Found problems: 24
2023 Indonesia Regional, 3
Find the maximum value of an integer $B$ such that for every 9 distinct natural number with the sum of $2023$, there must exist a sum of 4 of the number that is greater than or equal to $B$
2002 India Regional Mathematical Olympiad, 2
Solve for real $x$ : \[ (x^2 + x -2 )^3 + (2x^2 - x -1)^3 = 27(x^2 -1 )^3. \]
2024 Indonesia Regional, 1
Given a real number $C\leqslant 2$. Prove that for every positive real number $x,y$ with $xy=1$, the following inequality holds:
\[ \sqrt{\frac{x^2+y^2}{2}} + \frac{C}{x+y} \geqslant 1 + \frac{C}{2}.\]
[i]Proposed by Fajar Yuliawan, Indonesia[/i]
2019 India Regional Mathematical Olympiad, 5
In an acute angled triangle $ABC$, let $H$ be the orthocenter, and let $D,E,F$ be the feet of altitudes from $A,B,C$ to the opposite sides, respectively. Let $L,M,N$ be the midpoints of the segments $AH, EF, BC$ respectively. Let $X,Y$ be the feet of altitudes from $L,N$ on to the line $DF$ respectively. Prove that $XM$ is perpendicular to $MY$.
2019 India Regional Mathematical Olympiad, 1
Suppose $x$ is a non zero real number such that both $x^5$ and $20x+\frac{19}{x}$ are rational numbers. Prove that $x$ is a rational number.
2024 Indonesia Regional, 2
Given an $n \times n$ board which is divided into $n^2$ squares of size $1 \times 1$, all of which are white. Then, Aqua selects several squares from this board and colors them black. Ruby then places exactly one $1\times 2$ domino on the board, so that the domino covers exactly two squares on the board. Ruby can rotate the domino into a $2\times 1$ domino.
After Aqua colors, it turns out there are exactly $2024$ ways for Ruby to place a domino on the board so that it covers exactly $1$ black square and $1$ white square.
Determine the smallest possible value of $n$ so that Aqua and Ruby can do this.
[i]Proposed by Muhammad Afifurrahman, Indonesia [/i]
2024 Indonesia Regional, 4
Find the number of positive integer pairs $1\leqslant a,b \leqslant 2027$ that satisfy
\[ 2027 \mid a^6+b^5+b^2.\]
(Note: For integers $a$ and $b$, the notation $a \mid b$ means that there is an integer $c$ such that $ac=b$.)
[i]Proposed by Valentio Iverson, Indonesia[/i]
2008 India Regional Mathematical Olympiad, 4
Find the number of all $ 6$-digit natural numbers such that the sum of their digits is $ 10$ and each of the digits $ 0,1,2,3$ occurs at least once in them.
[14 points out of 100 for the 6 problems]
2022 Indonesia Regional, 4
Suppose $ABC$ is a triangle with circumcenter $O$. Point $D$ is the reflection of $A$ with respect to $BC$. Suppose $\ell$ is the line which is parallel to $BC$ and passes through $O$. The line through $B$ and parallel to $CD$ meets $\ell$ at $B_1$. Lines $CB_1$ and $BD$ intersect at point $B_2$. The line through $C$ parallel to $BD$ and $\ell$ meet at $C_1$. Finally, $BC_1$ and $CD$ intersects at point $C_2$. Prove that points $A, B_2, C_2, D$ lie on a circle.
2024 Indonesia Regional, 3
Given a triangle $ABC$, points $X,Y,$ and $Z$ are the midpoints of $BC,CA,$ and $AB$ respectively. The perpendicular bisector of $AB$ intersects line $XY$ and line $AC$ at $Z_1$ and $Z_2$ respectively. The perpendicular bisector of $AC$ intersects line $XZ$ and line $AB$ at $Y_1$ and $Y_2$ respectively. Let $K$ be a point such that $KZ_1 = KZ_2$ and $KY_1 = KY_2$. Prove that $KB=KC$.
2023 Indonesia Regional, 5
Given $\triangle ABC$ and points $D$ and $E$ at the line $BC$, furthermore there are points $X$ and $Y$ inside $\triangle ABC$. Let $P$ be the intersection of line $AD$ and $XE$, and $Q$ be the intersection of line $AE$ and $YD$.
If there exist a circle that passes through $X, Y, D, E$, and
$$\angle BXE + \angle BCA = \angle CYD + \angle CBA = 180^{\circ}$$
Prove that the line $BP$, $CQ$, and the perpendicular bisector of $BC$ intersect at one point.
2019 India Regional Mathematical Olympiad, 2
Let $ABC$ be a triangle with circumcircle $\Omega$ and let $G$ be the centroid of triangle $ABC$. Extend $AG, BG$ and $CG$ to meet the circle $\Omega$ again in $A_1, B_1$ and $C_1$. Suppose $\angle BAC = \angle A_1 B_1 C_1, \angle ABC = \angle A_1 C_1 B_1$ and $ \angle ACB = B_1 A_1 C_1$. Prove that $ABC$ and $A_1 B_1 C_1$ are equilateral triangles.
2019 India Regional Mathematical Olympiad, 4
Consider the following $3\times 2$ array formed by using the numbers $1,2,3,4,5,6$,
$$\begin{pmatrix} a_{11}& a_{12}\\a_{21}& a_{22}\\ a_{31}& a_{32}\end{pmatrix}=\begin{pmatrix}1& 6\\2& 5\\ 3& 4\end{pmatrix}.$$
Observe that all row sums are equal, but the sum of the square of the squares is not the same for each row. Extend the above array to a $3\times k$ array $(a_{ij})_{3\times k}$ for a suitable $k$, adding more columns, using the numbers $7,8,9,\dots ,3k$ such that $$\sum_{j=1}^k a_{1j}=\sum_{j=1}^k a_{2j}=\sum_{j=1}^k a_{3j}~~\text{and}~~\sum_{j=1}^k (a_{1j})^2=\sum_{j=1}^k (a_{2j})^2=\sum_{j=1}^k (a_{3j})^2$$
2022 Indonesia Regional, 1
Let $A$ and $B$ be sets such that there are exactly $144$ sets which are subsets of either $A$ or $B$. Determine the number of elements $A \cup B$ has.
2019 India Regional Mathematical Olympiad, 3
Find all triples of non-negative real numbers $(a,b,c)$ which satisfy the following set of equations
$$a^2+ab=c$$
$$b^2+bc=a$$
$$c^2+ca=b$$
2023 Indonesia Regional, 2
Let $K$ be a positive integer such that there exist a triple of positive integers $(x,y,z)$ such that
\[x^3+Ky , y^3 + Kz, \text{and } z^3 + Kx\]
are all perfect cubes.
(a) Prove that $K \ne 2$ and $K \ne 4$
(b) Find the minimum value of $K$ that satisfies.
[i]Proposed by Muhammad Afifurrahman[/i]
2013 India Regional Mathematical Olympiad, 2
Let $f(x)=x^3+ax^2+bx+c$ and $g(x)=x^3+bx^2+cx+a$, where $a,b,c$ are integers with $c\not=0$. Suppose that the following conditions hold:
[list=a][*]$f(1)=0$,
[*]the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$.[/list]
Find the value of $a^{2013}+b^{2013}+c^{2013}$.
2022 Indonesia Regional, 3
It is known that $x$ and $y$ are reals satisfying
\[ 5x^2 + 4xy + 11y^2 = 3. \]
Without using calculus (differentials/integrals), determine the maximum value of $xy - 2x + 5y$.
2022 Indonesia Regional, 2
(a) Determine a natural number $n$ such that $n(n+2022)+2$ is a perfect square.
[hide=Spoiler]In case you didn't realize, $n=1$ works lol[/hide]
(b) Determine all natural numbers $a$ such that for every natural number $n$, the number $n(n+a)+2$ is never a perfect square.
2023 Indonesia Regional, 1
Let $ABCD$ be a square with side length $43$ and points $X$ and $Y$ lies on sides $AD$ and $BC$ respectively such that the ratio of the area of $ABYX$ to the area of $CDXY$ is $20 : 23$ . Find the maximum possible length of $XY$.
2022 Indonesia Regional, 5
Numbers $1$ to $22$ are written on a board. A "move" is a procedure of picking two numbers $a,b$ on the board such that $b \geq a+2$, then erasing $a$ and $b$ to be replaced with $a+1$ and $b-1$. Determine the maximum possible number of moves that can be done on the board.
1990 India Regional Mathematical Olympiad, 8
If the circumcenter and centroid of a triangle coincide, prove that it must be equilateral.
1997 India Regional Mathematical Olympiad, 4
In a quadrilateral $ABCD$, it is given that $AB$ is parallel to $CD$ and the diagonals $AC$ and $BD$ are perpendicular to each other. Show that (a) $AD \cdot BC \geq AB \cdot CD$ (b) $AD + BC \geq AB + CD.$
2023 Indonesia Regional, 4
Find all irrational real numbers $\alpha$ such that
\[ \alpha^3 - 15 \alpha \text{ and } \alpha^4 - 56 \alpha \]
are both rational numbers.