Found problems: 85335
2022 Dutch BxMO TST, 2
Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude from $A$. The circle with centre $A$ passing through $D$ intersects the circumcircle of triangle $ABC$ in $X$ and $Y$ , in such a way that the order of the points on this circumcircle is: $A, X, B, C, Y$ . Show that $\angle BXD = \angle CYD$.
2001 Poland - Second Round, 1
Find all integers $n\ge 3$ for which the following statement is true:
Any arithmetic progression $a_1,\ldots ,a_n$ with $n$ terms for which $a_1+2a_2+\ldots+na_n$ is rational contains at least one rational term.
2019 Canada National Olympiad, 2
Let $a,b$ be positive integers such that $a+b^3$ is divisible by $a^2+3ab+3b^2-1$. Prove that $a^2+3ab+3b^2-1$ is divisible by the cube of an integer greater than 1.
2008 Abels Math Contest (Norwegian MO) Final, 2a
We wish to lay down boards on a floor with width $B$ in the direction across the boards. We have $n$ boards of width $b$, and $B/b$ is an integer, and $nb \le B$. There are enough boards to cover the floor, but the boards may have different lengths. Show that we can cut the boards in such a way that every board length on the floor has at most one join where two boards meet end to end.
[img]https://cdn.artofproblemsolving.com/attachments/f/f/24ce8ae05d85fd522da0e18c0bb8017ca3c8e8.png[/img]
2023 Thailand TST, 3
For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?
2012 India National Olympiad, 3
Define a sequence $<f_0 (x), f_1 (x), f_2 (x), \dots>$ of functions by $$f_0 (x) = 1$$ $$f_1(x)=x$$ $$(f_n(x))^2 - 1 = f_{n+1}(x) f_{n-1}(x)$$ for $n \ge 1$. Prove that each $f_n (x)$ is a polynomial with integer coefficients.
2000 Poland - Second Round, 3
On fields of $n \times n$ chessboard $n^2$ different integers have been arranged, one in each field. In each column, field with biggest number was colored in red. Set of $n$ fields of chessboard name [i]admissible[/i], if no two of that fields aren't in the same row and aren't in the same column. From all admissible sets, set with biggest sum of numbers in it's fields has been chosen. Prove that red field is in this set.
2006 AIME Problems, 9
Circles $\mathcal{C}_1$, $\mathcal{C}_2$, and $\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common internal tangent to $\mathcal{C}_2$ and $\mathcal{C}_3$ and has a negative slope. Given that lines $t_1$ and $t_2$ intersect at $(x,y)$, and that $x=p-q\sqrt{r}$, where $p$, $q$, and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.
2008 Vietnam National Olympiad, 2
Given a triangle with acute angle $ \angle BEC,$ let $ E$ be the midpoint of $ AB.$ Point $ M$ is chosen on the opposite ray of $ EC$ such that $ \angle BME \equal{} \angle ECA.$ Denote by $ \theta$ the measure of angle $ \angle BEC.$ Evaluate $ \frac{MC}{AB}$ in terms of $ \theta.$
2003 All-Russian Olympiad, 3
On a line are given $2k -1$ white segments and $2k -1$ black ones. Assume that each white segment intersects at least $k$ black segments, and each black segment intersects at least $k$ white ones. Prove that there are a black segment intersecting all the white ones, and a white segment intersecting all the black ones.
2020 Nigerian Senior MO Round 2, 1
Let $k$ be a real number. Define on the set of reals the operation $x*y$= $\frac{xy}{x+y+k}$ whenever $x+y$ does not equal $-k$. Let $x_1<x_2<x_3<x_4$ be the roots of $t^4=27(t^2+t+1)$.suppose that $[(x_1*x_2)*x_3]*x_4=1$. Find all possible values of $k$
2018 Purple Comet Problems, 26
Let $a, b$, and $c$ be real numbers. Let $u = a^2 + b^2 + c^2$ and $v = 2ab + 2bc + 2ca$. Suppose $2018u = 1001v + 1024$. Find the maximum possible value of $35a - 28b - 3c$.
2001 China Second Round Olympiad, 3
An $m\times n(m,n\in \mathbb{N}^*)$ rectangle is divided into some smaller squares. The sides of each square are all parallel to the corresponding sides of the rectangle, and the length of each side is integer. Determine the minimum of the sum of the sides of these squares.
2009 Junior Balkan Team Selection Test, 3
On each field of the board $ n\times n$ there is one figure, where $n\ge 2$. In one move we move every figure on one of its diagonally adjacent fields. After one move on one field there can be more than one figure. Find the least number of fields on which there can be all figures after some number of moves.
2013 China Team Selection Test, 3
Let $n>1$ be an integer and let $a_0,a_1,\ldots,a_n$ be non-negative real numbers. Definite $S_k=\sum_{i\equal{}0}^k \binom{k}{i}a_i$ for $k=0,1,\ldots,n$. Prove that\[\frac{1}{n} \sum_{k\equal{}0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k\equal{}0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.\]
2023 Stanford Mathematics Tournament, 1
There exists a unique real value of $x$ such that
\[(x+\sqrt{x})^2=16.\]
Compute $x$.
Durer Math Competition CD Finals - geometry, 2019.C5
$A, B, C, D$ are four distinct points such that triangles $ABC$ and $CBD$ are both equilateral. Find as many circles as you can, which are equidistant from the four points. How can these circles be constructed?
[i]Remark: The distance between a point $P$ and a circle c is measured as follows: we join $P$ and the centre of the circle with a straight line, and measure how much we need to travel along thisline (starting from $P$) to hit the perimeter of the circle. If $P$ is an internal point of the circle, the distance is the length of the shorter such segment. The distance between a circle and itscentre is the radius of the circle.[/i]
2017 Harvard-MIT Mathematics Tournament, 7
[b]O[/b]n a blackboard a stranger writes the values of $s_7(n)^2$ for $n=0,1,...,7^{20}-1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board.
2018 ASDAN Math Tournament, 3
In $\vartriangle ABC$, $AC > AB$. $B$ is reflected across $\overline{AC}$ to a point $D$, and $C$ is reflected across $\overline{AD}$ to a point $E$. Suppose that $AC = 6\sqrt3 + 6$, $BC = 6$, and $\overline{BC} \parallel \overline{AE}$. Compute $AB$.
Russian TST 2016, P1
Several people came to the congress, each of whom has a certain number of tattoos on both hands. There are $n{}$ types of tattoos, and each of the $n{}$ types is found on the hands of at least $k{}$ people. For which pairs $(n, k)$ is it always possible for each participant to raise one of their hands so that all $n{}$ types of tattoos are present on the raised hands?
2019 Bangladesh Mathematical Olympiad, 9
Let $ABCD$ is a convex quadrilateral.The internal angle bisectors of $\angle {BAC}$ and $\angle {BDC}$ meets at $P$.$\angle {APB}=\angle {CPD}$.Prove that $AB+BD=AC+CD$.
2023 Bulgaria National Olympiad, 5
For every positive integer $n$ determine the least possible value of the expression
\[|x_{1}|+|x_{1}-x_{2}|+|x_{1}+x_{2}-x_{3}|+\dots +|x_{1}+x_{2}+\dots +x_{n-1}-x_{n}|\]
given that $x_{1}, x_{2}, \dots , x_{n}$ are real numbers satisfying $|x_{1}|+|x_{2}|+\dots+|x_{n}| = 1$.
1995 Tuymaada Olympiad, 2
Let $x_1=a, x_2=a^{x_1}, ..., x_n=a^{x_{n-1}}$ where $a>1$. What is the maximum value of $a$ for which lim exists $\lim_{n\to \infty} x_n$ and what is this limit?
2023 Ukraine National Mathematical Olympiad, 9.7
You are given $n \ge 2$ distinct positive integers. Let's call a pair of these integers [i]elegant[/i] if their sum is an integer power of $2$. For every $n$ find the largest possible number of elegant pairs.
[i]Proposed by Oleksiy Masalitin[/i]
2002 Tournament Of Towns, 7
[list]
[*] A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this?
[*] Previous problem for the grid of $7\times 7$ lattice.[/list]