Olympiad problems

1999 Harvard-MIT Mathematics Tournament, 4

Tags:

You are given 16 pieces of paper numbered $16, 15, \ldots , 2, 1$ in that order. You want to put them in the order $1, 2, \ldots , 15, 16$ switching only two adjacent pieces of paper at a time. What is the minimum number of switches necessary?

1999 Gauss, 4

Tags: Gauss

$1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}$ is equal to $\textbf{(A)}\ \dfrac{15}{8} \qquad \textbf{(B)}\ 1\dfrac{3}{14} \qquad \textbf{(C)}\ \dfrac{11}{8} \qquad \textbf{(D)}\ 1\dfrac{3}{4} \qquad \textbf{(E)}\ \dfrac{7}{8}$

2019 JBMO Shortlist, G4

Tags: geometry , JBMO , perpendicular bisector , circumcircle , geometry solved , Balkan , Junior Balkan

Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.

1990 IMO Longlists, 9

Tags: number theory , partition , Ramsey Theory , Coloring , Extremal combinatorics , IMO Shortlist , Hi

Assume that the set of all positive integers is decomposed into $ r$ (disjoint) subsets $ A_1 \cup A_2 \cup \ldots \cup A_r \equal{} \mathbb{N}.$ Prove that one of them, say $ A_i,$ has the following property: There exists a positive $ m$ such that for any $ k$ one can find numbers $ a_1, a_2, \ldots, a_k$ in $ A_i$ with $ 0 < a_{j \plus{} 1} \minus{} a_j \leq m,$ $ (1 \leq j \leq k \minus{} 1)$.

1983 Putnam, B4

Tags: number theory , Putnam , function , Sequence

[b]Problem.[/b] Let $f:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ be a function defined as $$f(n)=n+\lfloor\sqrt{n}\rfloor~\forall~n\in\mathbb{R}_0^+.$$ Prove that for any positive integer $m,$ the sequence $$m,f(m),f(f(m)),f(f(f(m))),\ldots$$ contains a perfect square.

1986 Bundeswettbewerb Mathematik, 3

Tags: geometry , construction , ratio

The points $S$ lie on side $AB$, $T$ on side $BC$, and $U$ on side $CA$ of a triangle so that the following applies: $\overline{AS} : \overline{SB} = 1 : 2$, $\overline{BT} : \overline{TC} = 2 : 3$ and $\overline{CU} : \overline{UA} = 3 : 1$. Construct the triangle $ABC$ if only the points $S, T$ and $U$ are given.

JOM 2015 Shortlist, A5

Tags: Inequality , inequalities

Let $ a, b, c $ be the side length of a triangle, with $ ab + bc + ca = 18 $ and $ a, b, c > 1 $. Prove that $$ \sum_{cyc}\frac{1}{(a - 1)^3} > \frac{1}{a + b + c - 3} $$

2017 QEDMO 15th, 9

Tags: number theory , divisible

Let $p$ be a prime number and $h$ be a natural number smaller than $p$. We set $n = ph + 1$. Prove that if $2^{n-1}-1$, but not $2^h-1$, is divisible by $n$, then $n$ is a prime number.

2021 Saudi Arabia JBMO TST, 3

Tags: Junior , Balkan , shortlist , 2020 , algebra , Sequence

Consider the sequence $a_1, a_2, a_3, ...$ defined by $a_1 = 9$ and $a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}$ for $n \ge 1$. Find all natural numbers $n$ for which $a_n$ is a perfect square of an integer.

2012 Online Math Open Problems, 6

Tags: Online Math Open

Alice's favorite number has the following properties: [list] [*] It has 8 distinct digits. [*]The digits are decreasing when read from left to right. [*]It is divisible by 180.[/list] What is Alice's favorite number? [i]Author: Anderson Wang[/i]

1981 Romania Team Selection Tests, 1.

Tags: algebra

Consider the set $M$ of all sequences of integers $s=(s_1,\ldots,s_k)$ such that $0\leqslant s_i\leqslant n,\; i=1,2,\ldots,k$ and let $M(s)=\max\{s_1,\ldots,s_k\}$. Show that \[\sum_{s\in A} M(s)=(n+1)^{k+1}-(1^k+2^k+\ldots +(n+1)^k).\] [i]Ioan Tomescu[/i]

2021 Saudi Arabia Training Tests, 36

Tags: combinatorics

There are $330$ seats in the first row of the auditorium. Some of these seats are occupied by $25$ viewers. Prove that among the pairwise distances between these viewers, there are two equal.

2002 Junior Balkan Team Selection Tests - Moldova, 1

Tags: number theory , greatest common divisor

For any integer $n$ we define the numbers $a = n^5 + 6n^3 + 8n$ ¸ $b = n^4 + 4n^2 + 3$. Prove that the numbers $a$ and $b$ are relatively prime or have the greatest common factor of $3$.

2007 Purple Comet Problems, 10

Tags: trigonometry , complex numbers

For a particular value of the angle $\theta$ we can take the product of the two complex numbers $(8+i)\sin\theta+(7+4i)\cos\theta$ and $(1+8i)\sin\theta+(4+7i)\cos\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find the largest value for $a+b$.

2022 MIG, 12

Tags: 2022 B Individual

Out of a sample of $100$ people, $24$ do not like red or blue, $40$ like both red and blue, and $50$ people like red. How many people like blue but not red? $\textbf{(A) }24\qquad\textbf{(B) }26\qquad\textbf{(C) }48\qquad\textbf{(D) }64\qquad\textbf{(E) }76$

2016 Costa Rica - Final Round, G1

Tags: geometry , circumcircle

Let $\vartriangle ABC$ be isosceles with $AB = AC$. Let $\omega$ be its circumscribed circle and $O$ its circumcenter. Let $D$ be the second intersection of $CO$ with $\omega$. Take a point $E$ in $AB$ such that $DE \parallel AC$ and suppose that $AE: BE = 2: 1$. Show that $\vartriangle ABC$ is equilateral.

2017 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry , circumcircle , reflection , orthocenter

Let $P$ be a point in the interior of the acute-angled triangle $ABC$. Prove that if the reflections of $P$ with respect to the sides of the triangle lie on the circumcircle of the triangle, then $P$ is the orthocenter of $ABC$.

1971 IMO Longlists, 29

Tags: geometry , rhombus , geometry proposed

A rhombus with its incircle is given. At each vertex of the rhombus a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii $r_1,r_2$, while the incircle has radius $r$. Given that $r_1$ and $r_2$ are natural numbers and that $r_1r_2=r$, find $r_1,r_2,$ and $r$.

2004 Putnam, A6

Tags: Putnam , function , integration , calculus , college contests

Suppose that $f(x,y)$ is a continuous real-valued function on the unit square $0\le x\le1,0\le y\le1.$ Show that $\int_0^1\left(\int_0^1f(x,y)dx\right)^2dy + \int_0^1\left(\int_0^1f(x,y)dy\right)^2dx$ $\le\left(\int_0^1\int_0^1f(x,y)dxdy\right)^2 + \int_0^1\int_0^1\left[f(x,y)\right]^2dxdy.$

2018 Lusophon Mathematical Olympiad, 3

Tags: number theory , sum of digits , minimum

For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$.

1971 Czech and Slovak Olympiad III A, 5

Tags: geometry , Equilateral , Triangles , Locus

Let $ABC$ be a given triangle. Find the locus $\mathbf M$ of all vertices $Z$ such that triangle $XYZ$ is equilateral where $X$ is any point of segment $AB$ and $Y\neq X$ lies on ray $AC.$

2007 ITest, 38

Tags: geometry , 3D geometry , floor function , logarithms , inequalities

Find the largest positive integer that is equal to the cube of the sum of its digits.

2009 IMAR Test, 4

Tags: number theory , Perfect Square , number theory with sequences , Sequence

Given any $n$ positive integers, and a sequence of $2^n$ integers (with terms among them), prove there exists a subsequence made of consecutive terms, such that the product of its terms is a perfect square. Also show that we cannot replace $2^n$ with any lower value (therefore $2^n$ is the threshold value for this property).

LMT Accuracy Rounds, 2023 S2

Tags: algebra

Evaluate $2023^2 -2022^2 +2021^2 -2020^2$.

2014 Purple Comet Problems, 13

Tags: percent

A jar was filled with jelly beans so that $54\%$ of the beans were red, $30\%$ of the beans were green, and $16\%$ of the beans were blue. Alan then removed the same number of red jelly beans and green jelly beans from the jar so that now $20\%$ of the jelly beans in the jar are blue. What percent of the jelly beans in the jar are now red?