Olympiad problems

2009 Germany Team Selection Test, 1

Tags: geometry , combinatorics , extremal combinatorics , point set , bezout s identity , pigeonhole principle

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

2001 Czech-Polish-Slovak Match, 4

Tags: ratio , geometric transformation , homothety , geometry

Distinct points $A$ and $B$ are given on the plane. Consider all triangles $ABC$ in this plane on whose sides $BC,CA$ points $D,E$ respectively can be taken so that (i) $\frac{BD}{BC}=\frac{CE}{CA}=\frac{1}{3}$; (ii) points $A,B,D,E$ lie on a circle in this order. Find the locus of the intersection points of lines $AD$ and $BE$.

2013 Vietnam Team Selection Test, 2

Tags: divisibility , function , infinitely many solutions , perfect square , number theory

a. Prove that there are infinitely many positive integers $t$ such that both $2012t+1$ and $2013t+1$ are perfect squares. b. Suppose that $m,n$ are positive integers such that both $mn+1$ and $mn+n+1$ are perfect squares. Prove that $8(2m+1)$ divides $n$.

2000 Romania National Olympiad, 2b

Tags: geometry , inequalities , geometric inequality

If $a, b, c$ represent the lengths of the sides of a triangle, prove that: $$\frac{a}{b-a+c}+ \frac{b}{b-a+c}+ \frac{c}{b-a+c} \ge 3$$

2006 AMC 12/AHSME, 16

Tags: trigonometry , congruent triangles , geometry

Regular hexagon $ ABCDEF$ has vertices $ A$ and $ C$ at $ (0,0)$ and $ (7,1)$, respectively. What is its area? $ \textbf{(A) } 20\sqrt {3} \qquad \textbf{(B) } 22\sqrt {3} \qquad \textbf{(C) } 25\sqrt {3} \qquad \textbf{(D) } 27\sqrt {3} \qquad \textbf{(E) } 50$

1992 Cono Sur Olympiad, 2

Tags: trigonometry , circumcircle , geometry

In a $\triangle {ABC}$, consider a point $E$ in $BC$ such that $AE \perp BC$. Prove that $AE=\frac{bc}{2r}$, where $r$ is the radio of the circle circumscripte, $b=AC$ and $c=AB$.

2023 Indonesia TST, 1

Tags: number theory

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

2020 AMC 12/AHSME, 21

Tags: floor function

How many positive integers $n$ satisfy$$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$$(Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.) $\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$

2023 Princeton University Math Competition, 1

Tags: fe , algebra

1. Given $n \geq 1$, let $A_{n}$ denote the set of the first $n$ positive integers. We say that a bijection $f: A_{n} \rightarrow A_{n}$ has a hump at $m \in A_{n} \backslash\{1, n\}$ if $f(m)>f(m+1)$ and $f(m)>f(m-1)$. We say that $f$ has a hump at 1 if $f(1)>f(2)$, and $f$ has a hump at $n$ if $f(n)>f(n-1)$. Let $P_{n}$ be the probability that a bijection $f: A_{n} \rightarrow A_{n}$, when selected uniformly at random, has exactly one hump. For how many positive integers $n \leq 2020$ is $P_{n}$ expressible as a unit fraction?

2022 Korea National Olympiad, 4

Tags: inequalities , combinatorics

For positive integers $m, n$ ($m>n$), $a_{n+1}, a_{n+2}, ..., a_m$ are non-negative integers that satisfy the following inequality. $$ 2> \frac{a_{n+1}}{n+1} \ge \frac{a_{n+2}}{n+2} \ge \cdots \ge \frac{a_m}{m}$$ Find the number of pair $(a_{n+1}, a_{n+2}, \cdots, a_m)$.

2007 Today's Calculation Of Integral, 242

Tags: calculus , integration , function , geometry , calculus computations

A cubic function $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ touches a line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha$ and intersects $ x \equal{} \beta \ (\alpha \neq \beta)$. Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.

2003 Czech-Polish-Slovak Match, 4

Tags: geometry

Point $P$ lies on the median from vertex $C$ of a triangle $ABC$. Line $AP$ meets $BC$ at $X$, and line $BP$ meets $AC$ at $Y$ . Prove that if quadrilateral $ABXY$ is cyclic, then triangle $ABC$ is isosceles.

2006 Mediterranean Mathematics Olympiad, 2

Tags: inequalities , parallelogram , geometry

Let $P$ be a point inside a triangle $ABC$, and $A_1B_2,B_1C_2,C_1A_2$ be segments passing through $P$ and parallel to $AB, BC, CA$ respectively, where points $A_1, A_2$ lie on $BC, B_1, B_2$ on $CA$, and $C_1,C_2$ on $AB$. Prove that \[ \text{Area}(A_1A_2B_1B_2C_1C_2) \ge \frac{1}{2}\text{Area}(ABC)\]

2014 Ukraine Team Selection Test, 10

Tags: point , combinatorial geometry , combinatorics , convex hull

Find all positive integers $n \ge 4$ for which there are $n$ points in general position on the plane such that an arbitrary triangle with vertices belonging to the convex hull of these $n$ points, containing exactly one of $n - 3$ points inside remained.

2015 Princeton University Math Competition, A7/B8

Tags: algebra

We define the ridiculous numbers recursively as follows: [list=a] [*]1 is a ridiculous number. [*]If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers. [/list] A closed interval $I$ is ``boring'' if [list] [*]$I$ contains no ridiculous numbers, and [*]There exists an interval $[b,c]$ containing $I$ for which $b$ and $c$ are both ridiculous numbers. [/list] The smallest non-negative $l$ such that there does not exist a boring interval with length $l$ can be represented in the form $\dfrac{a + b\sqrt{c}}{d}$ where $a, b, c, d$ are integers, $\gcd(a, b, d) = 1$, and no integer square greater than 1 divides $c$. What is $a + b + c + d$?

2005 Tournament of Towns, 2

Tags: construction , geometry

A segment of length $\sqrt2 + \sqrt3 + \sqrt5$ is drawn. Is it possible to draw a segment of unit length using a compass and a straightedge? [i](3 points)[/i]

2019 Saint Petersburg Mathematical Olympiad, 7

Tags: geometry

Let $\omega$ and $O$ be respectively the circumcircle and the circumcenter of a triangle $ABC$. The line $AO$ intersects $\omega$ second time at $A'$. $M_B$ and $M_C$ are the midpoints of $AC$ and $AB$, respectively. The lines $A'M_B$ and $A'M_C$ intersect $\omega$ secondly at points $B'$ and $C$, and also intersect $BC$ at points $D_B$ and $D_C$, respectively. The circumcircles of $CD_BB'$ and $BD_CC'$ intersect at points $P$ and $Q$. Prove that $O$, $P$, $Q$ are collinear. [i] (М. Германсков)[/i] [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

2019 China Second Round Olympiad, 3

Tags: algebra , number theory

Let $m$ be an integer where $|m|\ge 2$. Let $a_1,a_2,\cdots$ be a sequence of integers such that $a_1,a_2$ are not both zero, and for any positive integer $n$, $a_{n+2}=a_{n+1}-ma_n$. Prove that if positive integers $r>s\ge 2$ satisfy $a_r=a_s=a_1$, then $r-s\ge |m|$.

1968 IMO Shortlist, 5

Tags: inequalities , geometry , polygon , geometric inequality

Let $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon ($n \geq 3$) inscribed in a circle of radius $r$. Prove the inequality \[(n + 1)h_n+1 - nh_n > r.\] Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n \geq 3.$

1974 Vietnam National Olympiad, 2

Tags: number theory , divisibility

i) How many integers $n$ are there such that $n$ is divisible by $9$ and $n+1$ is divisible by $25$? ii) How many integers $n$ are there such that $n$ is divisible by $21$ and $n+1$ is divisible by $165$? iii) How many integers $n$ are there such that $n$ is divisible by $9, n + 1$ is divisible by $25$, and $n + 2$ is divisible by $4$?

2003 JBMO Shortlist, 6

Tags: inequalities

Parallels to the sides of a triangle passing through an interior point divide the inside of a triangle into $6$ parts with the marked areas as in the figure. Show that $\frac{a}{\alpha}+\frac{b}{\beta}+\frac{c}{\gamma}\ge \frac{3}{2}$ [img]https://cdn.artofproblemsolving.com/attachments/a/a/b0a85df58f2994b0975b654df0c342d8dc4d34.png[/img]

2008 Purple Comet Problems, 20

Tags:

Find the least positive integer $n$ such that the decimal representation of the binomial coefficient $\dbinom{2n}{n}$ ends in four zero digits.

2014 Contests, 1

Tags: greatest common divisor , number theory

For $x, y$ positive integers, $x^2-4y+1$ is a multiple of $(x-2y)(1-2y)$. Prove that $|x-2y|$ is a square number.

2018 Bundeswettbewerb Mathematik, 1

Tags: digit , number theory , arithmetic

Find the largest positive integer with the property that each digit apart from the first and the last one is smaller than the arithmetic mean of her neighbours.

2007 Stanford Mathematics Tournament, 20

Tags:

Let there be $ 4n \plus{} 2$ distinct paths in space with exactly $ 2n^2 \plus{} 6n \plus{} 1$ points at which exactly two of the paths intersect. (A path never intersects itself.) What is the maximum number of points where exactly three paths intersect?