Found problems: 85335
2010 IMO, 3
Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$
[i]Proposed by Gabriel Carroll, USA[/i]
2015 PAMO, Problem 6
Let $ABCD$ be a quadrilateral (with non-perpendicular diagonals).
The perpendicular from $A$ to $BC$ meets $CD$ at $K$.
The perpendicular from $A$ to $CD$ meets $BC$ at $L$.
The perpendicular from $C$ to $AB$ meets $AD$ at $M$.
The perpendicular from $C$ to $AD$ meets $AB$ at $N$.
1. Prove that $KL$ is parallel to $MN$.
2. Prove that $KLMN$ is a parallelogram if $ABCD$ is cyclic.
2009 Germany Team Selection Test, 3
Let $ A,B,C,M$ points in the plane and no three of them are on a line. And let $ A',B',C'$ points such that $ MAC'B, MBA'C$ and $ MCB'A$ are parallelograms:
(a) Show that
\[ \overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} < \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.\]
(b) Assume segments $ AA', BB'$ and $ CC'$ have the same length. Show that $ 2 \left(\overline{MA} \plus{} \overline{MB} \plus{} \overline{MC} \right) \leq \overline{AA'} \plus{} \overline{BB'} \plus{} \overline{CC'}.$ When do we have equality?
2003 India Regional Mathematical Olympiad, 4
Find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying
(i) $x \leq y \leq z$
(ii) $x + y + z \leq 100.$
1992 IMO Longlists, 2
Let $m$ be a positive integer and $x_0, y_0$ integers such that $x_0, y_0$ are relatively prime, $y_0$ divides $x_0^2+m$, and $x_0$ divides $y_0^2+m$. Prove that there exist positive integers $x$ and $y$ such that $x$ and $y$ are relatively prime, $y$ divides $x^2 + m$, $x$ divides $y^2 + m$, and $x + y \leq m+ 1.$
2007 Hong kong National Olympiad, 1
Let $ABC$ be a triangle and $D$ be a point on $BC$ such that $AB+BD=AC+CD$. The line $AD$ intersects the incircle of triangle $ABC$ at $X$ and $Y$ where $X$ is closer to $A$ than $Y$ i. Suppose $BC$ is tangent to the incircle at $E$, prove that:
1) $EY$ is perpendicular to $AD$;
2) $XD=2IM$ where $I$ is the incentre and $M$ is the midpoint of $BC$.
2018 Online Math Open Problems, 7
Compute the number of ways to erase 24 letters from the string ``OMOMO$\cdots$OMO'' (with length 27), such that the three remaining letters are O, M and O in that order. Note that the order in which they are erased does not matter.
[i]Proposed by Yannick Yao
2002 China Team Selection Test, 3
There is a game. The magician let the participant think up a positive integer (at least two digits). For example, an integer $ \displaystyle\overline{a_1a_2 \cdots a_n}$ is rearranged as $ \overline{a_{i_1}a_{i_2} \cdots a_{i_n}}$, that is, $ i_1, i_2, \cdots, i_n$ is a permutation of $ 1,2, \cdots, n$. Then we get $ n!\minus{}1$ integers. The participant is asked to calculate the sum of the $ n!\minus{}1$ numbers, then tell the magician the sum $ S$. The magician claims to be able to know the original number when he is told the sum $ S$. Try to decide whether the magician can be successful or not.
2022 European Mathematical Cup, 2
We say that a positive integer $n$ is lovely if there exist a positive integer $k$ and (not necessarily distinct) positive integers $d_1$, $d_2$, $\ldots$, $d_k$ such that $n = d_1d_2\cdots d_k$ and $d_i^2 \mid n + d_i$ for $i=1,2,\ldots,k$.
a) Are there infinitely many lovely numbers?
b) Is there a lovely number, greater than $1$, which is a perfect square of an integer?
2014 Saudi Arabia BMO TST, 2
Let $\mathbb{N}$ denote the set of positive integers, and let $S$ be a set. There exists a function $f :\mathbb{N} \rightarrow S$ such that if $x$ and $y$ are a pair of positive integers with their difference being a prime number, then $f(x) \neq f(y)$. Determine the minimum number of elements in $S$.
2006 Purple Comet Problems, 11
Let $k$ be the product of every third positive integer from $2$ to $2006$, that is $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$. Find the number of zeros there are at the right end of the decimal representation for $k$.
1963 Vietnam National Olympiad, 3
Solve the equation $ \sin^3x \cos 3x \plus{} \cos^3x \sin 3x \equal{} \frac{3}{8}$.
2018 Estonia Team Selection Test, 10
A sequence of positive real numbers $a_1, a_2, a_3, ... $ satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \ge 3$. A sequence $b_1, b_2, b_3, ...$ is defined by equations
$b_1 = a_1$ ,
$b_n = a_n + (b_1 + b_3 + ...+ b_{n-1})$ for even $n > 1$ ,
$b_n = a_n + (b_2 + b_4 + ... +b_{n-1})$ for odd $n > 1$.
Prove that if $n\ge 3$, then $\frac13 < \frac{b_n}{n \cdot a_n} < 1$
Geometry Mathley 2011-12, 9.3
Let $ABCD$ be a quadrilateral inscribed in a circle $(O)$. Let $(O_1), (O_2), (O_3), (O_4)$ be the circles going through $(A,B), (B,C),(C,D),(D,A)$. Let $X, Y,Z, T$ be the second intersection of the pairs of the circles: $(O_1)$ and $(O_2), (O_2)$ and $(O_3), (O_3)$ and $(O_4), (O_4)$ and $(O_1)$.
(a) Prove that $X, Y,Z, T$ are on the same circle of radius $I$.
(b) Prove that the midpoints of the line segments $O_1O_3,O_2O_4,OI$ are collinear.
Nguyễn Văn Linh
1990 IMO Shortlist, 24
Let $ w, x, y, z$ are non-negative reals such that $ wx \plus{} xy \plus{} yz \plus{} zw \equal{} 1$.
Show that $ \frac {w^3}{x \plus{} y \plus{} z} \plus{} \frac {x^3}{w \plus{} y \plus{} z} \plus{} \frac {y^3}{w \plus{} x \plus{} z} \plus{} \frac {z^3}{w \plus{} x \plus{} y}\geq \frac {1}{3}$.
2017 Moldova Team Selection Test, 11
Find all ordered pairs of nonnegative integers $(x,y)$ such that
\[x^4-x^2y^2+y^4+2x^3y-2xy^3=1.\]
PEN S Problems, 24
A number $n$ is called a Niven number, named for Ivan Niven, if it is divisible by the sum of its digits. For example, $24$ is a Niven number. Show that it is not possible to have more than $20$ consecutive Niven numbers.
2022 Math Prize for Girls Olympiad, 4
Let $n > 1$ be an integer. Let $A$ denote the set of divisors of $n$ that are less than $\sqrt n$. Let $B$ denote the set of divisors of $n$ that are greater than $\sqrt n$. Prove that there exists a bijective function $f \colon A \to B$ such that $a$ divides $f(a)$ for all $a \in A$.
(We say $f$ is [i]bijective[/i] if for every $b \in B$ there exists a unique $a \in A$ with $f(a) = b$.)
2020-21 KVS IOQM India, 25
A five digit number $n= \overline{abcdc}$. Is such that when divided respectively by $2,3,4,5,6$ the remainders are $a,b,c,d,c$. What is the remainder when $n$ is divided by $100$?
2010 Princeton University Math Competition, 2
In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$ say $C_1$, and then the midpoint of $A_1D$ which is $A_2$. He continues in this fashion, indefinitely. The total length of his path if $AB=5$ and $BC=12$ is of the form $a + b\sqrt{c}$. Find $\displaystyle\frac{abc}{4}$.
2001 Korea - Final Round, 2
Let $P$ be a given point inside a convex quadrilateral $O_1O_2O_3O_4$. For each $i = 1,2,3,4$, consider the lines $l$ that pass through $P$ and meet the rays $O_iO_{i-1}$ and $O_iO_{i+1}$ (where $O_0 = O_4$ and $O_5 = O_1$) at distinct points $A_i(l)$ and $B_i(l)$, respectively. Denote $f_i(l) = PA_i(l) \cdot PB_i(l)$. Among all such lines $l$, let $l_i$ be the one that minimizes $f_i$. Show that if $l_1 = l_3$ and $l_2 = l_4$, then the quadrilateral $O_1O_2O_3O_4$ is a parallelogram.
1995 Iran MO (2nd round), 2
Let $ABC$ be an acute triangle and let $\ell$ be a line in the plane of triangle $ABC.$ We've drawn the reflection of the line $\ell$ over the sides $AB, BC$ and $AC$ and they intersect in the points $A', B'$ and $C'.$ Prove that the incenter of the triangle $A'B'C'$ lies on the circumcircle of the triangle $ABC.$
2002 Moldova National Olympiad, 4
Twelve teams participated in a soccer tournament. According to the rules, one team gets $ 2$ points for a victory, $ 1$ point for a draw and $ 0$ points for a defeat. When the tournament was over, all teams had distinct numbers of points, and the team ranked second had as many points as the teams ranked on the last five places in total. Who won the match between the fourth and the eighth place teams?
2024 Mongolian Mathematical Olympiad, 1
Find all triples $(a, b, c)$ of positive integers such that $a \leq b$ and \[a!+b!=c^4+2024\]
[i]Proposed by Otgonbayar Uuye.[/i]
2015 Estonia Team Selection Test, 4
Altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at $H$. Let $C_1 (H,HE)$ and $C_2(B,BE)$ be two circles tangent at $AC$ at point $E$. Let $P\ne E$ be the second point of tangency of the circle $C_1 (H,HE)$ with its tangent line going through point $C$, and $Q\ne E$ be the second point of tangency of the circle $C_2(B,BE)$ with its tangent line going through point $C$. Prove that points $D, P$, and $Q$ are collinear.