Found problems: 85335
2021 Bolivian Cono Sur TST, 2
Let $n$ be a posititve integer and let $M$ the set of all all integer cordinates $(a,b,c)$ such that $0 \le a,b,c \le n$. A frog needs to go from the point $(0,0,0)$ to the point $(n,n,n)$ with the following rules:
$\cdot$ The frog can jump only in points of $M$
$\cdot$ The frog can't jump more than $1$ time over the same point.
$\cdot$ In each jump the frog can go from $(x,y,z)$ to $(x+1,y,z)$, $(x,y+1,z)$, $(x,y,z+1)$ or $(x,y,z-1)$
In how many ways the Frog can make his target?
2021 New Zealand MO, 7
Let $a, b, c, d$ be integers such that $a > b > c > d \ge -2021$ and $$\frac{a + b}{b + c}=\frac{c + d}{d + a}$$ (and $b + c \ne 0 \ne d + a$). What is the maximum possible value of $ac$?
2018 Bulgaria National Olympiad, 4.
Let $ABCD$ be a quadrilateral ,circumscribed about a circle. Let $M$ be a point on the side $AB$. Let $I_{1}$,$I_{2}$ and $I_{3}$ be the incentres of triangles $AMD$, $CMD$ and $BMC$ respectively. Prove that $I_{1}I_{2}I_{3}M$ is circumscribed.
2023 JBMO TST - Turkey, 2
A marble is placed on each $33$ unit square of a $10*10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?
2014 Bulgaria JBMO TST, 8
Find the smallest positive integer $n,$ such that $3^k+n^k+ (3n)^k+ 2014^k$ is a perfect square for all natural numbers $k,$ but not a perfect cube, for all natural numbers $k.$
2006 Bundeswettbewerb Mathematik, 4
A positive integer is called [i]digit-reduced[/i] if at most nine different digits occur in its decimal representation (leading $0$s are omitted.) Let $M$ be a finite set of [i]digit-reduced[/i] numbers. Show that the sum of the reciprocals of the elements in $M$ is less than $180$.
2019 Online Math Open Problems, 30
For a positive integer $n$, we say an $n$-[i]transposition[/i] is a bijection $\sigma: \{1,2, \dots , n\} \rightarrow \{1,2, \dots , n\}$ such that there exist exactly two elements $i$ of $\{1,2, \dots , n\}$ such that $\sigma(i) \neq i$.
Fix some four pairwise distinct $n$-transpositions $\sigma_1,\sigma_2,\sigma_3,\sigma_4$. Let $q$ be any prime, and let $\mathbb{F}_q$ be the integers modulo $q$. Consider all functions $f:(\mathbb{F}_q^n)^n\to\mathbb{F}_q$ that satisfy, for all integers $i$ with $1 \leq i \leq n$ and all $x_1,\ldots x_{i-1},x_{i+1}, \dots ,x_n, y, z\in\mathbb{F}_q^n$, \[f(x_1, \ldots ,x_{i-1}, y, x_{i+1}, \ldots , x_n) +f(x_1, \ldots ,x_{i-1}, z, x_{i+1}, \ldots , x_n) = f(x_1, \ldots ,x_{i-1}, y+z, x_{i+1}, \ldots , x_n), \] and that satisfy, for all $x_1,\ldots,x_n\in\mathbb{F}_q^n$ and all $\sigma\in\{\sigma_1,\sigma_2,\sigma_3,\sigma_4\}$, \[f(x_1,\ldots,x_n)=-f(x_{\sigma(1)},\ldots,x_{\sigma(n)}).\]
(Note that the equalities in the previous sentence are in $\mathbb F_q$. Note that, for any $a_1,\ldots ,a_n, b_1, \ldots , b_n \in \mathbb{F}_q$, we have $(a_1,\ldots , a_n)+(b_1, \ldots, b_n)=(a_1+b_1,\ldots, a_n+b_n)$, where $a_1+b_1,\ldots , a_n+b_n \in \mathbb{F}_q$.)
For a given tuple $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$, let $g(x_1,\ldots,x_n)$ be the number of different values of $f(x_1,\ldots,x_n)$ over all possible functions $f$ satisfying the above conditions.
Pick $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$ uniformly at random, and let $\varepsilon(q,\sigma_1,\sigma_2,\sigma_3,\sigma_4)$ be the expected value of $g(x_1,\ldots,x_n)$. Finally, let \[\kappa(\sigma_1,\sigma_2,\sigma_3,\sigma_4)=-\lim_{q \to \infty}\log_q\left(-\ln\left(\frac{\varepsilon(q,\sigma_1,\sigma_2,\sigma_3,\sigma_4)-1}{q-1}\right)\right).\]
Pick four pairwise distinct $n$-transpositions $\sigma_1,\sigma_2,\sigma_3,\sigma_4$ uniformly at random from the set of all $n$-transpositions. Let $\pi(n)$ denote the expected value of $\kappa(\sigma_1,\ldots,\sigma_4)$. Suppose that $p(x)$ and $q(x)$ are polynomials with real coefficients such that $q(-3) \neq 0$ and such that $\pi(n)=\frac{p(n)}{q(n)}$ for infinitely many positive integers $n$. Compute $\frac{p\left(-3\right)}{q\left(-3\right)}$.
[i]Proposed by Gopal Goel[/i]
2024 Sharygin Geometry Olympiad, 13
Can an arbitrary polygon be cut into isosceles trapezoids?
2011 Peru IMO TST, 5
On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set.
[i]Proposed by Tonći Kokan, Croatia[/i]
2014 Iran MO (2nd Round), 2
Let $ABCD$ be a square. Let $N,P$ be two points on sides $AB, AD$, respectively such that $NP=NC$, and let $Q$ be a point on $AN$ such that $\angle QPN = \angle NCB$. Prove that \[ \angle BCQ = \dfrac{1}{2} \angle AQP .\]
1996 IMO Shortlist, 4
Find all positive integers $ a$ and $ b$ for which
\[ \left \lfloor \frac{a^2}{b} \right \rfloor \plus{} \left \lfloor \frac{b^2}{a} \right \rfloor \equal{} \left \lfloor \frac{a^2 \plus{} b^2}{ab} \right \rfloor \plus{} ab.\]
2005 Romania National Olympiad, 2
Let $f:[0,1)\to (0,1)$ a continous onto (surjective) function.
a) Prove that, for all $a\in(0,1)$, the function $f_a:(a,1)\to (0,1)$, given by $f_a(x) = f(x)$, for all $x\in(a,1)$ is onto;
b) Give an example of such a function.
2016 CCA Math Bonanza, T8
As $a$, $b$ and $c$ range over [i]all[/i] real numbers, let $m$ be the smallest possible value of $$2\left(a+b+c\right)^2+\left(ab-4\right)^2+\left(bc-4\right)^2+\left(ca-4\right)^2$$ and $n$ be the number of ordered triplets $\left(a,b,c\right)$ such that the above quantity is minimized. Compute $m+n$.
[i]2016 CCA Math Bonanza Team #8[/i]
2011 Peru IMO TST, 6
Let $a_1, a_2, \cdots , a_n$ be real numbers, with $n\geq 3,$ such that $a_1 + a_2 +\cdots +a_n = 0$ and $$ 2a_k\leq a_{k-1} + a_{k+1} \ \ \ \text{for} \ \ \ k = 2, 3, \cdots , n-1.$$ Find the least number $\lambda(n),$ such that for all $k\in \{ 1, 2, \cdots, n\} $ it is satisfied that $|a_k|\leq \lambda (n)\cdot \max \{|a_1|, |a_n|\} .$
2013 IMO Shortlist, N7
Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair of $(a,b)$ of positive integers is called [i]good[/i] if
\[a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu \right \rfloor = m.\] A good pair $(a,b)$ is called [i]excellent[/i] if neither of the pair $(a-b,b)$ and $(a,b-a)$ is good.
Prove that the number of excellent pairs is equal to the sum of the positive divisors of $m$.
1991 Romania Team Selection Test, 4
Let $S$ be the set of all polygonal areas in a plane. Prove that there is a function $f : S \to (0,1)$ which satisfies $f(S_1 \cup S_2) = f(S_1)+ f(S_2)$ for any $S_1,S_2 \in S$ which have common points only on their borders
2017 All-Russian Olympiad, 5
$P(x)$ is polynomial with degree $n\geq 2$ and nonnegative coefficients. $a,b,c$ - sides for some triangle. Prove, that $\sqrt[n]{P(a)},\sqrt[n]{P(b)},\sqrt[n]{P(c)}$ are sides for some triangle too.
1997 May Olympiad, 4
Joaquín and his brother Andrés go to class every day on the $62$ bus. Joaquín always pays for the tickets. Each ticket has a $5$-digit number printed on it. One day, Joaquín observes that the numbers on his tickets - his and his brother's - as well as being consecutive, are such that the sum of the ten digits is precisely $62$. Andrés asks him if the sum of the digits of any of the tickets is $35$ and, knowing the answer, he can directly say the number of each ticket. What were those numbers?
2007 Iran MO (3rd Round), 1
Let $ ABC$, $ l$ and $ P$ be arbitrary triangle, line and point. $ A',B',C'$ are reflections of $ A,B,C$ in point $ P$. $ A''$ is a point on $ B'C'$ such that $ AA''\parallel l$. $ B'',C''$ are defined similarly. Prove that $ A'',B'',C''$ are collinear.
2016 JBMO TST - Turkey, 2
A and B plays a game on a pyramid whose base is a $2016$-gon. In each turn, a player colors a side (which was not colored before) of the pyramid using one of the $k$ colors such that none of the sides with a common vertex have the same color. If A starts the game, find the minimal value of $k$ for which $B$ can guarantee that all sides are colored.
1997 Bulgaria National Olympiad, 1
Consider the polynomial
$P_n(x) = \binom {n}{2}+\binom {n}{5}x+\binom {n}{8}x^2 + \cdots + \binom {n}{3k+2}x^{3k}$
where $n \ge 2$ is a natural number and $k = \left\lfloor \frac{n-2}{3} \right \rfloor$
[b](a)[/b] Prove that $P_{n+3}(x)=3P_{n+2}(x)-3P_{n+1}(x)+(x+1)P_n(x)$
[b](b)[/b] Find all integer numbers $a$ such that $P_n(a^3)$ is divisible by $3^{ \lfloor \frac{n-1}{2} \rfloor}$ for all $n \ge 2$
MBMT Team Rounds, 2020.29
The center of circle $\omega_1$ of radius $6$ lies on circle $\omega_2$ of radius $6$. The circles intersect at points $K$ and $W$. Let point $U$ lie on the major arc $\overarc{KW}$ of $\omega_2$, and point $I$ be the center of the largest circle that can be inscribed in $\triangle KWU$. If $KI+WI=11$, find $KI\cdot WI$.
[i]Proposed by Bradley Guo[/i]
1997 IMO Shortlist, 7
The lengths of the sides of a convex hexagon $ ABCDEF$ satisfy $ AB \equal{} BC$, $ CD \equal{} DE$, $ EF \equal{} FA$. Prove that:
\[ \frac {BC}{BE} \plus{} \frac {DE}{DA} \plus{} \frac {FA}{FC} \geq \frac {3}{2}.
\]
2007 Hanoi Open Mathematics Competitions, 2
What is largest positive integer n satisfying the
following inequality:
$n^{2006}$ < $7^{2007}$?
2014 Regional Olympiad of Mexico Center Zone, 1
Find the smallest positive integer $n$ that satisfies that for any $n$ different integers, the product of all the positive differences of these numbers is divisible by $2014$.