This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 177

2007 Junior Balkan MO, 1
Tags: quadratics , inequalities , function , algebra , algebra solved , inequalities solved , JBMO
Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.

2008 JBMO Shortlist, 9
Tags: JBMO , number theory
Let $p$ be a prime number. Find all positive integers $a$ and $b$ such that: $\frac{4a + p}{b}+\frac{4b + p}{a}$ and $ \frac{a^2}{b}+\frac{b^2}{a}$ are integers.

2014 JBMO Shortlist, 2
Tags: geometry , JBMO , Spiral Similarity
Acute-angled triangle ${ABC}$ with ${AB<AC<BC}$ and let be ${c(O,R)}$ it’s circumcircle. Diameters ${BD}$ and ${CE}$ are drawn. Circle ${c_1(A,AE)}$ interescts ${AC}$ at ${K}$. Circle ${{c}_{2}(A,AD)}$ intersects ${BA}$ at ${L}$ .(${A}$ lies between ${B}$ and ${L}$). Prove that lines ${EK}$ and ${DL}$ intersect at circle $c$ . by Evangelos Psychas (Greece)

2007 JBMO Shortlist, 4
Tags: JBMO , algebra , positive integer
Let $a$ and $ b$ be positive integers bigger than $2$. Prove that there exists a positive integer $k$ and a sequence $n_1, n_2, ..., n_k$ consisting of positive integers, such that $n_1 = a,n_k = b$, and $(n_i + n_{i+1}) | n_in_{i+1}$ for all $i = 1,2,..., k - 1$

2016 JBMO Shortlist, 4
Tags: geometry , JBMO
Let ${ABC}$ be an acute angled triangle whose shortest side is ${BC}$. Consider a variable point ${P}$ on the side ${BC}$, and let ${D}$ and ${E}$ be points on ${AB}$ and ${AC}$, respectively, such that ${BD=BP}$ and ${CP=CE}$. Prove that, as ${P}$ traces ${BC}$, the circumcircle of the triangle ${ADE}$ passes through a fixed point.

2016 Greece JBMO TST, 2
Tags: geometry , JBMO
Let ${c\equiv c\left(O, R\right)}$ be a circle with center ${O}$ and radius ${R}$ and ${A, B}$ be two points on it, not belonging to the same diameter. The bisector of angle${\angle{ABO}}$ intersects the circle ${c}$ at point ${C}$, the circumcircle of the triangle $AOB$ , say ${c_1}$ at point ${K}$ and the circumcircle of the triangle $AOC$ , say ${{c}_{2}}$ at point ${L}$. Prove that point ${K}$ is the circumcircle of the triangle $AOC$ and that point ${L}$ is the incenter of the triangle $AOB$. Evangelos Psychas (Greece)

2021 JBMO Shortlist, C5
Tags: combinatorics , JBMO , JBMO Shortlist
Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, ..., 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$ we have $|a + b - c | > 10$. Determine the largest possible number of elements of $M$.

2009 JBMO Shortlist, 2
Tags: geometry , JBMO
In right trapezoid ${ABCD \left(AB\parallel CD\right)}$ the angle at vertex $B$ measures ${{75}^{{}^\circ }}$. Point ${H}$is the foot of the perpendicular from point ${A}$ to the line ${BC}$. If ${BH=DC}$ and${AD+AH=8}$, find the area of ${ABCD}$.

2009 JBMO Shortlist, 1
Tags: geometry , JBMO
Parallelogram ${ABCD}$ is given with ${AC>BD}$, and ${O}$ intersection point of ${AC}$ and ${BD}$. Circle with center at ${O}$and radius ${OA}$ intersects extensions of ${AD}$and ${AB}$at points ${G}$ and ${L}$, respectively. Let ${Z}$ be intersection point of lines ${BD}$and ${GL}$. Prove that $\angle ZCA={{90}^{{}^\circ }}$.

2022 Junior Balkan Mathematical Olympiad, 3
Tags: number theory , Diophantine equation , JBMO , JBMO Shortlist
Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$

2007 JBMO Shortlist, 2
Tags: JBMO , number theory
Prove that the equation $x^{2006} - 4y^{2006} -2006 = 4y^{2007} + 2007y$ has no solution in the set of the positive integers.

2021 JBMO Shortlist, A1
Tags: algebra , JBMO , JBMO Shortlist
Let $n$ ($n \ge 1$) be an integer. Consider the equation $2\cdot \lfloor{\frac{1}{2x}}\rfloor - n + 1 = (n + 1)(1 - nx)$, where $x$ is the unknown real variable. (a) Solve the equation for $n = 8$. (b) Prove that there exists an integer $n$ for which the equation has at least $2021$ solutions. (For any real number $y$ by $\lfloor{y} \rfloor$ we denote the largest integer $m$ such that $m \le y$.)

2008 JBMO Shortlist, 4
Tags: JBMO , combinatorics , Coloring
Every cell of table $4 \times 4$ is colored into white. It is permitted to place the cross (pictured below) on the table such that its center lies on the table (the whole fi gure does not need to lie on the table) and change colors of every cell which is covered into opposite (white and black). Find all $n$ such that after $n$ steps it is possible to get the table with every cell colored black.

2021 JBMO Shortlist, G1
Tags: geometry , JBMO , JBMO Shortlist
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.

2016 Junior Balkan MO, 2
Tags: inequalities , JBMO
Let $a,b,c $be positive real numbers.Prove that $\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}$.

2008 JBMO Shortlist, 8
Tags: geometry , JBMO
The side lengths of a parallelogram are $a, b$ and diagonals have lengths $x$ and $y$. Knowing that $ab = \frac{xy}{2}$, show that $\left( a,b \right)=\left( \frac{x}{\sqrt{2}},\frac{y}{\sqrt{2}} \right)$ or $\left( a,b \right)=\left( \frac{y}{\sqrt{2}},\frac{x}{\sqrt{2}} \right)$.

2008 JBMO Shortlist, 10
Tags: geometry , JBMO
Let $\Gamma$ be a circle of center $O$, and $\delta$. be a line in the plane of $\Gamma$, not intersecting it. Denote by $A$ the foot of the perpendicular from $O$ onto $\delta$., and let $M$ be a (variable) point on $\Gamma$. Denote by $\gamma$ the circle of diameter $AM$ , by $X$ the (other than M ) intersection point of $\gamma$ and $\Gamma$, and by $Y$ the (other than $A$) intersection point of $\gamma$ and $\delta$. Prove that the line $XY$ passes through a fixed point.

2016 JBMO Shortlist, 3
Tags: JBMO , number theory
Find all positive integers $n$ such that the number $A_n =\frac{ 2^{4n+2}+1}{65}$ is a) an integer, b) a prime.

2008 JBMO Shortlist, 2
Tags: JBMO , combinatorics
Kostas and Helene have the following dialogue: Kostas: I have in my mind three positive real numbers with product $1$ and sum equal to the sum of all their pairwise products. Helene: I think that I know the numbers you have in mind. They are all equal to $1$. Kostas: In fact, the numbers you mentioned satisfy my conditions, but I did not think of these numbers. The numbers you mentioned have the minimal sum between all possible solutions of the problem. Can you decide if Kostas is right? (Explain your answer).

2016 JBMO Shortlist, 2
Tags: JBMO , number theory , divides
Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions: a) No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$, and b) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$.

2004 JBMO Shortlist, 1
Tags: JBMO , geometry
Two circles $C_1$ and $C_2$ intersect in points $A$ and $B$. A circle $C$ with center in $A$ intersect $C_1$ in $M$ and $P$ and $C_2$ in $N$ and $Q$ so that $N$ and $Q$ are located on different sides wrt $MP$ and $AB> AM$. Prove that $\angle MBQ = \angle NBP$.

2007 JBMO Shortlist, 5
Tags: inequalities , JBMO , algebra
The real numbers $x,y,z, m, n$ are positive, such that $m + n \ge 2$. Prove that $x\sqrt{yz(x + my)(x + nz)} + y\sqrt{xz(y + mx)(y + nz)} + z\sqrt{xy(z + mx)(x + ny) }\le \frac{3(m + n)}{8} (x + y)(y + z)(z + x)$

2008 JBMO Shortlist, 4
Tags: JBMO , number theory
Find all integers $n$ such that $n^4 + 8n + 11$ is a product of two or more consecutive integers.

2019 JBMO Shortlist, N1
Tags: number theory , Junior , JBMO , Balkan , 2019 , prime numbers
Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number $x^p + y^p + z^p - x - y - z$ is a product of exactly three distinct prime numbers.

2004 JBMO Shortlist, 2
Tags: geometry , combinatorial geometry , area of triangle , JBMO
Let $E, F$ be two distinct points inside a parallelogram $ABCD$ . Determine the maximum possible number of triangles having the same area with three vertices from points $A, B, C, D, E, F$.