This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 230

2004 China Western Mathematical Olympiad, 4
Tags: function , euler , search , number theory , totient function , prime number , logarithm
Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$. Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$, smaller than $n$). Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that \[ d(n) + \varphi(n) = n+c , \] and for such $c$ find all values of $n$ satisfying the above relationship.

2023 Euler Olympiad, Round 2, 1
Tags: algebra , euler , 100 , power of 2 , residue , number theory
Consider a sequence of 100 positive integers. Each member of the sequence, starting from the second one, is derived by either multiplying the previous number by 2 or dividing it by 16. Is it possible for the sum of these 100 numbers to be equal to $2^{2023}$? [i]Proposed by Nika Glunchadze, Georgia[/i]

2013 All-Russian Olympiad, 3
Tags: circumcircle , geometric transformation , homothety , euler , symmetry , geometry
The incircle of triangle $ ABC $ has centre $I$ and touches the sides $ BC $, $ CA $, $ AB $ at points $ A_1 $, $ B_1 $, $ C_1 $, respectively. Let $ I_a $, $ I_b $, $ I_c $ be excentres of triangle $ ABC $, touching the sides $ BC $, $ CA $, $ AB $ respectively. The segments $ I_aB_1 $ and $ I_bA_1 $ intersect at $ C_2 $. Similarly, segments $ I_bC_1 $ and $ I_cB_1 $ intersect at $ A_2 $, and the segments $ I_cA_1 $ and $ I_aC_1 $ at $ B_2 $. Prove that $ I $ is the center of the circumcircle of the triangle $ A_2B_2C_2 $. [i]L. Emelyanov, A. Polyansky[/i]

PEN O Problems, 3
Tags: euler , induction , number theory , relatively prime
Prove that the set of integers of the form $2^{k}-3$ ($k=2,3,\cdots$) contains an infinite subset in which every two members are relatively prime.

2007 India IMO Training Camp, 1
Tags: euler , trigonometry , perpendicular bisector , geometry
Show that in a non-equilateral triangle, the following statements are equivalent: $(a)$ The angles of the triangle are in arithmetic progression. $(b)$ The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line.

2024 Euler Olympiad, Round 1, 5
Tags: ratio , euler , geometry
Let $ABCDEF$ be a convex hexagon. Segments $AE$ and $BF$ intersect at $X$ and segments $BD$ and $CE$ intersect in $Y.$ It's known that $$ \angle XBC = \angle XDE = \angle YAB = \angle YEF = 80^\circ \text{ and } \angle XCB = \angle XED = \angle YBA = \angle YFE = \angle 70^\circ.$$ Let $P$ and $Q$ be such points on line $XY$ that segments $PX$ and $AF$ intersect, segments $QY$ and $CD$ intersect and $\angle APF = \angle CQD = 30 ^\circ.$ Estimate the sum: \[ \frac{BX}{BF} + \frac{BY}{BD} + \frac{EX}{EA} + \frac{EY}{EC} + \frac{PX}{PY} + \frac{QY}{QX} \] [i]Proposed by Gogi Khimshiashvili, Georgia [/i]

2014 Korea - Final Round, 5
Tags: euler , abstract algebra , number theory , thues lemma
Let $p>5$ be a prime. Suppose that there exist integer $k$ such that $ k^2 + 5 $ is divisible by $p$. Prove that there exist two positive integers $m,n$ satisfying $ p^2 = m^2 + 5n^2 $.

2020 Switzerland - Final Round, 4
Tags: number theory , euler , totient function , function
Let $\varphi$ denote the Euler phi-function. Prove that for every positive integer $n$ $$2^{n(n+1)} | 32 \cdot \varphi \left( 2^{2^n} - 1 \right).$$

2005 CentroAmerican, 3
Tags: circumcircle , incenter , geometric transformation , homothety , euler , geometry
Let $ABC$ be a triangle. $P$, $Q$ and $R$ are the points of contact of the incircle with sides $AB$, $BC$ and $CA$, respectively. Let $L$, $M$ and $N$ be the feet of the altitudes of the triangle $PQR$ from $R$, $P$ and $Q$, respectively. a) Show that the lines $AN$, $BL$ and $CM$ meet at a point. b) Prove that this points belongs to the line joining the orthocenter and the circumcenter of triangle $PQR$. [i]Aarón Ramírez, El Salvador[/i]

2009 BMO TST, 3
Tags: function , modular arithmetic , euler , number theory
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 $\phi(\sigma(2^x))=2^x$.

2013 Bosnia Herzegovina Team Selection Test, 6
Tags: euler , circumcircle , geometry
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 $ABC$.

2008 Baltic Way, 9
Tags: euler , number theory
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.

2024 Euler Olympiad, Round 1, 7
Tags: euler , number theory
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]

1996 Bosnia and Herzegovina Team Selection Test, 2
Tags: euler , number theory , eulers function , greatest common divisor , function
$a)$ Let $m$ and $n$ be positive integers. If $m>1$ prove that $ n \mid \phi(m^n-1)$ where $\phi$ is Euler function $b)$ Prove that number of elements in sequence $1,2,...,n$ $(n \in \mathbb{N})$, which greatest common divisor with $n$ is $d$, is $\phi\left(\frac{n}{d}\right)$

2012 BMT Spring, 8
Tags: modular arithmetic , euler , probability , expected value
You are tossing an unbiased coin. The last $ 28 $ consecutive flips have all resulted in heads. Let $ x $ be the expected number of additional tosses you must make before you get $ 60 $ consecutive heads. Find the sum of all distinct prime factors in $ x $.

2005 National Olympiad First Round, 10
Tags: modular arithmetic , euler
Which of the following does not divide $n^{2225}-n^{2005}$ for every integer value of $n$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 23 $

2013 China Team Selection Test, 2
Tags: circumcircle , geometric transformation , homothety , trigonometry , euler , geometry
Let $P$ be a given point inside the triangle $ABC$. Suppose $L,M,N$ are the midpoints of $BC, CA, AB$ respectively and \[PL: PM: PN= BC: CA: AB.\] The extensions of $AP, BP, CP$ meet the circumcircle of $ABC$ at $D,E,F$ respectively. Prove that the circumcentres of $APF, APE, BPF, BPD, CPD, CPE$ are concyclic.

1976 IMO Longlists, 15
Tags: euler , ratio , symmetry , geometry
Let $ABC$ and $A'B'C'$ be any two coplanar triangles. Let $L$ be a point such that $AL || BC, A'L || B'C'$ , and $M,N$ similarly defined. The line $BC$ meets $B'C'$ at $P$, and similarly defined are $Q$ and $R$. Prove that $PL, QM, RN$ are concurrent.

2024 Euler Olympiad, Round 1, 4
Tags: euler , combinatorics
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]

2011 India Regional Mathematical Olympiad, 1
Tags: geometry , circumcircle , incenter , trigonometry , parallelogram , euler , geometric transformation
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 $ABC.$

2009 Indonesia TST, 4
Tags: euler , function , number theory
Given positive integer $ n > 1$ and define \[ S \equal{} \{1,2,\dots,n\}. \] Suppose \[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}. \] Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)

2007 India IMO Training Camp, 1
Tags: euler , trigonometry , perpendicular bisector , geometry
Show that in a non-equilateral triangle, the following statements are equivalent: $(a)$ The angles of the triangle are in arithmetic progression. $(b)$ The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line.

1950 Miklós Schweitzer, 7
Tags: integration , calculus , euler , function , logarithm , real analysis
Examine the behavior of the expression $ \sum_{\nu\equal{}1}^{n\minus{}1}\frac{\log(n\minus{}\nu)}{\nu}\minus{}\log^2 n$ as $ n\rightarrow \infty$

PEN E Problems, 11
Tags: euler , algebra , polynomial , modular arithmetic
In 1772 Euler discovered the curious fact that $n^2 +n+41$ is prime when $n$ is any of $0,1,2, \cdots, 39$. Show that there exist $40$ consecutive integer values of $n$ for which this polynomial is not prime.

2017 Thailand TSTST, 2
Tags: euler , function , modular arithmetic , quadratic , prime number , number theory
Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.