Found problems: 177
2016 Junior Balkan Team Selection Test, 2
Find minimal number of divisors that can number $|2016^m-36^n|$ have,where $m$ and $n$ are natural numbers.
2019 JBMO Shortlist, A6
Let $a, b, c$ be positive real numbers. Prove the inequality
$(a^2+ac+c^2) \left( \frac{1}{a+b+c}+\frac{1}{a+c} \right)+b^2 \left( \frac{1}{b+c}+\frac{1}{a+b} \right)>a+b+c$.
[i]Proposed by Tajikistan[/i]
2013 Junior Balkan MO, 4
Let $n$ be a positive integer. Two players, Alice and Bob, are playing the following game:
- Alice chooses $n$ real numbers; not necessarily distinct.
- Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are $\frac{n(n-1)}{2}$ such sums; not necessarily distinct.)
- Bob wins if he finds correctly the initial $n$ numbers chosen by Alice with only one guess.
Can Bob be sure to win for the following cases?
a. $n=5$
b. $n=6$
c. $n=8$
Justify your answer(s).
[For example, when $n=4$, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]
2008 JBMO Shortlist, 3
Let $s(a)$ denote the sum of digits of a given positive integer a. The sequence $a_1, a_2,..., a_n, ...$ of positive integers is such that $a_{n+1} = a_n+s(a_n)$ for each positive integer $n$. Find the greatest possible n for which it is possible to have $a_n = 2008$.
2022 JBMO Shortlist, A1
Find all pairs of positive integers $(a, b)$ such that $$11ab \le a^3 - b^3 \le 12ab.$$
2013 JBMO Shortlist, 3
Let $n$ be a positive integer. Two players, Alice and Bob, are playing the following game:
- Alice chooses $n$ real numbers; not necessarily distinct.
- Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are $\frac{n(n-1)}{2}$ such sums; not necessarily distinct.)
- Bob wins if he finds correctly the initial $n$ numbers chosen by Alice with only one guess.
Can Bob be sure to win for the following cases?
a. $n=5$
b. $n=6$
c. $n=8$
Justify your answer(s).
[For example, when $n=4$, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]
2009 JBMO Shortlist, 3
Parallelogram ${ABCD}$with obtuse angle $\angle ABC$ is given. After rotation of the triangle ${ACD}$ around the vertex ${C}$, we get a triangle ${CD'A'}$, such that points $B,C$ and ${D'}$are collinear. Extensions of median of triangle ${CD'A'}$ that passes through ${D'}$intersects the straight line ${BD}$ at point ${P}$. Prove that ${PC}$is the bisector of the angle $\angle BP{D}'$.
2008 JBMO Shortlist, 1
Find all the positive integers $x$ and $y$ that satisfy the equation $x(x - y) = 8y - 7$
2008 JBMO Shortlist, 9
Consider an integer $n \ge 4 $ and a sequence of real numbers $x_1, x_2, x_3,..., x_n$. An operation consists in eliminating all numbers not having the rank of the form $4k + 3$, thus leaving only the numbers $x_3. x_7. x_{11}, ...$(for example, the sequence $4,5,9,3,6, 6,1, 8$ produces the sequence $9,1$). Upon the sequence $1, 2, 3, ..., 1024 $ the operation is performed successively for $5$ times. Show that at the end only one number remains and fi nd this number.
2003 JBMO Shortlist, 5
Let $ABC$ be an isosceles triangle with $AB = AC$. A semi-circle of diameter $[EF] $ with $E, F \in [BC]$, is tangent to the sides $AB,AC$ in $M, N$ respectively and $AE$ intersects the semicircle at $P$. Prove that $PF$ passes through the midpoint of $[MN]$.
2023 JBMO Shortlist, G5
Let $D,E,F$ be the points of tangency of the incircle of a given triangle $ABC$ with sides $BC, CA, AB,$ respectively. Denote by $I$ the incenter of $ABC$, by $M$ the midpoint of $BC$ and by $G$ the foot of the perpendicular from $M$ to line $EF$. Prove that the line $ID$ is tangent to the circumcircle of the triangle $MGI$.
2005 JBMO Shortlist, 2
Let $ABCD$ be an isosceles trapezoid with $AB=AD=BC, AB//CD, AB>CD$. Let $E= AC \cap BD$ and $N$ symmetric to $B$ wrt $AC$. Prove that the quadrilateral $ANDE$ is cyclic.
2008 JBMO Shortlist, 6
Let $f : N \to R$ be a function, satisfying the following condition:
for every integer $n > 1$, there exists a prime divisor $p$ of $n$ such that $f(n) = f \big(\frac{n}{p}\big)-f(p)$.
If $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$, determine the value of $f(2007^2) + f(2008^3) + f(2009^5)$
2016 JBMO Shortlist, 5
Determine all four-digit numbers $\overline{abcd} $ such that
$(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) =\overline{abcd} $:
2023 JBMO Shortlist, G7
Let $D$ and $E$ be arbitrary points on the sides $BC$ and $AC$ of triangle $ABC$, respectively. The circumcircle of $\triangle ADC$ meets for the second time the circumcircle of $\triangle BCE$ at point $F$. Line $FE$ meets line $AD$ at point $G$, while line $FD$ meets line $BE$ at point $H$. Prove that lines $CF, AH$ and $BG$ pass through the same point.
2015 JBMO Shortlist, 2
The point ${P}$ is outside the circle ${\Omega}$. Two tangent lines, passing from the point ${P}$ touch the circle ${\Omega}$ at the points ${A}$ and ${B}$. The median${AM \left(M\in BP\right)}$ intersects the circle ${\Omega}$ at the point ${C}$ and the line ${PC}$ intersects again the circle ${\Omega}$ at the point ${D}$. Prove that the lines ${AD}$ and ${BP}$ are parallel.
(Moldova)
2022 Junior Balkan Mathematical Olympiad, 2
Let $ABC$ be an acute triangle such that $AH = HD$, where $H$ is the orthocenter of $ABC$ and $D \in BC$ is the foot of the altitude from the vertex $A$. Let $\ell$ denote the line through $H$ which is tangent to the circumcircle of the triangle $BHC$. Let $S$ and $T$ be the intersection points of $\ell$ with $AB$ and $AC$, respectively. Denote the midpoints of $BH$ and $CH$ by $M$ and $N$, respectively. Prove that the lines $SM$ and $TN$ are parallel.
2008 JBMO Shortlist, 1
Two perpendicular chords of a circle, $AM, BN$ , which intersect at point $K$, define on the circle four arcs with pairwise different length, with $AB$ being the smallest of them. We draw the chords $AD, BC$ with $AD // BC$ and $C, D$ different from $N, M$ . If $L$ is the intersection point of $DN, M C$ and $T$ the intersection point of $DC, KL,$ prove that $\angle KTC = \angle KNL$.
JBMO Geometry Collection, 2021
Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to the side $BC$. The lines $BC$ and $AO$ intersect at $E$. Let $s$ be the line through $E$ perpendicular to $AO$. The line $s$ intersects $AB$ and $AC$ at $K$ and $L$, respectively. Denote by $\omega$ the circumcircle of triangle $AKL$. Line $AD$ intersects $\omega$ again at $X$.
Prove that $\omega$ and the circumcircles of triangles $ABC$ and $DEX$ have a common point.
2019 Junior Balkan MO, 3
Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.
JBMO Geometry Collection, 2019
Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.
2022 JBMO Shortlist, N3
Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$
JBMO Geometry Collection, 1999
Let $ABC$ be a triangle with $AB=AC$. Also, let $D\in[BC]$ be a point such that $BC>BD>DC>0$, and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$. Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$).
[i]Greece[/i]
2022 JBMO Shortlist, G3
Let $ABC$ be an acute triangle such that $AH = HD$, where $H$ is the orthocenter of $ABC$ and $D \in BC$ is the foot of the altitude from the vertex $A$. Let $\ell$ denote the line through $H$ which is tangent to the circumcircle of the triangle $BHC$. Let $S$ and $T$ be the intersection points of $\ell$ with $AB$ and $AC$, respectively. Denote the midpoints of $BH$ and $CH$ by $M$ and $N$, respectively. Prove that the lines $SM$ and $TN$ are parallel.
2012 JBMO ShortLists, 5
Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.