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 Indonesia TST, 2
Tags: combinatorics , coin , invariant , monovariant
Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.

May Olympiad L1 - geometry, 2015.3
Tags: geometry
In the quadrilateral $ABCD$, we have $\angle C$ is triple of $\angle A$, let $P$ be a point in the side $AB$ such that $\angle DPA = 90º$ and let $Q$ be a point in the segment $DA$ where $\angle BQA = 90º$ the segments $DP$ and $CQ$ intersects in $O$ such that $BO = CO = DO$, find $\angle A$ and $\angle C$.

2016 Thailand TSTST, 1
Tags: algebra , polynomial , number theory , integer polynomial
Find all polynomials $P\in\mathbb{Z}[x]$ such that $$|P(x)-x|\leq x^2+1$$ for all real numbers $x$.

2012 Thailand Mathematical Olympiad, 5
Tags: algebra , functional , functional equation
Determine all functions $f : R \to R$ satisfying $f(f(x) + xf(y))= 3f(x) + 4xy$ for all real numbers $x,y$.

1997 China Team Selection Test, 1
Tags: polynomial , inequalities , vieta , induction , algebra
Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions: [b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0$; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

2010 District Olympiad, 2
Tags: remainder , number theory
Let $n$ be an integer, $n \ge 2$. Find the remainder of the division of the number $n(n + 1)(n + 2)$ by $n - 1$.

2023 Euler Olympiad, Round 2, 5
Tags: inequalities , algebra , euler
Find the smallest constant M, so that for any real numbers $a_1, a_2, \dots a_{2023} \in [4, 6]$ and $b_1, b_2, \dots b_{2023} \in [9, 12] $ following inequality holds: $$ \sqrt{a_1^2 + a_2^2 + \dots + a_{2023}^2} \cdot \sqrt{b_1^2 + b_2^2 + \dots + b_{2023}^2} \leq M \cdot \left ( a_1 b_1 + a_2 b_2 + \dots + a_{2023} b_{2023} \right) $$ [i]Proposed by Zaza Meliqidze, Georgia[/i]

1963 IMO, 3
Tags: inequalities , geometry , polygon , geometric inequality , angle
In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation \[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \] Prove that $a_{1}=a_{2}= \ldots= a_{n}$.

2016 Czech-Polish-Slovak Match, 2
Tags: combinatorics
Let $m,n > 2$ be even integers. Consider a board of size $m \times n$ whose every cell is colored either black or white. The Guesser does not see the coloring of the board but may ask the Oracle some questions about it. In particular, she may inquire about two adjacent cells (sharing an edge) and the Oracle discloses whether the two adjacent cells have the same color or not. The Guesser eventually wants to fi nd the parity of the number of adjacent cell-pairs whose colors are diff erent. What is the minimum number of inquiries the Guesser needs to make so that she is guaranteed to find her answer?(Czech Republic)

2010 Iran MO (3rd Round), 5
Tags: number theory
prove that if $p$ is a prime number such that $p=12k+\{2,3,5,7,8,11\}$($k \in \mathbb N \cup \{0\}$), there exist a field with $p^2$ elements.($\frac{100}{6}$ points)

2024 AMC 8 -, 2
Tags:
What is the value of this expression in decimal form? \[\dfrac{44}{11}+\dfrac{110}{44}+\dfrac{44}{1100}\] $\textbf{(A) }6.4\qquad\textbf{(B) }6.504\qquad\textbf{(C) }6.54\qquad\textbf{(D) }6.9\qquad\textbf{(E) }6.94$

1992 AMC 12/AHSME, 18
Tags: inequalities
The increasing sequence of positive integers $a_{1},a_{2},a_{3},\ldots$ has the property that $a_{n+2} = a_{n} + a_{n+1}$ for all $n \ge 1$. If $a_{7} = 120$, then $a_{8}$ is $ \textbf{(A)}\ 128\qquad\textbf{(B)}\ 168\qquad\textbf{(C)}\ 193\qquad\textbf{(D)}\ 194\qquad\textbf{(E)}\ 210 $

2017 Azerbaijan BMO TST, 2
Tags: divisibility , rmn
Determine all positive integers $n$ such that all positive integers less than or equal to $n$ and relatively prime to $n$ are pairwise coprime.

2000 Romania Team Selection Test, 2
Tags: circumcircle , geometry
Let $ABC$ be an acute-angled triangle and $M$ be the midpoint of the side $BC$. Let $N$ be a point in the interior of the triangle $ABC$ such that $\angle NBA=\angle BAM$ and $\angle NCA=\angle CAM$. Prove that $\angle NAB=\angle MAC$. [i]Gabriel Nagy[/i]

2003 Alexandru Myller, 3
Tags: algebra , function , analytic geometry , invariant
Let $ S $ be the first quadrant and $ T:S\longrightarrow S $ be a transformation that takes the reciprocal of the coordinates of the points that belong to its domain. Define an [i]S-line[/i] to be the intersection of a line with $ S. $ [b]a)[/b] Show that the fixed points of $ T $ lie on any fixed S-line of $ T. $ [b]b)[/b] Find all fixed S-lines of $ T. $ [i]Gabriel Popa[/i]

1994 Tournament Of Towns, (420) 1
Tags: combinatorics
Several boys and girls are dancing a waltz at a ball. Is it possible that each girl can always get to dance the next dance with a boy who is either more handsome or more clever than for the previous dance, and that each time one of the girls gets to dance the next dance with a boy who is more handsome and more clever? (The numbers of boys and girls are equal and all are dancing.) (AY Belov)

MOAA Individual Speed General Rounds, 2021.8
Tags: speed
Andrew chooses three (not necessarily distinct) integers $a$, $b$, and $c$ independently and uniformly at random from $\{1,2,3,4,5,6,7\}$. Let $p$ be the probability that $abc(a+b+c)$ is divisible by $4$. If $p$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$. [i]Proposed by Andrew Wen[/i]

2020 Simon Marais Mathematics Competition, A3
Tags: analysis , real analysis
Determine the set of real numbers $\alpha$ that can be expressed in the form \[\alpha=\sum_{n=0}^{\infty}\frac{x_{n+1}}{x_n^3}\] where $x_0,x_1,x_2,\dots$ is an increasing sequence of real numbers with $x_0=1$.

PEN S Problems, 31
Tags:
Is there a $3 \times 3$ magic square consisting of distinct Fibonacci numbers (both $f_{1}$ and $f_{2}$ may be used; thus two $1$s are allowed)?

2014 IFYM, Sozopol, 2
Tags: number theory , divisor
Find the least natural number $n$, which has at least 6 different divisors $1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.

1996 China National Olympiad, 3
Tags: geometry
In the triangle $ABC$, $\angle{C}=90^{\circ},\angle {A}=30^{\circ}$ and $BC=1$. Find the minimum value of the longest side of all inscribed triangles (i.e. triangles with vertices on each of three sides) of the triangle $ABC$.

2015 Online Math Open Problems, 22
Tags:
For a positive integer $n$ let $n\#$ denote the product of all primes less than or equal to $n$ (or $1$ if there are no such primes), and let $F(n)$ denote the largest integer $k$ for which $k\#$ divides $n$. Find the remainder when $F(1) + F(2) +F(3) + \dots + F(2015\#-1) + F(2015\#)$ is divided by $3999991$. [i]Proposed by Evan Chen[/i]

2006 Junior Balkan MO, 1
Tags: floor function , number theory
If $n>4$ is a composite number, then $2n$ divides $(n-1)!$.

2019 Turkey EGMO TST, 5
Tags: geometry
Let $D$ be the midpoint of $\overline{BC}$ in $\Delta ABC$. Let $P$ be any point on $\overline{AD}$. If the internal angle bisector of $\angle ABP$ and $\angle ACP$ intersect at $Q$. Prove that, if $BQ \perp QC$, then $Q$ lies on $AD$

2007 Hong Kong TST, 6
Tags: number theory
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 6 Determine all pairs $(x,y)$ of positive integers such that $\frac{x^{2}y+x+y}{xy^{2}+y+11}$ is an integer.