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

2000 Brazil Team Selection Test, Problem 2
Tags: functional equation , fe
Find all functions $f:\mathbb R\to\mathbb R$ such that (i) $f(0)=1$; (ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$; (iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.

2024 Euler Olympiad, Round 2, 1
Tags: euler , diophantine equation , number theory
Find all triples $(a, b,c) $ of positive integers, such that: \[ a! + b! = c!! \] where $(2k)!! = 2 \cdot 4 \cdot \ldots \cdot (2k)$ and $ (2k + 1)!! = 1 \cdot 3 \cdot \ldots \cdot (2k+1).$ [i]Proposed by Stijn Cambie, Belgium [/i]

2001 China Team Selection Test, 1
Tags: combinatorial geometry , graph theory , combinatorics
Given seven points on a plane, with no three points collinear. Prove that it is always possible to divide these points into the vertices of a triangle and a convex quadrilateral, with no shared parts between the two shapes.

2012 ISI Entrance Examination, 1
Tags: trigonometry
[b]i)[/b]If $X,Y,Z$ be the angles of a triangle then show that \[\tan {\frac{X}{2}}\tan {\frac{Y}{2}}+\tan {\frac{Y}{2}}\tan {\frac{Z}{2}}+\tan {\frac{Z}{2}}\tan {\frac{X}{2}}=1\] [b]ii)[/b] Prove using [b](i)[/b] or otherwise that \[\tan {\frac{X}{2}}\tan {\frac{Y}{2}}\tan {\frac{Z}{2}}\leq\frac {1}{3\sqrt{3}}\]

2013 Israel National Olympiad, 5
Tags: combinatorics , geometry , number theory , lattice points
A point in the plane is called [b]integral[/b] if both its $x$ and $y$ coordinates are integers. We are given a triangle whose vertices are integral. Its sides do not contain any other integral points. Inside the triangle, there are exactly 4 integral points. Must those 4 points lie on one line?

1997 Bosnia and Herzegovina Team Selection Test, 5
Tags: geometric mean , arithmetic mean , number theory , set , algebra
$a)$ Prove that for all positive integers $n$ exists a set $M_n$ of positive integers with exactly $n$ elements and: $i)$ Arithmetic mean of arbitrary non-empty subset of $M_n$ is integer $ii)$ Geometric mean of arbitrary non-empty subset of $M_n$ is integer $iii)$ Both arithmetic mean and geometry mean of arbitrary non-empty subset of $M_n$ is integer $b)$ Does there exist infinite set $M$ of positive integers such that arithmetic mean of arbitrary non-empty subset of $M$ is integer

1979 IMO Shortlist, 14
Tags: logarithm
Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist.

2024 Harvard-MIT Mathematics Tournament, 10
Tags:
Suppose point $P$ is inside quadrilateral $ABCD$ such that $\angle{PAB} = \angle{PDA}, \angle{PAD} = \angle{PDC}, \angle{PBA} = \angle{PCB}, \angle{PBC} = \angle{PCD}.$ If $PA = 4, PB = 5,$ and $PC = 10$, compute the perimeter of $ABCD$.

2007-2008 SDML (Middle School), 2
Tags:
How many positive divisors does $200$ have?

LMT Guts Rounds, 23
Tags:
In how many ways can six marbles be placed in the squares of a $6$-by-$6$ grid such that no two marbles lie in the same row or column?

2023 IFYM, Sozopol, 1
Tags: algebra
Solve the system of equations in integers: \[ ab + 1 = (c+1)(d+1), \quad cd + 1 = (a-1)(b-1). \]

2001 IMC, 5
Tags: function , functional equation
Prove that there is no function $f: \mathbb{R} \rightarrow \mathbb{R}$ with $f(0) >0$, and such that \[f(x+y) \geq f(x) +yf(f(x)) \text{ for all } x,y \in \mathbb{R}. \]

2024 Saint Petersburg Mathematical Olympiad, 7
Tags: combinatorics
The edges of a complete graph on $1000$ vertices are colored in three colors. Prove that this graph contains a non-self-intersecting single-color cycle whose length is odd and not less than $41$.

2014 Saudi Arabia Pre-TST, 2.2
Tags: algebra , polynomial
Let $a_1, a_2, a_3, a_4, a_5$ be nonzero real numbers. Prove that the polynomial $$P(x)= \prod_{k=0}^{4} a_{k+1}x^4 + a_{k+2}x^3 + a_{k+3}x^2 + a_{k+4}x + a_{k+5}$$, where $a_{5+i} = a_i$ for $i = 1,2, 3,4$, has a root with negative real part.

PEN A Problems, 45
Tags: induction , divisibility theory
Let $b,m,n\in\mathbb{N}$ with $b>1$ and $m\not=n$. Suppose that $b^{m}-1$ and $b^{n}-1$ have the same set of prime divisors. Show that $b+1$ must be a power of $2$.

2022 BMT, 10
Tags: number theory
Compute the number of ordered pairs $(a, b)$ of positive integers such that $a$ and $b$ divide $5040$ but share no common factors greater than $1$.

2023 Romania National Olympiad, 2
Tags: algebra , function
Determine functions $f : \mathbb{R} \rightarrow \mathbb{R},$ with property that \[ f(f(x)) + y \cdot f(x) \le x + x \cdot f(f(y)), \] for every $x$ and $y$ are real numbers.

2009 Junior Balkan Team Selection Tests - Romania, 3
Tags: segment , geometry , concurrency
Consider a regular polygon $A_0A_1...A_{n-1}, n \ge 3$, and $m \in\{1, 2, ..., n - 1\}, m \ne n/2$. For any number $i \in \{0,1, ... , n - 1\}$, let $r(i)$ be the remainder of $i + m$ at the division by $n$. Prove that no three segments $A_iA_{r(i)}$ are concurrent.

2014 IberoAmerican, 1
Tags: number theory
For each positive integer $n$, let $s(n)$ be the sum of the digits of $n$. Find the smallest positive integer $k$ such that \[s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k).\]

2024 Ecuador NMO (OMEC), 5
Tags: diophantine equation , number theory
Find all triples of non-negative integer numbers $(E, C, U)$ such that $EC \ge 1$ and: $$2^{3^E}+3^{2^C}=593 \cdot 5^U$$

2019 Romanian Master of Mathematics, 3
Tags: combinatorics , graph theory , probability , cycle
Given any positive real number $\varepsilon$, prove that, for all but finitely many positive integers $v$, any graph on $v$ vertices with at least $(1+\varepsilon)v$ edges has two distinct simple cycles of equal lengths. (Recall that the notion of a simple cycle does not allow repetition of vertices in a cycle.) [i]Fedor Petrov, Russia[/i]

2014 Belarus Team Selection Test, 2
Tags: geometry , midpoint , locus
Given a triangle $ABC$. Let $S$ be the circle passing through $C$, centered at $A$. Let $X$ be a variable point on $S$ and let $K$ be the midpoint of the segment $CX$ . Find the locus of the midpoints of $BK$, when $X$ moves along $S$. (I. Gorodnin)

1986 Tournament Of Towns, (131) 7
Tags: point , combinatorics , combinatorial geometry , arc , angle , circles
On the circumference of a circle are $21$ points. Prove that among the arcs which join any two of these points, at least $100$ of them must subtend an angle at the centre of the circle not exceeding $120^o$ . ( A . F . Sidorenko)

2001 AMC 10, 4
Tags:
What is the maximum number of possible points of intersection of a circle and a triangle? $ \textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$

2013 APMO, 1
Tags: equal area , ratio , geometry , trigonometry , circumcircle , parallelogram , analytic geometry
Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal areas.