Found problems: 85335
2010 CHMMC Winter, 2
In the following diagram, points $E, F, G, H, I$, and $J$ lie on a circle. The triangle $ABC$ has side lengths $AB = 6$, $BC = 7$, and $CA = 9$. The three chords have lengths $EF = 12$, $GH = 15$, and $IJ = 16$. Compute $6 \cdot AE + 7 \cdot BG + 9 \cdot CI$.
[img]https://cdn.artofproblemsolving.com/attachments/2/7/661b3d6a0f0baac0cd3b8d57c4cd4c62eeab46.png[/img]
2016 Iran Team Selection Test, 3
Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.
2018 India Regional Mathematical Olympiad, 3
For a rational number $r$, its *period* is the length of the smallest repeating block in its decimal expansion. for example, the number $r=0.123123123...$ has period $3$. If $S$ denotes the set of all rational numbers of the form $r=\overline{abcdefgh}$ having period $8$, find the sum of all elements in $S$.
2012 Bundeswettbewerb Mathematik, 3
The incircle of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ in points $A_1, B_1$ and $C_1$ respectively. $C_1D$ is a diameter of the incircle. Finally, let $E$ be the intersection of the lines $B_1C_1$ and $A_1D$. Prove that the segments $CE$ and $CB_1$ have equal length.
2009 Today's Calculation Of Integral, 446
Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$
2004 Federal Math Competition of S&M, 4
Baron Minchausen talked to a mathematician. Baron said that in his country from any town one can reach any other town by a road. Also, if one makes a circular trip from any town, one passes through an odd number of other towns. By this, as an answer to the mathematician’s question, Baron said that each town is counted as many times as it is passed through. Baron also added that the same number of roads start at each town in his country, except for the town where he was born, at which a smaller number of roads start. Then the mathematician said that baron lied. How did he conclude that?
2022 Baltic Way, 19
Find all triples $(x, y, z)$ of nonnegative integers such that
$$ x^5+x^4+1=3^y7^z $$
2023 IFYM, Sozopol, 3
Given a triangle $ABC$ ($AC < BC$) with circumcircle $k$ and orthocenter $H$, let $W$ be any point on segment $CH$. The circle with diameter $CW$ intersects $k$ a second time at point $K$ and intersects sides $BC$ and $AC$ at points $M$ and $N$, respectively. The line $KW$ intersects segment $AB$ at point $L$. Prove that the circumcircle of triangle $MNL$ passes through a fixed point, independent of the choice of $W$.
2011 Putnam, A4
For which positive integers $n$ is there an $n\times n$ matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd?
1990 IMO Longlists, 4
Find the minimal value of the function
\[\begin{array}{c}\ f(x) =\sqrt{15 - 12 \cos x} + \sqrt{4 -2 \sqrt 3 \sin x}+\sqrt{7-4\sqrt 3 \sin x} +\sqrt{10-4 \sqrt 3 \sin x - 6 \cos x}\end{array}\]
1999 AIME Problems, 3
Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.
2014 VJIMC, Problem 1
Let $f:(0,\infty)\to\mathbb R$ be a differentiable function. Assume that
$$\lim_{x\to\infty}\left(f(x)+\frac{f'(x)}x\right)=0.$$Prove that
$$\lim_{x\to\infty}f(x)=0.$$
2004 Pan African, 1
Do there exist positive integers $m$ and $n$ such that:
\[ 3n^2+3n+7=m^3 \]
2006 Estonia National Olympiad, 2
In a right triangle, the length of one side is a prime and the lengths of the other
side and the hypotenuse are integral. The ratio of the triangle perimeter and the incircle diameter is also an integer. Find all possible side lengths of the triangle.
MOAA Team Rounds, 2019.8
Suppose that $$\frac{(\sqrt2)^5 + 1}{\sqrt2 + 1} \times \frac{2^5 + 1}{2 + 1} \times \frac{4^5 + 1}{4 + 1} \times \frac{16^5 + 1}{16 + 1} =\frac{m}{7 + 3\sqrt2}$$ for some integer $m$. How many $0$’s are in the binary representation of $m$? (For example, the number $20 = 10100_2$ has three $0$’s in its binary representation.)
1971 IMO Longlists, 18
Let $a_1, a_2, \ldots, a_n$ be positive numbers, $m_g = \sqrt[n]{(a_1a_2 \cdots a_n)}$ their geometric mean, and $m_a = \frac{(a_1 + a_2 + \cdots + a_n)}{n}$ their arithmetic mean. Prove that
\[(1 + m_g)^n \leq (1 + a_1) \cdots(1 + a_n) \leq (1 + m_a)^n.\]
2005 Purple Comet Problems, 6
We glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted?
1957 Putnam, A5
Given $n$ points in the plane, show that the largest distance determined by these points cannot occur more than $n$ times.
2015 AMC 12/AHSME, 14
What is the value of $a$ for which $\frac1{\log_2a}+\frac1{\log_3a}+\frac1{\log_4a}=1$?
$\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }18\qquad\textbf{(D) }24\qquad\textbf{(E) }36$
1983 Spain Mathematical Olympiad, 8
In $1960$, the oldest of three brothers has an age that is the sum of the of his younger siblings. A few years later, the sum of the ages of two of brothers is double that of the other. A number of years have now passed since $1960$, which is equal to two thirds of the sum of the ages that the three brothers were at that year, and one of them has reached $21$ years. What is the age of each of the others two?
2007 Korea - Final Round, 1
Let $ O$ be the circumcenter of an acute triangle $ ABC$ and let $ k$ be the circle with center $ P$ that is tangent to $ O$ at $ A$ and tangent to side $ BC$ at $ D$. Circle $ k$ meets $ AB$ and $ AC$ again at $ E$ and $ F$ respectively. The lines $ OP$ and $ EP$ meet $ k$ again at $ I$ and $ G$. Lines $ BO$ and $ IG$ intersect at $ H$. Prove that $ \frac{{DF}^2}{AF}\equal{}GH$.
2006 Purple Comet Problems, 14
The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. This continued through the day when they caught more squirrels than rabbits. Up through that day how many rabbits did they catch in all?
Putnam 1939, B6
Do either $(1)$ or $(2)$:
$(1)$ $f$ is continuous on the closed interval $[a, b]$ and twice differentiable on the open interval $(a, b).$ Given $x_0 \in (a, b),$ prove that we can find $\xi \in (a, b)$ such that
$\dfrac{ ( \dfrac{(f(x_0) - f(a))}{(x_0 - a)} - \dfrac{(f(b) - f(a))}{(b - a)} )}{(x_0 - b)} = \dfrac{f''(\xi)}{2}.$
$(2)$ $AB$ and $CD$ are identical uniform rods, each with mass $m$ and length $2a.$ They are placed a distance $b$ apart, so that $ABCD$ is a rectangle. Calculate the gravitational attraction between them. What is the limiting value as a tends to zero?
2007 Indonesia TST, 2
Solve the equation \[ x\plus{}a^3\equal{}\sqrt[3]{a\minus{}x}\] where $ a$ is a real parameter.
2003 Abels Math Contest (Norwegian MO), 4a
$25$ boys and $25$ girls sit around a table. Show that there is a person who has a girl sitting on either side of them.