Found problems: 85335
2024 Tuymaada Olympiad, 4
A triangle $ABC$ is given. The segment connecting the points where the excircles touch $AB$ and $AC$ meets the bisector of angle $C$ at $X$. The segment connecting the points where the excircles touch $BC$ and $AC$ meets the bisector of angle $A$ at $Y$. Prove that the midpoint of $XY$ is equidistant from $A$ and $C$.
2004 Estonia Team Selection Test, 3
For which natural number $n$ is it possible to draw $n$ line segments between vertices of a regular $2n$-gon so that every vertex is an endpoint for exactly one segment and these segments have pairwise different lengths?
2005 MOP Homework, 2
Suppose that $n$ is s positive integer. Determine all the possible values of the first digit after the decimal point in the decimal expression of the number $\sqrt{n^3+2n^2+n}$
2000 Romania Team Selection Test, 3
Prove that for any positive integers $n$ and $k$ there exist positive integers $a>b>c>d>e>k$ such that
\[n=\binom{a}{3}\pm\binom{b}{3}\pm\binom{c}{3}\pm\binom{d}{3}\pm\binom{e}{3}\]
[i]Radu Ignat[/i]
2023 Mexican Girls' Contest, 4
A function $g$ is such that for all integer $n$:
$$g(n)=\begin{cases}
1\hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\geq 1 & \\
0 \hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\leq 0 &
\end{cases}$$
A function $f$ is such that for all integers $n\geq 0$ and $m\geq 0$:
$$f(0,m)=0 \hspace{0.5cm} \textrm{and}$$
$$f(n+1,m)=\Bigl(1-g(m)+g(m)\cdot g(m-1-f(n,m))\Bigr)\cdot\Bigl(1+f(n,m)\Bigr)$$
Find all the possible functions $f(m,n)$ that satisfies the above for all integers $n\geq0$ and $m\geq 0$
2006 Germany Team Selection Test, 3
Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$.
Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have
\[ n\mid a_i \minus{} b_i \minus{} c_i
\]
[i]Proposed by Ricky Liu & Zuming Feng, USA[/i]
2001 Croatia National Olympiad, Problem 4
Let $S$ be a set of $100$ positive integers less than $200$. Prove that there exists a nonempty subset $T$ of $S$ the product of whose elements is a perfect square.
2021 BMT, Tie 2
Real numbers $x$ and $y$ satisfy the equations $x^2 - 12y = 17^2$ and $38x - y^2 = 2 \cdot 7^3$. Compute $x + y$.
2021 Austrian MO Regional Competition, 1
Let $a$ and $b$ be positive integers and $c$ be a positive real number satisfying
$$\frac{a + 1}{b + c}=\frac{b}{a}.$$ Prove that $c \ge 1$ holds.
(Karl Czakler)
2000 Harvard-MIT Mathematics Tournament, 42
A $n$ by $n$ magic square contains numbers from $1$ to $n^2$ such that the sum of every row and every column is the same. What is this sum?
2010 AIME Problems, 8
Let $ N$ be the number of ordered pairs of nonempty sets $ \mathcal{A}$ and $ \mathcal{B}$ that have the following properties:
• $ \mathcal{A} \cup \mathcal{B} \equal{} \{1,2,3,4,5,6,7,8,9,10,11,12\}$,
• $ \mathcal{A} \cap \mathcal{B} \equal{} \emptyset$,
• The number of elements of $ \mathcal{A}$ is not an element of $ \mathcal{A}$,
• The number of elements of $ \mathcal{B}$ is not an element of $ \mathcal{B}$.
Find $ N$.
2018 Iran Team Selection Test, 1
Let $A_1, A_2, ... , A_k$ be the subsets of $\left\{1,2,3,...,n\right\}$ such that for all $1\leq i,j\leq k$:$A_i\cap A_j \neq \varnothing$. Prove that there are $n$ distinct positive integers $x_1,x_2,...,x_n$ such that for each $1\leq j\leq k$:
$$lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\}$$
[i]Proposed by Morteza Saghafian, Mahyar Sefidgaran[/i]
2004 Iran MO (3rd Round), 12
$\mathbb{N}_{10}$ is generalization of $\mathbb{N}$ that every hypernumber in $\mathbb{N}_{10}$ is something like: $\overline{...a_2a_1a_0}$ with $a_i \in {0,1..9}$
(Notice that $\overline {...000} \in \mathbb{N}_{10}$)
Also we easily have $+,*$ in $\mathbb{N}_{10}$.
first $k$ number of $a*b$= first $k$ nubmer of (first $k$ number of a * first $k$ number of b)
first $k$ number of $a+b$= first $k$ nubmer of (first $k$ number of a + first $k$ number of b)
Fore example $\overline {...999}+ \overline {...0001}= \overline {...000}$
Prove that every monic polynomial in $\mathbb{N}_{10}[x]$ with degree $d$ has at most $d^2$ roots.
2013 Stanford Mathematics Tournament, 5
A polygonal prism is made from a flexible material such that the two bases are regular $2^n$-gons $(n>1)$ of the same size. The prism is bent to join the two bases together without twisting, giving a figure with $2^n$ faces. The prism is then repeatedly twisted so that each edge of one base becomes aligned with each edge of the base exactly once. For an arbitrary $n$, what is the sum of the number of faces over all of these configurations (including the non-twisted case)?
2019 Hong Kong TST, 1
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
2005 Harvard-MIT Mathematics Tournament, 9
Compute \[ \displaystyle\sum_{k=0}^{\infty} \dfrac {4}{(4k)!}. \]
2005 Purple Comet Problems, 10
A jar contains $2$ yellow candies, $4$ red candies, and $6$ blue candies. Candies are randomly drawn out of the jar one-by-one and eaten. The probability that the $2$ yellow candies will be eaten before any of the red candies are eaten is given by the fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1972 Yugoslav Team Selection Test, Problem 2
If a convex set of points in the line has at least two diameters, say $AB$ and $CD$, prove that $AB$ and $CD$ have a common point.
2015 Kosovo Team Selection Test, 1
a)Prove that for every n,natural number exist natural numbers a and b such that
$(1-\sqrt{2})^n=a-b\sqrt{2}$ and $a^2-2b^2=(-1)^n$
b)Using first equation prove that for every n exist m such that
$(\sqrt{2}-1)^n=\sqrt{m}-\sqrt{m-1}$
2022 Belarusian National Olympiad, 9.7
Prove that for any positive integer $n$ there exist coprime numbers $a$ and $b$ such that for all $1 \leq k \leq n$ numbers $a+k$ and $b+k$ are not coprime.
2020 Taiwan TST Round 2, 4
Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$, $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$.
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$. In every turn, Alice picks a number $k$ from the set $\{1,2,\ldots,187\}$, and Bob will tell Alice the value of $f(k)$. Find the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns.
[i]Proposed by YaWNeeT[/i]
2017 Purple Comet Problems, 10
Find the number of rearrangements of the letters in the word MATHMEET that begin and end with the same letter such as TAMEMHET.
2010 Contests, 1
Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]
1986 IMO Longlists, 10
A set of $n$ standard dice are shaken and randomly placed in a straight line. If $n < 2r$ and $r < s$, then the probability that there will be a string of at least $r$, but not more than $s$, consecutive $1$'s can be written as $\frac{P}{6^{s+2}}$. Find an explicit expression for $P$.
2012 JBMO ShortLists, 2
Do there exist prime numbers $p$ and $q$ such that $p^2(p^3-1)=q(q+1)$ ?