Found problems: 85335
1972 Czech and Slovak Olympiad III A, 4
Show that there are infinitely many positive integers $a$ such that the number $n^4+a$ is composite for every positive integer $n.$ Give 5 (different) numbers $a$ with the mentioned property.
2006 Victor Vâlcovici, 1
Let be a nondegenerate and closed interval $ I $ of real numbers, a short map $ m:I\longrightarrow I, $ and a sequence of functions $ \left( x_n \right)_{n\ge 1} :I\longrightarrow\mathbb{R} $ such that $ x_1 $ is the identity map and
$$ 2x_{n+1}=x_n+m\circ x_n , $$
for any natural numbers $ n. $ Prove that:
[b]a)[/b] there exists a nondegenerate interval having the property that any point of it is a fixed point for $ m. $
[b]b)[/b] $ \left( x_n \right)_{n\ge 1} $ is pointwise convergent and that its limit function is a short map.
2021 AMC 12/AHSME Spring, 8
A sequence of numbers is defined by $D_0=0,D_1=0,D_2=1$ and $D_n=D_{n-1}+D_{n-3}$ for $n\ge 3$. What are the parities (evenness or oddness) of the triple of numbers $(D_{2021},D_{2022},D_{2023})$, where $E$ denotes even and $O$ denotes odd?
$\textbf{(A) }(O,E,O) \qquad \textbf{(B) }(E,E,O) \qquad \textbf{(C) }(E,O,E) \qquad \textbf{(D) }(O,O,E) \qquad \textbf{(E) }(O,O,O)$
2013 Online Math Open Problems, 48
$\omega$ is a complex number such that $\omega^{2013} = 1$ and $\omega^m \neq 1$ for $m=1,2,\ldots,2012$. Find the number of ordered pairs of integers $(a,b)$ with $1 \le a, b \le 2013$ such that \[ \frac{(1 + \omega + \cdots + \omega^a)(1 + \omega + \cdots + \omega^b)}{3} \] is the root of some polynomial with integer coefficients and leading coefficient $1$. (Such complex numbers are called [i]algebraic integers[/i].)
[i]Victor Wang[/i]
1990 AMC 8, 16
$ 1990-1980+1970-1960+\cdots-20+10 = $
$ \text{(A)}\ -990\qquad\text{(B)}\ -10\qquad\text{(C)}\ 990\qquad\text{(D)}\ 1000\qquad\text{(E)}\ 1990 $
2014 Harvard-MIT Mathematics Tournament, 17
Let $f:\mathbb{N}\to\mathbb{N}$ be a function satisfying the following conditions:
(a) $f(1)=1$.
(b) $f(a)\leq f(b)$ whenever $a$ and $b$ are positive integers with $a\leq b$.
(c) $f(2a)=f(a)+1$ for all positive integers $a$.
How many possible values can the $2014$-tuple $(f(1),f(2),\ldots,f(2014))$ take?
2022-2023 OMMC FINAL ROUND, 9
Let $\triangle ABC$ have incircle $\omega$. Let $\omega_1$, $\omega_2$, and $\omega_3$ be three circles centered at $A$, $B$, and $C$ respectively tangent to $\omega$ at points $D$, $E$, and $F$ respectively. Show there exists a circle $\Gamma \neq \omega$ tangent to circles $\omega_1$, $\omega_2$, and $\omega_3$ centered on the Euler line of $\triangle DEF$.
[i](Each of the three circles $\omega_1, \omega_2, \omega_3$ is allowed to be internally or externally tangent to $\omega$. They don't have to be all internally tangent or all externally tangent.)[/i]
2021 AIME Problems, 11
A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $6$, and a different number in $S$ was divisible by $7$. The teacher then asked if any of the students could deduce what $S$ is, but in unison, all of the students replied no.
However, upon hearing that all four students replied no, each student was able to determine the elements of $S$. Find the sum of all possible values of the greatest element of $S$.
2013 Taiwan TST Round 1, 2
A V-tromino is a diagram formed by three unit squares.(As attachment.)
(a)Is it possible to cover a $3\times 2013$ table by $3\times 671$ V-trominoes?
(b)Is it possible to cover a $5\times 2013$ table by $5\times 671$ V-trominoes?
2019 Jozsef Wildt International Math Competition, W. 43
Consider the sequence of polynomials $P_0(x) = 2$, $P_1(x) = x$ and $P_n(x) = xP_{n-1}(x) - P_{n-2}(x)$ for $n \geq 2$. Let $x_n$ be the greatest zero of $P_n$ in the the interval $|x| \leq 2$. Show that $$\lim \limits_{n \to \infty}n^2\left(4-2\pi +n^2\int \limits_{x_n}^2P_n(x)dx\right)=2\pi - 4-\frac{\pi^3}{12}$$
2013 Vietnam National Olympiad, 2
Let $ABC$ be a cute triangle.$(O)$ is circumcircle of $\triangle ABC$.$D$ is on arc $BC$ not containing $A$.Line $\triangle$ moved through $H$($H$ is orthocenter of $\triangle ABC$ cuts circumcircle of $\triangle ABH$,circumcircle $\triangle ACH$ again at $M,N$ respectively.
a.Find $\triangle$ satisfy $S_{AMN}$ max
b.$d_{1},d_{2}$ are the line through $M$ perpendicular to $DB$,the line through $N$ perpendicular to $DC$ respectively.
$d_{1}$ cuts $d_{2}$ at $P$.Prove that $P$ move on a fixed circle.
2021 Romania National Olympiad, 4
Determine all nonzero integers $a$ for which there exists two functions $f,g:\mathbb Q\to\mathbb Q$ such that
\[f(x+g(y))=g(x)+f(y)+ay\text{ for all } x,y\in\mathbb Q.\]
Also, determine all pairs of functions with this property.
[i]Vasile Pop[/i]
2011 HMNT, 4
Determine the number of quadratic polynomials $P(x) = p_1x^2 + p_2x - p_3$, where $p_1$, $p_2$, $p_3$ are not necessarily distinct (positive) prime numbers less than $50$, whose roots are distinct rational numbers.
2022 Francophone Mathematical Olympiad, 1
find all functions $f:\mathbb{Z} \to \mathbb{Z} $
such that $f(m+n)+f(m)f(n)=n^2(f(m)+1)+m^2(f(n)+1)+mn(2-mn)$ holds for all $m,n \in \mathbb{Z}$
1999 Gauss, 22
Forty-two cubes with 1 cm edges are glued together to form a solid rectangular block. If the perimeter of the base of the block is 18 cm, then the height, in cm, is
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \dfrac{7}{3} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$
2014 PUMaC Team, 14
Define the function $f_k(x)$ (where $k$ is a positive integer) as follows: \[f_k(x)=(\cos kx)(\cos x)^k+(\sin kx)(\sin x)^k-(\cos 2x)^k.\] Find the sum of all distinct value(s) of $k$ such that $f_k(x)$ is a constant function.
2016 Sharygin Geometry Olympiad, P7
Let all distances between the vertices of a convex $n$-gon ($n > 3$) be
different.
a) A vertex is called uninteresting if the closest vertex is adjacent to it. What is the
minimal possible number of uninteresting vertices (for a given $n$)?
b) A vertex is called unusual if the farthest vertex is adjacent to it. What is the maximal
possible number of unusual vertices (for a given $n$)?
[i](Proposed by B.Frenkin)[/i]
2013 Greece Team Selection Test, 4
Given are $n$ different concentric circles on the plane.Inside the disk with the smallest radius (strictly inside it),we consider two distinct points $A,B$.We consider $k$ distinct lines passing through $A$ and $m$ distinct lines passing through $B$.There is no line passing through both $A$ and $B$ and all the lines passing through $k$ intersect with all the lines passing through $B$.The intersections do not lie on some of the circles.Determine the maximum and the minimum number of regions formed by the lines and the circles and are inside the circles.
2005 MOP Homework, 7
Let $A$ be a finite subset of prime numbers and $a> 1$ be a positive integer. Show that the number of positive integers $m$ for which all prime divisors of $a^m-1$ are in $A$ is finite.
2009 Today's Calculation Of Integral, 488
For $ 0\leq x <\frac{\pi}{2}$, prove the following inequality.
$ x\plus{}\ln (\cos x)\plus{}\int_0^1 \frac{t}{1\plus{}t^2}\ dt\leq \frac{\pi}{4}$
2011 ELMO Shortlist, 4
Prove that for any convex pentagon $A_1A_2A_3A_4A_5$, there exists a unique pair of points $\{P,Q\}$ (possibly with $P=Q$) such that $\measuredangle{PA_i A_{i-1}} = \measuredangle{A_{i+1}A_iQ}$ for $1\le i\le 5$, where indices are taken $\pmod5$ and angles are directed $\pmod\pi$.
[i]Calvin Deng.[/i]
1972 Canada National Olympiad, 2
Let $a_1,a_2,\ldots,a_n$ be non-negative real numbers. Define $M$ to be the sum of all products of pairs $a_ia_j$ $(i<j)$, $\textit{i.e.}$, \[ M = a_1(a_2+a_3+\cdots+a_n)+a_2(a_3+a_4+\cdots+a_n)+\cdots+a_{n-1}a_n. \] Prove that the square of at least one of the numbers $a_1,a_2,\ldots,a_n$ does not exceed $2M/n(n-1)$.
2014 ELMO Shortlist, 1
In a non-obtuse triangle $ABC$, prove that
\[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]
2011 Dutch IMO TST, 1
Let $n \ge 2$ and $k \ge1$ be positive integers. In a country there are $n$ cities and between each pair of cities there is a bus connection in both directions. Let $A$ and $B$ be two different cities. Prove that the number of ways in which you can travel from $A$ to $B$ by using exactly $k$ buses is equal to $\frac{(n - 1)^k - (-1)^k}{n}$
.
1998 German National Olympiad, 6a
Find all real pairs $(x,y)$ that solve the system of equations \begin{align} x^5 &= 21x^3+y^3
\\ y^5 &= x^3+21y^3. \end{align}