Found problems: 85335
1985 Traian Lălescu, 1.1
$ n $ is a natural number, and $ S $ is the sum of all the solutions of the equations
$$ x^2+a_k\cdot x+a_k=0,\quad a_k\in\mathbb{R} ,\quad k\in\{ 1,2,...,n\} . $$
Show that if $ |S|>2n\left( \sqrt[n]{n} -1\right) , $ then at least one of the equations has real solutions.
1996 Balkan MO, 2
Let $ p$ be a prime number with $ p>5$. Consider the set $ X \equal{} \left\{p \minus{} n^2 \mid n\in \mathbb{N} ,\ n^2 < p\right\}$.
Prove that the set $ X$ has two distinct elements $ x$ and $ y$ such that $ x\neq 1$ and $ x\mid y$.
[i]Albania[/i]
1987 Federal Competition For Advanced Students, P2, 6
Determine all polynomials $ P_n(x)\equal{}x^n\plus{}a_1 x^{n\minus{}1}\plus{}...\plus{}a_{n\minus{}1} x\plus{}a_n$ with integer coefficients whose $ n$ zeros are precisely the numbers $ a_1,...,a_n$ (counted with their respective multiplicities).
1999 Argentina National Olympiad, 3
In a trick tournament $2k$ people sign up. All possible matches are played with the condition that in each match, each of the four players knows his partner and does not know any of his two opponents. Determine the maximum number of matches that can be in such a tournament.
2013 India Regional Mathematical Olympiad, 5
Let $ABC$ be a triangle with $\angle A=90^{\circ}$ and $AB=AC$. Let $D$ and $E$ be points on the segment $BC$ such that $BD:DE:EC = 1:2:\sqrt{3}$. Prove that $\angle DAE= 45^{\circ}$
1998 All-Russian Olympiad Regional Round, 9.7
Given a billiard in the form of a regular $1998$-gon $A_1A_2...A_{1998}$. A ball was released from the midpoint of side $A_1A_2$, which, reflected therefore from sides $A_2A_3$, $A_3A_4$, . . . , $A_{1998}A_1$ (according to the law, the angle of incidence is equal to the angle of reflection), returned to the starting point. Prove that the trajectory of the ball is a regular $1998$-gon.
2020 BMT Fall, 11
Equilateral triangle $ABC$ has side length $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the “lune” that is inside the semicircle but outside sector $ABC$ can be expressed in the form $\sqrt{p}-\frac{q\pi}{r}$, where $p, q$, and $ r$ are positive integers such that $q$ and $r$ are relatively prime. Compute $p + q + r$.
[img]https://cdn.artofproblemsolving.com/attachments/7/7/f349a807583a83f93ba413bebf07e013265551.png[/img]
1973 IMO Shortlist, 9
Let $Ox, Oy, Oz$ be three rays, and $G$ a point inside the trihedron $Oxyz$. Consider all planes passing through $G$ and cutting $Ox, Oy, Oz$ at points $A,B,C$, respectively. How is the plane to be placed in order to yield a tetrahedron $OABC$ with minimal perimeter ?
1997 Canadian Open Math Challenge, 1
In triangle ABC, $\angle$ A equals 120 degrees. A point D is inside the triangle such that $\angle$DBC = 2 $\times \angle $ABD and $\angle$DCB = 2 $\times \angle$ACD. Determine the measure, in degrees, of $\angle$ BDC.
[asy]
pair A = (5,4);
pair B = (0,0);
pair C = (10,0);
pair D = (5,2.5) ;
draw(A--B);
draw(B--C);
draw(C--A);
draw (B--D--C);
label ("A", A, dir(45));
label ("B", B, dir(45));
label ("C", C, dir(45));
label ("D", D, dir(45));
[/asy]
2002 Olympic Revenge, 3
Show that if $x,y,z,w$ are positive reals, then
\[
\frac{3}{2}\sqrt{(x^2+y^2)(w^2+z^2)} + \sqrt{(x^2+w^2)(y^2+z^2) - 3xyzw} \geq (x+z)(y+w)
\]
2023 India National Olympiad, 5
Euler marks $n$ different points in the Euclidean plane. For each pair of marked points, Gauss writes down the number $\lfloor \log_2 d \rfloor$ where $d$ is the distance between the two points. Prove that Gauss writes down less than $2n$ distinct values.
[i]Note:[/i] For any $d>0$, $\lfloor \log_2 d\rfloor$ is the unique integer $k$ such that $2^k\le d<2^{k+1}$.
[i]Proposed by Pranjal Srivastava[/i]
2004 Singapore Team Selection Test, 2
Let $0 < a, b, c < 1$ with $ab + bc + ca = 1$. Prove that
\[\frac{a}{1-a^2} + \frac{b}{1-b^2} + \frac{c}{1-c^2} \geq \frac {3 \sqrt{3}}{2}.\]
Determine when equality holds.
2022 AMC 10, 23
Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$
$\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{2}{3}\qquad\textbf{(E) }\frac{3}{4}$
1993 Nordic, 3
Find all solutions of the system of equations
$\begin{cases} s(x) + s(y) = x \\ x + y + s(z) = z \\ s(x) + s(y) + s(z) = y - 4 \end{cases}$
where $x, y$, and $z$ are positive integers, and $s(x), s(y)$, and $s(z)$ are the numbers of digits in the decimal representations of $x, y$, and $z$, respectively.
2017 Brazil Team Selection Test, 3
Let $A(n)$ denote the number of sequences $a_1\ge a_2\ge\cdots{}\ge a_k$ of positive integers for which $a_1+\cdots{}+a_k = n$ and each $a_i +1$ is a power of two $(i = 1,2,\cdots{},k)$. Let $B(n)$ denote the number of sequences $b_1\ge b_2\ge \cdots{}\ge b_m$ of positive integers for which $b_1+\cdots{}+b_m =n$ and each inequality $b_j\ge 2b_{j+1}$ holds $(j=1,2,\cdots{}, m-1)$. Prove that $A(n) = B(n)$ for every positive integer $n$.
[i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]
1994 IMO Shortlist, 2
Find all ordered pairs $ (m,n)$ where $ m$ and $ n$ are positive integers such that $ \frac {n^3 \plus{} 1}{mn \minus{} 1}$ is an integer.
2015 AMC 12/AHSME, 23
A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?
$ \textbf{(A) }4\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }21\qquad\textbf{(E) }26 $
2005 National Olympiad First Round, 9
Let $ABC$ be a triangle with circumradius $1$. If the center of the circle passing through $A$, $C$, and the orthocenter of $\triangle ABC$ lies on the circumcircle of $\triangle ABC$, what is $|AC|$?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ \dfrac 32
\qquad\textbf{(D)}\ \sqrt 2
\qquad\textbf{(E)}\ \sqrt 3
$
1996 North Macedonia National Olympiad, 4
A polygon is called [i]good [/i] if it satisfies the following conditions:
(i) All its angles are in $(0,\pi)$ or in $(\pi ,2\pi)$,
(ii) It is not self-intersecing,
(iii) For any three sides, two are parallel and equal.
Find all $n$ for which there exists a [i]good [/i] $n$-gon.
1961 Polish MO Finals, 4
Prove that if every side of a triangle is less than $ 1 $, then its area is less than $ \frac{\sqrt{3}}{4} $.
2020 LMT Spring, 5
For a positive integer $n$, let $\mathcal{D}(n)$ be the value obtained by, starting from the left, alternating between adding and subtracting the digits of $n$. For example, $\mathcal{D}(321)=3-2+1=2$, while $\mathcal{D}(40)=4-0=4$. Compute the value of the sum
\[\sum_{n=1}^{100}\mathcal{D}(n)=\mathcal{D}(1)+\mathcal{D}(2)+\dots+\mathcal{D}(100).\]
2012 USA TSTST, 4
In scalene triangle $ABC$, let the feet of the perpendiculars from $A$ to $BC$, $B$ to $CA$, $C$ to $AB$ be $A_1, B_1, C_1$, respectively. Denote by $A_2$ the intersection of lines $BC$ and $B_1C_1$. Define $B_2$ and $C_2$ analogously. Let $D, E, F$ be the respective midpoints of sides $BC, CA, AB$. Show that the perpendiculars from $D$ to $AA_2$, $E$ to $BB_2$ and $F$ to $CC_2$ are concurrent.
2007 Nicolae Coculescu, 2
[b]a)[/b] Prove that there exists two infinite sequences $ \left( a_n \right)_{n\ge 1} ,\left( b_n \right)_{n\ge 1} $ of nonnegative integers such that $ a_n>b_n $ and $ (2+\sqrt 3)^n =a_n (2+\sqrt 3) -b_n , $ for any natural numbers $ n. $
[b]b)[/b] Prove that the equation $ x^2-4xy+y^2=1 $ has infinitely many solutions in $ \mathbb{N}^2. $
[i]Florian Dumitrel[/i]
2010 HMNT, 5
There are 111 StarCraft programmers. The StarCraft team SKT starts with a given set of eleven programmers on it, and at the end of each season, it drops a progamer and adds a programmer (possibly the same one). At the start of the second season, SKT has to fi eld a team of five programmers to play the opening match. How many diff erent lineups of ve players could be fi elded if the order of players on the lineup matters?
2011 Indonesia TST, 4
Prove that there exists infinitely many positive integers $n$ such that $n^2+1$ has a prime divisor greater than $2n+\sqrt{5n+2011}$.