Olympiad problems

1961 Putnam, A4

Let $\Omega(n)$ be the number of prime factors of $n$. Define $f(1)=1$ and $f(n)=(-1)^{\Omega(n)}.$ Furthermore, let $$F(n)=\sum_{d|n} f(d).$$ Prove that $F(n)=0,1$ for all positive integers $n$. For which integers $n$ is $F(n)=1?$

2004 Miklós Schweitzer, 5

Tags: probability

Let $G$ be a non-solvable finite group and let $\varepsilon > 0$. Show that there exist a positive integer $k$ and a word $w\in F_k$ such that $w$ assumes the value $1$ with probability less than $\varepsilon$ when its $k$ arguments are considered to be independent and uniformly distributed random variables with values in $G$. (We write $F_k$ for the free group generated by $k$ elements.)

1989 All Soviet Union Mathematical Olympiad, 499

Tags: irrational , sum , sum of powers , rational , algebra

Do there exist two reals whose sum is rational, but the sum of their $n$ th powers is irrational for all $n > 1$? Do there exist two reals whose sum is irrational, but the sum of whose $n$ th powers is rational for all $n > 1$?

KoMaL A Problems 2022/2023, A. 834

Tags: geometry , octagon , conic section

Let $A_1A_2\ldots A_8$ be a convex cyclic octagon, and for $i=1,2\ldots,8$ let $B_i=A_iA_{i+3}\cap A_{i+1}A_{i+4}$ (indices are meant modulo 8). Prove that points $B_1,\ldots, B_8$ lie on the same conic section.

PEN A Problems, 51

Tags: divisibility theory

Let $a,b,c$ and $d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^{k}$ and $b+c=2^{m}$ for some integers $k$ and $m$, then $a=1$.

1999 Slovenia National Olympiad, Problem 4

Tags: combinatorics

Let be given three-element subsets $A_1,A_2,\ldots,A_6$ of a six-element set $X$. Prove that the elements of $X$ can be colored with two colors in such a way that none of the given subsets are monochromatic.

2019 PUMaC Algebra A, 3

Tags: quadratic , polynomial , algebra

Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i\geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.

2021 MOAA, 7

Tags: accuracy

Jeffrey rolls fair three six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

1989 IMO Longlists, 32

Tags: geometry

Given an acute triangle find a point inside the triangle such that the sum of the distances from this point to the three vertices is the least.

2016 Fall CHMMC, 3

Tags: probability

A gambler offers you a $2$ dollar ticket to play the following game: First, you pick a real number $0 \leq p \leq 1$, then you are given a weighted coin that comes up heads with probability $p$. If you receive $1$ dollar the [i]first[/i] time you flip a tail, and if you receive $2$ dollars [i]first[/i] time you flip a head, what is the optimal expected net winning of flipping the coin twice?

2014 Harvard-MIT Mathematics Tournament, 10

Tags: algebra , polynomial , function

For an integer $n$, let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing $n$. Suppose that $m$ is a positive integer and $b_1,b_2,\ldots,b_m$ are real numbers such that $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$ for all $n>m$. Find the smallest possible value of $m$.

2014 Contests, 2

Tags: number theory , divisor

Find the least natural number $n$, which has at least 6 different divisors $1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.

2012 NIMO Problems, 1

Tags:

Compute the largest integer $N \le 2012$ with four distinct digits. [i]Proposed by Evan Chen[/i]

2002 Italy TST, 3

Tags: algebra , polynomial , vieta , relatively prime , number theory

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$

Fractal Edition 2, P1

Tags:

Viorel claims that for any natural number $n$ greater than $2024$, the number $2024^n + 1$ is prime. Is Viorel's statement true?

2018 BMT Spring, Tie 1

Tags: combinatorics

Every face of a cube is colored one of $3$ colors at random. What is the expected number of edges that lie along two faces of different colors?

1988 China Team Selection Test, 2

Tags: trapezoid , geometry

Let $ABCD$ be a trapezium $AB // CD,$ $M$ and $N$ are fixed points on $AB,$ $P$ is a variable point on $CD$. $E = DN \cap AP$, $F = DN \cap MC$, $G = MC \cap PB$, $DP = \lambda \cdot CD$. Find the value of $\lambda$ for which the area of quadrilateral $PEFG$ is maximum.

1993 AMC 12/AHSME, 5

Tags: percent , percent increase

Last year a bicycle cost $\$160$ and a cycling helmet cost $ \$ 40$. This year the cost of the bicycle increased by $5\%$, and the cost of the helmet increased by $10\%$. The percent increase in the combined cost of the bicycle and the helmet is $ \textbf{(A)}\ 6\% \qquad\textbf{(B)}\ 7\% \qquad\textbf{(C)}\ 7.5\% \qquad\textbf{(D)}\ 8\% \qquad\textbf{(E)}\ 15\% $