Found problems: 85335
1968 Polish MO Finals, 6
Consider a set of $n > 3$ points in the plane, no three of which are collinear, and a natural number $k < n$. Prove the following statements:
(a) If $k \le \frac{n}{2}$, then each point can be connected with at least k other points by segments so that no three segments form a triangle.
(b) If $k \ge \frac{n}{2}$, and each point is connected with at least k other points by segments, then some three segments form a triangle.
1986 Traian Lălescu, 2.2
We know that the function $ f: \left[ 0,\frac{\pi }{2}\right]\longrightarrow [a,b], f(x)=\sqrt[n]{\cos x } +\sqrt[n]{\sin x} , $ is surjective for a given natural number $ n\ge 2. $ Determine the numbers $ a,b, $ and the monotony of $ f. $
2024 Ecuador NMO (OMEC), 1
Find all real solutions:
$$\begin{cases}a^3=2024bc \\ b^3=2024cd \\ c^3=2024da \\ d^3=2024ab \end{cases}$$
2017 Hanoi Open Mathematics Competitions, 12
Does there exist a sequence of $2017$ consecutive integers which contains exactly $17$ primes?
2024 Miklos Schweitzer, 8
Prove that for any finite bipartite planar graph, a circle can be assigned to each vertex so that all circles are coplanar, the circles assigned to any two adjacent vertices are tangent to one another, while the circles assigned to any two distinct, non-adjacent vertices intersect in two points.
2025 Euler Olympiad, Round 2, 5
We are given an infinite row of cells extending infinitely in both directions. Some cells contain one or more stones. The total number of stones is finite. At each move, the player performs one of the following three operations:
[b]1. [/b]Take three stones from some cell, and add one stone to the cells located one cell to the left and one cell to the right, each skipping one cell in between.
[b]2. [/b]Take two stones from some cell, and add one stone to the cell one cell to the left, skipping one cell and one stone to the adjacent cell to the right.
[b]3.[/b] Take one stone from each of two adjacent cells, and add one stone to the cell to the right of these two cells.
The process ends when no moves are possible. Prove that the process always terminates and the final distribution of stones does not depend on the choices of moves made by the player.
[img]https://i.imgur.com/IjcIDOa.png[/img]
[i]Proposed by Luka Tsulaia, Georgia[/i]
2013 Today's Calculation Of Integral, 867
Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$
2003 Flanders Math Olympiad, 2
Two circles $C_1$ and $C_2$ intersect at $S$.
The tangent in $S$ to $C_1$ intersects $C_2$ in $A$ different from $S$.
The tangent in $S$ to $C_2$ intersects $C_1$ in $B$ different from $S$.
Another circle $C_3$ goes through $A, B, S$.
The tangent in $S$ to $C_3$ intersects $C_1$ in $P$ different from $S$ and $C_2$ in $Q$ different from $S$.
Prove that the distance $PS$ is equal to the distance $QS$.
1988 Romania Team Selection Test, 16
The finite sets $A_1$, $A_2$, $\ldots$, $A_n$ are given and we denote by $d(n)$ the number of elements which appear exactly in an odd number of sets chosen from $A_1$, $A_2$, $\ldots$, $A_n$. Prove that for any $k$, $1\leq k\leq n$ the number \[{ d(n) - \sum\limits^n_{i=1} |A_i| + 2\sum\limits_{ i<j} |A_i \cap A_j | - \cdots + (-1)^k2^{k-1} \sum\limits_{i_1 <i_2 <\cdots < i_k} | A_{i_1} \cap A_{i_2} \cap \cdots \cap A_{i_k}}| \] is divisible by $2^k$.
[i]Ioan Tomescu, Dragos Popescu[/i]
2014 IMO Shortlist, G6
Let $ABC$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $AC$ and $AB$, respectively, and let $M$ be the midpoint of $EF$ . Let the perpendicular bisector of $EF$ intersect the line $BC$ at $K$, and let the perpendicular bisector of $MK$ intersect the lines $AC$ and $AB$ at $S$ and $T$ , respectively. We call the pair $(E, F )$ $\textit{interesting}$, if the quadrilateral $KSAT$ is cyclic.
Suppose that the pairs $(E_1 , F_1 )$ and $(E_2 , F_2 )$ are interesting. Prove that $\displaystyle\frac{E_1 E_2}{AB}=\frac{F_1 F_2}{AC}$
[i]Proposed by Ali Zamani, Iran[/i]
2016 ELMO Problems, 4
Big Bird has a polynomial $P$ with integer coefficients such that $n$ divides $P(2^n)$ for every positive integer $n$. Prove that Big Bird's polynomial must be the zero polynomial.
[i]Ashwin Sah[/i]
2009 Brazil National Olympiad, 1
Emerald writes $ 2009^2$ integers in a $ 2009\times 2009$ table, one number in each entry of the table. She sums all the numbers in each row and in each column, obtaining $ 4018$ sums. She notices that all sums are distinct. Is it possible that all such sums are perfect squares?
Kyiv City MO Juniors Round2 2010+ geometry, 2017.8.2
Triangle $ABC$ is right-angled and isosceles with a right angle at the vertex $C$. On rays $CB$ on vertex $B$ is selected point F, on rays $BA$ on vertex $A$ is selected point G so that $AG = BF.$ The ray $GD$ is drawn so that it intersects with ray $AC$ at point $D$ with $\angle FGD = 45^o$. Find $\angle FDG$.
(Bogdan Rublev)
1998 AMC 12/AHSME, 4
Define $[a,b,c]$ to mean $\frac{a+b}{c},$ where $c \neq 0$. What is the value of \[[[60,30,90],[2,1,3],[10,5,15]]?\]
$\text{(A)} \ 0 \qquad \text{(B)} \ 0.5 \qquad \text{(C)} \ 1 \qquad \text{(D)} \ 1.5 \qquad \text{(E)} \ 2$
1992 Putnam, A1
Find all functions $ f : Z\rightarrow Z$ for which we have $ f (0) \equal{} 1$ and $ f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n$, for every natural number $ n$.
2016 India Regional Mathematical Olympiad, 8
At some integer points a polynomial with integer coefficients take values $1, 2$ and $3$. Prove that there exist not more than one integer at which the polynomial is equal to $5$.
KoMaL A Problems 2017/2018, A. 720
We call a positive integer [i]lively[/i] if it has a prime divisor greater than $10^{10^{100}}$. Prove that if $S$ is an infinite set of lively positive integers, then it has an infinite subset $T$ with the property that the sum of the elements in any finite nonempty subset of $T$ is a lively number.
2006 Thailand Mathematical Olympiad, 2
From a point $P$ outside a circle, two tangents are drawn touching the circle at points $A$ and $C$. Let $B$ be a point on segment $AC$, and let segment $PB$ intersect the circle at point $Q$. The angle bisector of $\angle AQC$ intersects segment $AC$ at $R$. Show that $$\frac{AB}{BC} =\left(\frac{ AR}{RC}\right)^2$$
1996 Cono Sur Olympiad, 1
In the following figure, the largest square is divided into two squares and three rectangles, as shown:
The area of each smaller square is equal to $a$ and the area of each small rectangle is equal to $b$. If $a+b=24$ and the root square of $a$ is a natural number, find all possible values for the area of the largest square.
[img]https://cdn.artofproblemsolving.com/attachments/f/a/0b424d9c293889b24d9f31b1531bed5081081f.png[/img]
2019 Jozsef Wildt International Math Competition, W. 41
For $n \in \mathbb{N}$, consider in $\mathbb{R}^3$ the regular tetrahedron with vertices $O(0, 0, 0)$, $A(n, 9n, 4n)$, $B(9n, 4n, n)$ and $C(4n, n, 9n)$. Show that the number $N$ of points $(x, y, z)$, $[x, y, z \in \mathbb{Z}]$ inside or on the boundary of the tetrahedron $OABC$ is given by$$N=\frac{343n^3}{3}+\frac{35n^2}{2}+\frac{7n}{6}+1$$
2001 Swedish Mathematical Competition, 2
Show that $\sqrt[3]{\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5}$ is rational.
2017 Ukraine Team Selection Test, 1
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]
2019 JBMO Shortlist, C4
We have a group of $n$ kids. For each pair of kids, at least one has sent a message to
the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly
$25 \% $ have sent a message to $A$. How many possible two-digit values of $n$ are there?
[i]Proposed by Bulgaria[/i]
2016 Iran Team Selection Test, 1
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
2018 Bosnia And Herzegovina - Regional Olympiad, 5
Board with dimesions $2018 \times 2018$ is divided in unit cells $1 \times 1$. In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain black chips. If $W$ is number of remaining white chips, and $B$ number of remaining black chips on board and $A=min\{W,B\}$, determine maximum of $A$