This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2023 NMTC Junior, P6
Tags: inequalities
The sum of squares of four reals $x,y,z,u$ is $1$. Find the minimum value of the expression $E=(x-y)(y-z)(z-u)(u-x)$. Find also the minimum values of $x$, $y$, $z$ and $u$ when this minimum occurs.

2016 IFYM, Sozopol, 4
Tags: geometry
Let $ABCD$ be a convex quadrilateral. The circle $\omega_1$ is tangent to $AB$ in $S$ and the continuations after $A$ and $B$ of sides $DA$ and $CB$, circle $\omega_2$ with center $I$ is tangent to $BC$ and the continuations after $B$ and $C$ of sides $AB$ and $DC$, circle $\omega_3$ is tangent to $CD$ in $T$ and the continuations after $C$ and $D$ of sides $BC$ and $AD$, and circle $\omega_4$ with center $J$ is tangent to $DA$ and the continuations after $D$ and $A$ of sides $CD$ and $BA$. Prove that points $S$ and $T$ are on equal distance from the middle point of segment $IJ$.

2006 Czech-Polish-Slovak Match, 4
Tags: modular arithmetic , number theory
Show that for every integer $k \ge 1$ there is a positive integer $n$ such that the decimal representation of $2^n$ contains a block of exactly $k$ zeros, i.e. $2^n = \dots a00 \dots 0b \cdots$ with $k$ zeros and $a, b \ne 0$.

2013 India Regional Mathematical Olympiad, 3
Tags: perpendicular bisector , geometry
In an acute-angled triangle $ABC$ with $AB < AC$, the circle $\omega$ touches $AB$ at $B$ and passes through $C$ intersecting $AC$ again at $D$. Prove that the orthocentre of triangle $ABD$ lies on $\omega$ if and only if it lies on the perpendicular bisector of $BC$.

2007 Iran Team Selection Test, 3
Tags: geometry , incenter , geometric transformation , homothety , trapezoid , analytic geometry , reflection
Let $\omega$ be incircle of $ABC$. $P$ and $Q$ are on $AB$ and $AC$, such that $PQ$ is parallel to $BC$ and is tangent to $\omega$. $AB,AC$ touch $\omega$ at $F,E$. Prove that if $M$ is midpoint of $PQ$, and $T$ is intersection point of $EF$ and $BC$, then $TM$ is tangent to $\omega$. [i]By Ali Khezeli[/i]

2011 Bulgaria National Olympiad, 3
Tags: induction , combinatorial geometry , combinatorics
In the interior of the convex 2011-gon are $2011$ points, such that no three among the given $4022$ points (the interior points and the vertices) are collinear. The points are coloured one of two different colours and a colouring is called "good" if some of the points can be joined in such a way that the following conditions are satisfied: 1) Each segment joins two points of the same colour. 2) None of the line segments intersect. 3) For any two points of the same colour there exists a path of segments connecting them. Find the number of "good" colourings.

2018 ELMO Shortlist, 2
Tags: algebra
Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$ [i]Proposed by Carl Schildkraut[/i]

1988 Irish Math Olympiad, 3
Tags: combinatorics
A city has a system of bus routes laid out in such a way that (a) there are exactly $11$ bus stops on each route; (b) it is possible to travel between any two bus stops without changing routes; (c) any two bus routes have exactly one bus stop in common. What is the number of bus routes in the city?

2006 Costa Rica - Final Round, 1
Tags: algebra
Consider the set $S=\{1,2,...,n\}$. For every $k\in S$, define $S_{k}=\{X \subseteq S, \ k \notin X, X\neq \emptyset\}$. Determine the value of the sum \[S_{k}^{*}=\sum_{\{i_{1},i_{2},...,i_{r}\}\in S_{k}}\frac{1}{i_{1}\cdot i_{2}\cdot...\cdot i_{r}}\] [hide]in fact, this problem was taken from an austrian-polish[/hide]

2018 Estonia Team Selection Test, 8
Tags: number theory , divisor
Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors $1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$.

2022 Polish MO Finals, 3
Tags: combinatorics
One has marked $n$ points on a circle and has drawn a certain number of chords whose endpoints are the marked points. It turned out that the following property is satisfied: whenever any $2021$ drawn chords are removed one can join any two marked points by a broken line composed of some of the remaining drawn chords. Prove that one can remove some of the drawn chords so that at most $2022n$ chords remain and the property described above is preserved.

ICMC 2, 1
Tags:
This questions comprises two independent parts. (i) Let \(g:\mathbb{R}\to\mathbb{R}\) be continuous and such that \(g(0)=0\) and \(g(x)g(-x)>0\) for any \(x > 0\). Find all solutions \(f : \mathbb{R}\to\mathbb{R}\) to the functional equation \[g(f(x+y))=g(f(x))+g(f(y)),\ x,y\in\mathbb{R}\] (ii) Find all continuously differentiable functions \(\phi : [a, \infty) \to \mathbb{R}\), where \(a > 0\), that satisfies the equation \[(\phi(x))^2=\int_a^x \left(\left|\phi(y)\right|\right)^2+\left(\left|\phi'(y)\right|\right)^2\mathrm{d}y -(x-a)^3,\ \forall x\geq a.\]

2019 Iran MO (3rd Round), 2
Tags: combinatorics
Let $T$ be a triangulation of a $100$-gon.We construct $P(T)$ by copying the same $100$-gon and drawing a diagonal if it was not drawn in $T$ an there is a quadrilateral with this diagonal and two other vertices so that all the sides and diagonals(Except the one we are going to draw) are present in $T$.Let $f(T)$ be the number of intersections of diagonals in $P(T)$.Find the minimum and maximum of $f(T)$.

2002 Miklós Schweitzer, 1
Tags: function
For an arbitrary ordinal number $\alpha$ let $H(\alpha)$ denote the set of functions $f\colon \alpha \rightarrow \{ -1,0,1\}$ that map all but finitely many elements of $\alpha$ to $0$. Order $H(\alpha)$ according to the last difference, that is, for $f, g\in H(\alpha)$ let $f\prec g$ if $f(\beta) < g(\beta)$ holds for the maximum ordinal number $\beta < \alpha$ with $f(\beta) \neq g(\beta)$. Prove that the ordered set $(H(\alpha), \prec)$ is scattered (i.e. it doesn't contain a subset isomorphic to the set of rational numbers with the usual order), and that any scattered order type can be embedded into some $(H(\alpha), \prec)$.

2020 LMT Spring, 3
Tags:
Let $LMT$ represent a 3-digit positive integer where $L$ and $M$ are nonzero digits. Suppose that the 2-digit number $MT$ divides $LMT$. Compute the difference between the maximum and minimum possible values of $LMT$.

2018 USA TSTST, 5
Tags: geometry
Let $ABC$ be an acute triangle with circumcircle $\omega$, and let $H$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be the points on $\omega$ with $PA = PH$ and $QA = QH$. The tangent to $\omega$ at $P$ intersects lines $AC$ and $AB$ at $E_1$ and $F_1$ respectively; the tangent to $\omega$ at $Q$ intersects lines $AC$ and $AB$ at $E_2$ and $F_2$ respectively. Show that the circumcircles of $\triangle AE_1F_1$ and $\triangle AE_2F_2$ are congruent, and the line through their centers is parallel to the tangent to $\omega$ at $A$. [i]Ankan Bhattacharya and Evan Chen[/i]

2005 Cuba MO, 7
Tags: number theory , lcm , gcd , gcd and lcm
Determine all triples of positive integers $(x, y, z)$ that satisfy $$x < y < z, \ \ gcd(x, y) = 6, \ \ gcd(y, z) = 10, \ \ gcd(z, x) = 8 \ \ and \ \ lcm(x, y, z) = 2400.$$

1986 IMO Shortlist, 8
Tags: combinatorics , graph theory , permutation , extremal graph theory
From a collection of $n$ persons $q$ distinct two-member teams are selected and ranked $1, \cdots, q$ (no ties). Let $m$ be the least integer larger than or equal to $2q/n$. Show that there are $m$ distinct teams that may be listed so that : [b](i)[/b] each pair of consecutive teams on the list have one member in common and [b](ii)[/b] the chain of teams on the list are in rank order. [i]Alternative formulation.[/i] Given a graph with $n$ vertices and $q$ edges numbered $1, \cdots , q$, show that there exists a chain of $m$ edges, $m \geq \frac{2q}{n}$ , each two consecutive edges having a common vertex, arranged monotonically with respect to the numbering.

MIPT student olimpiad spring 2023, 3
Tags: geometry , 3d geometry , sphere
Prove that if a set $X\subset S^n$ takes up more than half a Riemannian volume of a unit sphere $S^n$, then the set of all possible geodesic segments length less than $\pi$ with endpoints in the set $X$ covers the entire sphere. Geodetic on sphere $S^n$ is a curve lying on some circle of intersection of the sphere $S^n\subset R^{n+1}$ two-dimensional linear subspace $L \subset R^{n+1}$

1999 National High School Mathematics League, 4
Tags:
Statement 1: Line $a\in\alpha$, line $b\in\beta$, and $a,b$ are skew lines. If $c=\alpha\cap\beta$, then $c$ intersects at most one of $a,b$. Statement 2: It's impossible to find infintely many lines, any two of them are skew lines. $\text{(A)}$ Statement 1 is true, Statement 2 is false. $\text{(B)}$ Statement 2 is true, Statement 1 is false. $\text{(C)}$ Both are true. $\text{(D)}$ Neither is true.

2017 USAJMO, 5
Tags: geometry
Let $O$ and $H$ be the circumcenter and the orthocenter of an acute triangle $ABC$. Points $M$ and $D$ lie on side $BC$ such that $BM=CM$ and $\angle BAD = \angle CAD$. Ray $MO$ intersects the circumcircle of triangle $BHC$ in point $N$. Prove that $\angle ADO = \angle HAN$.

2013 Danube Mathematical Competition, 1
Tags: algebra , sum , reciprocal sum
Determine the natural numbers $n\ge 2$ for which exist $x_1,x_2,...,x_n \in R^*$, such that $$x_1+x_2+...+x_n=\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}=0$$

2007 Grigore Moisil Intercounty, 4
Tags: inequalities , solve inequality , algebra
Solve in the set of real numbers the fractional part inequality $ \{ x \}\le\{ nx \} , $ where $ n $ is a fixed natural number.

Russian TST 2018, P3
Tags: algebra , function
A function $f:\mathbb{R} \to \mathbb{R}$ has the following property: $$\text{For every } x,y \in \mathbb{R} \text{ such that }(f(x)+y)(f(y)+x) > 0, \text{ we have } f(x)+y = f(y)+x.$$ Prove that $f(x)+y \leq f(y)+x$ whenever $x>y$.

1990 IMO Longlists, 11
Tags: extremal combinatorics , extremal graph theory , graph theory , vertex degree , combinatorics
In a group of mathematicians, every mathematician has some friends (the relation of friend is reciprocal). Prove that there exists a mathematician, such that the average of the number of friends of all his friends is no less than the average of the number of friends of all these mathematicians.