Found problems: 85335
2018 India PRMO, 3
Consider all $6$-digit numbers of the form $abccba$ where $b$ is odd. Determine the number of all such $6$-digit numbers that are divisible by $7$.
2020 LMT Fall, 26
Let $\omega_1$ and $\omega_2$ be two circles with centers $O_1$ and $O_2$. The two circles intersect at $A$ and $B$. $\ell$ is the circles' common external tangent that is closer to $B$, and it meets $\omega_1$ at $T_1$ and $\omega_2$ at $T_2$. Let $C$ be the point on line $AB$ not equal to $A$ that is the same distance from $\ell$ as $A$ is. Given that $O_1O_2=15$, $AT_1=5$ and $AT_2=12$, find $AC^2+{T_1T_2}^2$.
[i]Proposed by Zachary Perry[/i]
2012 Middle European Mathematical Olympiad, 1
Let $ \mathbb{R} ^{+} $ denote the set of all positive real numbers. Find all functions $ \mathbb{R} ^{+} \to \mathbb{R} ^{+} $ such that
\[ f(x+f(y)) = yf(xy+1)\]
holds for all $ x, y \in \mathbb{R} ^{+} $.
1955 Putnam, A1
Prove that there is no set of integers $m, n, p$ except $0, 0, 0$ for which \[m + n \sqrt2 + p \sqrt3 = 0.\]
2009 USAMTS Problems, 4
The Rational Unit Jumping Frog starts at $(0, 0)$ on the Cartesian plane, and each minute jumps a distance of exactly $1$ unit to a point with rational coordinates.
(a) Show that it is possible for the frog to reach the point $\left(\frac15,\frac{1}{17}\right)$ in a finite amount of time.
(b) Show that the frog can never reach the point $\left(0,\frac14\right)$.
2023 Chile Junior Math Olympiad, 5
$1600$ bananas are distributed among $100$ monkeys (it is possible that some monkeys do not receive bananas). Prvove that at least four monkeys receive the same amount of bananas.
2008 Princeton University Math Competition, A7/B9
Let $\mathcal{H}$ be the region of points $(x, y)$, such that $(1, 0), (x, y), (-x, y)$, and $(-1,0)$ form an isosceles trapezoid whose legs are shorter than the base between $(x, y)$ and $(-x,y)$. Find the least possible positive slope that a line could have without intersecting $\mathcal{H}$.
1977 Germany Team Selection Test, 2
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
2010 Romania Team Selection Test, 4
Let $X$ and $Y$ be two finite subsets of the half-open interval $[0, 1)$ such that $0 \in X \cap Y$ and $x + y = 1$ for no $x \in X$ and no $y \in Y$. Prove that the set $\{x + y - \lfloor x + y \rfloor : x \in X \textrm{ and } y \in Y\}$ has at least $|X| + |Y| - 1$ elements.
[i]***[/i]
1971 IMO Longlists, 37
Let $S$ be a circle, and $\alpha =\{A_1,\ldots ,A_n\}$ a family of open arcs in $S$. Let $N(\alpha )=n$ denote the number of elements in $\alpha$. We say that $\alpha$ is a covering of $S$ if $\bigcup_{k=1}^n A_k\supset S$.
Let $\alpha=\{A_1,\ldots ,A_n\}$ and $\beta =\{B_1,\ldots ,B_m\}$ be two coverings of $S$.
Show that we can choose from the family of all sets $A_i\cap B_j,\ i=1,2,\ldots ,n,\ j=1, 2,\ldots ,m,$ a covering $\gamma$ of $S$ such that $N(\gamma )\le N(\alpha)+N(\beta)$.
2004 AIME Problems, 4
A square has sides of length $2$. Set $S$ is the set of all line segments that have length $2$ and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$.
2012 Online Math Open Problems, 4
Let $\text{lcm} (a,b)$ denote the least common multiple of $a$ and $b$. Find the sum of all positive integers $x$ such that $x\le 100$ and $\text{lcm}(16,x) = 16x$.
[i]Ray Li.[/i]
1988 ITAMO, 1
Players $A$ and $B$ play the following game: $A$ tosses a coin $n$ times, and $B$ does $n+1$ times. The player who obtains more ”heads” wins; or in the case of equal balances, $A$ is assigned victory. Find the values of $n$ for which this game is fair (i.e. both players have equal chances for victory).
2000 Romania National Olympiad, 4
Let $ I $ be the center of the incircle of a triangle $ ABC. $ Shw that, if for any point $ M $ on the segment $ AB $ (extremities excluded) there exist two points $ N,P $ on $ BC, $ respectively, $ AC $ (both excluding the extremities) such that the center of mass of $ MNP $ coincides with $ I, $ then $ ABC $ is equilateral.
2010 Postal Coaching, 5
Find the first integer $n > 1$ such that the average of $1^2 , 2^2 ,\cdots, n^2$ is itself a perfect square.
2023 CUBRMC, 4
Let square $ABCD$ and circle $\Omega$ be on the same plane, and $AA'$, $BB'$, $CC'$, $DD'$ be tangents to $\Omega$. Let $WXY Z$ be a convex quadrilateral with side lengths $WX = AA'$, $XY = BB'$, $Y Z = CC'$, and $ZW = DD'$. If $WXY Z$ has an inscribed circle, prove that the diagonals $WY$ and $XZ$ are perpendicular to each other.
1988 All Soviet Union Mathematical Olympiad, 474
In the triangle $ABC$, $\angle C$ is obtuse and $D$ is a fixed point on the side $BC$, different from $B$ and $C$. For any point $M$ on the side $BC$, different from $D$, the ray $AM$ intersects the circumcircle $S$ of $ABC$ at $N$. The circle through $M, D$ and $N$ meets $S$ again at $P$, different from $N$. Find the location of the point $M$ which minimises $MP$.
2010 CHMMC Fall, 5
The three positive integers $a, b, c$ satisfy the equalities $gcd(ab, c^2) = 20$, $gcd(ac, b^2) = 18$, and $gcd(bc, a^2) = 75$. Compute the minimum possible value of $a + b + c$.
2014 Harvard-MIT Mathematics Tournament, 2
Find the integer closest to
\[\frac{1}{\sqrt[4]{5^4+1}-\sqrt[4]{5^4-1}}\]
2019 USA TSTST, 2
Let $ABC$ be an acute triangle with circumcircle $\Omega$ and orthocenter $H$. Points $D$ and $E$ lie on segments $AB$ and $AC$ respectively, such that $AD = AE$. The lines through $B$ and $C$ parallel to $\overline{DE}$ intersect $\Omega$ again at $P$ and $Q$, respectively. Denote by $\omega$ the circumcircle of $\triangle ADE$.
[list=a]
[*] Show that lines $PE$ and $QD$ meet on $\omega$.
[*] Prove that if $\omega$ passes through $H$, then lines $PD$ and $QE$ meet on $\omega$ as well.
[/list]
[i]Merlijn Staps[/i]
2018 Hanoi Open Mathematics Competitions, 12
Let $ABC$ be an acute triangle with $AB < AC$, and let $BE$ and $CF$ be the altitudes. Let the median $AM$ intersect $BE$ at point $P$, and let line $CP$ intersect $AB$ at point $D$ (see Figure 2). Prove that $DE \parallel BC$, and $AC$ is tangent to the circumcircle of $\vartriangle DEF$.
[img]https://cdn.artofproblemsolving.com/attachments/f/7/bbad9f6019a77c6aa46c3a821857f06233cb93.png[/img]
2005 Danube Mathematical Olympiad, 4
Let $k$ and $n$ be positive integers. Consider an array of $2\left(2^n-1\right)$ rows by $k$ columns. A $2$-coloring of the elements of the array is said to be [i]acceptable[/i] if any two columns agree on less than $2^n-1$ entries on the same row.
Given $n$, determine the maximum value of $k$ for an acceptable $2$-coloring to exist.
2019 Azerbaijan BMO TST, 3
Let $ a, b, c$ be positive real numbers such that $ abc = 1. $ Prove that:
$$ 2 (a^ 2 + b^ 2 + c^ 2) \left (\frac 1 {a^ 2} + \frac 1{b^ 2}+ \frac 1{c^2}\right)\geq 3(a+ b + c + ab + bc + ca).$$
2019 Tournament Of Towns, 3
The product of two positive integers $m$ and $n$ is divisible by their sum. Prove that $m + n \le n^2$.
(Boris Frenkin)
PEN A Problems, 96
Find all positive integers $n$ that have exactly $16$ positive integral divisors $d_{1},d_{2} \cdots, d_{16}$ such that $1=d_{1}<d_{2}<\cdots<d_{16}=n$, $d_6=18$, and $d_{9}-d_{8}=17$.