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

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Found problems: 85335

2015 Princeton University Math Competition, A8
Tags: algebra
Let $P(x)$ be a polynomial with positive integer coefficients and degree 2015. Given that there exists some $\omega \in \mathbb{C}$ satisfying $$\omega^{73} = 1\quad \text{and}$$ $$P(\omega^{2015}) + P(\omega^{2015^2}) + P(\omega^{2015^3}) + \ldots + P(\omega^{2015^{72}}) = 0,$$ what is the minimum possible value of $P(1)$?

2022 Kazakhstan National Olympiad, 3
Tags: functional equation , algebra
Given $m\in\mathbb{N}$. Find all functions $f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ such that $$f(f(x)+y)-f(x)=\left( \frac{f(y)}{y}-1\right)x+f^m(y)$$ holds for all $x,y\in\mathbb{R^{+}}.$ ($f^m(x) =$ $f$ applies $m$ times.)

2008 Switzerland - Final Round, 10
Tags: algebra , inequalities
Find all pairs$ (a, b)$ of positive real numbers with the following properties: (i) For all positive real numbers $x, y, z,w$ holds $x + y^2 + z^3 + w^6 \ge a (xyzw)^{b}$ . (ii) There is a quadruple $(x, y, z,w)$ of positive real numbers such that in equality (i) applies.

1980 Canada National Olympiad, 5
Tags: perimeter , geometry
A parallelepiped has the property that all cross sections, which are parallel to any fixed face $F$, have the same perimeter as $F$. Determine whether or not any other polyhedron has this property. Typesetter's Note: I believe that proof of existence or non-existence suffices.

2022 Princeton University Math Competition, B1
Tags: geometry
A triangle $\vartriangle ABC$ is situated on the plane and a point $E$ is given on segment $AC$. Let $D$ be a point in the plane such that lines $AD$ and $BE$ are parallel. Suppose that $\angle EBC = 25^o$, $\angle BCA = 32^o$, and $\angle CAB = 60^o$. Find the smallest possible value of $\angle DAB$ in degrees.

2018 239 Open Mathematical Olympiad, 10-11.4
Tags: combinatorics
In a $9\times 9$ table, all cells contain zeros. The following operations can be performed on the table: 1. Choose an arbitrary row, add one to all the numbers in that row, and shift all these numbers one cell to the right (and place the last number in the first position). 2. Choose an arbitrary column, subtract one from all its numbers, and shift all these numbers one cell down (and place the bottommost number in the top cell). Is it possible to obtain a table in which all cells, except two, contain zeros, with 1 in the bottom-left cell and -1 in the top-right cell after several such operations? [i]Proposed by N. Vlasova[/i]

2018 Sharygin Geometry Olympiad, 21
Tags: geometry
In the plane a line $l$ and a point $A$ outside it are given. Find the locus of the incenters of acute-angled triangles having a vertex $A$ and an opposite side lying on $l$.

2017 Sharygin Geometry Olympiad, 5
Tags: geometry
10.5 Let $BB'$, $CC'$ be the altitudes of an acute triangle $ABC$. Two circles through $A$ and $C'$ are tangent to $BC$ at points $P$ and $Q$. Prove that $A, B', P, Q$ are concyclic.

2005 Today's Calculation Of Integral, 67
Tags: calculus , integration , geometry , calculus computations
Evaluate \[\frac{2005\displaystyle \int_0^{1002}\frac{dx}{\sqrt{1002^2-x^2}+\sqrt{1003^2-x^2}}+\int_{1002}^{1003}\sqrt{1003^2-x^2}dx}{\displaystyle \int_0^1\sqrt{1-x^2}dx}\]

2019 Irish Math Olympiad, 4
Tags: algebra , equation
Find the set of all quadruplets $(x,y, z,w)$ of non-zero real numbers which satisfy $$1 +\frac{1}{x}+\frac{2(x + 1)}{xy}+\frac{3(x + 1)(y + 2)}{xyz}+\frac{4(x + 1)(y + 2)(z + 3)}{xyzw}= 0$$

2004 India IMO Training Camp, 3
Tags: trigonometry , function , functional equation , system of equations , algebra
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that \[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.

2020 Taiwan TST Round 1, 1
Tags: combinatorics , induction , local argument
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

2014-2015 SDML (High School), 4
Tags:
What is the maximum number of points that can be placed in the interior of an equilateral triangle of side length $2$ such that the distance between any two points is greater than one? $\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) }6\qquad\text{(E) }7$

2020 Kazakhstan National Olympiad, 1
Tags: divide , number theory
Find all pairs $ (m, n) $ of natural numbers such that $ n ^ 4 \ | \ 2m ^ 5 - 1 $ and $ m ^ 4 \ | \ 2n ^ 5 + 1 $.

2020 AMC 12/AHSME, 8
Tags: algebra
How many ordered pairs of integers $(x, y)$ satisfy the equation$$x^{2020}+y^2=2y?$$ $\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{infinitely many}$

1951 Putnam, A6
Tags:
Determine the position of a normal chord of a parabola such that it cuts off of the parabola a segment of minimum area.

1973 IMO Longlists, 6
Tags: geometry , analytic geometry , linear algebra , matrix , pigeonhole principle , combinatorics
Let $P_i (x_i, y_i)$ (with $i = 1, 2, 3, 4, 5$) be five points with integer coordinates, no three collinear. Show that among all triangles with vertices at these points, at least three have integer areas.

2017 ASDAN Math Tournament, 20
Tags:
Let $\alpha$ and $\beta$ be positive rational numbers so that $\alpha+\beta\sqrt{5}$ is a root of some polynomial $x^2+ax+b$ where $a$ and $b$ are integers. What is the smallest possible value of $\alpha\beta$?

1991 Arnold's Trivium, 64
Tags: algebra , function , domain
Does the Cauchy problem $u|_{y=x^2}=1$, $(\nabla u)^2=1$ have a smooth solution in the domain $y\ge x^2$? In the domain $y\le x^2$?

2013 IMO Shortlist, G2
Tags: geometry , circumcircle , trapezoid , symmetry
Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.

2016 European Mathematical Cup, 3
Tags: algebra
Determine all functions $f:\mathbb R\to\mathbb R$ such that equality $$f(x + y + yf(x)) = f(x) + f(y) + xf(y)$$ holds for all real numbers $x$, $y$. Proposed by Athanasios Kontogeorgis

1997 Brazil National Olympiad, 2
Tags: combinatorics
Let $A$ be a set of $n$ non-negative integers. We say it has property $\mathcal P$ if the set $\{x + y \mid x, y \in A\}$ has $\binom{n}{2}$ elements. We call the largest element of $A$ minus the smallest element, the diameter of $A$. Let $f(n)$ be the smallest diameter of any set $A$ with property $\mathcal P$. Show that $n^2 \leq 4 f(n) < 4 n^3$. [hide="Comment"](If you have some amount of time, try a best estimative for $f(n)$, such that $f(p)<2p^2$ for prime $p$).[/hide]

2007 Tournament Of Towns, 2
Tags: perimeter , geometry
$K, L, M$ and $N$ are points on sides $AB, BC, CD$ and $DA$, respectively, of the unit square $ABCD$ such that $KM$ is parallel to $BC$ and $LN$ is parallel to $AB$. The perimeter of triangle $KLB$ is equal to $1$. What is the area of triangle $MND$?

2020 Jozsef Wildt International Math Competition, W6
Tags: calculus , integration
Determine the functions $f:(0,\pi)\to\mathbb R$ which satisfy $$f'(x)=\frac{\cos2020x}{\sin x}$$ for any real $x\in(0,\pi)$. [i]Proposed by D.M. Bătinețu-Giurgiu and Neculai Stanciu[/i]

1978 Chisinau City MO, 160
Tags: factor , polynomial , algebra
Factor the polynomial $P (x) = 1 + x +x^2+...+x^{2^k-1}$