Olympiad problems

1977 IMO Longlists, 28

Tags: algebra , recurrence relation , Sequence , equation , floor function , IMO Shortlist

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

1980 IMO Shortlist, 16

Tags: induction , combinatorics , Summation , counting , IMO Shortlist

Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)

2003 Vietnam National Olympiad, 3

Tags: function , induction , limit , algebra proposed , algebra

Let $\mathcal{F}$ be the set of all functions $f : (0,\infty)\to (0,\infty)$ such that $f(3x) \geq f( f(2x) )+x$ for all $x$. Find the largest $A$ such that $f(x) \geq A x$ for all $f\in\mathcal{F}$ and all $x$.

1988 Balkan MO, 2

Tags: algebra , polynomial , algebra proposed

Find all polynomials of two variables $P(x,y)$ which satisfy \[P(a,b) P(c,d) = P (ac+bd, ad+bc), \forall a,b,c,d \in \mathbb{R}.\]

KoMaL A Problems 2018/2019, A. 742

Tags: geometry

Convex quadrilateral $ABCD$ is inscribed in circle $\Omega$. Its sides $AD$ and $BC$ intersect at point $E$. Let $M$ and $N$ be the midpoints of the circle arcs $AB$ and $CD$ not containing the other vertices, and let $I$, $J$, $K$, $L$ denote the incenters of triangles $ABD$, $ABC$, $BCD$, $CDA$, respectively. Suppose $\Omega$ intersects circles $IJM$ and $KLN$ for the second time at points $U \neq M$ and $V \neq N$. Show that the points $E$, $U$, and $V$ are collinear.

2011 Czech-Polish-Slovak Match, 1

Tags: algebra , polynomial , algebra unsolved

A polynomial $P(x)$ with integer coefficients satisfies the following: if $F(x)$, $G(x)$, and $Q(x)$ are polynomials with integer coefficients satisfying $P\Big(Q(x)\Big)=F(x)\cdot G(x)$, then $F(x)$ or $G(x)$ is a constant polynomial. Prove that $P(x)$ is a constant polynomial.

2016 Harvard-MIT Mathematics Tournament, 5

Tags:

Let $a$, $b$, $c$, $d$, $e$, $f$ be integers selected from the set $\{1,2,\dots,100\}$, uniformly and at random with replacement. Set \[ M = a + 2b + 4c + 8d + 16e + 32f. \] What is the expected value of the remainder when $M$ is divided by $64$?

2014 Contests, 3

Tags: number theory , prime divisor , polynomial , IMO Shortlist , imo shortlist n3 , Hi

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

2012 Danube Mathematical Competition, 1

Tags: lattice points , square , covering , combinatorial geometry , combinatorics

Given a positive integer $n$, determine the maximum number of lattice points in the plane a square of side length $n +\frac{1}{2n+1}$ may cover.

2010 South East Mathematical Olympiad, 3

Tags: inequalities unsolved , inequalities

Let $n$ be a positive integer. The real numbers $a_1,a_2,\cdots,a_n$ and $r_1,r_2,\cdots,r_n$ are such that $a_1\leq a_2\leq \cdots \leq a_n$ and $0\leq r_1\leq r_2\leq \cdots \leq r_n$. Prove that $\sum_{i=1}^n\sum_{j=1}^n a_i a_j \min (r_i,r_j)\geq 0$

2014 Sharygin Geometry Olympiad, 23

Tags: geometry , harmonic division , Concyclic

Let $A, B, C$ and $D$ be a triharmonic quadruple of points, i.e $AB\cdot CD = AC \cdot BD = AD \cdot BC.$ Let $A_1$ be a point distinct from $A$ such that the quadruple $A_1, B, C$ and $D$ is triharmonic. Points $B_1, C_1$ and $D_1$ are defined similarly. Prove that a) $A, B, C_1, D_1$ are concyclic; b) the quadruple $A_1, B_1, C_1, D_1$ is triharmonic.

1997 Estonia Team Selection Test, 1

Tags: geometry , concurrency

In a triangle $ABC$ points $A_1,B_1,C_1$ are the midpoints of $BC,CA,AB$ respectively,and $A_2,B_2,C_2$ are the midpoints of the altitudes from $A,B,C$ respectively. Show that the lines $A_1A_2,B_1B_2,C_1,C_2$ are concurrent.

PEN G Problems, 13

Tags: Irrational numbers

It is possible to show that $ \csc\frac{3\pi}{29}\minus{}\csc\frac{10\pi}{29}\equal{} 1.999989433...$. Prove that there are no integers $ j$, $ k$, $ n$ with odd $ n$ satisfying $ \csc\frac{j\pi}{n}\minus{}\csc\frac{k\pi}{n}\equal{} 2$.

1990 Poland - Second Round, 4

Tags: algebra , trigonometry

For each pair of even natural numbers $ k $, $ m $determine all real numbers $ x $that satisfy the equation $$ (\sin x)^k + (\cos x)^{-m} = (\cos x)^k + (\sin x)^{-m}$$

2023 Assam Mathematics Olympiad, 15

Tags:

Let $f(x)$ be a polynomial of degree $3$ with real coefficients satisfying $|f(x)| = 12$ for $x = 1, 2, 3, 5, 6, 7$. Find $|f(0)|$.

2002 Croatia National Olympiad, Problem 1

Tags: geometry

The length of the middle line of a trapezoid is $4$ and the angles at one of the bases are $40^\circ$ and $50^\circ$. Determine the lengths of the bases if the distance between their midpoints is $1$.

2019-2020 Fall SDPC, 3

Tags: algebra , polynomial

Find all polynomials $P$ with integer coefficients such that for all positive integers $x,y$, $$\frac{P(x)-P(y)}{x^2+y^2}$$ evaluates to an integer (in particular, it can be zero).

2008 239 Open Mathematical Olympiad, 4

Tags: combinatorics

For what natural number $n> 100$ can $n$ pairwise distinct numbers be arranged on a circle such that each number is either greater than $100$ numbers following it clockwise or less than all of them? and would any property be violated when deleting any of those numbers?

2021 Olimphíada, 2

Tags: geometry

Let $P$, $A$, $B$ and $C$ be points on a line $r$, in that order, so that $AB = BC$. Let $H$ be a point that does not belong to this line and let $S$ be the other intersection of the circles $(HPB)$ and $(HAC)$. Let $I$ be the second intersection of the circle with diameter $HB$ and $(HAC)$. Show that the points $P$, $H$, $I$ lie on the same line if and only if $HS$ is perpendicular to $r$.

2019 All-Russian Olympiad, 5

Tags: rectangle , number theory , prime numbers

In a kindergarten, a nurse took $n$ congruent cardboard rectangles and gave them to $n$ kids, one per each. Each kid has cut its rectangle into congruent squares(the squares of different kids could be of different sizes). It turned out that the total number of the obtained squares is a prime number. Prove that all the initial squares were in fact squares.

2010 Today's Calculation Of Integral, 624

Tags: calculus , integration , function , trigonometry , real analysis , calculus computations

Find the continuous function $f(x)$ such that the following equation holds for any real number $x$. \[\int_0^x \sin t \cdot f(x-t)dt=f(x)-\sin x.\] [i]1977 Keio University entrance exam/Medicine[/i]

2006 Tournament of Towns, 6

Tags: geometry

Let us call a pentagon curved, if all its sides are arcs of some circles. Are there exist a curved pentagon $P$ and a point $A$ on its boundary so that any straight line passing through $A$ divides perimeter of $P$ into two parts of the same length? [i](7 points)[/i]

2012 Belarus Team Selection Test, 1

Tags: combinatorial geometry , geometry , combinatorics , regular polygon , Regular

Let $m,n,k$ be pairwise relatively prime positive integers greater than $3$. Find the minimal possible number of points on the plane with the following property: there are $x$ of them which are the vertices of a regular $x$-gon for $x = m, x = n, x = k$. (E.Piryutko)

2009 Today's Calculation Of Integral, 409

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^1 \sqrt{\frac{x\plus{}\sqrt{x^2\plus{}1}}{x^2\plus{}1}}\ dx$.

2021 Belarusian National Olympiad, 8.5

Tags: function , algebra

Let $f(x)$ be a linear function and $k,l,m$ - pairwise different real numbers. It is known that $f(k)=l^3+m^3$, $f(l)=m^3+k^3$ and $f(m)=k^3+l^3$. Find the value of $k+l+m$.