Olympiad problems

2006 MOP Homework, 3

Prove for every irrational real number a, there are irrational numbers b and b' such that a+b and ab' are rational while a+b' and ab are irrational.

2004 Canada National Olympiad, 1

Tags: algebra , system of equations

Find all ordered triples $ (x,y,z)$ of real numbers which satisfy the following system of equations: \[ \left\{\begin{array}{rcl} xy & \equal{} & z \minus{} x \minus{} y \\ xz & \equal{} & y \minus{} x \minus{} z \\ yz & \equal{} & x \minus{} y \minus{} z \end{array} \right. \]

1985 IMO Longlists, 52

Tags: geometry , geometry proposed

In the triangle $ABC$, let $B_1$ be on $AC, E$ on $AB, G$ on $BC$, and let $EG$ be parallel to $AC$. Furthermore, let $EG$ be tangent to the inscribed circle of the triangle $ABB_1$ and intersect $BB_1$ at $F$. Let $r, r_1$, and $r_2$ be the inradii of the triangles $ABC, ABB_1$, and $BFG$, respectively. Prove that $r = r_1 + r_2.$

2009 Ukraine National Mathematical Olympiad, 2

Tags: combinatorics proposed , combinatorics

There is a knight in the left down corner of $2009 \times 2009$ chessboard. The row and the column containing this corner are painted. The knight cannot move into painted cell and after its move new row and column that contains a square with knight become painted. Is it possible to paint all rows and columns of the chessboard?

2011 LMT, 12

Tags:

In a round robin tournament of $7$ people, each person plays every other person exactly once in a game of table tennis. For each game played, the winner is given $2$ points, the loser is given $0$ points, and in the event of a tie, each player gets $1$ point. At the end of the tournament, what is the average score of the $7$ people?

2020 Tournament Of Towns, 5

Tags: geometry , concurrency , concurrent , cyclic quadrilateral , circles

Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent. Maxim Didin

2007 Mongolian Mathematical Olympiad, Problem 2

Tags: geometry

Given $101$ segments in a line, prove that there exists $11$ segments meeting in $1$ point or $11$ segments such that every two of them are disjoint.

1970 AMC 12/AHSME, 15

Tags: analytic geometry , arithmetic sequence , AMC

Lines in the xy-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $(-4,5)$ and $(5,-1).$ One of these lines has the equation $\textbf{(A) }3x-2y-1=0\qquad\textbf{(B) }4x-5y+8=0\qquad\textbf{(C) }5x+2y-23=0\qquad$ $\textbf{(D) }x+7y-31=0\qquad \textbf{(E) }x-4y+13=0$

2018 PUMaC Algebra A, 8

Tags: algebra

$$\frac{p}{q} = \sum_{n = 1}^\infty \frac{1}{2^{n + 6}} \frac{(10 - 4\cos^2(\frac{\pi n}{24})) (1 - (-1)^n) - 3\cos(\frac{\pi n}{24}) (1 + (-1)^n)}{25 - 16\cos^2(\frac{\pi n}{24})}$$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

2010 Indonesia TST, 2

Tags: modular arithmetic , number theory , number theory proposed

Let $ A\equal{}\{n: 1 \le n \le 2009^{2009},n \in \mathbb{N} \}$ and let $ S\equal{}\{n: n \in A,\gcd \left(n,2009^{2009}\right)\equal{}1\}$. Let $ P$ be the product of all elements of $ S$. Prove that \[ P \equiv 1 \pmod{2009^{2009}}.\] [i]Nanang Susyanto, Jogjakarta[/i]

1984 AIME Problems, 5

Tags: logarithms , symmetry

Determine the value of $ab$ if $\log_8 a + \log_4 b^2 = 5$ and $\log_8 b + \log_4 a^2 = 7$.

1968 German National Olympiad, 5

Tags: algebra , inequalities , trigonometry

Prove that for all real numbers $x$ of the interval $0 < x <\pi$ the inequality $$\sin x +\frac12 \sin 2x +\frac13 \sin 3x > 0$$ holds.

2015 AIME Problems, 3

Tags: number theory , modular arithmetic

Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$.

2015 May Olympiad, 5

Tags: combinatorics

Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a friend to everyone. How many people is Zoila a friend of, person number twenty-six? Clarification: If $A$ is a friend of $B$ then $B$ is a friend of $A$.

2013 District Olympiad, 2

Tags: superior algebra , superior algebra unsolved

Problem 2. A group $\left( G,\cdot \right)$ has the propriety$\left( P \right)$, if, for any automorphism f for G,there are two automorphisms g and h in G, so that $f\left( x \right)=g\left( x \right)\cdot h\left( x \right)$, whatever $x\in G$would be. Prove that: (a) Every group which the property $\left( P \right)$ is comutative. (b) Every commutative finite group of odd order doesn’t have the $\left( P \right)$ property. (c) No finite group of order $4n+2,n\in \mathbb{N}$, doesn’t have the $\left( P \right)$property. (The order of a finite group is the number of elements of that group).

2005 National Olympiad First Round, 26

Tags:

For every positive integer $n$, $f(2n+1)=2f(2n)$, $f(2n)=f(2n-1)+1$, and $f(1)=0$. What is the remainder when $f(2005)$ is divided by $5$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

2013 Bundeswettbewerb Mathematik, 1

Tags:

Suppose $m$ and $n$ are positive integers such that $m^2+n^2+m$ is divisible by $mn$. Prove that $m$ is a square number.

Kvant 2021, M2648

Tags: geometry , inradius , Kvant

The triangle $ABC$ is given. Consider the point $C'{}$ on the side $AB$ such that the segment $CC'$ divides the triangle into two triangles with equal radii of inscribed circles. Denote by $t_c$ the length of the segment $CC'$. Similarly, we define $t_a$ and $t_b$. Express the area of triangle $ABC$ in terms of $t_a,t_b$ and $t_c$. [i]Proposed by K. Mosevich[/i]

2014 Saudi Arabia GMO TST, 4

Tags: rational , set , algebra

Let $X$ be a set of rational numbers satisfying the following two conditions: (a) The set $X$ contains at least two elements, (b) For any $x, y$ in $X$, if $x \ne y$ then there exists $z$ in $X$ such that either $\left| \frac{x- z}{y - z} \right|= 2$ or $\left| \frac{y -z}{x - z} \right|= 2$ . Prove that $X$ contains infinitely many elements.

Cono Sur Shortlist - geometry, 2005.G3.4

Tags: geometry , incenter , geometric transformation , reflection , symmetry , perpendicular bisector , geometry proposed

Let $ABC$ be a isosceles triangle, with $AB=AC$. A line $r$ that pass through the incenter $I$ of $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. Let $F$ and $G$ be points on $BC$ such that $BF=CE$ and $CG=BD$. Show that the angle $\angle FIG$ is constant when we vary the line $r$.

2008 Mathcenter Contest, 10

Tags: combinatorics , this problem is a nightmare

One test is a multiple choice test with $5$ questions, each with $4$ options, $2000$ candidates, each choosing only one answer for each item.Find the smallest possible integer $n$ that gives a student's answer sheet the following properties: In the student's answer sheet $n$, there are four sheets in it. Any two of the four tiles have exactly the same three answers. [i](tatari/nightmare)[/i]

1960 AMC 12/AHSME, 18

Tags: AMC

The pair of equations $3^{x+y}=81$ and $81^{x-y}=3$ has: $ \textbf{(A)}\ \text{no common solution} \qquad\textbf{(B)}\ \text{the solution} \text{ } x=2, y=2\qquad$ $\textbf{(C)}\ \text{the solution} \text{ } x=2\frac{1}{2}, y=1\frac{1}{2} \qquad$ $\textbf{(D)}\text{ a common solution in positive and negative integers} \qquad$ $\textbf{(E)}\ \text{none of these} $

2018 Iran MO (1st Round), 6

Tags: number theory , modular arithmetic

Let $n$ be the smallest positive integer such that the remainder of $3n+45$, when divided by $1060$, is $16$. Find the remainder of $18n+17$ upon division by $1920$.

2014 Mid-Michigan MO, 10-12

Tags: Mid-Michigan MO , algebra , geometry , combinatorics , number theory

[b]p1.[/b] The length of the side $AB$ of the trapezoid with bases $AD$ and $BC$ is equal to the sum of lengths $|AD|+|BC|$. Prove that bisectors of angles $A$ and $B$ do intersect at a point of the side $CD$. [b]p2.[/b] Polynomials $P(x) = x^4 + ax^3 + bx^2 + cx + 1$ and $Q(x) = x^4 + cx^3 + bx^2 + ax + 1$ have two common roots. Find these common roots of both polynomials. [b]p3.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace: $\bullet$ two candies in the box with one chocolate bar, $\bullet$ two muffins in the box with one chocolate bar, $\bullet$ two chocolate bars in the box with one candy and one muffin, $\bullet$ one candy and one chocolate bar in the box with one muffin, $\bullet$ one muffin and one chocolate bar in the box with one candy. Is it possible that after some time it remains only one object in the box? [b]p4.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points? [b]p5.[/b] It is known that $x$ is a real number such that $x+\frac{1}{x}$ is an integer. Prove that $x^n+\frac{1}{x^n}$ is an integer for any positive integer $n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2011 Postal Coaching, 1

Tags: geometry , incenter , geometric transformation , circumcircle , reflection , inradius , IMO Shortlist

Let $I$ be the incentre of a triangle $ABC$ and $\Gamma_a$ be the excircle opposite $A$ touching $BC$ at $D$. If $ID$ meets $\Gamma_a$ again at $S$, prove that $DS$ bisects $\angle BSC$.