Found problems: 230
2024 Euler Olympiad, Round 1, 7
Anna took a number \(N\), which is written in base 10 and has fewer than 9 digits, and duplicated it by writing another \(N\) to its left, creating a new number with twice as many digits. Bob computed the sum of all integers from 1 to \(N\). It turned out that Anna's new number is 7 times as large as the sum computed by Bob. Determine \(N\).
[i]Proposed by Bachana Kutsia, Georgia [/i]
2012 Putnam, 6
Let $p$ be an odd prime number such that $p\equiv 2\pmod{3}.$ Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x)\equiv x^3\pmod{p}.$ Show that $\pi$ is an even permutation if and only if $p\equiv 3\pmod{4}.$
2005 Vietnam Team Selection Test, 1
Let $(I),(O)$ be the incircle, and, respectiely, circumcircle of $ABC$. $(I)$ touches $BC,CA,AB$ in $D,E,F$ respectively. We are also given three circles $\omega_a,\omega_b,\omega_c$, tangent to $(I),(O)$ in $D,K$ (for $\omega_a$), $E,M$ (for $\omega_b$), and $F,N$ (for $\omega_c$).
[b]a)[/b] Show that $DK,EM,FN$ are concurrent in a point $P$;
[b]b)[/b] Show that the orthocenter of $DEF$ lies on $OP$.
1970 IMO Longlists, 21
In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$
2024 Euler Olympiad, Round 2, 2
Find all pairs of function $f : Q \rightarrow R$ and $g : Q \rightarrow R,$ for which equations
\begin{align*}
f(x+y) &= f(x) f(y) + g(x) g(y) \\
g(x+y) &= f(x)g(y) + g(x)f(y) + g(x)g(y)
\end{align*}
holds for all rational numbers $x$ and $y.$
[i]Proposed by Gurgen Asatryan, Armenia [/i]
2024 Euler Olympiad, Round 1, 4
Find the number of ordered pairs $(a, b, c, d)$ of positive integers satisfying the equation:
\[a + 2b + 3c + 1000d = 2024.\]
[i]Proposed by Irakli Khutsishvili, Georgia [/i]
2024 Euler Olympiad, Round 2, 4
Three numbers are initially written on the board: 2023, 2024, and 2025. In each move, you can increase any two of these numbers by 1 and decrease the third one by 2.
a) Determine whether it is possible to perform a sequence of operations such that the board eventually contains two numbers that are equal.
b) Calculate the number of all possible ordered triples of positive integers that can be obtained by performing such operations some number of times.
[i]Proposed by Giorgi Arabidze, Georgia [/i]
2023 Euler Olympiad, Round 1, 2
A student took a rectangular piece of paper with length equal to one meter and width equal to five centimeters. The student brought the ends together, turning one end 180 degrees and gluing the surfaces to create a figure called a Möbius strip. On one side of this strip, the student placed a flea and an ant. It is known that if the flea and the ant move in different directions on the Möbius strip, they will meet each other in 2 minutes. However, if they move in the same direction, they will meet in 7 minutes. Given that the flea is faster than the ant and both move at constant speeds, determine the speed of the flea.
[i]Proposed by Lia Chitishvili, Georgia[/i]
2009 BMO TST, 3
For the give functions in $\mathbb{N}$:
[b](a)[/b] Euler's $\phi$ function ($\phi(n)$- the number of natural numbers smaller than $n$ and coprime with $n$);
[b](b)[/b] the $\sigma$ function such that the $\sigma(n)$ is the sum of natural divisors of $n$.
solve the equation $\phi(\sigma(2^x))=2^x$.
2002 Putnam, 2
Consider a polyhedron with at least five faces such that exactly three edges emerge from each of its vertices. Two players play the following game: Each, in turn, signs his or her name on a previously unsigned face. The winner is the player who first succeeds in signing three faces that share a common vertex. Show that the player who signs first will always win by playing as well as possible.
2023 UMD Math Competition Part I, #4
Euler is selling Mathematician cards to Gauss. Three Fermat cards plus $5$ Newton cards costs $95$ Euros, while $5$ Fermat cards plus $2$ Newton cards also costs $95$ Euros. How many Euroes does one Fermat card cost?
$$
\mathrm a. ~ 10\qquad \mathrm b.~15\qquad \mathrm c. ~20 \qquad \mathrm d. ~30 \qquad \mathrm e. ~35
$$
1984 Balkan MO, 3
Show that for any positive integer $m$, there exists a positive integer $n$ so that in the decimal representations of the numbers $5^{m}$ and $5^{n}$, the representation of $5^{n}$ ends in the representation of $5^{m}$.
1998 All-Russian Olympiad, 4
A connected graph has $1998$ points and each point has degree $3$. If $200$ points, no two of them joined by an edge, are deleted, show that the result is a connected graph.
2013 Bosnia Herzegovina Team Selection Test, 6
In triangle $ABC$, $I$ is the incenter. We have chosen points $P,Q,R$ on segments $IA,IB,IC$ respectively such that $IP\cdot IA=IQ \cdot IB=IR\cdot IC$.
Prove that the points $I$ and $O$ belong to Euler line of triangle $PQR$ where $O$ is circumcenter of $ABC$.
1994 Irish Math Olympiad, 5
If a square is partitioned into $ n$ convex polygons, determine the maximum possible number of edges in the obtained figure.
(You may wish to use the following theorem of Euler: If a polygon is partitioned into $ n$ polygons with $ v$ vertices and $ e$ edges in the resulting figure, then $ v\minus{}e\plus{}n\equal{}1$.)
2003 Canada National Olympiad, 2
Find the last three digits of the number $2003^{{2002}^{2001}}$.
2008 Baltic Way, 9
Suppose that the positive integers $ a$ and $ b$ satisfy the equation $ a^b\minus{}b^a\equal{}1008$ Prove that $ a$ and $ b$ are congruent modulo 1008.
1989 Austrian-Polish Competition, 8
$ABC$ is an acute-angled triangle and $P$ a point inside or on the boundary. The feet of the perpendiculars from $P$ to $BC, CA, AB$ are $A', B', C'$ respectively. Show that if $ABC$ is equilateral, then $\frac{AC'+BA'+CB'}{PA'+PB'+PC'}$ is the same for all positions of $P$, but that for any other triangle it is not.
PEN A Problems, 10
Let $n$ be a positive integer with $n \ge 3$. Show that \[n^{n^{n^{n}}}-n^{n^{n}}\] is divisible by $1989$.
2007 Nicolae Coculescu, 4
Prove that $ p $ divides $ \varphi (1+a^p) , $ where $ a\ge 2 $ is a natural number, $ p $ is a prime, and $ \varphi $ is Euler's totient.
[i]Cristinel Mortici[/i]
2006 Germany Team Selection Test, 1
For any positive integer $n$, let $w\left(n\right)$ denote the number of different prime divisors of the number $n$. (For instance, $w\left(12\right)=2$.) Show that there exist infinitely many positive integers $n$ such that $w\left(n\right)<w\left(n+1\right)<w\left(n+2\right)$.
2006 Germany Team Selection Test, 1
For any positive integer $n$, let $w\left(n\right)$ denote the number of different prime divisors of the number $n$. (For instance, $w\left(12\right)=2$.) Show that there exist infinitely many positive integers $n$ such that $w\left(n\right)<w\left(n+1\right)<w\left(n+2\right)$.
2005 Iran MO (3rd Round), 2
Suppose $O$ is circumcenter of triangle $ABC$. Suppose $\frac{S(OAB)+S(OAC)}2=S(OBC)$. Prove that the distance of $O$ (circumcenter) from the radical axis of the circumcircle and the 9-point circle is \[\frac {a^2}{\sqrt{9R^2-(a^2+b^2+c^2)}}\]
2005 China Team Selection Test, 2
In acute angled triangle $ABC$, $BC=a$,$CA=b$,$AB=c$, and $a>b>c$. $I,O,H$ are the incentre, circumcentre and orthocentre of $\triangle{ABC}$ respectively. Point $D \in BC$, $E \in CA$ and $AE=BD$, $CD+CE=AB$. Let the intersectionf of $BE$ and $AD$ be $K$. Prove that $KH \parallel IO$ and $KH = 2IO$.
2011 India Regional Mathematical Olympiad, 1
Let $ABC$ be an acute angled scalene triangle with circumcentre $O$ and orthocentre $H.$ If $M$ is the midpoint of $BC,$ then show that $AO$ and $HM$ intersect on the circumcircle of $ABC.$