Found problems: 85335
2020 Balkan MO Shortlist, N4
Let $a_1=2$ and, for every positive integer $n$, let $a_{n+1}$ be the smallest integer strictly greater than $a_n$ that has more positive divisors than $a_n$. Prove that $2a_{n+1}=3a_n$ only for finitely many indicies $n$.
[i] Proposed by Ilija Jovčevski, North Macedonia[/i]
2009 Kosovo National Mathematical Olympiad, 2
If $x_1$ and $x_2$ are the solutions of the equation
$x^2-(m+3)x+m+2=0$
Find all real values of $m$ such that the following inequations are valid
$\frac {1}{x_1}+\frac {1}{x_2}>\frac{1}{2}$
and
$x_1^2+x_2^2<5$
1970 AMC 12/AHSME, 26
The number of distinct points in the xy-plane common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$ is
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4$
2009 Middle European Mathematical Olympiad, 4
Determine all integers $ k\ge 2$ such that for all pairs $ (m$, $ n)$ of different positive integers not greater than $ k$, the number $ n^{n\minus{}1}\minus{}m^{m\minus{}1}$ is not divisible by $ k$.
2014 China National Olympiad, 2
Let $f:X\rightarrow X$, where $X=\{1,2,\ldots ,100\}$, be a function satisfying:
1) $f(x)\neq x$ for all $x=1,2,\ldots,100$;
2) for any subset $A$ of $X$ such that $|A|=40$, we have $A\cap f(A)\neq\emptyset$.
Find the minimum $k$ such that for any such function $f$, there exist a subset $B$ of $X$, where $|B|=k$, such that $B\cup f(B)=X$.
2025 Kosovo EGMO Team Selection Test, P1
Let $ABC$ be an acute triangle. Let $D$ and $E$ be the feet of the altitudes of the triangle $ABC$ from $A$ and $B$, respectively. Let $F$ be the reflection of the point $A$ over $BC$. Let $G$ be a point such that the quadrilateral $ABCG$ is a parallelogram. Show that the circumcircles of triangles $BCF$ , $ACG$ and $CDE$ are concurrent on a point different from $C$.
2000 AMC 8, 25
The area of rectangle $ABCD$ is $72$. If point $A$ and the midpoints of $\overline{BC}$ and $\overline{CD}$ are joined to form a triangle, the area of that triangle is
[asy]
pair A,B,C,D;
A = (0,8); B = (9,8); C = (9,0); D = (0,0);
draw(A--B--C--D--A--(9,4)--(4.5,0)--cycle);
label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,SE);
label("$D$",D,SW);
[/asy]
$\text{(A)}\ 21 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 40$
2019 China Team Selection Test, 1
$AB$ and $AC$ are tangents to a circle $\omega$ with center $O$ at $B,C$ respectively. Point $P$ is a variable point on minor arc $BC$. The tangent at $P$ to $\omega$ meets $AB,AC$ at $D,E$ respectively. $AO$ meets $BP,CP$ at $U,V$ respectively. The line through $P$ perpendicular to $AB$ intersects $DV$ at $M$, and the line through $P$ perpendicular to $AC$ intersects $EU$ at $N$. Prove that as $P$ varies, $MN$ passes through a fixed point.
2021 Kosovo National Mathematical Olympiad, 1
Each of the spots in a $8\times 8$ chessboard is occupied by either a black or white “horse”. At most how many black horses can be on the chessboard so that none of the horses attack more than one black horse?
[b]Remark:[/b] A black horse could attack another black horse.
2013 Danube Mathematical Competition, 1
Given six points on a circle, $A, a, B, b, C, c$, show that the Pascal lines of the hexagrams $AaBbCc, AbBcCa, AcBaCb$ are concurrent.
JOM 2015 Shortlist, N1
Prove that there exists an infinite sequence of positive integers $ a_1, a_2, ... $ such that for all positive integers $ i $, \\
i) $ a_{i + 1} $ is divisible by $ a_{i} $.\\
ii) $ a_i $ is not divisible by $ 3 $.\\
iii) $ a_i $ is divisible by $ 2^{i + 2} $ but not $ 2^{i + 3} $.\\
iv) $ 6a_i + 1 $ is a prime power.\\
v) $ a_i $ can be written as the sum of the two perfect squares.
2001 Bundeswettbewerb Mathematik, 1
10 vertices of a regular 100-gon are coloured red and ten other (distinct) vertices are coloured blue. Prove that there is at least one connection edge (segment) of two red which is as long as the connection edge of two blue points.
[hide="Hint"]Possible approaches are pigeon hole principle, proof by contradiction, consider turns (bijective congruent mappings) which maps red in blue points.
[/hide]
2019 Saudi Arabia JBMO TST, 1
Real nonzero numbers $x, y, z$ are such that $x+y +z = 0$. Moreover, it is known that $$A =\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}+ 1$$Determine $A$.
2018 Miklós Schweitzer, 4
Let $P$ be a finite set of points in the plane. Assume that the distance between any two points is an integer. Prove that $P$ can be colored by three colors such that the distance between any two points of the same color is an even number.
2016 ASDAN Math Tournament, 13
Ash writes the positive integers from $1$ to $2016$ inclusive as a single positive integer $n=1234567891011\dots2016$. What is the result obtained by successively adding and subtracting the digits of $n$? (In other words, compute $1-2+3-4+5-6+7-8+9-1+0-1+\dots$.)
2017 Argentina National Olympiad, 4
For a positive integer $n$ we denote $D_2(n)$ to the number of divisors of $n$ which are perfect squares and $D_3(n)$ to the number of divisors of $n$ which are perfect cubes. Prove that there exists such that $D_2(n)=999D_3(n).$
Note. The perfect squares are $1^2,2^2,3^2,4^2,…$ , the perfect cubes are $1^3,2^3,3^3,4^3,…$ .
2013 Stanford Mathematics Tournament, 7
Find all real $x$ that satisfy $\sqrt[3]{20x+\sqrt[3]{20x+13}}=13$.
2001 Swedish Mathematical Competition, 2
Show that $\sqrt[3]{\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5}$ is rational.
2010 AMC 8, 13
The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shorter side is $30\%$ of the perimeter. What is the length of the longest side?
$ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 $
2009 AMC 10, 10
Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD\equal{}3$, and $ DC\equal{}4$. What is the area of $ \triangle{ABC}$?
[asy]unitsize(5mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21));
pair D=foot(B,A,C);
pair[] ps={B,C,A,D};
draw(A--B--C--cycle);
draw(B--D);
draw(rightanglemark(B,D,C));
dot(ps);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,NE);
label("$3$",midpoint(A--D),NE);
label("$4$",midpoint(D--C),NE);[/asy]$ \textbf{(A)}\ 4\sqrt3 \qquad
\textbf{(B)}\ 7\sqrt3 \qquad
\textbf{(C)}\ 21 \qquad
\textbf{(D)}\ 14\sqrt3 \qquad
\textbf{(E)}\ 42$
1987 Austrian-Polish Competition, 2
Let $n$ be the square of an integer whose each prime divisor has an even number of decimal digits. Consider $P(x) = x^n - 1987x$. Show that if $x,y$ are rational numbers with $P(x) = P(y)$, then $x = y$.
1998 IMO Shortlist, 4
For any two nonnegative integers $n$ and $k$ satisfying $n\geq k$, we define the number $c(n,k)$ as follows:
- $c\left(n,0\right)=c\left(n,n\right)=1$ for all $n\geq 0$;
- $c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right)$ for $n\geq k\geq 1$.
Prove that $c\left(n,k\right)=c\left(n,n-k\right)$ for all $n\geq k\geq 0$.
2015 All-Russian Olympiad, 1
Real numbers $a$ and $b$ are chosen so that each of two quadratic trinomials $x^2+ax+b$ and $x^2+bx+a$ has two distinct real roots,and the product of these trinomials has exactly three distinct real roots.Determine all possible values of the sum of these three roots. [i](S.Berlov)[/i]
2021 BMT, 5
Anthony the ant is at point $A$ of regular tetrahedron $ABCD$ with side length $4$. Anthony wishes to crawl on the surface of the tetrahedron to the midpoint of $\overline{BC}$. However, he does not want to touch the interior of face $\vartriangle ABC$, since it is covered with lava. What is the shortest distance Anthony must travel?
2006 Grigore Moisil Urziceni, 1
Consider two quadrilaterals $ A_1B_1C_1D_1,A_2B_2C_2D_2 $ and the points $ M,N,P,Q,E_1,F_1,E_2,F_2 $ representing the middle of the segments $ A_1A_2,B_1B_2,C_1C_2,D_1D_2,B_1D_1,A_1C_1,B_2D_2,A_2,C_2, $ respectively. Show that $ MNPQ $ is a parallelogram if and only if $ E_1F_1E_2F_2 $ is a parallelogram.
[i]Cristinel Mortici[/i]