Found problems: 85335
2008 iTest Tournament of Champions, 5
For positive integers $m,n\geq 3$, let $h(m,n)$ be the maximum (finite) number of intersection points between a simple $m$-gon and a simple $n$-gon. (Note: a polygon is simple if it does not intersect itself.) Evaluate \[\sum_{m=3}^6\sum_{n=3}^{12}h(m,n).\]
2020 Stars of Mathematics, 4
Let $a_0 = 1, \ a_1 = 2,$ and $a_2 = 10,$ and define $a_{k+2} = a_{k+1}^3+a_k^2+a_{k-1}$ for all positive integers $k.$ Is it possible for some $a_x$ to be divisible by $2021^{2021}?$
[i]Flavian Georgescu[/i]
2013 Sharygin Geometry Olympiad, 5
The altitude $AA'$, the median $BB'$, and the angle bisector $CC'$ of a triangle $ABC$ are concurrent at point $K$. Given that $A'K = B'K$, prove that $C'K = A'K$.
2023 Chile Classification NMO Juniors, 2
There are 2023 points on the plane. Prove that there exists a circle that contains 2000 points inside it and leaves the remaining 23 outside.
For example, if we had 5 points on the plane, we could find a circle that contains 4 of them inside and leaves 1 outside. Similarly, for 10 points, there exists a circle that contains 7 inside and leaves 3 outside. This reasoning extends to 2023 points, ensuring that such a division is always possible.
2020 Brazil National Olympiad, 4
A positive integer is [i]brazilian[/i] if the first digit and the last digit are equal. For instance, $4$ and $4104$ are brazilians, but $10$ is not brazilian. A brazilian number is [i]superbrazilian[/i] if it can be written as sum of two brazilian numbers. For instance, $101=99+2$ and $22=11+11$ are superbrazilians, but $561=484+77$ is not superbrazilian, because $561$ is not brazilian. How many $4$-digit numbers are superbrazilians?
1985 Greece National Olympiad, 4
Consider function $f:\mathbb{R}\to \mathbb{R}$ with $f(x)=\frac{4^x}{4^x+2},$ for any $x\in \mathbb{R}$
a) Prove that $f(x)+f(1-x)=1,$
b) Claculate the sum $$f\left(\frac{1}{1986} \right)+f\left(\frac{2}{1986} \right)+\cdots f\left(\frac{1986}{1986} \right).$$
2025 Harvard-MIT Mathematics Tournament, 1
Let $a,b,$ and $c$ be pairwise distinct positive integers such that $\tfrac{1}{a}, \tfrac{1}{b}, \tfrac{1}{c}$ is an increasing arithmetic sequence in that order. Prove that $\gcd(a,b)>1.$
2019 Korea National Olympiad, 1
The sequence ${a_1, a_2, ..., a_{2019}}$ satisfies the following condition.
$a_1=1, a_{n+1}=2019a_{n}+1$
Now let $x_1, x_2, ..., x_{2019}$ real numbers such that $x_1=a_{2019}, x_{2019}=a_1$ (The others are arbitary.)
Prove that $\sum_{k=1}^{2018} (x_{k+1}-2019x_k-1)^2 \ge \sum_{k=1}^{2018} (a_{2019-k}-2019a_{2020-k}-1)^2$
1995 Flanders Math Olympiad, 1
Four couples play chess together. For the match, they're paired as follows: ("man Clara" indicates the husband of Clara, etc.)
\[Bea \Longleftrightarrow Eddy\]
\[An \Longleftrightarrow man\ Clara\]
\[Freddy \Longleftrightarrow woman\ Guy\]
\[Debby \Longleftrightarrow man\ An\]
\[Guy \Longleftrightarrow woman\ Eddy\]
Who is $Hubert$ married to?
2019 Purple Comet Problems, 19
Rectangle $ABCD$ has sides $AB = 10$ and $AD = 7$. Point $G$ lies in the interior of $ABCD$ a distance $2$ from side $\overline{CD}$ and a distance $2$ from side $\overline{BC}$. Points $H, I, J$, and $K$ are located on sides $\overline{BC}, \overline{AB}, \overline{AD}$, and $\overline{CD}$, respectively, so that the path $GHIJKG$ is as short as possible. Then $AJ = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2023 All-Russian Olympiad Regional Round, 9.4
Let $a, b, c$ be positive integers such that no number divides some other number. If $ab-b+1 \mid abc+1$, prove that $c \geq b$.
2012 CHMMC Fall, 1
Let $[n] = \{1, 2, 3, ... ,n\}$ and for any set $S$, let$ P(S)$ be the set of non-empty subsets of $S$. What is the last digit of $|P(P([2013]))|$?
2015 Peru Cono Sur TST, P1
$A$ writes, at his choice, $8$ ones and $8$ twos on a $4\times 4$ board. Then $B$ covers the board with $8$ dominoes and for each domino she finds the smaller of the two numbers that that domino covers. Finally, $A$ adds these $8$ numbers and the result is her score. What is the highest score $A$ can secure, no matter how $B$ plays?
Clarification: A domino is a $1\times 2$ or $2\times 1$ rectangle that covers exactly two squares on the board.
1989 China Team Selection Test, 3
Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$
2008 JBMO Shortlist, 5
Is it possible to arrange the numbers $1^1, 2^2,..., 2008^{2008}$ one after the other, in such a way that the obtained number is a perfect square? (Explain your answer.)
2011 Cuba MO, 5
Determine all functions $f : R \to R$ such that
$$f(x)f(y) = 2f(x + y) + 9xy \ \ \forall x, y \in R.$$
2023 AMC 10, 23
An arithmetic sequence has $n \geq 3$ terms, initial term $a$ and common difference $d > 1$. Carl wrote down all the terms in this sequence correctly except for one term which was off by $1$. The sum of the terms was $222$. What was $a + d + n$
$\textbf{(A) } 24 \qquad \textbf{(B) } 20 \qquad \textbf{(C) } 22 \qquad \textbf{(D) } 28 \qquad \textbf{(E) } 26$
2007 Kyiv Mathematical Festival, 1
Is it possible to cut the table of size $2007\times2007$ into figures shown here, if one has to use
at least one figure of each sort? $\begin{picture}(45,25) \put(5,5){\put(0,0){\line(1,0){16}}\put(0,8){\line(1,0){24}}\put(0,16){\line(1,0){24}}\put(8,24){\line(1,0){16}}\put(0,0){\line(0,1){16}}\put(8,0){\line(0,1){24}}\put(16,0){\line(0,1){24}}\put(24,8){\line(0,1){16}}}\put(35,5){\put(0,0){\line(1,0){8}}\put(0,8){\line(1,0){8}}\put(0,16){\line(1,0){8}}\put(0,24){\line(1,0){8}}\put(0,0){\line(0,1){24}}\put(8,0){\line(0,1){24}}}\end{picture}$
1957 AMC 12/AHSME, 14
If $ y \equal{} \sqrt{x^2 \minus{} 2x \plus{} 1} \plus{} \sqrt{x^2 \plus{} 2x \plus{} 1}$, then $ y$ is:
$ \textbf{(A)}\ 2x\qquad
\textbf{(B)}\ 2(x \plus{} 1)\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ |x \minus{} 1| \plus{} |x \plus{} 1|\qquad
\textbf{(E)}\ \text{none of these}$
2018 Sharygin Geometry Olympiad, 1
The altitudes $AH, CH$ of an acute-angled triangle $ABC$ meet the internal bisector of angle $B$ at points $L_1, P_1$, and the external bisector of this angle at points $L_2, P_2$. Prove that the orthocenters of triangles $HL_1P_1, HL_2P_2$ and the vertex $B$ are collinear.
2024 Nigerian MO Round 2, Problem 5
Let the centroid of the triangle $ABC$ be $G$ and let the line parallel to $\overline{BC}$ that passes through $A$ be $l$. Define a point, $D$ on $l$ such that $\angle DGC=90^o$. Prove that
\[2[ADCG]\leq AB\cdot DC\]
For clarification, [ADGC] represents the area of the quadrilateral ADGC.
Kvant 2022, M2689
There are 1000 gentlemen listed in the register of a city with numbers from 1 to 1000. Any 720 of them form a club. The mayor wants to impose a tax on each club, which is paid by all club members in equal shares (the tax is an arbitrary non-negative real number). At the same time, the total tax paid by a gentleman should not exceed his number in the register. What is the largest tax the mayor can collect?
[i]Proposed by I. Bogdanov[/i]
1985 IMO Longlists, 21
Let $A$ be a set of positive integers such that for any two distinct elements $x, y\in A$ we have $|x-y| \geq \frac{xy}{25}.$ Prove that $A$ contains at most nine elements. Give an example of such a set of nine elements.
Oliforum Contest V 2017, 2
Find all quadrilaterals which can be covered (without overlappings) with squares with side $ 1$ and equilateral triangles with side $ 1$.
(Emanuele Tron)
1989 Irish Math Olympiad, 5
(i): Prove that if $n$ is a positive integer, then $$\binom{2n}{n}=\frac{(2n)!}{(n!)^2}$$ is a positive integer that is divisible by all prime numbers $p$ with $n<p\le 2n$, and that $$\binom{2n}{n}<2^{2n}.$$
(ii): For $x$ a positive real number, let $\pi(x)$ denote the number of prime numbers $p \le x$. [Thus, $\pi(10) = 4$ since there are $4$ primes, viz., $2$, $3$, $5$, and $7$, not exceeding $10$.]Prove that if $n \ge 3$ is an integer, then
(a)$$\pi(2n) < \pi(n) + {{2n}\over{\log_2(n)}};$$(b)$$\pi(2^n) < {{2^{n+1}\log_2(n-1)}\over{n}};$$(c) Deduce that, for all real numbers $x \ge 8$,$$\pi(x) < {{4x \log_2(\log_2(x))}\over{\log_2(x)}}.$$