Found problems: 19
2023 Azerbaijan Senior NMO, 4
To open the magic chest, one needs to say a magic code of length $n$ consisting of digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$ Each time Griphook tells the chest a code it thinks up, the chest's talkative guardian responds by saying the number of digits in that code that match the magic code. (For example, if the magic code is $0423$ and Griphook says $3442,$ the chest's talkative guard will say $1$). Prove that there exists a number $k$ such that for any natural number $n \geq k,$ Griphook can find the magic code by checking at most $4n-2023$ times, regardless of what the magic code of the box is.
2023 Azerbaijan Senior NMO, 1
The teacher calculates and writes on the board all the numbers $a^b$ that satisfy the condition $n = a\times b$ for the natural number $n.$ Here $a$ and $b$ are natural numbers. Is there a natural number $n$ such that each of the numbers $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ is the last digit of one of the numbers written by the teacher on the board? Justify your opinion.
2024 Azerbaijan Senior NMO, 5
At the beginning of the academic year, the Olympic Center must accept a certain number of talented students for the 2024 different classes it offers. Although the admitted students are given freedom of choice in classes, there are certain rules. So, any student must take at least one class and cannot take all the classes. At the same time, there cannot be a common class that all students take, and any class must be taken by at least one student. As a final addition to the center's rules, for any student and any class that this student did not enroll in (call this type of class A), the number of students in each A must be greater than the number of classes this student enrolled. At least how many students must the center accept for these rules to be valid?
2023 Azerbaijan Senior NMO, 5
The incircle of the acute-angled triangle $ABC$ is tangent to the sides $AB, BC, CA$ at points $C_1, A_1, B_1,$ respectively, and $I$ is the incenter. Let the midpoint of side $BC$ be $M.$ Let $J$ be the foot of the altitude drawn from $M$ to $C_1B_1.$ The tangent drawn from $B$ to the circumcircle of $\triangle BIC$ intersects $IJ$ at $X.$ If the circumcircle of $\triangle AXI$ intersects $AB$ at $Y,$ prove that $BY = BM.$
2023 Azerbaijan Senior NMO, 3
Let $m$ be a positive integer. Find polynomials $P(x)$ with real coefficients such that $$(x-m)P(x+2023) = xP(x)$$
is satisfied for all real numbers $x.$
2023 Azerbaijan Senior NMO, 2
Find all the integer solutions of the equation:
$$\sqrt{x} + \sqrt{y} = \sqrt{x+2023}$$
2025 Azerbaijan Senior NMO, 3
You are given a positive integer $n$. $n^2$ amount of people stand on coordinates $(x;y)$ where $x,y\in\{0;1;2;...;n-1\}$. Every person got a water cup and two people are considered to be neighbour if the distance between them is $1$. At the first minute, the person standing on coordinates $(0;0)$ got $1$ litres of water, and the other $n^2-1$ people's water cup is empty. Every minute, two neighbouring people are chosen that does not have the same amount of water in their water cups, and they equalize the amount of water in their water cups.
Prove that, no matter what, the person standing on the coordinates $(x;y)$ will not have more than $\frac1{x+y+1}$ litres of water.
2021 Azerbaijan Senior NMO, 3
In $\triangle ABC\ T$ is a point lies on the internal angle bisector of $B$. Let $\omega$ be circle with diameter $BT$.
$\omega$ intersects with $BA$ and $BC$ at $P$ and $Q$,respectively. A circle passes through $A$ and tangent to $\omega$ at $P$ intersects with $AC$ again at $X$ . A circle passes through $B$ and tangent to $\omega$ at $Q$ intersects with $AC$ again at $Y$ . Prove that $TX=TY$
2025 Azerbaijan Senior NMO, 2
Find all the positive reals $x,y,z$ satisfying the following equations: $$y=\frac6{(2x-1)^2}$$ $$z=\frac6{(2y-1)^2}$$ $$x=\frac6{(2z-1)^2}$$
2024 Azerbaijan Senior NMO, 1
Numbers from 1 to 100 are written on the board in ascending order to make the following large number: 12345678910111213...9899100. Then 100 digits of this number are deleted to get the largest possible number. Find the first 10 digits of the number after deletion.
2024 Azerbaijan Senior NMO, 4
Let $P(x)$ be a polynomial with the coefficients being $0$ or $1$ and degree $2023$. If $P(0)=1$, then prove that every real root of this polynomial is less than $\frac{1-\sqrt{5}}{2}$.
2025 Azerbaijan Senior NMO, 5
A 9-digit number $N$ is given, whose digits are non-zero and all different.The sums of all consecutive three-digit segments in the decimal representation of number $N$ are calculated and arranged in increasing order.Is it possible to obtain the following sequences as a result of this operation?
$\text{a)}$ $11,15,16,18,19,21,22$
$\text{b)}$ $11,15,16,18,19,21,23$
2025 Azerbaijan Senior NMO, 4
Prove that for any $p>2$ prime number, there exists only one positive number $n$ that makes the equation $n^2-np$ a perfect square of a positive integer
2024 Azerbaijan Senior NMO, 2
Let $d(n)$ denote the number of positive divisors of the natural number $n$. Find all the natural numbers $n$ such that $$d(n) = \frac{n}{5}$$.
2024 Azerbaijan National Mathematical Olympiad, 5
In a scalene triangle $ABC$, the points $E$ and $F$ are the foot of altitudes drawn from $B$ and $C$, respectively. The points $X$ and $Y$ are the reflections of the vertices $B$ and $C$ to the line $EF$, respectively. Let the circumcircles of the $\triangle ABC$ and $\triangle AEF$ intersect at $T$ for the second time. Show that the four points $A, X, Y, T$ lie on a single circle.
2024 Azerbaijan Senior NMO, 3
In a scalene triangle $ABC$, the points $E$ and $F$ are the foot of altitudes drawn from $B$ and $C$, respectively. The points $X$ and $Y$ are the reflections of the vertices $B$ and $C$ to the line $EF$, respectively. Let the circumcircles of the $\triangle ABC$ and $\triangle AEF$ intersect at $T$ for the second time. Show that the four points $A, X, Y, T$ lie on a single circle.
2025 Azerbaijan Senior NMO, 1
Alice creates a sequence: For the first $2025$ terms of this sequence, she writes a random permutation of $\{1;2;3;...;2025\}$. To define the following terms, she does the following: She takes the last $2025$ terms of the sequence, and takes its median. How many values could this sequence's $3000$'th term could get?
(Note: To find the median of $2025$ numbers, you write them in an increasing order,and take the number in the middle)
2021 Azerbaijan Junior NMO, 5
In $\triangle ABC\ T$ is a point lies on the internal angle bisector of $B$. Let $\omega$ be circle with diameter $BT$.
$\omega$ intersects with $BA$ and $BC$ at $P$ and $Q$,respectively. A circle passes through $A$ and tangent to $\omega$ at $P$ intersects with $AC$ again at $X$ . A circle passes through $B$ and tangent to $\omega$ at $Q$ intersects with $AC$ again at $Y$ . Prove that $TX=TY$
2025 Azerbaijan Senior NMO, 6
In an acute triangle $ABC$ with $AB<AC$, the foot of altitudes from $A,B,C$ to the sides $BC,CA,AB$ are $D,E,F$, respectively. $H$ is the orthocenter. $M$ is the midpoint of segment $BC$. Lines $MH$ and $EF$ intersect at $K$. Let the tangents drawn to circumcircle $(ABC)$ from $B$ and $C$ intersect at $T$. Prove that $T;D;K$ are colinear