miklós Schweitzer, 9

Tags: college contests , Miklos Schweitzer , topology , vector , function , invariant

Let $M$ be a connected, compact $C^{\infty}$-differentiable manifold, and denote the vector space of smooth real functions on $M$ by $C^{\infty}(M)$. Let the subspace $V\le C^{\infty}(M)$ be invariant under $C^{\infty}$-diffeomorphisms of $M$, that is, let $f\circ h\in V$ for every $f\in V$ and for every $C^{\infty}$-diffeomorphism $h\colon M\rightarrow M$. Prove that if $V$ is different from the subspaces $\{ 0\}$ and $C^{\infty}(M)$ then $V$ only contains the constant functions.

2017 Korea USCM, 7

Tags: series , real analysis , trigonometric series , inequalities , college contests , trigonometry

Prove the following inequality holds if $\{a_n\}$ is a deceasing sequence of positive reals, and $0<\theta<\frac{\pi}{2}$. $$\left|\sum_{n=1}^{2017} a_n \cos n\theta \right| \leq \frac{\pi a_1}{\theta}$$

2010 Putnam, A1

Tags: Putnam , ceiling function , pigeonhole principle , floor function , college contests , Putnam easy

Given a positive integer $n,$ what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same? [When $n=8,$ the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is [i]at least[/i] 3.]

2004 Putnam, B1

Tags: Putnam , algebra , polynomial , number theory , relatively prime , Rational Root Theorem , college contests

Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers $c_nr, c_nr^2+c_{n-1}r, c_nr^3+c_{n-1}r^2+c_{n-1}r, \dots, c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$ are all integers.

1979 Putnam, B2

Tags: college contests

Let $0<a<b.$ Evaluate $$\lim_{t\to 0} \{ \int_{0}^{1} [bx+a(1-x)]^t dx\}^{1/t}.$$ [i][The final answer should not involve any other operations other than addition, subtraction, multiplication, division and exponentiation.][/i]