Found problems: 85335
1997 Flanders Math Olympiad, 4
Thirteen birds arrive and sit down in a plane. It's known that from each 5-tuple of birds, at least four birds sit on a circle. Determine the greatest $M \in \{1, 2, ..., 13\}$ such that from these 13 birds, at least $M$ birds sit on a circle, but not necessarily $M + 1$ birds sit on a circle. (prove that your $M$ is optimal)
1985 All Soviet Union Mathematical Olympiad, 402
Given unbounded strictly increasing sequence $a_1, a_2, ... , a_n, ...$ of positive numbers. Prove that
a) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid:
$$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1$$
b) there exists a number $k_0$ such that for all $k>k_0$ the following inequality is valid:
$$\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1985$$
2012 Dutch BxMO/EGMO TST, 1
Do there exist quadratic polynomials $P(x)$ and $Q(x)$ with real coeffcients such that the polynomial $P(Q(x))$ has precisely the zeros $x = 2, x = 3, x =5$ and $x = 7$?
2016 Estonia Team Selection Test, 10
Let $m$ be an integer, $m \ge 2$. Each student in a school is practising $m$ hobbies the most. Among any $m$ students there exist two students who have a common hobby. Find the smallest number of students for which there must exist a hobby which is practised by at least $3$ students .
2011 Putnam, B3
Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0,$ with $g$ nonzero and continuous at $0.$ If $fg$ and $f/g$ are differentiable at $0,$ must $f$ be differentiable at $0?$
2020 Dutch Mathematical Olympiad, 3
Given is a parallelogram $ABCD$ with $\angle A < 90^o$ and $|AB| < |BC|$. The angular bisector of angle $A$ intersects side $BC$ in $M$ and intersects the extension of $DC$ in $N$. Point $O$ is the centre of the circle through $M, C$, and $N$. Prove that $\angle OBC = \angle ODC$.
[asy]
unitsize (1.2 cm);
pair A, B, C, D, M, N, O;
A = (0,0);
B = (2,0);
D = (1,3);
C = B + D - A;
M = extension(A, incenter(A,B,D), B, C);
N = extension(A, incenter(A,B,D), D, C);
O = circumcenter(C,M,N);
draw(D--A--B--C);
draw(interp(D,N,-0.1)--interp(D,N,1.1));
draw(A--interp(A,N,1.1));
draw(circumcircle(M,C,N));
label("$\circ$", A + (0.45,0.15));
label("$\circ$", A + (0.25,0.35));
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, dir(90));
dot("$D$", D, dir(90));
dot("$M$", M, SE);
dot("$N$", N, dir(90));
dot("$O$", O, SE);
[/asy]
2018 India PRMO, 24
If $N$ is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (in degrees), what is $\frac{N}{100}$?
2008 Bulgarian Autumn Math Competition, Problem 11.4
a) Prove that $\lfloor x\rfloor$ is odd iff $\Big\lfloor 2\{\frac{x}{2}\}\Big\rfloor=1$ ($\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ and $\{x\}=x-\lfloor x\rfloor$).
b) Let $n$ be a natural number. Find the number of [i]square free[/i] numbers $a$, such that $\Big\lfloor\frac{n}{\sqrt{a}}\Big\rfloor$ is odd. (A natural number is [i]square free[/i] if it's not divisible by any square of a prime number).
2022 Latvia Baltic Way TST, P6
The numbers $1,2,3,\ldots ,n$ are written in a row. Two players, Maris and Filips, take turns making moves with Maris starting. A move consists of crossing out a number from the row which has not yet been crossed out. The game ends when there are exactly two uncrossed numbers left in the row. If the two remaining uncrossed numbers are coprime, Maris wins, otherwise Filips is the winner. For each positive integer $n\ge 4$ determine which player can guarantee a win.
1969 IMO Longlists, 65
$(USS 2)$ Prove that for $a > b^2,$ the identity ${\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+\cdots}}}}=\sqrt{a-\frac{3}{4}b^2}-\frac{1}{2}b}$
2016 ASDAN Math Tournament, 3
Compute
$$\int_0^\pi\frac{1-\sin x}{1+\sin x}dx.$$
2018 Online Math Open Problems, 19
Players $1,2,\ldots,10$ are playing a game on Christmas. Santa visits each player's house according to a set of rules:
-Santa first visits player $1$. After visiting player $i$, Santa visits player $i+1$, where player $11$ is the same as player $1$.
-Every time Santa visits someone, he gives them either a present or a piece of coal (but not both).
-The absolute difference between the number of presents and pieces of coal that Santa has given out is at most $3$ at every point in time.
-If Santa has a choice between giving out a present and a piece of coal, he chooses with equal probability.
Let $p$ be the probability that player $1$ gets a present before player $2$ does. If $p=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $100m+n$.
[i]Proposed by Tristan Shin
2024 Kazakhstan National Olympiad, 2
Given an integer $n>1$. The board $n\times n$ is colored white and black in a chess-like manner. We call any non-empty set of different cells of the board as a [i]figure[/i]. We call figures $F_1$ and $F_2$ [i]similar[/i], if $F_1$ can be obtained from $F_2$ by a rotation with respect to the center of the board by an angle multiple of $90^\circ$ and a parallel transfer. (Any figure is similar to itself.) We call a figure $F$ [i]connected[/i] if for any cells $a,b\in F$ there is a sequence of cells $c_1,\ldots,c_m\in F$ such that $c_1 = a$, $c_m = b$, and also $c_i$ and $c_{i+1}$ have a common side for each $1\le i\le m - 1$. Find the largest possible value of $k$ such that for any connected figure $F$ consisting of $k$ cells, there are figures $F_1,F_2$ similar to $F$ such that $F_1$ has more white cells than black cells and $F_2$ has more black cells than white cells in it.
2017 Nordic, 2
Let $a, b, \alpha, \beta$ be real numbers such that $0 \leq a, b \leq 1$, and $0 \leq \alpha, \beta \leq \frac{\pi}{2}$. Show that if \[ ab\cos(\alpha - \beta) \leq \sqrt{(1-a^2)(1-b^2)}, \] then \[ a\cos\alpha + b\sin\beta \leq 1 + ab\sin(\beta - \alpha). \]
2025 Romania National Olympiad, 4
Let $A,B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $A+B=AB+BA$. Prove that:
a) if $n$ is odd, then $\det(AB-BA)=0$;
b) if $\text{tr}(A)\neq \text{tr}(B)$, then $\det(AB-BA)=0$.
2020-2021 OMMC, 10
How many ways are there to arrange the numbers $1$ through $8$ into a $2$ by $4$ grid such that the sum of the numbers in each of the two rows are all multiples of $6,$ and the sum of the numbers in each of the four columns are all multiples of $3$?
1995 Belarus National Olympiad, Problem 1
Mark six points in a plane so that any three of them are vertices of a nondegenerate isosceles triangle.
2021 Stanford Mathematics Tournament, R6
[b]p21[/b]. If $f = \cos(\sin (x))$. Calculate the sum $\sum^{2021}_{n=0} f'' (n \pi)$.
[b]p22.[/b] Find all real values of $A$ that minimize the difference between the local maximum and local minimum of $f(x) = \left(3x^2 - 4\right)\left(x - A + \frac{1}{A}\right)$.
[b]p23.[/b] Bessie is playing a game. She labels a square with vertices labeled $A, B, C, D$ in clockwise order. There are $7$ possible moves: she can rotate her square $90$ degrees about the center, $180$ degrees about the center, $270$ degrees about the center; or she can flip across diagonal $AC$, flip across diagonal $BD$, flip the square horizontally (flip the square so that vertices A and B are switched and vertices $C$ and $D$ are switched), or flip the square vertically (vertices $B$ and $C$ are switched, vertices $A$ and $D$ are switched). In how many ways can Bessie arrive back at the original square for the first time in $3$ moves?
[b]p24.[/b] A positive integer is called [i]happy [/i] if the sum of its digits equals the two-digit integer formed by its two leftmost digits. Find the number of $5$-digit happy integers.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2022 Harvard-MIT Mathematics Tournament, 2
Find, with proof, the maximum positive integer $k$ for which it is possible to color $6k$ cells of $6 \times 6$ grid such that, for any choice of three distinct rows $R_1$, $R_2$, $R_3$ and three distinct columns $C_1$, $C_2$, $C_3$, there exists an uncolored cell $c$ and integers $1 \le i, j \le 3$ so that $c$ lies in $R_i$ and $C_j$
1985 AIME Problems, 10
How many of the first 1000 positive integers can be expressed in the form
\[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \]
where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?
Kyiv City MO 1984-93 - geometry, 1984.7.3
On the extension of the largest side $AC$ of the triangle $ABC$ set aside the segment $CM$ such that $CM = BC$. Prove that the angle $ABM$ is obtuse or right.
2023 JBMO Shortlist, C5
Consider an increasing sequence of real numbers $a_1<a_2<\ldots<a_{2023}$ such that all pairwise sums of the elements in the sequence are different. For such a sequence, denote by $M$ the number of pairs $(a_i,a_j)$ such that $a_i<a_j$ and $a_i+a_j<a_2+a_{2022}$. Find the minimal and the maximal possible value of $M$.
LMT Speed Rounds, 6
Blue rolls a fair $n$-sided die that has sides its numbered with the integers from $1$ to $n$, and then he flips a coin. Blue knows that the coin is weighted to land heads either $\dfrac{1}{3}$ or $\dfrac{2}{3}$ of the time. Given that the probability of both rolling a $7$ and flipping heads is $\dfrac{1}{15}$, find $n$.
[i]Proposed by Jacob Xu[/i]
[hide=Solution][i]Solution[/i]. $\boxed{10}$
The chance of getting any given number is $\dfrac{1}{n}$
, so the probability of getting $7$ and heads is either $\dfrac{1}{n} \cdot \dfrac{1}{3}=\dfrac{1}{3n}$ or $\dfrac{1}{n} \cdot \dfrac{2}{3}=\dfrac{2}{3n}$. We get that either $n = 5$ or $n = 10$, but since rolling a $7$ is possible, only $n = \boxed{10}$ is a solution.[/hide]
1992 IMO, 1
In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.
2022 Iran MO (3rd Round), 1
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for all $x,y,z\in\mathbb{R}^+$
$$f(x+f(y)+f(f(z)))=z+f(y+f(x))$$