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

1997 Cono Sur Olympiad, 5
Tags: combinatorics , number theory
Let $n$ be a natural number $n>3$. Show that in the multiples of $9$ less than $10^n$, exist more numbers with the sum of your digits equal to $9(n - 2)$ than numbers with the sum of your digits equal to $9(n - 1)$.

PEN Q Problems, 12
Tags: polynomial , algebra , gauss
Prove that if the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are all distinct, then the polynomial \[(x-a_{1})^{2}(x-a_{2})^{2}\cdots (x-a_{n})^{2}+1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

2007 Alexandru Myller, 4
Tags: discrete math
At a math contest which has $ 5 $ problems, each candidate has solved $ 3 $ problems. Among these candidates, for any group of $ 5 $ candidates that we might choose, we see that there is a problem which all members of the group had solved it. Prove that there is a problem solved by all candidates.

1986 ITAMO, 2
Tags: recurrence relation , sequence , algebra
Determine the general term of the sequence ($a_n$) given by $a_0 =\alpha > 0$ and $a_{n+1} =\frac{a_n}{1+a_n}$ .

2000 Mediterranean Mathematics Olympiad, 2
Tags: geometry
Suppose that in the exterior of a convex quadrilateral $ABCD$ equilateral triangles $XAB,YBC,ZCD,WDA$ with centroids $S_1,S_2,S_3,S_4$ respectively are constructed. Prove that $S_1S_3\perp S_2S_4$ if and only if $AC=BD$.

2021 Estonia Team Selection Test, 3
Tags: inequalities , geometric inequality , geometry , disks
In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

2020 LMT Fall, 27
Tags:
A list consists of all positive integers from $1$ to $2020$, inclusive, with each integer appearing exactly once. Define a move as the process of choosing four numbers from the current list and replacing them with the numbers $1,2,3,4$. If the expected number of moves before the list contains exactly two $4$'s can be expressed as $\frac{a}{b}$ for relatively prime positive integers, evaluate $a+b$. [i]Proposed by Richard Chen and Taiki Aiba[/i]

2018 International Zhautykov Olympiad, 2
Tags: incenter , geometry
Let $N,K,L$ be points on $AB,BC,CA$ such that $CN$ bisector of angle $\angle ACB$ and $AL=BK$.Let $BL\cap AK=P$.If $I,J$ be incenters of triangles $\triangle BPK$ and $\triangle ALP$ and $IJ\cap CN=Q$ prove that $IQ=JP$

1985 IMO Longlists, 94
Tags: geometry , circumcircle , angle , triangle
A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.

1997 Putnam, 4
Tags: function
Let $G$ be group with identity $e$ and $\phi :G\to G$ be a function such that : \[ \phi(g_1)\cdot \phi(g_2)\cdot \phi(g_3)=\phi(h_1)\cdot \phi(h_2)\cdot \phi(h_3) \] Whenever $g_1\cdot g_2\cdot g_3=e=h_1\cdot h_2\cdot h_3$ Show there exists $a\in G$ such that $\psi(x)=a\phi(x)$ is a homomorphism. (that is $\psi(x\cdot y)=\psi (x)\cdot \psi(y)$ for all $x,y\in G$ )

2012 Ukraine Team Selection Test, 9
Tags: geometry , collinear , incircle , centroid
The inscribed circle $\omega$ of the triangle $ABC$ touches its sides $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$, respectively. Let $S$ be the intersection point of lines passing through points $B$ and $C$ and parallel to $A_1C_1$ and $A_1B_1$ respectively, $A_0$ be the foot of the perpendicular drawn from point $A_1$ on $B_1C_1$, $G_1$ be the centroid of triangle $A_1B_1C_1$, $P$ be the intersection point of the ray $G_1A_0$ with $\omega$. Prove that points $S, A_1$, and $P$ lie on a straight line.

2016 ASDAN Math Tournament, 9
Tags:
Compute $$\int_0^\infty\frac{\ln\left(\frac{1+x^{11}}{1+x^3}\right)}{(1+x^2)\ln x}dx.$$

2021 Purple Comet Problems, 3
Tags:
The diagram shows a semicircle with diameter $20$ and the circle with greatest diameter that fits inside the semicircle. The area of the shaded region is $N\pi$, where $N$ is a positive integer. Find $N$.

2023 Brazil Undergrad MO, 5
Tags: modular arithmetic , generating functions , recurrence relation , algebra
A drunken horse moves on an infinite board whose squares are numbered in pairs $(a, b) \in \mathbb{Z}^2$. In each movement, the 8 possibilities $$(a, b) \rightarrow (a \pm 1, b \pm 2),$$ $$(a, b) \rightarrow (a \pm 2, b \pm 1)$$ are equally likely. Knowing that the knight starts at $(0, 0)$, calculate the probability that, after $2023$ moves, it is in a square $(a, b)$ with $a \equiv 4 \pmod 8$ and $b \equiv 5 \pmod 8$.

2014 Contests, 2
Tags: quick nt
2. What’s the closest number to $169$ that’s divisible by $9$?

2010 Canada National Olympiad, 2
Tags:
Let $A,B,P$ be three points on a circle. Prove that if $a,b$ are the distances from $P$ to the tangents at $A,B$ respectively, and $c$ is the distance from $P$ to the chord $AB$, then $c^2 =ab$.

2018 IFYM, Sozopol, 6
Tags: number theory
Find all sets $(a, b, c)$ of different positive integers $a$, $b$, $c$, for which: [b]*[/b] $2a - 1$ is a multiple of $b$; [b]*[/b] $2b - 1$ is a multiple of $c$; [b]*[/b] $2c - 1$ is a multiple of $a$.

1978 Polish MO Finals, 2
Tags: combinatorics , combinatorial geometry
In a coordinate plane, consider the set of points with integer cooedinates at least one of which is not divisible by $4$. Prove that these points cannot be partitioned into pairs such that the distance between points in each pair equals $1$. In other words, an infinite chessboard, whose cells with both coordinates divisible by $4$ are cut out, cannot be tiled by dominoes.

2002 Dutch Mathematical Olympiad, 3
Tags: analytic geometry , geometry , perimeter
$A, B$ and $C$ are points in the plane with integer coordinates. The lengths of the sides of triangle $ABC$ are integer numbers. Prove that the perimeter of the triangle is an even number.

VII Soros Olympiad 2000 - 01, 11.2
Tags: parameterization , algebra , trinomial
For all valid values ​​of $a, b$, and $c$, solve the equation $$\frac{a (x-b) (x-c) }{(a-b) (a-c)} + \frac{b (x-c) (x-a)}{(b-c) (b-a)} +\frac{c (x-a) (x-b) }{(c-a ) (c-b)} = x^2$$

1988 Brazil National Olympiad, 3
Tags: number theory , function
Find all functions $f:\mathbb{N}^* \rightarrow \mathbb{N}$ such that [list] [*] $f(x \cdot y) = f(x) + f(y)$ [*] $f(30) = 0$ [*] $f(x)=0$ always when the units digit of $x$ is $7$ [/list]

1995 IMC, 10
Tags: function , approximation , real analysis
a) Prove that for every $\epsilon>0$ there is a positive integer $n$ and real numbers $\lambda_{1},\dots,\lambda_{n}$ such that $$\max_{x\in [-1,1]}|x-\sum_{k=1}^{n}\lambda_{k}x^{2k+1}|<\epsilon.$$ b) Prove that for every odd continuous function $f$ on $[-1,1]$ and for every $\epsilon>0$ there is a positive integer $n$ and real numbers $\mu_{1},\dots,\mu_{n}$ such that $$\max_{x\in [-1,1]}|f(x)-\sum_{k=1}^{n}\mu_{k}x^{2k+1}|<\epsilon.$$

2021 New Zealand MO, 5
Tags: diophantine equation , number theory , diophantine
Find all pairs of integers $x, y$ such that $y^5 + 2xy = x^2 + 2y^4.$ .

1980 USAMO, 3
Tags: trigonometry , calculus , integration , induction
Let $F_r=x^r\sin{rA}+y^r\sin{rB}+z^r\sin{rC}$, where $x,y,z,A,B,C$ are real and $A+B+C$ is an integral multiple of $\pi$. Prove that if $F_1=F_2=0$, then $F_r=0$ for all positive integral $r$.

2024 JBMO TST - Turkey, 5
Tags: perfect square , number theory , divisibility
Find all positive integer values of $n$ such that the value of the $$\frac{2^{n!}-1}{2^n-1}$$ is a square of an integer.