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.

AND:
OR:
NO:

Found problems: 85335

2015 IFYM, Sozopol, 1
Tags: functional equation , algebra
Determine all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy the following equations: a) $f(f(n))=4n+3$ $\forall$ $n \in \mathbb{Z}$; b) $f(f(n)-n)=2n+3$ $\forall$ $n \in \mathbb{Z}$.

2019 Saudi Arabia JBMO TST, 3
Tags: algebra
Let $S$ be a set of real numbers such that: i) $1$ is from $S$; ii) for any $a, b$ from $S$ (not necessarily different), we have that $a-b$ is also from $S$; iii) for any $a$ from $S$ ($a$ is different from $0$), we have that $1/a$ is from $S$. Show that for every $a, b$ from $S$, we have that $ab$ is from $S$.

2007 India National Olympiad, 2
Tags: number theory
Let $ n$ be a natural number such that $ n \equal{} a^2 \plus{} b^2 \plus{}c^2$ for some natural numbers $ a,b,c$. Prove that \[ 9n \equal{} (p_1a\plus{}q_1b\plus{}r_1c)^2 \plus{} (p_2a\plus{}q_2b\plus{}r_2c)^2 \plus{} (p_3a\plus{}q_3b\plus{}r_3c)^2\] where $ p_j$'s , $ q_j$'s , $ r_j$'s are all [b]nonzero[/b] integers. Further, if $ 3$ does [b]not[/b] divide at least one of $ a,b,c,$ prove that $ 9n$ can be expressed in the form $ x^2\plus{}y^2\plus{}z^2$, where $ x,y,z$ are natural numbers [b]none[/b] of which is divisible by $ 3$.

2014 Iran MO (3rd Round), 3
Tags: polynomial , complex analysis , algebra
Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$. [i]Proposed by Mohammad Ahmadi[/i]

1989 IMO Longlists, 14
Tags: geometry , geometric inequality , area of a triangle , circumcircle
For a triangle $ ABC,$ let $ k$ be its circumcircle with radius $ r.$ The bisectors of the inner angles $ A, B,$ and $ C$ of the triangle intersect respectively the circle $ k$ again at points $ A', B',$ and $ C'.$ Prove the inequality \[ 16Q^3 \geq 27 r^4 P,\] where $ Q$ and $ P$ are the areas of the triangles $ A'B'C'$ and $ABC$ respectively.

2021 Baltic Way, 12
Tags: incenter , geometry
Let $I$ be the incentre of a triangle $ABC$. Let $F$ and $G$ be the projections of $A$ onto the lines $BI$ and $CI$, respectively. Rays $AF$ and $AG$ intersect the circumcircles of the triangles $CFI$ and $BGI$ for the second time at points $K$ and $L$, respectively. Prove that the line $AI$ bisects the segment $KL$.

2003 Denmark MO - Mohr Contest, 5
Tags: combinatorics , number theory
For which natural numbers $n\ge 2$ can the numbers from $1$ to $16$ be lined up in a square scheme so that the four row sums and the four column sums are all mutually different and divisible by $n$?

2007 AIME Problems, 15
Tags: geometry , ratio , trigonometry , quadratic
Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA=5$ and $CD=2$. Point $E$ lies on side $CA$ such that $\angle DEF = 60^\circ$. The area of triangle $DEF$ is $14\sqrt{3}$. The two possible values of the length of side $AB$ are $p \pm q\sqrt{r}$, where $p$ and $q$ are rational, and $r$ is an integer not divisible by the square of a prime. Find $r$.

2007 Junior Balkan Team Selection Tests - Romania, 2
Tags: calculus , inequalities
Let $x, y, z \ge 0$ be real numbers. Prove that: \[\frac{x^{3}+y^{3}+z^{3}}{3}\ge xyz+\frac{3}{4}|(x-y)(y-z)(z-x)| .\] [hide="Additional task"]Find the maximal real constant $\alpha$ that can replace $\frac{3}{4}$ such that the inequality is still true for any non-negative $x,y,z$.[/hide]

2000 Federal Competition For Advanced Students, Part 2, 2
Tags: number theory
Find all pairs of integers $(m, n)$ such that \[ \left| (m^2 + 2000m+ 999999)- (3n^3 + 9n^2 + 27n) \right|= 1.\]

2016 Costa Rica - Final Round, LR1
Tags: combinatorics , tiles , rectangle
With $21$ tiles, some white and some black, a $3 \times 7$ rectangle is formed. Show that there are always four tokens of the same color located at the vertices of a rectangle.

2019 India IMO Training Camp, P2
Tags: geometry , rectangle
Determine all positive integers $m$ satisfying the condition that there exists a unique positive integer $n$ such that there exists a rectangle which can be decomposed into $n$ congruent squares and can also be decomposed into $m+n$ congruent squares.

1989 All Soviet Union Mathematical Olympiad, 502
Tags: lattice points , polygon , rectangle , tiling
Show that for each integer $n > 0$, there is a polygon with vertices at lattice points and all sides parallel to the axes, which can be dissected into $1 \times 2$ (and / or $2 \times 1$) rectangles in exactly $n$ ways.

2012 Thailand Mathematical Olympiad, 10
Tags: algebra , number theory , irrational , inequalities
Let $x$ be an irrational number. Show that there are integers $m$ and $n$ such that $\frac{1}{2555}< mx + n <\frac{1}{2012}$

2016 Germany National Olympiad (4th Round), 6
Tags: algebra , cyclic function , function , inequality , maximum
Let \[ f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3 \] be defined for non-negative real numbers $x_1,x_2,\dots,x_7$ with sum $1$. Prove that $f(x_1,x_2,\dots,x_7)$ has a maximum value and find that value.

2012 Czech And Slovak Olympiad IIIA, 1
Tags: number theory , prime number , prime
Find all integers for which $n$ is $n^4 -3n^2 + 9$ prime

2009 District Olympiad, 4
Tags: complex number , geometry , absolute value
[b]a)[/b] Let $ z_1,z_2,z_3 $ be three complex numbers of same absolute value, and $ 0=z_1+z_2+z_3. $ Show that these represent the affixes of an equilateral triangle. [b]b)[/b] Find all subsets formed by roots of the same unity that have the property that any three elements of every such, doesn’t represent the vertices of an equilateral triangle.

1992 IMO Longlists, 56
Tags: path , combinatorics , graph theory
A directed graph (any two distinct vertices joined by at most one directed line) has the following property: If $x, u,$ and $v$ are three distinct vertices such that $x \to u$ and $x \to v$, then $u \to w$ and $v \to w$ for some vertex $w$. Suppose that $x \to u \to y \to\cdots \to z$ is a path of length $n$, that cannot be extended to the right (no arrow goes away from $z$). Prove that every path beginning at $x$ arrives after $n$ steps at $z.$

2021 MMATHS, 4
Tags:
Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!" Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I'd know for certain what it is?" Cat says, "Nope! However, if I now told you the units digit of my favorite number, you'd know which one it is!" Claire says, "Now I know your favorite number!" What is Cat's favorite number? [i]Proposed by Andrew Wu[/i]

2013 BMT Spring, 9
Tags: combinatorics
$2013$ people sit in a circle, playing a ball game. When one player has a ball, he may only pass it to another player $3$, $11$, or $61$ seats away (in either direction). If $f(A,B)$ represents the minimal number of passes it takes to get the ball from Person $A$ to Person $B$, what is the maximal possible value of $f$?

2017 Saudi Arabia JBMO TST, 3
Tags: combinatorics
On the table, there are $1024$ marbles and two students, $A$ and $B$, alternatively take a positive number of marble(s). The student $A$ goes first, $B$ goes after that and so on. On the first move, $A$ takes $k$ marbles with $1 < k < 1024$. On the moves after that, $A$ and $B$ are not allowed to take more than $k$ marbles or $0$ marbles. The student that takes the last marble(s) from the table wins. Find all values of $k$ the student $A$ should choose to make sure that there is a strategy for him to win the game.

2023 Kazakhstan National Olympiad, 6
Tags: geometry
Inside an equilateral triangle with side $3$ there are two rhombuses with sides $1,061$ and acute angles $60^\circ$. Prove that these two rhombuses intersect each other. (The vertices of the rhombus are strictly inside the triangle.)

1996 Iran MO (3rd Round), 2
Tags: inequalities , similar triangles , geometry
Let $ABCD$ be a convex quadrilateral. Construct the points $P,Q,R,$ and $S$ on continue of $AB,BC,CD,$ and $DA$, respectively, such that \[BP=CQ=DR=AS.\] Show that if $PQRS$ is a square, then $ABCD$ is also a square.

2014 Sharygin Geometry Olympiad, 6
Tags: perpendicular bisector , geometry
Given a circle with center $O$ and a point $P$ not lying on it, let $X$ be an arbitrary point on this circle and $Y$ be a common point of the bisector of angle $POX$ and the perpendicular bisector to segment $PX$. Find the locus of points $Y$.

2011 Uzbekistan National Olympiad, 2
Tags: circumcircle , parameterization , trigonometry , geometry
Let triangle ABC with $ AB=c$ $AC=b$ $BC=a$ $R$ circumradius, $p$ half peremetr of $ABC$. I f $\frac{acosA+bcosB+ccosC}{asinA+bsinB+csinC}=\frac{p}{9R}$ then find all value of $cosA$.