Found problems: 85335
2017 QEDMO 15th, 12
Let $a$ be a real number such that $\left(a + \frac{1}{a}\right)^2=11$. For which $n\in N$ is $a^n + \frac{1}{a^n}$ an integer? Does this depend on the exact value of $a$?
2010 Saudi Arabia Pre-TST, 4.3
Let $a, b, c$ be positive real numbers such that $abc = 8$. Prove that
$$\frac{a-2}{a+1}+\frac{b-2}{b+1}+\frac{c-2}{c+1} \le 0$$
2014 India PRMO, 13
For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer?
2015 FYROM JBMO Team Selection Test, 5
$A$ and $B$ are two identical convex polygons, each with an area of $2015$. The polygon $A$ is divided into polygons $A_1, A_2,...,A_{2015}$, while $B$ is divided into polygons $B_1, B_2,...,B_{2015}$. Each of these smaller polygons has a positive area. Furthermore, $A_1, A_2,...,A_{2015}$ and $B_1, B_2,...,B_{2015}$ are colored in $2015$ distinct colors, such that $A_i$ and $A_j$ are differently colored for every distinct $i$ and $j$ and $B_i$ and $B_j$ are also differently colored for every distinct $i$ and $j$. After $A$ and $B$ overlap, we calculate the sum of the areas with the same colors. Prove that we can color the polygons such that this sum is at least $1$.
2007 Italy TST, 3
Let $p \geq 5$ be a prime.
(a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$
(b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that:
\[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]
2015 Bulgaria National Olympiad, 5
In a triangle $\triangle ABC$ points $L, P$ and $Q$ lie on the segments $AB, AC$ and $BC$, respectively, and are such that $PCQL$ is a parallelogram. The circle with center the midpoint $M$ of the segment $AB$ and radius $CM$ and the circle of diameter $CL$ intersect for the second time at the point $T$. Prove that the lines $AQ, BP$ and $LT$ intersect in a point.
2010 Dutch BxMO TST, 5
For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.
2007 China Western Mathematical Olympiad, 3
Let $ P$ be an interior point of an acute angled triangle $ ABC$. The lines $ AP,BP,CP$ meet $ BC,CA,AB$ at points $ D,E,F$ respectively. Given that triangle $ \triangle DEF$ and $ \triangle ABC$ are similar, prove that $ P$ is the centroid of $ \triangle ABC$.
2014 Contests, 3
Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$.
[i]Proposed by Mohammad Ahmadi[/i]
2019 Jozsef Wildt International Math Competition, W. 11
Let $(s_n)_{n\geq 1}$ be a sequence given by $s_n=-2\sqrt{n}+\sum \limits_{k=1}^n\frac{1}{\sqrt{k}}$ with $\lim \limits_{n \to \infty}s_n=s=$Ioachimescu constant and $(a_n)_{n\geq 1}$ , $(b_n)_{n\geq 1}$ be a positive real sequences such that $$\lim \limits_{n\to \infty}\frac{a_{n+1}}{na_n}=a\in \mathbb{R}^*_+, \lim \limits_{n\to \infty}\frac{b_{n+1}}{b_n\sqrt{n}}=b\in \mathbb{R}^*_+$$Compute$$\lim \limits_{n\to \infty}\left(1+e^{s_n}-e^{s_{n+1}}\right)^{\sqrt[n]{a_nb_n}}$$
2020 European Mathematical Cup, 4
Let \(a,b,c\) be positive real numbers such that \(ab+bc+ac = a+b+c\). Prove the following inequality:
\[\sqrt{a+\frac{b}{c}} + \sqrt{b+\frac{c}{a}} + \sqrt{c+\frac{a}{b}} \leq \sqrt{2} \cdot \min \left\{ \frac{a}{b}+\frac{b}{c}+\frac{c}{a},\ \frac{b}{a}+\frac{c}{b}+\frac{a}{c} \right\}.\] \\ \\ [i]Proposed by Dorlir Ahmeti.[/i]
2018 CIIM, Problem 1
Show that there exists a $2 \times 2$ matrix of order 6 with rational entries, such that the sum of its entries is 2018.
Note: The order of a matrix (if it exists) is the smallest positive integer $n$ such that $A^n = I$, where $I$ is the identity matrix.
2017 ASDAN Math Tournament, 3
Let $f(x)=x^4+2x+1$. Find the slope of the tangent line to the curve at $(0,1)$.
2023 MOAA, 3
After the final exam, Mr. Liang asked each of his 17 students to guess the average final exam score. David, a very smart student, received a 100 and guessed the average would be 97. Each of the other 16 students guessed $30+\frac{n}{2}$ where $n$ was that student’s score. If the average of the final exam scores was the same as the average of the guesses, what was the average score on the final exam?
[i]Proposed by Eric Wang[/i]
LMT Speed Rounds, 2010.9
Let $ABC$ and $BCD$ be equilateral triangles, such that $AB=1,$ and $A \neq D.$ Find the area of triangle $ABD.$
1999 Bundeswettbewerb Mathematik, 3
In the plane are given a segment $AC$ and a point $B$ on the segment. Let us draw the positively oriented isosceles triangles $ABS_1, BCS_2$, and $CAS_3$ with the angles at $S_1,S_2,S_3$ equal to $120^o$. Prove that the triangle $S_1S_2S_3$ is equilateral.
2012 National Olympiad First Round, 8
In how many different ways can one select two distinct subsets of the set $\{1,2,3,4,5,6,7\}$, so that one includes the other?
$ \textbf{(A)}\ 2059 \qquad \textbf{(B)}\ 2124 \qquad \textbf{(C)}\ 2187 \qquad \textbf{(D)}\ 2315 \qquad \textbf{(E)}\ 2316$
2011 All-Russian Olympiad, 1
A quadratic trinomial $P(x)$ with the $x^2$ coefficient of one is such, that $P(x)$ and $P(P(P(x)))$ share a root. Prove that $P(0)*P(1)=0$.
2013 Princeton University Math Competition, 3
Let $x_1=\sqrt{10}$ and $y_1=\sqrt3$. For all $n\geq 2$, let \begin{align*}x_n&=x_{n-1}\sqrt{77}+15y_{n-1}\\y_n&=5x_{n-1}+y_{n-1}\sqrt{77}\end{align*} Find $x_5^6+2x_5^4-9x_5^4y_5^2-12x_5^2y_5^2+27x_5^2y_5^4+18y_5^4-27y_5^6.$
LMT Team Rounds 2010-20, A26
Jeff has planted $7$ radishes, labelled $R$, $A$, $D$, $I$, $S$, $H$, and $E$. Taiki then draws circles through $S,H,I,E,D$, then through $E,A,R,S$, and then through $H,A,R,D$, and notices that lines drawn through $SH$, $AR$, and $ED$ are parallel, with $SH = ED$. Additionally, $HER$ is equilateral, and $I$ is the midpoint of $AR$. Given that $HD = 2$, $HE$ can be written as $\frac{-\sqrt{a} + \sqrt{b} + \sqrt{1+\sqrt{c}}}{2}$, where $a,b,$ and $c$ are integers, find $a+b+c$.
[i]Proposed by Jeff Lin[/i]
1990 Swedish Mathematical Competition, 2
The points $A_1, A_2,.. , A_{2n}$ are equally spaced in that order along a straight line with $A_1A_2 = k$. $P$ is chosen to minimise $\sum PA_i$. Find the minimum.
1995 Brazil National Olympiad, 3
For any positive integer $ n>1$, let $ P\left(n\right)$ denote the largest prime divisor of $ n$. Prove that there exist infinitely many positive integers $ n$ for which
\[ P\left(n\right)<P\left(n\plus{}1\right)<P\left(n\plus{}2\right).\]
MathLinks Contest 2nd, 6.2
A triangle $ABC$ is located in a cartesian plane $\pi$ and has a perimeter of $3 + 2\sqrt3$. It is known that the triangle $ABC$ has the property that any triangle in the plane $\pi$, congruent with it, contains inside or on the boundary at least one lattice point (a point with both coordinates integers). Prove that the triangle $ABC$ is equilateral.
2025 All-Russian Olympiad, 11.1
$777$ pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers \(a\) and \(b\) from the board such that:
\[
a^2 + b^2 + 1 = 2ab
\]
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers \(c\) and \(d\) from the board such that:
\[
c^2 + d^2 + 2025 = 2cd
\]
2011 JBMO Shortlist, 2
Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$.