Found problems: 85335
1957 Moscow Mathematical Olympiad, 370
* Three equal circles are tangent to each other externally and to the fourth circle internally. Tangent lines are drawn to the circles from an arbitrary point on the fourth circle. Prove that the sum of the lengths of two tangent lines equals the length of the third tangent.
1961 All Russian Mathematical Olympiad, 003
Prove that among $39$ sequential natural numbers there always is a number with the sum of its digits divisible by $11$.
2012 Tuymaada Olympiad, 2
Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$.
[i]Proposed by A. Golovanov, M. Ivanov, K. Kokhas[/i]
2014 District Olympiad, 2
Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a differentiable function, with continuous derivative, and let
\[ s_{n}=\sum_{k=1}^{n}f\left( \frac{k}{n}\right) \]
Prove that the sequence $(s_{n+1}-s_{n})_{n\in{\mathbb{N}}^{\ast}}$ converges to $\int_{0}^{1}f(x)\mathrm{d}x$.
2008 Harvard-MIT Mathematics Tournament, 1
Four students from Harvard, one of them named Jack, and five students from MIT, one of them named Jill, are going to see a Boston Celtics game. However, they found out that only $ 5$ tickets remain, so $ 4$ of them must go back. Suppose that at least one student from each school must go see the game, and at least one of Jack and Jill must go see the game, how many ways are there of choosing which $ 5$ people can see the game?
2020 AMC 12/AHSME, 24
Suppose that $\triangle ABC$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP = 1$, $BP = \sqrt{3}$, and $CP = 2$. What is $s?$
$\textbf{(A) } 1 + \sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3} \qquad \textbf{(D) } \sqrt{5 + \sqrt{5}} \qquad \textbf{(E) } 2\sqrt{2}$
2021 CHMMC Winter (2021-22), 2
A prefrosh is participating in Caltech’s “Rotation.” They must rank Caltech’s $8$ houses, which are Avery, Page, Lloyd, Venerable, Ricketts, Blacker, Dabney, and Fleming, each a distinct integer rating from $1$ to $8$ inclusive. The conditions are that the rating $x$ they give to Fleming is at most the average rating $y$ given to Ricketts, Blacker, and Dabney, which is in turn at most the average rating $z$ given to Avery, Page, Lloyd, and Venerable. Moreover $x, y, z$ are all integers. How many such rankings can the prefrosh provide?
1990 Dutch Mathematical Olympiad, 3
A polynomial $ f(x)\equal{}ax^4\plus{}bx^3\plus{}cx^2\plus{}dx$ with $ a,b,c,d>0$ is such that $ f(x)$ is an integer for $ x \in \{ \minus{}2,\minus{}1,0,1,2 \}$ and $ f(1)\equal{}1$ and $ f(5)\equal{}70$.
$ (a)$ Show that $ a\equal{}\frac{1}{24}, b\equal{}\frac{1}{4},c\equal{}\frac{11}{24},d\equal{}\frac{1}{4}$.
$ (b)$ Prove that $ f(x)$ is an integer for all $ x \in \mathbb{Z}$.
2013-2014 SDML (High School), 10
The sum $$\frac{1}{1+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}+\cdots+\frac{1}{\sqrt{2n-1}+\sqrt{2n+1}}$$ is a root of the quadratic $x^2+x+c$. What is $c$ in terms of $n$?
$\text{(A) }-\frac{n}{2}\qquad\text{(B) }2n\qquad\text{(C) }-2n\qquad\text{(D) }n+\frac{1}{2}\qquad\text{(E) }n-2$
1955 AMC 12/AHSME, 27
If $ r$ and $ s$ are the roots of $ x^2\minus{}px\plus{}q\equal{}0$, then $ r^2\plus{}s^2$ equals:
$ \textbf{(A)}\ p^2\plus{}2q \qquad
\textbf{(B)}\ p^2\minus{}2q \qquad
\textbf{(C)}\ p^2\plus{}q^2 \qquad
\textbf{(D)}\ p^2\minus{}q^2 \qquad
\textbf{(E)}\ p^2$
1978 Dutch Mathematical Olympiad, 3
There are $1978$ points in the flat plane. Each point has a circular disk with that point as its center and the radius is the distance to a fixed point. Prove that there are five of these circular disks, which together cover all $1978$ points (circular disk means: the circle and its inner area).
Novosibirsk Oral Geo Oly IX, 2022.6
Triangle $ABC$ is given. On its sides $AB$, $BC$ and $CA$, respectively, points $X, Y, Z$ are chosen so that $$AX : XB =BY : YC = CZ : ZA = 2:1.$$ It turned out that the triangle $XYZ$ is equilateral. Prove that the original triangle $ABC$ is also equilateral.
2017 VTRMC, 7
Find all pairs $(m, n)$ of nonnegative integers for which $ m ^ { 2 } + 2 \cdot 3 ^ { n } = m \left( 2 ^ { n + 1 } - 1 \right) $.
Ukraine Correspondence MO - geometry, 2017.11
Inside the parallelogram $ABCD$, choose a point $P$ such that $\angle APB+ \angle CPD= \angle BPC+ \angle APD$. Prove that there exists a circle tangent to each of the circles circumscribed around the triangles $APB$, $BPC$, $CPD$ and $APD$.
2007 Estonia Math Open Senior Contests, 7
Does there exist a natural number $ n$ such that $ n>2$ and the sum of squares of
some $ n$ consecutive integers is a perfect square?
2010 Brazil National Olympiad, 1
Let $ABCD$ be a convex quadrilateral, and $M$ and $N$ the midpoints of the sides $CD$ and $AD$, respectively. The lines perpendicular to $AB$ passing through $M$ and to $BC$ passing through $N$ intersect at point $P$. Prove that $P$ is on the diagonal $BD$ if and only if the diagonals $AC$ and $BD$ are perpendicular.
2010 Laurențiu Panaitopol, Tulcea, 2
Find the strictly monotone functions $ f:\{ 0\}\cup\mathbb{N}\longrightarrow\{ 0\}\cup\mathbb{N} $ that satisfy the following two properties:
$ \text{(i)} f(2n)=n+f(n), $ for any nonnegative integers $ n. $
$ \text{(ii)} f(n) $ is a perfect square if and only if $ n $ is a perfect square.
2015 Estonia Team Selection Test, 8
Find all positive integers $n$ for which it is possible to partition a regular $n$-gon into triangles with diagonals not intersecting inside the $n$-gon such that at every vertex of the $n$-gon an odd number of triangles meet.
2016 AMC 10, 21
Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$?
$\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}$
2015 Azerbaijan JBMO TST, 2
There are some real numbers on the board (at least two). In every step we choose two of them, for example $a$ and $b$, and then we replace them with $\frac{ab}{a+b}$. We continue until there is one number. Prove that the last number does not depend on which order we choose the numbers to erase.
1997 Italy TST, 1
Let $x,y,z,t$ be real numbers with $x,y,z$ not all equal such that \[x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}=t.\] Find all possible values of $ t$ such that $xyz+t=0$.
1989 IberoAmerican, 3
Let $a,b$ and $c$ be the side lengths of a triangle. Prove that:
\[\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}<\frac{1}{16}\]
2015 China Girls Math Olympiad, 8
Let $n\geq 2$ be a given integer. Initially, we write $n$ sets on the blackboard and do a sequence of moves as follows: choose two sets $A$ and $B$ on the blackboard such that none of them is a subset of the other, and replace $A$ and $B$ by $A\cap B$ and $A\cup B$. This is called a $\textit{move}$.
Find the maximum number of moves in a sequence for all possible initial sets.
2018 Canada National Olympiad, 5
Let $k$ be a given even positive integer. Sarah first picks a positive integer $N$ greater than $1$ and proceeds to alter it as follows: every minute, she chooses a prime divisor $p$ of the current value of $N$, and multiplies the current $N$ by $p^k -p^{-1}$ to produce the next value of $N$. Prove that there are infinitely many even positive integers $k$ such that, no matter what choices Sarah makes, her number $N$ will at some point be divisible by $2018$.
Novosibirsk Oral Geo Oly IX, 2016.3
A square is drawn on a sheet of grid paper on the sides of the cells $ABCD$ with side $8$. Point $E$ is the midpoint of side $BC$, $Q$ is such a point on the diagonal $AC$ such that $AQ: QC = 3: 1$. Find the angle between straight lines $AE$ and $DQ$.