Found problems: 85335
Durer Math Competition CD Finals - geometry, 2011.D2
In an right isosceles triangle $ABC$, there are two points on the hypotenuse $AB, K$ and $M$, respectively, such that $KCM$ angle is $45^o$ (point $K$ lies between $A$ and $M$). Prove that $AK^2 + MB^2 = KM^2$
[img]https://cdn.artofproblemsolving.com/attachments/2/c/e7c57e0651e5a4c492cc4ae4b115bf68a7a833.png[/img]
2013 ELMO Shortlist, 11
Let $\triangle ABC$ be a nondegenerate isosceles triangle with $AB=AC$, and let $D, E, F$ be the midpoints of $BC, CA, AB$ respectively. $BE$ intersects the circumcircle of $\triangle ABC$ again at $G$, and $H$ is the midpoint of minor arc $BC$. $CF\cap DG=I, BI\cap AC=J$. Prove that $\angle BJH=\angle ADG$ if and only if $\angle BID=\angle GBC$.
[i]Proposed by David Stoner[/i]
2013 Dutch IMO TST, 5
Let $a, b$, and $c$ be positive real numbers satisfying $abc = 1$.
Show that $a + b + c \ge \sqrt{\frac13 (a + 2)(b + 2)(c + 2)}$
1986 China National Olympiad, 5
Given a sequence $1,1,2,2,3,3,\ldots,1986,1986$, determine, with proof, if we can rearrange the sequence
so that for any integer $1\le k \le 1986$ there are exactly $k$ numbers between the two “$k$”s.
2016 Saint Petersburg Mathematical Olympiad, 5
Points $A$ and $P$ are marked in the plane not lying on the line $\ell$. For all right triangles $ABC$ with hypotenuse on $\ell$, show that the circumcircle of triangle $BPC$ passes through a fixed point other than $P$.
2004 Greece National Olympiad, 4
Let $M\subset \Bbb{N}^*$ such that $|M|=2004.$
If no element of $M$ is equal to the sum of any two elements of $M,$
find the least value that the greatest element of $M$ can take.
2020 Germany Team Selection Test, 2
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.
(Nigeria)
2024 Dutch IMO TST, 3
Let $a,b,c$ be real numbers such that $0 \le a \le b \le c$ and $a+b+c=1$. Show that
\[ab\sqrt{b-a}+bc\sqrt{c-b}+ac\sqrt{c-a}<\frac{1}{4}.\]
2023 CUBRMC, 1
Ben starts with an integer greater than $9$ and subtracts the sum of its digits from it to get a new integer. He repeats this process with each new integer he gets until he gets a positive $1$-digit integer. Find all possible $1$-digit integers Ben can end with from this process.
2003 AMC 12-AHSME, 8
What is the probability that a randomly drawn positive factor of $ 60$ is less than $ 7$?
$ \textbf{(A)}\ \frac{1}{10} \qquad
\textbf{(B)}\ \frac{1}{6} \qquad
\textbf{(C)}\ \frac{1}{4} \qquad
\textbf{(D)}\ \frac{1}{3} \qquad
\textbf{(E)}\ \frac{1}{2}$
2016 Serbia National Math Olympiad, 3
Let $ABC$ be a triangle and $O$ its circumcentre. A line tangent to the circumcircle of the triangle $BOC$ intersects sides $AB$ at $D$ and $AC$ at $E$. Let $A'$ be the image of $A$ under $DE$. Prove that the circumcircle of the triangle $A'DE$ is tangent to the circumcircle of triangle $ABC$.
2011 Kosovo National Mathematical Olympiad, 5
Let $n>1$ be an integer and $S_n$ the set of all permutations $\pi : \{1,2,\cdots,n \} \to \{1,2,\cdots,n \}$ where $\pi$ is bijective function. For every permutation $\pi \in S_n$ we define:
\[ F(\pi)= \sum_{k=1}^n |k-\pi(k)| \ \ \text{and} \ \ M_{n}=\frac{1}{n!}\sum_{\pi \in S_n} F(\pi) \]
where $M_n$ is taken with all permutations $\pi \in S_n$. Calculate the sum $M_n$.
2005 Federal Math Competition of S&M, Problem 4
Inside a circle $k$ of radius $R$ some round spots are made. The area of each spot is $1$. Every radius of circle $k$, as well as every circle concentric with $k$, meets in no more than one spot. Prove that the total area of all the spots is less than
$$\pi\sqrt R+\frac12R\sqrt R.$$
2012 Putnam, 3
A round-robin tournament among $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once?
2017 Singapore MO Open, 1
The incircle of $\vartriangle ABC$ touches the sides $BC,CA,AB$ at $D,E,F$ respectively. A circle through $A$ and $B$ encloses $\vartriangle ABC$ and intersects the line $DE$ at points $P$ and $Q$. Prove that the midpoint of $AB$ lies on the circumircle of $\vartriangle PQF$.
2021 China National Olympiad, 3
Let $n$ be positive integer such that there are exactly 36 different prime numbers that divides $n.$ For $k=1,2,3,4,5,$ $c_n$ be the number of integers that are mutually prime numbers to $n$ in the interval $[\frac{(k-1)n}{5},\frac{kn}{5}] .$ $c_1,c_2,c_3,c_4,c_5$ is not exactly the same.Prove that$$\sum_{1\le i<j\le 5}(c_i-c_j)^2\geq 2^{36}.$$
1960 Putnam, A2
Show that if three points are inside are closed square of unit side, then some pair of them are within $\sqrt{6}-\sqrt{2}$ units apart.
2011 Lusophon Mathematical Olympiad, 2
A non-negative integer $n$ is said to be [i]squaredigital[/i] if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital.
2009 Singapore MO Open, 5
Find all integers x,y,z with $2\leq x\leq y\leq z$ st
$xy \equiv 1 $(mod z) $xz\equiv 1$(mod y) $yz \equiv 1$ (mod x)
2012 Germany Team Selection Test, 2
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.
[i]Proposed by Härmel Nestra, Estonia[/i]
2022 Junior Balkan Team Selection Tests - Moldova, 12
Let $p$ and $q$ be two distinct integers. The square trinomial $x^2 + px + q$ is written on the board. At each step, a number is deleted: or the coefficient next to $x$, or the free term, and instead of the deleted number, a number is written, which is obtained from the deleted number by adding or subtracting the number $1$. After several such steps on the board, the square trinomial $x^2 + qx + p$ appeared. Show that at one stage a square trinomial was written on the board, both roots of which are integers.
2006 China Team Selection Test, 1
Two positive valued sequences $\{ a_{n}\}$ and $\{ b_{n}\}$ satisfy:
(a): $a_{0}=1 \geq a_{1}$, $a_{n}(b_{n+1}+b_{n-1})=a_{n-1}b_{n-1}+a_{n+1}b_{n+1}$, $n \geq 1$.
(b): $\sum_{i=1}^{n}b_{i}\leq n^{\frac{3}{2}}$, $n \geq 1$.
Find the general term of $\{ a_{n}\}$.
2011 IMAR Test, 4
Given an integer number $n \ge 3$, show that the number of lists of jointly coprime positive integer numbers that sum to $n$ is divisible by $3$.
(For instance, if $n = 4$, there are six such lists: $(3, 1), (1, 3), (2, 1, 1), (1, 2, 1), (1, 1, 2)$ and $(1, 1, 1, 1)$.)
2020 Taiwan TST Round 1, 1
Let $a$, $b$, $c$, $d$ be real numbers satisfying
\begin{align*}
(a + c)(b + d) = \sqrt{2}(ac - 2bd - 1).
\end{align*}
Show that
\begin{align*}
(ab - 1)^2 + (bc - 1)^2 + (cd - 1)^2 + (da - 1)^2 + (ac - 1)^2 + (2bd + 1)^2 \ge 4.
\end{align*}
2022 Belarusian National Olympiad, 10.8
A sequence $a_1,\ldots,a_n$ of positive integers is given. For each $l$ from $1$ to $n-1$ the array $(gcd(a_1,a_{1+l}),\ldots,gcd(a_n,a_{n+l}))$ is considered, where indices are taken modulo $n$. It turned out that all this arrays consist of the same $n$ pairwise distinct numbers and differ only,possibly, by their order.
Can $n$ be a) $21$ b) $2021$