Olympiad problems

2007 Junior Macedonian Mathematical Olympiad, 4

Tags: Junior , JMMO , 2007 , Macedonia , algebra , Inequality

The numbers $a_{1}, a_{2}, ..., a_{20}$ satisfy the following conditions: $a_{1} \ge a_{2} \ge ... \ge a_{20} \ge 0$ $a_{1} + a_{2} = 20$ $a_{3} + a_{4} + ... + a_{20} \le 20$ . What is maximum value of the expression: $a_{1}^2 + a_{2}^2 + ... + a_{20}^2$ ? For which values of $a_{1}, a_{2}, ..., a_{20}$ is the maximum value achieved?

2017 Macedonia JBMO TST, Source

Tags: Macedonia , JBMO TST , 2017 , number theory , geometry , inequalities , rectangle

[url=https://artofproblemsolving.com/community/c675693][b]Macedonia JBMO TST 2017[/b][/url] [url=http://artofproblemsolving.com/community/c6h1663908p10569198][b]Problem 1[/b][/url]. Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$. [url=http://artofproblemsolving.com/community/c6h1663916p10569261][b]Problem 2[/b][/url]. In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that $$C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$$ Show that $TB_2AC_2$ is a rectangle. [url=http://artofproblemsolving.com/community/c6h1663918p10569305][b]Problem 3[/b][/url]. Let $x,y,z$ be positive reals such that $xyz=1$. Show that $$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$ When does equality happen? [url=http://artofproblemsolving.com/community/c6h1663920p10569326][b]Problem 4[/b][/url]. In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$ [url=http://artofproblemsolving.com/community/c6h1663922p10569370][b]Problem 5[/b][/url]. Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.

2016 Macedonia JBMO TST, 4

Tags: JMMO , Macedonia , Junior , 2016 , algebra , Inequality

Let $x$, $y$, and $z$ be positive real numbers. Prove that $\sqrt {\frac {xy}{x^2 + y^2 + 2z^2}} + \sqrt {\frac {yz}{y^2 + z^2 + 2x^2}}+\sqrt {\frac {zx}{z^2 + x^2 + 2y^2}} \le \frac{3}{2}$. When does equality hold?

2019 Macedonia Junior BMO TST, 4

Tags: JMMO , Macedonia , 2019 , Junior , algebra

Let the real numbers $a$, $b$, and $c$ satisfy the equations $(a+b)(b+c)(c+a)=abc$ and $(a^9+b^9)(b^9+c^9)(c^9+a^9)=(abc)^9$. Prove that at least one of $a$, $b$, and $c$ equals $0$.

2016 Macedonia JBMO TST, 5

Tags: Junior , JMMO , 2016 , number theory , Macedonia , greatest common divisor

Solve the following equation in the set of positive integers $x + y^2 + (GCD(x, y))^2 = xy \cdot GCD(x, y)$.

2019 Macedonia National Olympiad, 1

Tags: MMO 2019 , Macedonia , geometry , circumcircle

In an acute-angled triangle $ABC$, point $M$ is the midpoint of side $BC$ and the centers of the $M$- excircles of triangles $AMB$ and $AMC$ are $D$ and $E$, respectively. The circumcircle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. The circumcircle of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF\hspace{0.25mm} = \hspace{0.25mm} CG$ .

2007 Junior Macedonian Mathematical Olympiad, 2

Tags: Junior , Macedonia , JMMO , 2007 , geometry , parallelogram

Let $ABCD$ be a parallelogram and let $E$ be a point on the side $AD$, such that $\frac{AE}{ED} = m$. Let $F$ be a point on $CE$, such that $BF \perp CE$, and the point $G$ is symmetrical to $F$ with respect to $AB$. If point $A$ is the circumcenter of triangle $BFG$, find the value of $m$.

2016 Macedonia JBMO TST, 1

Tags: JMMO , Junior , Macedonia , 2016 , number theory

Solve the following equation in the set of integers $x_{1}^4 + x_{2}^4 +...+ x_{14}^4=2016^3 - 1$.

2007 Junior Macedonian Mathematical Olympiad, 5

Tags: Junior , JMMO , Macedonia , 2007 , geometry , combinatorics

We are given an arbitrary $\bigtriangleup ABC$. a) Can we dissect $\bigtriangleup ABC$ in $4$ pieces, from which we can make two triangle similar to $\bigtriangleup ABC$ (each piece can be used only once)? Justify your answer! b) Is it possible that for every positive integer $n \ge 2$ , we are able to dissect $\bigtriangleup ABC$ in $2n$ pieces, from which we can make two triangles similar to $\bigtriangleup ABC$ (each piece can be used only once)? Justify your answer!

2019 Macedonia Junior BMO TST, 1

Tags: JMMO , Junior , Macedonia , number theory , prime numbers

Determine all prime numbers of the form $1 + 2^p + 3^p +...+ p^p$ where $p$ is a prime number.

2016 Macedonia JBMO TST, 3

Tags: JMMO , Junior , 2016 , Macedonia , combinatorics

We are given a $4\times4$ square, consisting of $16$ squares with side length of $1$. Every $1\times1$ square inside the square has a non-negative integer entry such that the sum of any five squares that can be covered with the figures down below (the figures can be moved and rotated) equals $5$. What is the greatest number of different numbers that can be used to cover the square?

2019 Macedonia Junior BMO TST, 5

Tags: JMMO , 2019 , Junior , Macedonia , number theory , prime numbers

Let $p_{1}$, $p_{2}$, ..., $p_{k}$ be different prime numbers. Determine the number of positive integers of the form $p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}...p_{k}^{\alpha_{k}}$, $\alpha_{i}$ $\in$ $\mathbb{N}$ for which $\alpha_{1} \alpha_{2}...\alpha_{k}=p_{1}p_{2}...p_{k}$.

2007 Junior Macedonian Mathematical Olympiad, 3

Tags: Junior , JMMO , 2007 , Macedonia , algebra , Inequality

Let $a$, $b$, $c$ be real numbers such that $0 < a \le b \le c$. Prove that $(a + 3b)(b + 4c)(c + 2a) \ge 60abc$. When does equality hold?

2019 Macedonia National Olympiad, 4

Tags: Macedonia , functions , number theory , 2019

Determine all functions $f: \mathbb {N} \to \mathbb {N}$ such that $n!\hspace{1mm} +\hspace{1mm} f(m)!\hspace{1mm} |\hspace{1mm} f(n)!\hspace{1mm} +\hspace{1mm} f(m!)$, for all $m$, $n$ $\in$ $\mathbb{N}$.

2016 Macedonia JBMO TST, 2

Tags: 2016 , JMMO , Macedonia , Junior , geometry , parallelogram

Let $ABCD$ be a parallelogram and let $E$, $F$, $G$, and $H$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$, respectively. If $BH \cap AC = I$, $BD \cap EC = J$, $AC \cap DF = K$, and $AG \cap BD = L$, prove that the quadrilateral $IJKL$ is a parallelogram.

2018 Macedonia JBMO TST, 5

Tags: JMMO , Junior , Macedonia , 2018 , combinatorics

A regular $2018$-gon is inscribed in a circle. The numbers $1, 2, ..., 2018$ are arranged on the vertices of the $2018$-gon, with each vertex having one number on it, such that the sum of any $2$ neighboring numbers ($2$ numbers are neighboring if the vertices they are on lie on a side of the polygon) equals the sum of the $2$ numbers that are on the antipodes of those $2$ vertices (with respect to the given circle). Determine the number of different arrangements of the numbers. (Two arrangements are identical if you can get from one of them to the other by rotating around the center of the circle).

2018 Macedonia JBMO TST, 3

Tags: JMMO , 2018 , Macedonia , algebra , Inequality , inequalities

Let $x$, $y$, and $z$ be positive real numbers such that $x + y + z = 1$. Prove that $\frac{(x+y)^3}{z} + \frac{(y+z)^3}{x} + \frac{(z+x)^3}{y} + 9xyz \ge 9(xy + yz + zx)$. When does equality hold?

2018 Macedonia JBMO TST, 1

Tags: JMMO , Junior , 2018 , Macedonia , number theory

Determine all positive integers $n>2$, such that $n = a^3 + b^3$, where $a$ is the smallest positive divisor of $n$ greater than $1$ and $b$ is an arbitrary positive divisor of $n$.

2019 Macedonia Junior BMO TST, 2

Tags: JMMO , Macedonia , Junior , 2019 , geometry , circumcircle

Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $A$ and $B$. Let $t_{1}$ and $t_{2}$ be the tangents to $\omega_{1}$ and $\omega_{2}$, respectively, at point $A$. Let the second intersection of $\omega_{1}$ and $t_{2}$ be $C$, and let the second intersection of $\omega_{2}$ and $t_{1}$ be $D$. Points $P$ and $E$ lie on the ray $AB$, such that $B$ lies between $A$ and $P$, $P$ lies between $A$ and $E$, and $AE = 2 \cdot AP$. The circumcircle to $\bigtriangleup BCE$ intersects $t_{2}$ again at point $Q$, whereas the circumcircle to $\bigtriangleup BDE$ intersects $t_{1}$ again at point $R$. Prove that points $P$, $Q$, and $R$ are collinear.

2019 Macedonia National Olympiad, 2

Tags: Macedonia , MMO 2019 , number theory

Let $n$ be a positive integer. If $r\hspace{0.25mm} \equiv \hspace{1mm} n\hspace{1mm} (mod\hspace{1mm} 2)$ and $r\hspace{0.10mm} \in \hspace{0.10mm} \{ 0,\hspace{0.10mm} 1 \} $, find the number of integer solutions to the system of equations $\left\{\begin{array}{l}x+y+z = r \\ \mid x \mid + \mid y \mid + \mid z \mid = n \end{array}\right.$