Found problems: 85335
2018 PUMaC Live Round, 1.2
Define a function given the following $2$ rules:
$\qquad$ 1) for prime $p$, $f(p)=p+1$.
$\qquad$ 2) for positive integers $a$ and $b$, $f(ab)=f(a)\cdot f(b)$.
For how many positive integers $n\leq 100$ is $f(n)$ divisible by $3$?
1967 IMO Longlists, 24
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?
Russian TST 2014, P2
Let $n$ be a positive integer, and let $A$ be a subset of $\{ 1,\cdots ,n\}$. An $A$-partition of $n$ into $k$ parts is a representation of n as a sum $n = a_1 + \cdots + a_k$, where the parts $a_1 , \cdots , a_k $ belong to $A$ and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set $\{ a_1 , a_2 , \cdots , a_k \} $.
We say that an $A$-partition of $n$ into $k$ parts is optimal if there is no $A$-partition of $n$ into $r$ parts with $r<k$. Prove that any optimal $A$-partition of $n$ contains at most $\sqrt[3]{6n}$ different parts.
2002 AMC 12/AHSME, 17
Several sets of prime numbers, such as $ \{ 7, 83, 421, 659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
$ \textbf{(A)}\ 193\qquad\textbf{(B)}\ 207\qquad\textbf{(C)}\ 225\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 447$
1988 All Soviet Union Mathematical Olympiad, 464
$ABCD$ is a convex quadrilateral. The midpoints of the diagonals and the midpoints of $AB$ and $CD$ form another convex quadrilateral $Q$. The midpoints of the diagonals and the midpoints of $BC$ and $CA$ form a third convex quadrilateral $Q'$. The areas of $Q$ and $Q'$ are equal. Show that either $AC$ or $BD$ divides $ABCD$ into two parts of equal area.
2010 Contests, 3
Find all non-zero real numbers $ x, y, z$ which satisfy the system of equations:
\[ (x^2 \plus{} xy \plus{} y^2)(y^2 \plus{} yz \plus{} z^2)(z^2 \plus{} zx \plus{} x^2) \equal{} xyz\]
\[ (x^4 \plus{} x^2y^2 \plus{} y^4)(y^4 \plus{} y^2z^2 \plus{} z^4)(z^4 \plus{} z^2x^2 \plus{} x^4) \equal{} x^3y^3z^3\]
2018 Moldova EGMO TST, 4
Find all sets of positive integers $A=\big\{ a_1,a_2,...a_{19}\big\}$ which satisfy the following:
$1\big) a_1+a_2+...+a_{19}=2017;$
$2\big) S(a_1)=S(a_2)=...=S(a_{19})$ where $S\big(n\big)$ denotes digit sum of number $n$.
2022 AMC 8 -, 22
A bus takes $2$ minutes to drive from one stop to the next, and waits $1$ minute at each stop to let passengers board. Zia takes $5$ minutes to walk from one bus stop to the next. As Zia reaches a bus stop, if the bus is at the previous stop or has already left the previous stop, then she will wait for the bus. Otherwise she will start walking toward the next stop. Suppose the bus and Zia start at the same time toward the library, with the bus $3$ stops behind. After how many minutes will Zia board the bus?
$\textbf{(A)} ~17\qquad\textbf{(B)} ~19\qquad\textbf{(C)} ~20\qquad\textbf{(D)} ~21\qquad\textbf{(E)} ~23$
1951 Miklós Schweitzer, 9
Let $ \{m_1,m_2,\dots\}$ be a (finite or infinite) set of positive integers. Consider the system of congruences
(1) $ x\equiv 2m_i^2 \pmod{2m_i\minus{}1}$ ($ i\equal{}1,2,...$ ).
Give a necessary and sufficient condition for the system (1) to be solvable.
Brazil L2 Finals (OBM) - geometry, 2003.5
Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.
1991 All Soviet Union Mathematical Olympiad, 547
$ABC$ is an acute-angled triangle with circumcenter $O$. The circumcircle of $ABO$ intersects$ AC$ and $BC$ at $M$ and $N$. Show that the circumradii of $ABO$ and $MNC$ are the same.
2013 Saudi Arabia BMO TST, 2
The base-$7$ representation of number $n$ is $\overline{abc}_{(7)}$, and the base-$9$ representation of number $n$ is $\overline{cba}_{(9)}$. What is the decimal (base-$10$) representation of $n$?
2018 Bangladesh Mathematical Olympiad, 2
BdMO National 2018 Higher Secondary P2
$AB$ is a diameter of a circle and $AD$ & $BC$ are two tangents of that circle.$AC$ & $BD$ intersect on a point of the circle.$AD=a$ & $BC=b$.If $a\neq b$ then $AB=?$
2010 China Team Selection Test, 3
Given integer $n\geq 2$ and real numbers $x_1,x_2,\cdots, x_n$ in the interval $[0,1]$. Prove that there exist real numbers $a_0,a_1,\cdots,a_n$ satisfying the following conditions:
(1) $a_0+a_n=0$;
(2) $|a_i|\leq 1$, for $i=0,1,\cdots,n$;
(3) $|a_i-a_{i-1}|=x_i$, for $i=1,2,\cdots,n$.
2005 Harvard-MIT Mathematics Tournament, 8
Compute \[ \displaystyle\sum_{n=0}^{\infty} \dfrac {n}{n^4 + n^2 + 1}. \]
II Soros Olympiad 1995 - 96 (Russia), 10.2
Without using a calculator, find out what is greater: $\sin 28^o$ or $tg21^o$?
LMT Team Rounds 2010-20, 2020.S11
Let set $\mathcal{S}$ contain all positive integers less than or equal to $2020$ that can be written in the form $n(n+1)$ for some positive integer $n$. Compute the number of ordered pairs $(a,b)$ such that $a, b\in \mathcal{S}$ and $a-b$ is a power of two.
2006 Purple Comet Problems, 23
We have two positive integers both less than $1000$. The arithmetic mean and the geometric mean of these numbers are consecutive odd integers. Find the maximum possible value of the difference of the two integers.
1997 Brazil National Olympiad, 6
$f$ is a plane map onto itself such that points at distance 1 are always taken at point at distance 1.
Show that $f$ preserves distances.
2019 Tournament Of Towns, 5
Basil has an unrestricted supply of straight bricks $1 \times 1 \times 3$ and Γ-shape bricks made of three cubes $1\times 1\times 1$. Basil filled a whole box $m \times n \times k$ with these bricks, where $m, n$ and $k$ are integers greater than $1$. Prove that it was sufficient to use only Γ-shape bricks.
(Mikhail Evdokimov)
2016 SDMO (Middle School), 2
Let $AB$ be a diameter of a circle and let $C$ be a point on $AB$ with $2\cdot AC=BC$. Let $D$ and $E$ be points on the circle such that $DC\perp AB$ and $DE$ is a second diameter. What is the ratio of the area of $\triangle{DCE}$ to the area of $\triangle{ABD}$?
1961 All-Soviet Union Olympiad, 3
Consider $n$ points, some of them connected by segments. These segments do not intersect each other. You can reach every point from any every other one in exactly one way by traveling along the segments. Prove that the total number of segments is $n-1$.
2023 Korea Junior Math Olympiad, 3
Positive integers $a_1, a_2, \dots, a_{2023}$ satisfy the following conditions.
[list]
[*] $a_1 = 5, a_2 = 25$
[*] $a_{n + 2} = 7a_{n+1}-a_n-6$ for each $n = 1, 2, \dots, 2021$
[/list]
Prove that there exist integers $x, y$ such that $a_{2023} = x^2 + y^2.$
1981 USAMO, 3
If $A,B,C$ are the angles of a triangle, prove that
\[-2 \le \sin{3A}+\sin{3B}+\sin{3C} \le \frac{3\sqrt{3}}{2}\]
and determine when equality holds.
2008 Bosnia and Herzegovina Junior BMO TST, 3
Point $ M$ is given in the interior of parallelogram $ ABCD$, and the point $ N$ inside triangle $ AMD$ is chosen so that $ < MNA \plus{} < MCB \equal{} MND \plus{} < MBC \equal{} 180^0$. Prove that $ MN$ is parallel to $ AB$.