Olympiad problems

2022 Putnam, A4

Suppose that $X_1, X_2, \ldots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S=\sum_{i=1}^kX_i/2^i,$ where $k$ is the least positive integer such that $X_k<X_{k+1},$ or $k=\infty$ if there is no such integer. Find the expected value of $S.$

2003 Putnam, 1

Tags: Putnam , LaTeX , inequalities , algorithm , floor function , college contests , partition

Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers, \[n = a_1 + a_2 + \cdots a_k\] with $k$ an arbitrary positive integer and $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$? For example, with $n = 4$, there are four ways: $4$, $2 + 2$, $1 + 1 + 2$, $1 + 1 + 1 + 1$.

2023 Putnam, A5

Tags: Putnam , Putnam 2023

For a nonnegative integer $k$, let $f(k)$ be the number of ones in the base 3 representation of $k$. Find all complex numbers $z$ such that $$ \sum_{k=0}^{3^{1010}-1}(-2)^{f(k)}(z+k)^{2023}=0 $$

1996 Putnam, 2

Tags: Putnam , inequalities , induction , logarithms , geometry , rectangle , integration

Prove the inequality for all positive integer $n$ : \[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]

2000 Putnam, 2

Tags: Putnam , number theory , greatest common divisor , college contests

Prove that the expression \[ \dfrac {\text {gcd}(m, n)}{n} \dbinom {n}{m} \] is an integer for all pairs of integers $ n \ge m \ge 1 $.

2018 Putnam, A5

Tags: Putnam , Putnam 2018 , Taylor series , college contests , real analysis

Let $f: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1) = 1$, and $f(x) \ge 0$ for all $x \in \mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x) < 0$.

1961 Putnam, B4

Tags: Putnam , absolute value , inequalities

Let $x_1 , x_2 ,\ldots, x_n$ be real numbers in $[0,1].$ Determine the maximum value of the sum of the $\frac{n(n-1)}{2}$ terms: $$\sum_{i<j}|x_i -x_j |.$$

Russian TST 2016, P2

Tags: Putnam , Putnam 2015 , Putnam number theory

Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $0<a<q/4$ and $\gcd(a,q)=1.$ Show that $N_q$ is odd if and only if $q$ is of the form $p^k$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8.$

1996 Putnam, 6

Tags: Putnam , function , college contests

Let $c\ge 0$ be a real number. Give a complete description with proof of the set of all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that $f(x)=f(x^2+c)$ for all $x\in \mathbb{R}$.

1957 Putnam, A6

Tags: Putnam , limit , logarithms

Let $a>0$, $S_1 =\ln a$ and $S_n = \sum_{i=1 }^{n-1} \ln( a- S_i )$ for $n >1.$ Show that $$ \lim_{n \to \infty} S_n = a-1.$$

2000 Putnam, 4

Tags: Putnam , function , trigonometry , induction , college contests

Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\le x \le 1$.

1975 Putnam, B6

Tags: Putnam , harmonic series , inequalities

Let $H_n=\sum_{r=1}^{n} \frac{1}{r}$. Show that $$n-(n-1)n^{-1\slash (n-1)}>H_n>n(n+1)^{1\slash n}-n$$ for $n>2$.

1969 Putnam, B1

Tags: Putnam , number theory , sum of divisors

Let $n$ be a positive integer such that $24\mid n+1$. Prove that the sum of the positive divisors of $n$ is divisble by 24.

1993 Putnam, A5

Tags: Putnam , real analysis

Let U be the set formed as the union of three open intervals, $U = (-100, -10) \cup (1/101, 1/11) \cup (101/100, 11/10)$. Show that $\int_{U} \frac{(x^2-x)^2}{(x^3-3x+1)^2} dx$ is rational.

1951 Putnam, A5

Tags: Putnam

Consider in the plane the network of points having integral coordinates. For lines having rational slope show that: (i) the line passes through no points of the network or through infinitely many; (ii) there exists for each line a positive number $d$ having the property that no point of the network, except such as may be on the line, is closer to the line than the distance $d.$

2020 Putnam, B6

Tags: Putnam , Putnam 2020

Let $n$ be a positive integer. Prove that $$\sum_{k=1}^n (-1)^{\lfloor k (\sqrt{2} - 1) \rfloor} \geq 0.$$ (As usual, $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)

1946 Putnam, A5

Tags: Putnam , analytic geometry

Find the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1.$$

1940 Putnam, B2

Tags: Putnam , geometry

A cylindrical hole of radius $r$ is bored through a cylinder of radiues $R$ ($r\leq R$) so that the axes intersect at right angles. i) Show that the area of the larger cylinder which is inside the smaller one can be expressed in the form $$S=8r^2\int_{0}^{1} \frac{1-v^{2}}{\sqrt{(1-v^2)(1-m^2 v^2)}}\;dv,\;\; \text{where} \;\; m=\frac{r}{R}.$$ ii) If $K=\int_{0}^{1} \frac{1}{\sqrt{(1-v^2)(1-m^2 v^2)}}\;dv$ and $E=\int_{0}^{1} \sqrt{\frac{1-m^2 v^2}{1-v^2 }}dv$. show that $$S=8[R^2 E - (R^2 - r^2 )K].$$

2020 Putnam, B5

Tags: Putnam , Putnam 2020

For $j \in \{ 1,2,3,4\}$, let $z_j$ be a complex number with $| z_j | = 1$ and $z_j \neq 1$. Prove that $$3 - z_1 - z_2 - z_3 - z_4 + z_1z_2z_3z_4 \neq 0.$$

1952 Putnam, A5

Tags: Putnam

Let $a_j (j = 1, 2, \ldots, n)$ be entirely arbitrary numbers except that no one is equal to unity. Prove \[ a_1 + \sum^n_{i=2} a_i \prod^{i-1}_{j=1} (1 - a_j) = 1 - \prod^n_{j=1} (1 - a_j).\]

1995 Putnam, 4

Tags: Putnam , college contests

Suppose we have a necklace of $n$ beads. Each bead is labelled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_1, x_2,\cdots , x_n$ satisfy \[ \sum_{i=1}^{k}x_i\le k-1\quad \forall \;\;1\le k\le n \]

2016 Putnam, B4

Tags: Putnam , Putnam 2016 , Putnam matrices

Let $A$ be a $2n\times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be $0$ or $1,$ each with probability $1/2.$ Find the expected value of $\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A.$

1960 Putnam, A3

Tags: Putnam , inequalities

Show that if $t_1 , t_2, t_3, t_4, t_5$ are real numbers, then $$\sum_{j=1}^{5} (1-t_j )\exp \left( \sum_{k=1}^{j} t_k \right) \leq e^{e^{e^{e}}}.$$

2005 Taiwan TST Round 1, 1

Tags: Putnam , Diophantine equation , number theory solved , number theory

Prove that there exists infinitely many positive integers $n$ such that $n, n+1$, and $n+2$ can be written as the sum of two perfect squares.

1974 Putnam, A2

Tags: Putnam , geometry , shortest distance

A circle stands in a plane perpendicular to the ground and a point $A$ lies in this plane exterior to the circle and higher than its bottom. A particle starting from rest at $A$ slides without friction down an inclined straight line until it reaches the circle. Which straight line allows descent in the shortest time?