Found problems: 85335
1969 IMO Shortlist, 60
$(SWE 3)$ Find the natural number $n$ with the following properties:
$(1)$ Let $S = \{P_1, P_2, \cdots\}$ be an arbitrary finite set of points in the plane, and $r_j$ the distance from $P_j$ to the origin $O.$ We assign to each $P_j$ the closed disk $D_j$ with center $P_j$ and radius $r_j$. Then some $n$ of these disks contain all points of $S.$
$(2)$ $n$ is the smallest integer with the above property.
2017 ASDAN Math Tournament, 17
For $\triangle ABC$, $AB=BC=5$, and $AC=6$. Circle $O$ is inscribed in $\triangle ABC$, and circle $P$ is tangent to circle $O$, $AB$, and $AC$. Compute the area of $\triangle ABC$ not covered by circles $O$ and $P$.
2012 Waseda University Entrance Examination, 4
For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions:
(1) Find $f'(x)$.
(2) Sketch the graph of $y=f(x)$.
(3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.
2020 Australian Maths Olympiad, 2
Amy and Bec play the following game. Initially, there are three piles, each containing $2020$ stones. The players take turns to make a move, with Amy going first. Each move consists of choosing one of the piles available, removing the unchosen pile(s) from the game, and then dividing the chosen pile into $2$ or $3$ non-empty piles. A player loses the game if he/she is unable to make a move.
Prove that Bec can always win the game, no matter how Amy plays.
Oliforum Contest IV 2013, 4
Let $p,q$ be integers such that the polynomial $x^2+px+q+1$ has two positive integer roots. Show that $p^2+q^2$ is composite.
1992 Baltic Way, 5
It is given that $ a^2\plus{}b^2\plus{}(a\plus{}b)^2\equal{}c^2\plus{}d^2\plus{}(c\plus{}d)^2$. Prove that $ a^4\plus{}b^4\plus{}(a\plus{}b)^4\equal{}c^4\plus{}d^4\plus{}(c\plus{}d)^4$.
2013 AMC 12/AHSME, 18
Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?
$ \textbf{(A)} \ \sqrt{2} \qquad \textbf{(B)} \ \frac{3}{2} \qquad \textbf{(C)} \ \frac{5}{3} \qquad \textbf{(D)} \ \sqrt{3} \qquad \textbf{(E)} \ 2$
2016 Czech-Polish-Slovak Junior Match, 5
Determine the smallest integer $j$ such that it is possible to fill the fields of the table $10\times 10$ with numbers from $1$ to $100$ so that every $10$ consecutive numbers lie in some of the $j\times j$ squares of the table.
Czech Republic
2008 JBMO Shortlist, 8
The side lengths of a parallelogram are $a, b$ and diagonals have lengths $x$ and $y$. Knowing that $ab = \frac{xy}{2}$, show that $\left( a,b \right)=\left( \frac{x}{\sqrt{2}},\frac{y}{\sqrt{2}} \right)$ or $\left( a,b \right)=\left( \frac{y}{\sqrt{2}},\frac{x}{\sqrt{2}} \right)$.
2008 Baltic Way, 13
For an upcoming international mathematics contest, the participating countries were asked to choose from nine combinatorics problems. Given how hard it usually is to agree, nobody was surprised that the following happened:
[b]i)[/b] Every country voted for exactly three problems.
[b]ii)[/b] Any two countries voted for different sets of problems.
[b]iii)[/b] Given any three countries, there was a problem none of them voted for.
Find the maximal possible number of participating countries.
2010 Peru IMO TST, 7
Let $a, b, c$ be positive real numbers such that $a + b + c = 1.$ Prove that $$ \displaystyle{\frac{1}{a + b}+\frac{1}{b + c}+\frac{1}{c + a}+ 3(ab + bc + ca) \geq \frac{11}{2}.}$$
2017 India PRMO, 2
Suppose $a, b$ are positive real numbers such that $a\sqrt{a} + b\sqrt{b} = 183, a\sqrt{b} + b\sqrt{a} = 182$. Find $\frac95 (a + b)$.
MBMT Team Rounds, 2020.2
Daniel, Clarence, and Matthew split a \$20.20 dinner bill so that Daniel pays half of what Clarence pays. If Daniel pays \$6.06, what is the ratio of Clarence's pay to Matthew's pay?
[i]Proposed by Henry Ren[/i]
Dumbest FE I ever created, 1.
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $m$ and $n$,
$$ m^{\phi(n)}+n^{\phi(m)} \mid f(m)^n + f(n)^m$$
1972 Spain Mathematical Olympiad, 1
Let $K$ be a ring with unit and $M$ the set of $2 \times 2$ matrices constituted with elements of $K$. An addition and a multiplication are defined in $M$ in the usual way between arrays. It is requested to:
a) Check that $M$ is a ring with unit and not commutative with respect to the laws of defined composition.
b) Check that if $K$ is a commutative field, the elements of$ M$ that have inverse they are characterized by the condition $ad - bc \ne 0$.
c) Prove that the subset of $M$ formed by the elements that have inverse is a multiplicative group.
2011 Purple Comet Problems, 12
Find the area of the region in the coordinate plane satisfying the three conditions
$\star$ $x \le 2y$
$\star$ $y \le 2x$
$\star$ $x + y \le 60.$
2014 International Zhautykov Olympiad, 2
Let $U=\{1, 2,\ldots, 2014\}$. For positive integers $a$, $b$, $c$ we denote by $f(a, b, c)$ the number of ordered 6-tuples of sets $(X_1,X_2,X_3,Y_1,Y_2,Y_3)$ satisfying the following conditions:
(i) $Y_1 \subseteq X_1 \subseteq U$ and $|X_1|=a$;
(ii) $Y_2 \subseteq X_2 \subseteq U\setminus Y_1$ and $|X_2|=b$;
(iii) $Y_3 \subseteq X_3 \subseteq U\setminus (Y_1\cup Y_2)$ and $|X_3|=c$.
Prove that $f(a,b,c)$ does not change when $a$, $b$, $c$ are rearranged.
[i]Proposed by Damir A. Yeliussizov, Kazakhstan[/i]
2004 Canada National Olympiad, 1
Find all ordered triples $ (x,y,z)$ of real numbers which satisfy the following system of equations:
\[ \left\{\begin{array}{rcl} xy & \equal{} & z \minus{} x \minus{} y \\
xz & \equal{} & y \minus{} x \minus{} z \\
yz & \equal{} & x \minus{} y \minus{} z \end{array} \right.
\]
2010 Romania Team Selection Test, 3
Given a positive integer $a$, prove that $\sigma(am) < \sigma(am + 1)$ for infinitely many positive integers $m$. (Here $\sigma(n)$ is the sum of all positive divisors of the positive integer number $n$.)
[i]Vlad Matei[/i]
2021 JBMO Shortlist, G3
Let $ABC$ be an acute triangle with circumcircle $\omega$ and circumcenter $O$. The perpendicular from $A$ to $BC$ intersects $BC$ and $\omega$ at $D$ and $E$, respectively. Let $F$ be a point on the segment $AE$, such that $2 \cdot FD = AE$. Let $l$ be the perpendicular to $OF$ through $F$. Prove that $l$, the tangent to $\omega$ at $E$, and the line $BC$ are concurrent.
Proposed by [i] Stefan Lozanovski, Macedonia[/i]
2016 Germany National Olympiad (4th Round), 3
Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$.
Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.
2015 Switzerland Team Selection Test, 6
Find all polynomial function $P$ of real coefficients such that for all $x \in \mathbb{R}$ $$P(x)P(x+1)=P(x^2+2)$$
2015 Math Prize for Girls Problems, 6
In baseball, a player's [i]batting average[/i] is the number of hits divided by the number of at bats, rounded to three decimal places. Danielle's batting average is $.399$. What is the fewest number of at bats that Danielle could have?
2017 Czech-Polish-Slovak Junior Match, 5
In each square of the $100\times 100$ square table, type $1, 2$, or $3$. Consider all subtables $m \times n$, where $m = 2$ and $n = 2$. A subtable will be called [i]balanced [/i] if it has in its corner boxes of four identical numbers boxes . For as large a number $k$ prove, that we can always find $k$ balanced subtables, of which no two overlap, i.e. do not have a common box.
1966 IMO Shortlist, 30
Let $n$ be a positive integer, prove that :
[b](a)[/b] $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$
[b](b)[/b] $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$