Olympiad problems

1998 Estonia National Olympiad, 5

Tags: point , circles , coloring , combinatorics , combinatorial geometry

A circle is divided into $n$ equal arcs by $n$ points. Assume that, no matter how we color the $n$ points in two colors, there always exists an axis of symmetry of the set of points such that any two of the $n$ points which are symmetric with respect to that axis have the same color. Find all possible values of $n$.

1985 IMO Shortlist, 2

Tags: 3d geometry , polyhedron , euler s theorem , angle , geometry

A polyhedron has $12$ faces and is such that: [b][i](i)[/i][/b] all faces are isosceles triangles, [b][i](ii)[/i][/b] all edges have length either $x$ or $y$, [b][i](iii)[/i][/b] at each vertex either $3$ or $6$ edges meet, and [b][i](iv)[/i][/b] all dihedral angles are equal. Find the ratio $x/y.$

2014 Federal Competition For Advanced Students, P2, 5

Tags: inequalities , three variable inequality , 3-variable inequality , algebra

Show that the inequality $(x^2 + y^2z^2) (y^2 + x^2z^2) (z^2 + x^2y^2) \ge 8xy^2z^3$ is valid for all integers $x, y$ and $z$.When does equality apply?

1976 Yugoslav Team Selection Test, Problem 1

Tags: combinatorics , geometry , combinatorial geometry

Prove that for a given convex polygon of area $A$ and perimeter $P$ there exists a circle of radius $\frac AP$ that is contained in the interior of the polygon.

2004 Korea Junior Math Olympiad, 1

Tags: geometry , inequalities

For positive reals $a_1, a_2, ..., a_5$ such that $a^2_1+a^2_2+...+a^2_5=2$, consider five squares with sides $a_1, a_2, ..., a_5$ respectively. Show that these squares can be placed inside (including boundaries) a square with side length of $2$ so that the square themselves do not overlap each other.

2025 Malaysian IMO Training Camp, 2

Tags: algebra

Determine all infinite sequences of nonnegative integers $a_1,a_2,\ldots$ such that: 1. Every positive integer appears in the sequence at least once, and; 2. $a_i$ is the smallest integer $j$ such that $a_{j+2}=i$, for all $i\ge 1$. [i](Proposed by Ho Janson)[/i]

2017 District Olympiad, 1

Tags: number theory

Let be a natural number $ n\ge 3 $ with the property that $ 1+3n $ is a perfect square. Show that there are three natural numbers $ a,b,c, $ such that the number $$ 1+\frac{3n+3}{a^2+b^2+c^2} $$ is a perfect square.

1966 Polish MO Finals, 3

Tags: geometry , 3d geometry , parallelepiped , cube

Prove that the sum of the squares of the areas of the projections of the faces of a rectangular parallelepiped on a plane is the same for all positions of the plane if and only if the parallelepiped is a cube.

2023-IMOC, A2

Tags: algebra

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(f(x)+y)(x-f(y)) = f(x)^2-f(y^2).$$

2012 China Team Selection Test, 3

Tags: vector , geometry , geometric transformation , analytic geometry , combinatorics

In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle $B$. For all possible starting and ending configurations, find the maximum length of the sum of the [i]translation vectors[/i] of all beetles.

2002 VJIMC, Problem 1

Tags: algebra , complex number , system of equations

Find all complex solutions to the system \begin{align*} (a+ic)^3+(ia+b)^3+(-b+ic)^3&=-6,\\ (a+ic)^2+(ia+b)^2+(-b+ic)^2&=6,\\ (1+i)a+2ic&=0.\end{align*}

2013 BMT Spring, 3

Tags: geometry

Given a regular tetrahedron $ABCD$ with center $O$, find $\sin \angle AOB$.

1995 Irish Math Olympiad, 2

Tags: abstract algebra , trigonometry , algebra

Let $ a,b,c$ be complex numbers. Prove that if all the roots of the equation $ x^3\plus{}ax^2\plus{}bx\plus{}c\equal{}0$ are of module $ 1$, then so are the roots of the equation $ x^3\plus{}|a|x^2\plus{}|b|x\plus{}|c|\equal{}0$.

2005 International Zhautykov Olympiad, 2

Tags: geometry , perimeter , analytic geometry , linear algebra , parallelogram , conic

Let the circle $ (I; r)$ be inscribed in the triangle $ ABC$. Let $ D$ be the point of contact of this circle with $ BC$. Let $ E$ and $ F$ be the midpoints of $ BC$ and $ AD$, respectively. Prove that the three points $ I$, $ E$, $ F$ are collinear.

2009 HMNT, 6

Tags:

Find the maximum value of $x+y$, given $ x^2 + y^2 - 3y - 1 = 0 $.

2012 All-Russian Olympiad, 2

Tags: geometry , incenter , symmetry , geometric transformation , homothety , circumcircle , power of a point

The points $A_1,B_1,C_1$ lie on the sides $BC,CA$ and $AB$ of the triangle $ABC$ respectively. Suppose that $AB_1-AC_1=CA_1-CB_1=BC_1-BA_1$. Let $O_A,O_B$ and $O_C$ be the circumcentres of triangles $AB_1C_1,A_1BC_1$ and $A_1B_1C$ respectively. Prove that the incentre of triangle $O_AO_BO_C$ is the incentre of triangle $ABC$ too.

2015 German National Olympiad, 2

Tags: equation , number theory

A positive integer $n$ is called [i]smooth[/i] if there exist integers $a_1,a_2,\dotsc,a_n$ satisfying \[a_1+a_2+\dotsc+a_n=a_1 \cdot a_2 \cdot \dotsc \cdot a_n=n.\] Find all smooth numbers.

2007 National Olympiad First Round, 30

Tags: modular arithmetic

Let $(a_n)_{n=1}^{\infty}$ be an integer sequence such that $a_{n+48} \equiv a_n \pmod {35}$ for every $n \geq 1$. Let $i$ and $j$ be the least numbers satisfying the conditions $a_{n+i} \equiv a_n \pmod {5}$ and $a_{n+j} \equiv a_n \pmod {7}$ for every $n\geq 1$. Which one below cannot be an $(i,j)$ pair? $ \textbf{(A)}\ (16,4) \qquad\textbf{(B)}\ (3,16) \qquad\textbf{(C)}\ (8,6) \qquad\textbf{(D)}\ (1,48) \qquad\textbf{(E)}\ (16,18) $

2012 Princeton University Math Competition, B7

Tags: geometry

Assume the earth is a perfect sphere with a circumference of $60$ units. A great circle is a circle on a sphere whose center is also the center of the sphere. There are three train tracks on three great circles of the earth. One is along the equator and the other two pass through the poles, intersecting at a $90$ degree angle. If each track has a train of length $L$ traveling at the same speed, what is the maximum value of $L$ such that the trains can travel without crashing?

1951 AMC 12/AHSME, 50

Tags: geometry

Tom, Dick and Harry started out on a $ 100$-mile journey. Tom and Harry went by automobile at the rate of $ 25$ mph, while Dick walked at the rate of $ 5$ mph. After a certain distance, Harry got off and walked on at $ 5$ mph, while Tom went back for Dick and got him to the destination at the same time that Harry arrived. The number of hours required for the trip was: $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \text{none of these answers}$

1993 Vietnam National Olympiad, 3

Tags: function , number theory

Find a function $f(n)$ on the positive integers with positive integer values such that $f( f(n) ) = 1993 n^{1945}$ for all $n$.

2013 Stanford Mathematics Tournament, 4

Tags:

Given that $f(x)+2f(8-x)=x^2$ for all real $x$, compute $f(2)$.

2013 HMNT, 3

Tags: combinatorics

The digits $1,2,3,4, 5,6$ are randomly chosen (without replacement) to form the three-digit numbers $M = \overline{ABC}$ and $N = \overline{DEF}$. For example, we could have $M = 413$ and $N = 256$. Find the expected value of $M \cdot N$.

2021 Science ON all problems, 2

Tags: rings , abstract algebra , superior algebra

Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\ $\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\ $\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\ $\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true? \\ \\ [i] (Bogdan Blaga)[/i]

2001 Baltic Way, 13

Tags: floor function , algebra

Let $a_0, a_1, a_2,\ldots $ be a sequence of real numbers satisfying $a_0=1$ and $a_n=a_{\lfloor 7n/9\rfloor}+a_{\lfloor n/9\rfloor}$ for $n=1, 2,\ldots $ Prove that there exists a positive integer $k$ with $a_k<\frac{k}{2001!}$.