Found problems: 85335
Brazil L2 Finals (OBM) - geometry, 2007.1
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be the intersection of straight lines $BO$ and $AC$ and $\omega$ be the circumcircle of triangle $AOP$. Suppose that $BO = AP$ and that the measure of the arc $OP$ in $\omega$, that does not contain $A$, is $40^o$. Determine the measure of the angle $\angle OBC$.
[img]https://3.bp.blogspot.com/-h3UVt-yrJ6A/XqBItXzT70I/AAAAAAAAL2Q/7LVv0gmQWbo1_3rn906fTn6wosY1-nIfwCK4BGAYYCw/s1600/2007%2Bomb%2Bl2.png[/img]
2010 Sharygin Geometry Olympiad, 8
Given is a regular polygon. Volodya wants to mark $k$ points on its perimeter so that any another regular polygon (maybe having a different number of sides) doesn’t contain all marked points on its perimeter. Find the minimal $k$ sufficient for any given polygon.
2022 Harvard-MIT Mathematics Tournament, 6
Let f be a function from $\{1, 2, . . . , 22\}$ to the positive integers such that $mn | f(m) + f(n)$ for all $m, n \in \{1, 2, . . . , 22\}$. If $d$ is the number of positive divisors of $f(20)$, compute the minimum possible value of $d$.
1983 IMO Longlists, 5
Consider the set $\mathbb Q^2$ of points in $\mathbb R^2$, both of whose coordinates are rational.
[b](a)[/b] Prove that the union of segments with vertices from $\mathbb Q^2$ is the entire set $\mathbb R^2$.
[b](b)[/b] Is the convex hull of $\mathbb Q^2$ (i.e., the smallest convex set in $\mathbb R^2$ that contains $\mathbb Q^2$) equal to $\mathbb R^2$ ?
2018 Istmo Centroamericano MO, 4
Let $t$ be an integer. Suppose the equation $$x^2 + (4t - 1) x + 4t^2 = 0$$ has at least one positive integer solution $n$. Show that $n$ is a perfect square.
2013 Greece Team Selection Test, 4
Let $n$ be a positive integer. An equilateral triangle with side $n$ will be denoted by $T_n$ and is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two).
Let also $m$ be a positive integer with $m<n$ and suppose that $T_n$ and $T_m$ can be tiled with "trapezoids".
Prove that, if from $T_n$ we remove a $T_m$ with the same orientation, then the rest can be tiled with "trapezoids".
2012 Saint Petersburg Mathematical Olympiad, 1
$\begin{cases} x^3-ax^2+b^3=0 \\x^3-bx^2+c^3=0 \\ x^3-cx^2+a^3=0 \end{cases}$
Prove that system hasn`t solutions if $a,b,c$ are different.
1958 Miklós Schweitzer, 1
[b]1.[/b] Find the groups every generating system of which contains a basis. (A basis is a set of elements of the group such that the direct product of the cyclic groups generated by them is the group itself.) [b](A. 14)[/b]
1987 IMO Longlists, 9
In the set of $20$ elements $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C, D, J, K, L, U, X, Y , Z\}$ we have made a random sequence of $28$ throws. What is the probability that the sequence $CUBA \ JULY \ 1987$ appears in this order in the sequence already thrown?
2022 Stanford Mathematics Tournament, 8
For all positive integers $m>10^{2022}$, determine the maximum number of real solutions $x>0$ of the equation $mx=\lfloor x^{11/10}\rfloor$.
2007 AMC 12/AHSME, 1
One ticket to a show costs $ \$20$ at full price. Susan buys $ 4$ tickets using a coupon that gives her a $ 25\%$ discount. Pam buys $ 5$ tickets using a coupon that gives her a $ 30\%$ discount. How many more dollars does Pam pay than Susan?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 20$
2012 AMC 8, 17
A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square?
$\textbf{(A)}\hspace{.05in}3 \qquad \textbf{(B)}\hspace{.05in}4 \qquad \textbf{(C)}\hspace{.05in}5 \qquad \textbf{(D)}\hspace{.05in}6 \qquad \textbf{(E)}\hspace{.05in}7 $
2021 Junior Balkan Team Selection Tests - Romania, P3
Let $p,q$ be positive integers. For any $a,b\in\mathbb{R}$ define the sets $$P(a)=\bigg\{a_n=a \ + \ n \ \cdot \ \frac{1}{p} : n\in\mathbb{N}\bigg\}\text{ and }Q(b)=\bigg\{b_n=b \ + \ n \ \cdot \ \frac{1}{q} : n\in\mathbb{N}\bigg\}.$$
The [i]distance[/i] between $P(a)$ and $Q(b)$ is the minimum value of $|x-y|$ as $x\in P(a), y\in Q(b)$. Find the maximum value of the distance between $P(a)$ and $Q(b)$ as $a,b\in\mathbb{R}$.
2024 HMNT, 8
Compute the unique real numbers $x<3$ such that $$\sqrt{(3-x)(4-x)}+\sqrt{(4-x)(6-x)}+\sqrt{(6-x)(3-x)}=x.$$
2019 Iran Team Selection Test, 6
$x,y$ and $z$ are real numbers such that $x+y+z=xy+yz+zx$. Prove that
$$\frac{x}{\sqrt{x^4+x^2+1}}+\frac{y}{\sqrt{y^4+y^2+1}}+\frac{z}{\sqrt{z^4+z^2+1}}\geq \frac{-1}{\sqrt{3}}.$$
[i]Proposed by Navid Safaei[/i]
2005 All-Russian Olympiad, 4
$w_B$ and $w_C$ are excircles of a triangle $ABC$. The circle $w_B'$ is symmetric to $w_B$ with respect to the midpoint of $AC$, the circle $w_C'$ is symmetric to $w_C$ with respect to the midpoint of $AB$. Prove that the radical axis of $w_B'$ and $w_C'$ halves the perimeter of $ABC$.
2013 Sharygin Geometry Olympiad, 22
The common perpendiculars to the opposite sidelines of a nonplanar quadrilateral are mutually orthogonal. Prove that they intersect.
2019 Harvard-MIT Mathematics Tournament, 5
Isosceles triangle $ABC$ with $AB = AC$ is inscibed is a unit circle $\Omega$ with center $O$. Point $D$ is the reflection of $C$ across $AB$. Given that $DO = \sqrt{3}$, find the area of triangle $ABC$.
2008 Alexandru Myller, 2
Find all natural numbers $ n\ge 3 $ and real numbers $ a $ which have the property that the polynomial $ X^n-aX-1 $ admits a monic quadratic integer polynomial.
[i]Mihai Bălună[/i]
2014 NIMO Problems, 1
You drop a 7 cm long piece of mechanical pencil lead on the floor. A bully takes the lead and breaks it at a random point into two pieces. A piece of lead is unusable if it is 2 cm or shorter. If the expected value of the number of usable pieces afterwards is $\frac{m}n$ for relatively prime positive integers $m$ and $n$, compute $100m + n$.
[i]Proposed by Aaron Lin[/i]
2003 Korea Junior Math Olympiad, 4
When any $11$ integers are given, prove that you can always choose $6$ integers among them so that the sum of the chosen numbers is a multiple of $6$. The $11$ integers aren't necessarily different.
2019 Brazil EGMO TST, 2
In a sequence of positive integers, a inversion is a pair of positions, where the number in left is greater than the number in right. For example in the sequence $2, 5, 3, 1, 3$ has $5$ inversions{(5,1),(3,1),(5,3),(2,1),(5,3)}. Find the greatest number of inversions in a sequence where the sum of elements is $n$
a) where $n=7$
b) where $n=2019$
2019 European Mathematical Cup, 4
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that
$$f(x)+f(yf(x)+f(y))=f(x+2f(y))+xy$$for all $x,y\in \mathbb{R}$.
[i]Proposed by Adrian Beker[/i]
2018 China Team Selection Test, 4
Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.
2001 AMC 10, 23
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
$ \displaystyle \textbf{(A)} \ \frac {3}{10} \qquad \textbf{(B)} \ \frac {2}{5} \qquad \textbf{(C)} \ \frac {1}{2} \qquad \textbf{(D)} \ \frac {3}{5} \qquad \textbf{(E)} \ \frac {7}{10}$