Olympiad problems

2010 Today's Calculation Of Integral, 654

A function $f(x)$ defined in $x\geq 0$ satisfies $\lim_{x\to\infty} \frac{f(x)}{x}=1$. Find $\int_0^{\infty} \{f(x)-f'(x)\}e^{-x}dx$. [i]1997 Hokkaido University entrance exam/Science[/i]

2006 Croatia Team Selection Test, 2

Tags: inequalities

Assume that $a, b,$ and $c$ are positive real numbers for which $(a+b)(a+c)(b+c) = 1$. Prove that $ab+bc+ca \leq\frac{3 }{4}.$

2005 JBMO Shortlist, 3

Tags: circumcircle , trigonometry , angle bisector , geometry

Let $ABCDEF$ be a regular hexagon and $M\in (DE)$, $N\in(CD)$ such that $m (\widehat {AMN}) = 90^\circ$ and $AN = CM \sqrt {2}$. Find the value of $\frac{DM}{ME}$.

1967 Miklós Schweitzer, 4

Tags: inequalities , logarithm , real analysis

Let $ a_1,a_2,...,a_N$ be positive real numbers whose sum equals $ 1$. For a natural number $ i$, let $ n_i$ denote the number of $ a_k$ for which $ 2^{1-i} \geq a_k \geq 2^{-i}$ holds. Prove that \[ \sum_{i=1}^{\infty} \sqrt{n_i2^{-i}} \leq 4+\sqrt{\log_2 N}.\] [i]L. Leinder[/i]

2007 USAMO, 2

Tags: geometry , combinatorics , combinatorial geometry , soddy

A square grid on the Euclidean plane consists of all points $(m,n)$, where $m$ and $n$ are integers. Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least $5$?

2020-IMOC, A5

Tags: inequalities

Let $0<c<1$ be a given real number. Determine the least constant $K$ such that the following holds: For all positive real $M$ that is greater than $1$, there exists a strictly increasing sequence $x_0, x_1, \ldots, x_n$ (of arbitrary length) such that $x_0=1, x_n\geq M$ and \[\sum_{i=0}^{n-1}\frac{\left(x_{i+1}-x_i\right)^c}{x_i^{c+1}}\leq K.\] (From 2020 IMOCSL A5. I think this problem is particularly beautiful so I want to make a separate thread for it :D )

2023 Serbia National Math Olympiad, 1

Tags: geometry

Given is a triangle $ABC$ with circumcenter $O$ and orthocenter $H$. If $O_a, O_b, O_c$ denote the circumcenters of $\triangle AOH$, $\triangle BOH$, $\triangle COH$, then prove that $AO_a, BO_b, CO_c$ are concurrent.

2011 IMC, 1

Tags: topology , real analysis

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function. A point $x$ is called a [i]shadow[/i] point if there exists a point $y\in \mathbb{R}$ with $y>x$ such that $f(y)>f(x).$ Let $a<b$ be real numbers and suppose that $\bullet$ all the points of the open interval $I=(a,b)$ are shadow points; $\bullet$ $a$ and $b$ are not shadow points. Prove that a) $f(x)\leq f(b)$ for all $a<x<b;$ b) $f(a)=f(b).$ [i]Proposed by José Luis Díaz-Barrero, Barcelona[/i]

2001 Paraguay Mathematical Olympiad, 3

Tags: number theory , sum of digits , divisible

Find a $10$-digit number, in which no digit is zero, that is divisible by the sum of their digits.

2009 AMC 10, 7

Tags: percent

A carton contains milk that is $ 2\%$ fat, and amount that is $ 40\%$ less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk? $ \textbf{(A)}\ \frac{12}{5} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac{10}{3} \qquad \textbf{(D)}\ 38 \qquad \textbf{(E)}\ 42$

2005 ISI B.Stat Entrance Exam, 5

Tags: geometry , incenter , triangle

Consider an acute angled triangle $PQR$ such that $C,I$ and $O$ are the circumcentre, incentre and orthocentre respectively. Suppose $\angle QCR, \angle QIR$ and $\angle QOR$, measured in degrees, are $\alpha, \beta$ and $\gamma$ respectively. Show that \[\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}>\frac{1}{45}\]

2020 Peru Cono Sur TST., P4

Tags: perfect square , number theory , phi function , euler s phi function

Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

2007 Ukraine Team Selection Test, 3

Tags: polynomial , algebra , logarithm

It is known that $ k$ and $ n$ are positive integers and \[ k \plus{} 1\leq\sqrt {\frac {n \plus{} 1}{\ln(n \plus{} 1)}}.\] Prove that there exists a polynomial $ P(x)$ of degree $ n$ with coefficients in the set $ \{0,1, \minus{} 1\}$ such that $ (x \minus{} 1)^{k}$ divides $ P(x)$.

2015 Estonia Team Selection Test, 3

Tags: number theory , rational

Let $q$ be a fixed positive rational number. Call number $x$ [i]charismatic [/i] if there exist a positive integer $n$ and integers $a_1, a_2, . . . , a_n$ such that $x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} ...(q + n)^{a_n}$. a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic. b) Is it true for every $q$ that, for every charismatic number $x$, the number $x + 1$ is charismatic, too?

1955 Czech and Slovak Olympiad III A, 2

Tags: geometry , 3d geometry , sphere , prism

Let $\mathsf{S}_1,\mathsf{S}_2$ be concentric spheres with radii $a,b$ respectively, where $a<b.$ Denote $ABCDA'B'C'D'$ a square cuboid ($ABCD,A'B'C'D$ are the squares and $AA'\parallel BB'\parallel CC'\parallel DD'$) such that $A,B,C,D\in\mathsf{S}_2$ and the plane $A'B'C'D'$ is tangent to $\mathsf{S}_1.$ Finally assume that \[\frac{AB}{AA'}=\frac ab.\] Compute the lengths $AB,AA'.$ How many of such cuboids exist (up to a congruence)?

2014 Online Math Open Problems, 10

Tags: geometry

Let $A_1A_2 \dots A_{4000}$ be a regular $4000$-gon. Let $X$ be the foot of the altitude from $A_{1986}$ onto diagonal $A_{1000}A_{3000}$, and let $Y$ be the foot of the altitude from $A_{2014}$ onto $A_{2000}A_{4000}$. If $XY = 1$, what is the area of square $A_{500}A_{1500}A_{2500}A_{3500}$? [i]Proposed by Evan Chen[/i]

2010 Contests, 1

Tags: inequalities , floor function , combinatorics

Some of $n$ towns are connected by two-way airlines. There are $m$ airlines in total. For $i = 1, 2, \cdots, n$, let $d_i$ be the number of airlines going from town $i$. If $1\le d_i \le 2010$ for each $i = 1, 2,\cdots, 2010$, prove that \[\displaystyle\sum_{i=1}^n d_i^2\le 4022m- 2010n\] Find all $n$ for which equality can be attained. [i]Proposed by Aleksandar Ilic[/i]

LMT Team Rounds 2021+, 4

Tags: combinatorics

Jeff has a deck of $12$ cards: $4$ $L$s, $4$ $M$s, and $4$ $T$s. Armaan randomly draws three cards without replacement. The probability that he takes $3$ $L$s can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m +n$.

1998 Gauss, 7

Tags: gauss

A rectangular field is 80 m long and 60 m wide. If fence posts are placed at the corners and are 10 m apart along the 4 sides of the field, how many posts are needed to completely fence the field? $\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 26 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 32$

2001 Saint Petersburg Mathematical Olympiad, 9.3

Tags: geometry , pentagon , circumscribed

A convex pentagon $ABCDE$ is given with $AB=BC$, $CD=DE$ and $\angle A=\angle C=\angle E>90^{\circ}$. Prove that the pentagon is circumscribed [I]Proposed by F. Baharev[/i]

2000 Baltic Way, 10

Tags: function , combinatorics

Two positive integers are written on the blackboard. Initially, one of them is $2000$ and the other is smaller than $2000$. If the arithmetic mean $ m$ of the two numbers on the blackboard is an integer, the following operation is allowed: one of the two numbers is erased and replaced by $ m$. Prove that this operation cannot be performed more than ten times. Give an example where the operation is performed ten times.

1972 IMO Longlists, 26

Tags: algebra , inequality system

Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 \\ (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 \\ (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 \\ (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 \\ (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \\ \end{eqnarray*}

2019 Mathematical Talent Reward Programme, MCQ: P 1

Tags: function

Let $f : (0, \infty) \to \mathbb{R}$ is differentiable such that $\lim \limits_{x \to \infty} f(x)=2019$ Then which of the following is correct? [list=1] [*] $\lim \limits_{x \to \infty} f'(x)$ always exists but not necessarily zero. [*] $\lim \limits_{x \to \infty} f'(x)$ always exists and is equal to zero. [*] $\lim \limits_{x \to \infty} f'(x)$ may not exist. [*] $\lim \limits_{x \to \infty} f'(x)$ exists if $f$ is twice differentiable. [/list]

2015 CCA Math Bonanza, L1.4

Tags:

How many digits are in the base $10$ representation of $3^{30}$ given $\log 3 = 0.47712$? [i]2015 CCA Math Bonanza Lightning Round #1.4[/i]

2024 Serbia Team Selection Test, 4

Tags: algebra

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a bijection and let $k$ be a positive integer such that $|f(x+1)-f(x)| \leq k$ for all positive integers $x$. Show that there exists an integer $d$, such that $f(x)=x+d$ for infinitely many positive integers $x$.