Found problems: 85335
1989 Chile National Olympiad, 6
The function $f$, with domain on the set of non-negative integers, is defined by the following :
$\bullet$ $f (0) = 2$
$\bullet$ $(f (n + 1) -1)^2 + (f (n)-1) ^2 = 2f (n) f (n + 1) + 4$, taking $f (n)$ the largest possible value.
Determine $f (n)$.
2004 AMC 8, 14
What is the area enclosed by the geoboard quadrilateral below?
[asy]
int i,j;
for(i=0; i<11; i=i+1) {
for(j=0; j<11; j=j+1) {
dot((i,j));
}
}
draw((0,5)--(4,0)--(10,10)--(3,4)--cycle, linewidth(0.7));
[/asy]
$\textbf{(A)} 15\qquad
\textbf{(B)} 18\tfrac12\qquad
\textbf{(C)} 22\tfrac12\qquad
\textbf{(D)} 27\qquad
\textbf{(E)} 41\qquad$
1962 All Russian Mathematical Olympiad, 025
Given $a_0, a_1, ... , a_n$. It is known that $$a_0=a_n=0, a_{k-1}-2a_k+a_{k+1}\ge 0$$ for all $k = 1, 2, ... , k-1$.Prove that all the numbers are nonnegative.
2019 Moroccan TST, 4
Let $p$ be a prime number. Find all the positive integers $n$ such that $p+n$ divides $pn$
1991 Arnold's Trivium, 51
Calculate the integral
\[\int_{-\infty}^{+\infty}e^{ikx}\frac{1-e^x}{1+e^x}dx\]
2017 OMMock - Mexico National Olympiad Mock Exam, 5
Let $k$ be a positive real number. Determine all functions $f:[-k, k]\rightarrow[0, k]$ satisfying the equation
$$f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2$$
for any $x, y\in[-k, k]$ such that $x+y\in[-k, k]$.
[i]Proposed by Maximiliano Sánchez[/i]
2007 IMO Shortlist, 7
Let $ \alpha < \frac {3 \minus{} \sqrt {5}}{2}$ be a positive real number. Prove that there exist positive integers $ n$ and $ p > \alpha \cdot 2^n$ for which one can select $ 2 \cdot p$ pairwise distinct subsets $ S_1, \ldots, S_p, T_1, \ldots, T_p$ of the set $ \{1,2, \ldots, n\}$ such that $ S_i \cap T_j \neq \emptyset$ for all $ 1 \leq i,j \leq p$
[i]Author: Gerhard Wöginger, Austria[/i]
2018 India PRMO, 4
The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation.
1998 Estonia National Olympiad, 1
Let $d_1$ and $d_2$ be divisors of a positive integer $n$. Suppose that the greatest common divisor of $d_1$ and $n/d_2$ and the greatest common divisor of $d_2$ and $n/d_1$ are equal. Show that $d_1 = d_2$.
2021 Israel TST, 1
Let $ABCDEFGHIJ$ be a regular $10$-gon. Let $T$ be a point inside the $10$-gon, such that the $DTE$ is isosceles: $DT = ET$ , and its angle at the apex is $72^\circ$. Prove that there exists a point $S$ such that $FTS$ and $HIS$ are both isosceles, and for both of them the angle at the apex is $72^\circ$.
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 \[ \displaystyle\sum_{n=0}^{\infty} \dfrac {n}{n^4 + n^2 + 1}. \]