Found problems: 230
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.
PEN D Problems, 6
Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[2, \; 2^{2}, \; 2^{2^{2}}, \; 2^{2^{2^{2}}}, \cdots \pmod{n}\] is eventually constant.
2007 China Team Selection Test, 2
Let $ ABCD$ be the inscribed quadrilateral with the circumcircle $ \omega$.Let $ \zeta$ be another circle that internally tangent to
$ \omega$ and to the lines $ BC$ and $ AD$ at points $ M,N$ respectively.Let $ I_1,I_2$ be the incenters of the $ \triangle ABC$ and $ \triangle ABD$.Prove that $ M,I_1,I_2,N$ are collinear.
2007 China Team Selection Test, 2
After multiplying out and simplifying polynomial $ (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007})$ $ (mod$ $ x^{2008}).$
1991 Arnold's Trivium, 38
Calculate the integral of the Gaussian curvature of the surface
\[z^4+(x^2+y^2-1)(2x^2+3y^2-1)=0\]
2010 Contests, 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 $OH$.
2014 Korea - Final Round, 5
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 $.
2014 Bundeswettbewerb Mathematik, 1
Show that for all positive integers $n$, the number $2^{3^n}+1$ is divisible by $3^{n+1}$.
1996 All-Russian Olympiad, 5
Show that in the arithmetic progression with first term 1 and ratio 729, there are infinitely many powers of 10.
[i]L. Kuptsov[/i]
2014 NIMO Summer Contest, 14
Let $ABC$ be a triangle with circumcenter $O$ and let $X$, $Y$, $Z$ be the midpoints of arcs $BAC$, $ABC$, $ACB$ on its circumcircle. Let $G$ and $I$ denote the centroid of $\triangle XYZ$ and the incenter of $\triangle ABC$.
Given that $AB = 13$, $BC = 14$, $CA = 15$, and $\frac {GO}{GI} = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Evan Chen[/i]
2023 Euler Olympiad, Round 1, 7
Tsrutsuna starts in the bottom left cell of a 7 × 7 square table, while Tsuna is in the upper right cell. The center cell of the table contains cheese. Tsrutsuna wants to reach Tsuna and bring a piece of cheese with him. From a cell Tsrutsuna can only move to the right or the top neighboring cell. Determine the number of different paths Tsrutsuna can take from the lower left cell to the upper right cell, such that he passes through the center cell.
[i]Proposed by Giorgi Arabidze, Georgia[/i]
2006 AMC 12/AHSME, 25
A sequence $ a_1, a_2, \ldots$ of non-negative integers is defined by the rule $ a_{n \plus{} 2} \equal{} |a_{n \plus{} 1} \minus{} a_n|$ for $ n\ge 1$. If $ a_1 \equal{} 999, a_2 < 999,$ and $ a_{2006} \equal{} 1$, how many different values of $ a_2$ are possible?
$ \textbf{(A) } 165 \qquad \textbf{(B) } 324 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 499 \qquad \textbf{(E) } 660$
2007 Ukraine Team Selection Test, 5
$ AA_{3}$ and $ BB_{3}$ are altitudes of acute-angled $ \triangle ABC$. Points $ A_{1}$ and $ B_{1}$ are second points of intersection lines $ AA_{3}$ and $ BB_{3}$ with circumcircle of $ \triangle ABC$ respectively. $ A_{2}$ and $ B_{2}$ are points on $ BC$ and $ AC$ respectively. $ A_{1}A_{2}\parallel AC$, $ B_{1}B_{2}\parallel BC$. Point $ M$ is midpoint of $ A_{2}B_{2}$. $ \angle BCA \equal{} x$. Find $ \angle A_{3}MB_{3}$.
2014 AIME Problems, 3
Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and denominator have a sum of $1000$.
2013 Online Math Open Problems, 19
Let $\sigma(n)$ be the number of positive divisors of $n$, and let $\operatorname{rad} n$ be the product of the distinct prime divisors of $n$. By convention, $\operatorname{rad} 1 = 1$. Find the greatest integer not exceeding \[ 100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}. \][i]Proposed by Michael Kural[/i]
2020 GQMO, 8
Let $ABC$ be an acute scalene triangle, with the feet of $A,B,C$ onto $BC,CA,AB$ being $D,E,F$ respectively. Let $W$ be a point inside $ABC$ whose reflections over $BC,CA,AB$ are $W_a,W_b,W_c$ respectively. Finally, let $N$ and $I$ be the circumcenter and the incenter of $W_aW_bW_c$ respectively. Prove that, if $N$ coincides with the nine-point center of $DEF$, the line $WI$ is parallel to the Euler line of $ABC$.
[i]Proposed by Navneel Singhal, India and Massimiliano Foschi, Italy[/i]
2023 Euler Olympiad, Round 2, 1
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]
2007 China Team Selection Test, 2
Let $ ABCD$ be the inscribed quadrilateral with the circumcircle $ \omega$.Let $ \zeta$ be another circle that internally tangent to
$ \omega$ and to the lines $ BC$ and $ AD$ at points $ M,N$ respectively.Let $ I_1,I_2$ be the incenters of the $ \triangle ABC$ and $ \triangle ABD$.Prove that $ M,I_1,I_2,N$ are collinear.
2024 Euler Olympiad, Round 1, 10
Find all $x$ that satisfy the following equation: \[ \sqrt {1 + \frac {20}x } = \sqrt {1 + 24x} + 2 \]
[i]Proposed by Andria Gvaramia, Georgia [/i]
2020 Switzerland - Final Round, 4
Let $\varphi$ denote the Euler phi-function. Prove that for every positive integer $n$
$$2^{n(n+1)} | 32 \cdot \varphi \left( 2^{2^n} - 1 \right).$$
2016 USA TSTST, 4
Suppose that $n$ and $k$ are positive integers such that \[ 1 = \underbrace{\varphi( \varphi( \dots \varphi(}_{k\ \text{times}} n) \dots )). \] Prove that $n \le 3^k$.
Here $\varphi(n)$ denotes Euler's totient function, i.e. $\varphi(n)$ denotes the number of elements of $\{1, \dots, n\}$ which are relatively prime to $n$. In particular, $\varphi(1) = 1$.
[i]Proposed by Linus Hamilton[/i]
2023 Euler Olympiad, Round 1, 5
Consider a 3 × 4 rectangular table where each cell can be colored using one of three available colors. Determine the number of different ways the table can be colored such that no two cells sharing a common side have the same color. It is not necessary to use all three colors in each coloring.
[i]Proposed by Prudencio Guerrero Fernández, Cuba[/i]
2010 Postal Coaching, 2
Suppose $\triangle ABC$ has circumcircle $\Gamma$, circumcentre $O$ and orthocentre $H$. Parallel lines $\alpha, \beta, \gamma$ are drawn through the vertices $A, B, C$, respectively. Let $\alpha ', \beta ', \gamma '$ be the reflections of $\alpha, \beta, \gamma$ in the sides $BC, CA, AB$, respectively.
$(a)$ Show that $\alpha ', \beta ', \gamma '$ are concurrent if and only if $\alpha, \beta, \gamma$ are parallel to the Euler line $OH$.
$(b)$ Suppose that $\alpha ', \beta ' , \gamma '$ are concurrent at the point $P$ . Show that $\Gamma$ bisects $OP$ .
2008 District Olympiad, 3
For any real $ a$ define $ f_a : \mathbb{R} \rightarrow \mathbb{R}^2$ by the law $ f_a(t) \equal{} \left( \sin(t), \cos(at) \right)$.
a) Prove that $ f_{\pi}$ is not periodic.
b) Determine the values of the parameter $ a$ for which $ f_a$ is periodic.
[b]Remark[/b]. L. Euler proved in $ 1737$ that $ \pi$ is irrational.
2014 NIMO Problems, 4
Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer. Compute the remainder when $3^n$ is divided by $1000$.