Found problems: 16
2025 Al-Khwarizmi IJMO, 8
There are $100$ cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from $1$ to $100$. In a move, Madina is allowed to choose any $3$ cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?
[i]Shubin Yakov, Russia[/i]
2025 Al-Khwarizmi IJMO, 5
Sevara writes in red $8$ distinct positive integers and then writes in blue the $28$ sums of each two red numbers. At most how many of the blue numbers can be prime?
[i]Marin Hristov, Bulgaria[/i]
2025 Al-Khwarizmi IJMO, 6
Let $a,b,c$ be real numbers such that \[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\] Find the smallest possible value of $a^2 + b^2 + c^2$.
[i]Binh Luan and Nhan Xet, Vietnam[/i]
2025 Al-Khwarizmi IJMO, 3
On a circle are arranged $100$ baskets, each containing at least one candy. The total number of candies is $780$. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad takes all the candies from $9$ consecutive non-empty baskets, while Sevinch takes all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least $700$ candies, regardless of the initial distribution of candies and Sevinch's actions.
[i] Shubin Yakov, Russia [/i]
2024 Al-Khwarizmi IJMO, 1
We have triangle $ABC$ with area $S$. In one step we can move only one vertex at a time so that the area of the triangle during movement remains constant. Prove that we can move this triangle into any other arbitrary triangle $DEF$ with area $S$.
[i]Proposed by Marek Maruin, Slovakia[/i]
2025 Al-Khwarizmi IJMO, 7
Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$, such that $CD$ is not a diameter of its circumcircle. The lines $AD$ and $BC$ intersect at point $P$, so that $A$ lies between $D$ and $P$, and $B$ lies between $C$ and $P$. Suppose triangle $PCD$ is acute and let $H$ be its orthocenter. The points $E$ and $F$ on the lines $BC$ and $AD$, respectively, are such that $BD \parallel HE$ and $AC\parallel HF$. The line through $E$, perpendicular to $BC$, intersects $AD$ at $L$, and the line through $F$, perpendicular to $AD$, intersects $BC$ at $K$. Prove that the points $K$, $L$, $O$ are collinear.
[i]Amir Parsa Hosseini Nayeri, Iran[/i]
2024 Al-Khwarizmi IJMO, 7
Two circles with centers $O_{1}$ and $O_{2}$ intersect at $P$ and $Q$. Let $\omega$ be the circumcircle of the triangle $P O_{1} O_{2}$; the circle $\omega$ intersect the circles centered at $O_{1}$ and $O_{2}$ at points $A$ and $B$, respectively. The point $Q$ is inside triangle $P A B$ and $P Q$ intersects $\omega$ at $M$. The point $E$ on $\omega$ is such that $P Q=Q E$. Let $M E$ and $A B$ meet at $L$, prove that $\angle Q L A=\angle M L A$.
[i]Proposed by Amir Parsa Hoseini Nayeri, Iran[/i]
2024 Al-Khwarizmi IJMO, 6
Let $a, b, c$ be distinct real numbers such that $a+b+c=0$ and $$
a^{2}-b=b^{2}-c=c^{2}-a.
$$
Evaluate all the possible values of $a b+a c+b c$.
[i]Proposed by Nguyen Anh Vu, Vietnam[/i]
2024 Al-Khwarizmi IJMO, 5
At a party, every guest is a friend of exactly fourteen other guests (not including him or her). Every two friends have exactly six other attending friends in common, whereas every pair of non-friends has only two friends in common. How many guests are at the party? Please explain your answer with proof.
[i]Proposed by Alexander Slavik, Czech Republic[/i]
2025 Al-Khwarizmi IJMO, 2
Let $ABCD$ be a convex quadrilateral with \[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\] The line through \(C\), parallel to \(AD\), intersects the external angle bisector of \(\angle ABC\) at point \(T\). Prove that the angles $\angle ATB$, $\angle TBC$, $\angle BCD$, $\angle CDA$, $\angle DAT$ can be divided into two groups, so that the angles in each group have a sum of $270^{\circ}$.
[i] Miroslav Marinov, Bulgaria [/i]
2024 Al-Khwarizmi IJMO, 8
Three positive integers are written on the board. In every minute, instead of the numbers $a, b, c$, Elbek writes $a+\gcd(b,c), b+\gcd(a,c), c+\gcd(a,b)$ . Prove that there will be two numbers on the board after some minutes, such that one is divisible by the other.
Note. $\gcd(x,y)$ - Greatest common divisor of numbers $x$ and $y$
[i]Proposed by Sergey Berlov, Russia[/i]
2025 Al-Khwarizmi IJMO, 1
Determine the largest integer $c$ for which the following statement holds: there exists at least one triple $(x,y,z)$ of integers such that
\begin{align*} x^2 + 4(y + z) = y^2 + 4(z + x) = z^2 + 4(x + y) = c \end{align*}
and all triples $(x,y,z)$ of real numbers, satisfying the equations, are such that $x,y,z$ are integers.
[i]Marek Maruin, Slovakia [/i]
2025 Al-Khwarizmi IJMO, 4
For two sets of integers $X$ and $Y$ we define $X\cdot Y$ as the set of all products of an element of $X$ and an element of $Y$. For example, if $X=\{1, 2, 4\}$ and $Y=\{3, 4, 6\}$ then $X\cdot Y=\{3, 4, 6, 8, 12, 16, 24\}.$ We call a set $S$ of positive integers [i] good [/i] if there do not exist sets $A,B$ of positive integers, each with at least two elements and such that the sets $A\cdot B$ and $S$ are the same. Prove that the set of perfect powers greater than or equal to $2025$ is good.
([i]In any of the sets $A$, $B$, $A\cdot B$ no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form $n^k$, where $n>1$ and $k > 1$ are integers.[/i])
[i] Lajos Hajdu and Andras Sarkozy, Hungary [/i]
2024 Al-Khwarizmi IJMO, 3
Find all $x, y, z \in \left (0, \frac{1}{2}\right )$ such that
$$
\begin{cases}
(3 x^{2}+y^{2}) \sqrt{1-4 z^{2}} \geq z; \\
(3 y^{2}+z^{2}) \sqrt{1-4 x^{2}} \geq x; \\
(3 z^{2}+x^{2}) \sqrt{1-4 y^{2}} \geq y.
\end{cases}
$$
[i]Proposed by Ngo Van Trang, Vietnam[/i]
2024 Al-Khwarizmi IJMO, 4
We call a permutation of the set of real numbers $\{a_1,\cdots,a_n\}$, $n\in\mathbb{N}$ [i]average increasing[/i] if the arithmetic mean of its first $k$ elements for $k=1,\cdots ,n$ form a strictly increasing sequence.
1) Depending on $n$, determine the smallest number that can be the last term of some average increasing permutation of the numbers $\{1,\cdots,n\}$;
2) Depending on $n$, determine the lowest position (in some general order) that the number $n$ can be achieved in some average increasing permutation of the numbers $\{1,\cdots,n\}.$
[i] Proposed by David Hruska, Czech Republic[/i]
2024 Al-Khwarizmi IJMO, 2
For how many $x \in \{1,2,3,\dots, 2024\}$ is it possible that [i]Bekhzod[/i] summed $2024$ non-negative consecutive integers, [i]Ozod[/i] summed $2024+x$ non-negative consecutive integers and they got the same result?
[i]Proposed by Marek Maruin, Slovakia[/i]