Olympiad problems

1994 Putnam, 6

For $a\in \mathbb{Z}$ define \[ n_a=101a-100\cdot 2^a \] Show that, for $0\le a,b,c,d\le 99$ \[ n_a+n_b\equiv n_c+n_d\pmod{10100}\implies \{a,b\}=\{c,d\} \]

Alice and the Mad Hatter are playing a game. At the start of the game, three $2024$’s are written on the blackboard. Then, Alice and the Mad Hatter alternate turns, with the Mad Hatter starting. On the Mad Hatter’s turn, he must pick one of the numbers on the blackboard and increase it by $1$. On Alice’s turn, she must: - pick one of the numbers on the blackboard and decrease it by 1, and then - replace the two numbers $a$ and $b$ on the blackboard which were not chosen by the Mad Hatter on the previous turn with $\sqrt{ab}$. Alice wins if, on the start of her turn, any of the three numbers are less than $1$. Can the Mad Hatter prevent Alice from winning?

1998 Putnam, 6

Tags: Putnam , geometry , college contests

Let $A,B,C$ denote distinct points with integer coefficients in $\mathbb{R}^2$. Prove that if \[(|AB|+|BC|)^2<8\cdot[ABC]+1\] then $A,B,C$ are three vertices of a square. Here $|XY|$ is the length of segment $XY$ and $[ABC]$ is the area of triangle $ABC$.

2018 Brazil Undergrad MO, 3

Tags: permutations , Brazilian Math Olympiad , college contests , number theory

How many permutations $a_1, a_2, a_3, a_4$ of $1, 2, 3, 4$ satisfy the condition that for $k = 1, 2, 3,$ the list $a_1,. . . , a_k$ contains a number greater than $k$?

2015 IMC, 5

Tags: college contests , geometry , vector geometry , IMC2015

Let $n\ge2$, let $A_1,A_2,\ldots,A_{n+1}$ be $n+1$ points in the $n$-dimensional Euclidean space, not lying on the same hyperplane, and let $B$ be a point strictly inside the convex hull of $A_1,A_2,\ldots,A_{n+1}$. Prove that $\angle A_iBA_j>90^\circ$ holds for at least $n$ pairs $(i,j)$ with $\displaystyle{1\le i<j\le n+1}$. Proposed by Géza Kós, Eötvös University, Budapest

1990 Putnam, A5

Tags: Putnam , college contests

If $\mathbf{A}$ and $\mathbf{B}$ are square matrices of the same size such that $\mathbf{ABAB}=\mathbf{0}$, does it follow that $\mathbf{BABA}=\mathbf{0}$.

2019 Korea USCM, 1

Tags: linear algebra , too easy , rank , matrix , college contests

$A = \begin{pmatrix} 2019 & 2020 & 2021 \\ 2020 & 2021 & 2022 \\ 2021 & 2022 & 2023 \end{pmatrix}$. Find $\text{rank}(A)$.

2003 IMC, 1

Tags: limit , IMC , college contests

(a) Let $a_1,a_2,...$ be a sequenceof reals with $a_1=1$ and $a_{n+1}>\frac32 a_n$ for all $n$. Prove that $\lim_{n\rightarrow\infty}\frac{a_n}{\left(\frac32\right)^{n-1}}$ exists. (finite or infinite) (b) Prove that for all $\alpha>1$ there is a sequence $a_1,a_2,...$ with the same properties such that $\lim_{n\rightarrow\infty}\frac{a_n}{\left(\frac32\right)^{n-1}}=\alpha$

2003 Putnam, 6

Tags: Putnam , function , integration , college contests

Let $f(x)$ be a continuous real-valued function deﬁned on the interval $[0, 1]$. Show that \[\int_0^1\int_0^1|f(x)+f(y)|dx \; dy \ge \int_0^1 |f(x)|dx\]

1968 Putnam, B6

Tags: Putnam , topology , compact set , college contests , real analysis

Show that one cannot find compact sets $A_1, A_2, A_3, \ldots$ in $\mathbb{R}$ such that (1) All elements of $A_n$ are rational. (2) Any compact set $K\subset \mathbb{R}$ which only contains rational numbers is contained in some $A_{m}$.

2006 IMC, 4

Tags: function , algebra , polynomial , rational function , IMC , college contests

Let f be a rational function (i.e. the quotient of two real polynomials) and suppose that $f(n)$ is an integer for infinitely many integers n. Prove that f is a polynomial.

1996 Putnam, 3

Tags: Putnam , college contests

Suppose that each of $20$ students has made a choice of anywhere from $0$ to $6$ courses from a total of $6$ courses offered. Prove or disprove : there are $5$ students and $2$ courses such that all $5$ have chosen both courses or all $5$ have chosen neither course.

2017 Korea USCM, 8

Tags: differential equation , ordinary differential equation , college contests , trigonometry

$u(t)$ is solution of the following initial value problem. $$\begin{cases} u''(t) + u'(t) = \sin u(t) &\;\;(t>0),\\ u(0)=1,\;\; u'(0)=0 & \end{cases}$$ (1) Show that $u(t)$ and $u'(t)$ are bounded on $t>0$. (2) Find $\lim\limits_{t\to\infty} u(t)$ with proof.

1960 Miklós Schweitzer, 5

Tags: college contests

[b]5.[/b] Define the sequence $\{c_n\}_{n=1}^{\infty}$ as follows: $c_1= \frac {1}{2}$, $c_{n+1}= c_{n}-c_{n}^2$($n\geq 1$). Prove that $\lim_{n \to \infty} nc_n= 1$ [b](S.12)[/b]

1977 Putnam, A1

Tags: college contests

Consider all lines which meet the graph of $$y=2x^4+7x^3+3x-5$$ in four distinct points, say $(x_i,y_i), i=1,2,3,4.$ Show that $$\frac{x_1+x_2+x_3+x_4}{4}$$ is independent of the line and find its value.

2005 IMC, 5

Tags: IMC , college contests

Find all $ r > 0$ such that when $ f: \mathbb R^{2}\to \mathbb R$ is differentiable, $ \|\textrm{grad} \; f(0,0)\| \equal{} 1$, $ \|\textrm{grad} \; f(u) \minus{} \textrm{grad} \; f(v)\| \leq \| u \minus{} v\|$, then the max of $ f$ on the disk $ \|u\|\leq r$, is attained at exactly one point.

2011 Putnam, B3

Tags: Putnam , function , limit , college contests

Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0,$ with $g$ nonzero and continuous at $0.$ If $fg$ and $f/g$ are differentiable at $0,$ must $f$ be differentiable at $0?$

2013 IMC, 4

Tags: number theory , IMC , college contests

Does there exist an infinite set $\displaystyle{M}$ consisting of positive integers such that for any $\displaystyle{a,b \in M}$ with $\displaystyle{a < b}$ the sum $\displaystyle{a + b}$ is square-free? [b]Note.[/b] A positive integer is called square-free if no perfect square greater than $\displaystyle{1}$ divides it. [i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]

2004 Miklós Schweitzer, 7

Tags: college contests , Miklos Schweitzer , 3D geometry , sphere , polynomial , topology , geometry

Suppose that the closed subset $K$ of the sphere $$S^2=\{ (x,y,z)\in \mathbb{R}^3\colon x^2+y^2+z^2=1 \}$$ is symmetric with respect to the origin and separates any two antipodal points in $S^2 \backslash K$. Prove that for any positive $\varepsilon$ there exists a homogeneous polynomial $P$ of odd degree such that the Hausdorff distance between $$Z(P)=\{ (x,y,z)\in S^2 \colon P(x,y,z)=0\}$$ and $K$ is less than $\varepsilon$.

2013 IMC, 1

Tags: linear algebra , matrix , vector , inequalities , IMC , college contests

Let $\displaystyle{A}$ and $\displaystyle{B}$ be real symmetric matrixes with all eigenvalues strictly greater than $\displaystyle{1}$. Let $\displaystyle{\lambda }$ be a real eigenvalue of matrix $\displaystyle{{\rm A}{\rm B}}$. Prove that $\displaystyle{\left| \lambda \right| > 1}$. [i]Proposed by Pavel Kozhevnikov, MIPT, Moscow.[/i]

2004 Putnam, A3

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

Define a sequence $\{u_n\}_{n=0}^{\infty}$ by $u_0=u_1=u_2=1,$ and thereafter by the condition that $\det\begin{vmatrix} u_n & u_{n+1} \\ u_{n+2} & u_{n+3} \end{vmatrix}=n!$ for all $n\ge 0.$ Show that $u_n$ is an integer for all $n.$ (By convention, $0!=1$.)

2023 Miklós Schweitzer, 2

Tags: hausdorff space , topology , college contests , Miklos Schweitzer

Let $G_0, G_1,\ldots$ be infinite open subsets of a Hausdorff space. Prove that there exist some infinite pairwise disjoint open sets $V_0,V_1,\ldots$ and some indices $n_0<n_1<\cdots$ such that $V_i\subseteq G_{n_i}$ for every $i\geqslant 0.$

2004 Putnam, B4

Tags: Putnam , rotation , analytic geometry , geometry , geometric transformation , college contests , Putnam complex

Let $n$ be a positive integer, $n \ge 2$, and put $\theta=\frac{2\pi}{n}$. Define points $P_k=(k,0)$ in the [i]xy[/i]-plane, for $k=1,2,\dots,n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$. Let $R$ denote the map obtained by applying in order, $R_1$, then $R_2$, ..., then $R_n$. For an arbitrary point $(x,y)$, find and simplify the coordinates of $R(x,y)$.

2008 IMC, 2

Tags: vector , algebra , polynomial , IMC , college contests

Denote by $\mathbb{V}$ the real vector space of all real polynomials in one variable, and let $\gamma :\mathbb{V}\to \mathbb{R}$ be a linear map. Suppose that for all $f,g\in \mathbb{V}$ with $\gamma(fg)=0$ we have $\gamma(f)=0$ or $\gamma(g)=0$. Prove that there exist $c,x_0\in \mathbb{R}$ such that \[ \gamma(f)=cf(x_0)\quad \forall f\in \mathbb{V}\]

2017 Miklós Schweitzer, 4

Tags: number fields , number theory , college contests , Miklos Schweitzer

Let $K$ be a number field which is neither $\mathbb{Q}$ nor a quadratic imaginary extension of $\mathbb{Q}$. Denote by $\mathcal{L}(K)$ the set of integers $n\ge 3$ for which we can find units $\varepsilon_1,\ldots,\varepsilon_n\in K$ for which $$\varepsilon_1+\dots+\varepsilon_n=0,$$but $\displaystyle\sum_{i\in I}\varepsilon_i\neq 0$ for any nonempty proper subset $I$ of $\{1,2,\dots,n\}$. Prove that $\mathcal{L}(K)$ is infinite, and that its smallest element can be bounded from above by a function of the degree and discriminant of $K$. Further, show that for infinitely many $K$, $\mathcal{L}(K)$ contains infinitely many even and infinitely many odd elements.