Olympiad problems

1981 USAMO, 4

Tags: geometry

The sum of the measures of all the face angles of a given complex polyhedral angle is equal to the sum of all its dihedral angles. Prove that the polyhedral angle is a trihedral angle. $\mathbf{Note:}$ A convex polyhedral angle may be formed by drawing rays from an exterior point to all points of a convex polygon.

2019 Centroamerican and Caribbean Math Olympiad, 5

Tags: algebra , inequalities

Let $a,\ b$ and $c$ be positive real numbers so that $a+b+c=1$. Show that $$a\sqrt{a^2+6bc}+b\sqrt{b^2+6ac}+c\sqrt{c^2+6ab}\leq\frac{3\sqrt{2}}{4}$$

2015 IMO Shortlist, G4

Tags: geometry

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

1998 Brazil Team Selection Test, Problem 2

Tags: arithmetic sequence , combinatorics

Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.

2011 Indonesia TST, 1

Tags: algebra , functional equation , functional equation in q , functional

Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions: (i) $f(x)$ is an integer if and only if $x$ is an integer; (ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.

2002 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , symmetry , combinatorics

Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$. Find the least number $k$ such that $s(M,N)\le k$, for all points $M,N$. [i]Dinu Șerbănescu[/i]

2018 Nepal National Olympiad, 3b

Tags: geometry

[b] Problem Section #3 NOTE: Neglect that HF and CD.

2006 Irish Math Olympiad, 5

Tags: geometry , modular arithmetic , combinatorics , circles , intersection , double counting

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

2021 LMT Spring, B3

Tags: combinatorics

Aidan rolls a pair of fair, six sided dice. Let$ n$ be the probability that the product of the two numbers at the top is prime. Given that $n$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers, find $a +b$. [i]Proposed by Aidan Duncan[/i]

1994 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 7

Tags:

We have a hexagon such that all its edges touch a circle. If five of the edges have lengths 1,2,3,4, and 5 as on the figure, how long is the last edge? [img]http://i250.photobucket.com/albums/gg265/geometry101/HexagonImage.jpg[/img] A. 1 B. 3 C. 15/8 D. $ \sqrt{15}$ E. Not uniquely determined, more than one possibility

2004 Bundeswettbewerb Mathematik, 4

Tags: number theory

Prove that there exist infinitely many pairs $\left(x;\;y\right)$ of different positive rational numbers, such that the numbers $\sqrt{x^2+y^3}$ and $\sqrt{x^3+y^2}$ are both rational.

2001 National Olympiad First Round, 21

Tags: geometry , trigonometry

Let $b$ be the length of the largest diagonal and $c$ be the length of the smallest diagonal of a regular nonagon with side length $a$. Which one of the followings is true? $ \textbf{(A)}\ b=\dfrac{a+c}2 \qquad\textbf{(B)}\ b=\sqrt {ac} \qquad\textbf{(C)}\ b^2=\dfrac{a^2+c^2}2 \\ \textbf{(D)}\ c=a+b \qquad\textbf{(E)}\ c^2=a^2+b^2 $

2001 Bundeswettbewerb Mathematik, 1

Tags: algebra

On a table there is a pile with $ T$ tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process? a.) $ T \equal{} 1000$ (Cono Sur) b.) $ T \equal{} 2001$ (BWM)

1971 AMC 12/AHSME, 4

Tags:

After simple interest for two months at $5\%$ per annum was credited, a Boy Scout Troop had a total of $\textdollar 255.31$ in the Council Treasury. The interest credited was a number of dollars plus the following number of cents $\textbf{(A) }11\qquad\textbf{(B) }12\qquad\textbf{(C) }13\qquad\textbf{(D) }21\qquad \textbf{(E) }31$

2012 Hanoi Open Mathematics Competitions, 9

Tags:

[b]Q9.[/b] Evaluate the integer part of the number \[H= \sqrt{1+2011^2+ \frac{2011^2}{2012^2}}+ \frac{2011}{2012}.\]

2005 Grigore Moisil Urziceni, 2

Tags: geometric sequence , number theory , inequalities

Find all triples $ (x,y,z) $ of natural numbers that are in geometric progression and verify the inequalities $$ 4016016\le x<y<z\le 4020025. $$

2001 Abels Math Contest (Norwegian MO), 4

Tags: combinatorics

At a two-day team competition in chess, three schools with $15$ pupils each attend. Each student plays one game against each player on the other two teams, ie a total of $30$ chess games per student. a) Is it possible for each student to play exactly $15$ games after the first day? b) Show that it is possible for each student to play exactly $16$ games after the first day. c) Assume that each student has played exactly $16$ games after the first day. Show that there are three students, one from each school, who have played their three parties

2015 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: algebra , inequality

Let $a$, $b$ and $c$ be positive real numbers such that $abc=2015$. Prove that $$\frac{a+b}{a^2+b^2}+\frac{b+c}{b^2+c^2}+\frac{c+a}{c^2+a^2} \leq \frac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{\sqrt{2015}}$$

2012 Saint Petersburg Mathematical Olympiad, 7

Tags: combinatorics , number theory

We have $2012$ sticks with integer length, and sum of length is $n$. We need to have sticks with lengths $1,2,....,2012$. For it we can break some sticks ( for example from stick with length $6$ we can get $1$ and $4$). For what minimal $n$ it is always possible?

2012 Kyrgyzstan National Olympiad, 3

Tags: complex number , geometry

Prove that if the diagonals of a convex quadrilateral are perpendicular, then the feet of perpendiculars dropped from the intersection point of diagonals on the sides of this quadrilateral lie on one circle. Is the converse true?

1999 All-Russian Olympiad, 5

Tags: complementary counting , combinatorics

An equilateral triangle of side $n$ is divided into equilateral triangles of side $1$. Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle.

2015 AMC 10, 23

Tags: quadratic , function , polynomial , vieta , sfft , algebra

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

2014 IMO Shortlist, G2

Tags: geometry

Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$. [i]Proposed by Estonia[/i]

2000 AMC 10, 1

Tags:

In the year $ 2001$, the United States will host the International Mathematical Olympiad. Let $ I$, $ M$, and $ O$ be distinct positive integers such that the product $ I\cdot M \cdot O \equal{} 2001$. What's the largest possible value of the sum $ I \plus{} M \plus{} O$? $ \textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\ 111 \qquad \textbf{(E)}\ 671$

2019 Saudi Arabia Pre-TST + Training Tests, 2.1

Tags: number theory , perfect square

Suppose that $a, b, c,d$ are pairwise distinct positive integers such that $a+b = c+d = p$ for some odd prime $p > 3$ . Prove that $abcd$ is not a perfect square.