Found problems: 230
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?
$$
\mathrm a. ~ 10\qquad \mathrm b.~15\qquad \mathrm c. ~20 \qquad \mathrm d. ~30 \qquad \mathrm e. ~35
$$
2006 Italy TST, 2
Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$.
a) Find all $n$ such that $A_{n}\neq \emptyset$
b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero.
c) Is there $n$ such that $|{A_{n}}| = 130$?
1996 Baltic Way, 5
Let $ABCD$ be a cyclic convex quadrilateral and let $r_a,r_b,r_c,r_d$ be the radii of the circles inscribed in the triangles $BCD, ACD, ABD, ABC$, respectively. Prove that $r_a+r_c=r_b+r_d$.
2000 All-Russian Olympiad, 6
A perfect number, greater than $28$ is divisible by $7$. Prove that it is also divisible by $49$.
2010 APMO, 4
Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.
2008 Vietnam Team Selection Test, 2
Let $ k$ be a positive real number. Triangle ABC is acute and not isosceles, O is its circumcenter and AD,BE,CF are the internal bisectors. On the rays AD,BE,CF, respectively, let points L,M,N such that $ \frac {AL}{AD} \equal{} \frac {BM}{BE} \equal{} \frac {CN}{CF} \equal{} k$. Denote $ (O_1),(O_2),(O_3)$ be respectively the circle through L and touches OA at A, the circle through M and touches OB at B, the circle through N and touches OC at C.
1) Prove that when $ k \equal{} \frac{1}{2}$, three circles $ (O_1),(O_2),(O_3)$ have exactly two common points, the centroid G of triangle ABC lies on that common chord of these circles.
2) Find all values of k such that three circles $ (O_1),(O_2),(O_3)$ have exactly two common points
2017 Thailand TSTST, 2
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 $(m,n)>1$.
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]
2004 APMO, 2
Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Prove that the area of one of the triangles $AOH$, $BOH$ and $COH$ is equal to the sum of the areas of the other two.
2024 Euler Olympiad, Round 2, 3
Consider a convex quadrilateral \(ABCD\) with \(AC > BD\). In the plane of this quadrilateral, points \(M\) and \(N\) are chosen such that triangles \(ABM\) and \(CDN\) are equilateral, and segments \(MD\) and \(NA\) intersect lines \(AB\) and \(CD\) respectively. Similarly, points \(P\) and \(Q\) are chosen such that triangles \(ADP\) and \(BCQ\) are equilateral, but here segments \(PB\) and \(QA\) do not intersect lines \(AD\) and \(BC\) respectively.
Prove that \(MN = AC + BD\) if and only if \(PQ = AC - BD\).
[i]Proposed by Zaza Meliqidze, Georgia [/i]
2010 Sharygin Geometry Olympiad, 2
Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$
2019 Bulgaria EGMO TST, 1
Determine the length of $BC$ in an acute triangle $ABC$ with $\angle ABC = 45^{\circ}$, $OG = 1$ and $OG \parallel BC$. (As usual $O$ is the circumcenter and $G$ is the centroid.)
2015 AMC 8, 15
At Euler Middle School, $198$ students voted on two issues in a school referendum with the following results: $149$ voted in favor of the first issue and $119$ voted in favor of the second issue. If there were exactly $29$ students who voted against both issues, how many students voted in favor of both issues?
$\textbf{(A) }49\qquad\textbf{(B) }70\qquad\textbf{(C) }79\qquad\textbf{(D) }99\qquad \textbf{(E) }149$
2023 Euler Olympiad, Round 2, 2
Let $n$ be a positive integer. The Georgian folk dance team consists of $2n$ dancers, with $n$ males and $n$ females. Each dancer, both male and female, is assigned a number from 1 to $n$. During one of their dances, all the dancers line up in a single line. Their wish is that, for every integer $k$ from 1 to $n$, there are exactly $k$ dancers positioned between the $k$th numbered male and the $k$th numbered female. Prove the following statements:
a) If $n \equiv 1 \text{ or } 2 \mod{4}$, then the dancers cannot fulfill their wish.
b) If $n \equiv 0 \text{ or } 3 \mod{4}$, then the dancers can fulfill their wish.
[i]Proposed by Giorgi Arabidze, 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]
PEN O Problems, 3
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.
1996 Bosnia and Herzegovina Team Selection Test, 2
$a)$ 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 $\phi\left(\frac{n}{d}\right)$
2014 Greece National Olympiad, 3
For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once).
Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\frac{S_1}{S_2}=\frac{39}{64}.$
2005 Romania Team Selection Test, 2
On the edges of a convex polyhedra we draw arrows such that from each vertex at least an arrow is pointing in and at least one is pointing out.
Prove that there exists a face of the polyhedra such that the arrows on its edges form a circuit.
[i]Dan Schwartz[/i]
2007 Today's Calculation Of Integral, 172
Evaluate $\int_{-1}^{0}\sqrt{\frac{1+x}{1-x}}dx.$
1950 Miklós Schweitzer, 7
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$
2000 Taiwan National Olympiad, 1
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 $(m,n)>1$.
1999 Mongolian Mathematical Olympiad, Problem 4
Maybe well known:
$p$ a prime number, $n$ an integer. Prove that $n$ divides $\phi(p^n-1)$ where $\phi(x)$ is the Euler function.
2013 Korea - Final Round, 5
Two coprime positive integers $ a, b $ are given. Integer sequence $ \{ a_n \}, \{b_n \} $ satisties
\[ (a+b \sqrt2 )^{2n} = a_n + b_n \sqrt2 \]
Find all prime numbers $ p $ such that there exist positive integer $ n \le p $ satisfying $ p | b_n $.
2006 Italy TST, 2
Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$.
a) Find all $n$ such that $A_{n}\neq \emptyset$
b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero.
c) Is there $n$ such that $|{A_{n}}| = 130$?