Found problems: 85335
1999 National Olympiad First Round, 8
If the polynomial $ P\left(x\right)$ satisfies $ 2P\left(x\right) \equal{} P\left(x \plus{} 3\right) \plus{} P\left(x \minus{} 3\right)$ for every real number $ x$, degree of $ P\left(x\right)$ will be at most
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$
MMPC Part II 1958 - 95, 1994
[b]p1.[/b] Al usually arrives at the train station on the commuter train at $6:00$, where his wife Jane meets him and drives him home. Today Al caught the early train and arrived at $5:00$. Rather than waiting for Jane, he decided to jog along the route he knew Jane would take and hail her when he saw her. As a result, Al and Jane arrived home $12$ minutes earlier than usual. If Al was jogging at a constant speed of $5$ miles per hour, and Jane always drives at the constant speed that would put her at the station at $6:00$, what was her speed, in miles per hour?
[b]p2.[/b] In the figure, points $M$ and $N$ are the respective midpoints of the sides $AB$ and $CD$ of quadrilateral $ABCD$. Diagonal $AC$ meets segment $MN$ at $P$, which is the midpoint of $MN$, and $AP$ is twice as long as $PC$. The area of triangle $ABC$ is $6$ square feet.
(a) Find, with proof, the area of triangle $AMP$.
(b) Find, with proof, the area of triangle $CNP$.
(c) Find, with proof, the area of quadrilateral $ABCD$.
[img]https://cdn.artofproblemsolving.com/attachments/a/c/4bdcd8390bae26bc90fc7eae398ace06900a67.png[/img]
[b]p3.[/b] (a) Show that there is a triangle whose angles have measure $\tan^{-1}1$, $\tan^{-1}2$ and $\tan^{-1}3$.
(b) Find all values of $k$ for which there is a triangle whose angles have measure $\tan^{-1}\left(\frac12 \right)$, $\tan^{-1}\left(\frac12 +k\right)$, and $\tan^{-1}\left(\frac12 +2k\right)$
[b]p4.[/b] (a) Find $19$ consecutive integers whose sum is as close to $1000$ as possible.
(b) Find the longest possible sequence of consecutive odd integers whose sum is exactly $1000$, and prove that your sequence is the longest.
[b]p5.[/b] Let $AB$ and $CD$ be chords of a circle which meet at a point $X$ inside the circle.
(a) Suppose that $\frac{AX}{BX}=\frac{CX}{DX}$. Prove that $|AB|=|CD|$.
(b) Suppose that $\frac{AX}{BX}>\frac{CX}{DX}>1$. Prove that $|AB|>|CD|$.
($|PQ|$ means the length of the segment $PQ$.)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2018 CCA Math Bonanza, T3
In the game of Avalon, there are $10$ players, $4$ of which are bad. A [i]quest[/i] is a subset of those players. A quest fails if it contains at least one bad player. A randomly chosen quest of $3$ players happens to fail. What is the probability that there is exactly one bad player in the failed quest?
[i]2018 CCA Math Bonanza Team Round #3[/i]
2010 Romania Team Selection Test, 1
Given a positive integer number $n$, determine the minimum of
\[\max \left\{\dfrac{x_1}{1 + x_1},\, \dfrac{x_2}{1 + x_1 + x_2},\, \cdots,\, \dfrac{x_n}{1 + x_1 + x_2 + \cdots + x_n}\right\},\]
as $x_1, x_2, \ldots, x_n$ run through all non-negative real numbers which add up to $1$.
[i]Kvant Magazine[/i]
2019-2020 Winter SDPC, 1
Six people sit at a circular table (in the shape of a regular hexagon) such that no two friends sit next to or across from each other. Find, with proof, the maximum number of unordered pairs of people that can be friends.
2011 Today's Calculation Of Integral, 701
Evaluate
\[\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{(1+\cos x)\{1-\tan ^ 2 \frac{x}{2}\tan (x+\sin x)\tan (x-\sin x)\}}{\tan (x+\sin x)}\ dx\]
2014 Regional Olympiad of Mexico Center Zone, 4
Let $ABCD$ be a square and let $M$ be the midpoint of $BC$. Let $C ^ \prime$ be the reflection of $C$ wrt to $DM$. The parallel to $AB$ passing through $C ^ \prime$ intersects $AD$ at $R$ and $BC$ at $S$. Show that $$\frac {RC ^ \prime} {C ^\prime S} = \frac {3} {2}$$
1950 AMC 12/AHSME, 19
If $ m$ men can do a job in $ d$ days, then $ m\plus{}r$ men can do the job in:
$\textbf{(A)}\ d+r\text{ days} \qquad
\textbf{(B)}\ d-r\text{ days} \qquad
\textbf{(C)}\ \dfrac{md}{m+r}\text{ days} \qquad
\textbf{(D)}\ \dfrac{d}{m+r}\text{ days} \qquad
\textbf{(E)}\ \text{None of these}$
2022 China Girls Math Olympiad, 7
Let $n \geqslant 3$ be integer. Given convex $n-$polygon $\mathcal{P}$. A $3-$coloring of the vertices of $\mathcal{P}$ is called [i]nice[/i] such that every interior point of $\mathcal{P}$ is inside or on the bound of a triangle formed by polygon vertices with pairwise distinct colors. Determine the number of different nice colorings.
([I]Two colorings are different as long as they differ at some vertices. [/i])
2014 Brazil Team Selection Test, 1
For $m$ and $n$ positive integers that are prime to each other, determine the possible values of
$$\gcd (5^m + 7^m, 5^n + 7^n)$$
2013 Miklós Schweitzer, 5
A subalgebra $\mathfrak{h}$ of a Lie algebra $\mathfrak g$ is said to have the $\gamma$ property with respect to a scalar product ${\langle \cdot,\cdot \rangle}$ given on ${\mathfrak g}$ if ${X \in \mathfrak{h}}$ implies ${\langle [X,Y],X\rangle =0}$ for all ${Y \in \mathfrak g}$. Prove that the maximum dimension of ${\gamma}$-property subalgebras of a given ${2}$ step nilpotent Lie algebra with respect to a scalar product is independent of the selection of the scalar product.
[i]Proposed by Péter Nagy Tibor[/i]
2009 Tournament Of Towns, 1
We only know that the password of a safe consists of $7$ different digits. The safe will open if we enter $7$ different digits, and one of them matches the corresponding digit of the password. Can we open this safe in less than $7$ attempts?
[i](5 points for Juniors and 4 points for Seniors)[/i]
2008 Sharygin Geometry Olympiad, 6
(B.Frenkin) Consider the triangles such that all their vertices are vertices of a given regular 2008-gon. What triangles are more numerous among them: acute-angled or obtuse-angled?
1975 AMC 12/AHSME, 11
Let $ P$ be an interior point of circle $ K$ other than the center of $ K$. Form all chords of $ K$ which pass through $ P$, and determine their midpoints. The locus of these midpoints is
$ \textbf{(A)}\ \text{a circle with one point deleted} \qquad$
$ \textbf{(B)}\ \text{a circle if the distance from } P \text{ to the center of } K \text{ is less than}$
$ \text{one half the radius of } K \text{; otherwise a circular arc of less than}$
$ 360^{\circ}\qquad$
$ \textbf{(C)}\ \text{a semicircle with one point deleted} \qquad$
$ \textbf{(D)}\ \text{a semicircle} \qquad$
$ \textbf{(E)}\ \text{a circle}$
2000 Moldova National Olympiad, Problem 2
Prove that if a,b,c are integers with $a+b+c=0$, then $2a^4+2b^4+2c^4$ is a perfect square.
1954 Moscow Mathematical Olympiad, 277
The map of a town shows a plane divided into equal equilateral triangles. The sides of these triangles are streets and their vertices are intersections; $6$ streets meet at each junction. Two cars start simultaneously in the same direction and at the same speed from points $A$ and $B$ situated on the same street (the same side of a triangle). After any intersection an admissible route for each car is either to proceed in its initial direction or turn through $120^o$ to the right or to the left. Can these cars meet? (Either prove that these cars won’t meet or describe a route by which they will meet.)
[img]https://cdn.artofproblemsolving.com/attachments/2/d/2c934bcd0c7fc3d9dca9cee0b6f015076abbdb.png[/img]
2009 Princeton University Math Competition, 4
We divide up the plane into disjoint regions using a circle, a rectangle and a triangle. What is the greatest number of regions that we can get?
1968 AMC 12/AHSME, 19
Let $n$ be the number of ways that $10$ dollars can be changed into dimes and quarters, with at least one of each coin being used. Then $n$ equals:
$\textbf{(A)}\ 40 \qquad
\textbf{(B)}\ 38 \qquad
\textbf{(C)}\ 21 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 19 $
2017 BMT Spring, 8
In a $1024$ person randomly seeded single elimination tournament bracket, each player has a unique skill rating. In any given match, the player with the higher rating has a $\frac34$ chance of winning the match. What is the probability the second lowest rated player wins the tournament?
2010 ISI B.Math Entrance Exam, 4
If $a,b,c\in (0,1)$ satisfy $a+b+c=2$ , prove that
$\frac{abc}{(1-a)(1-b)(1-c)}\ge 8$
2013 Silk Road, 2
Circle with center $I$, inscribed in a triangle $ABC$ , touches the sides $BC$ and $AC$ at points $A_1$ and $B_1$ respectively. On rays $A_1I$ and $B_1I$, respectively, let be the points $A_2$ and $B_2$ such that $IA_2=IB_2=R$, where $R$is the radius of the circumscribed circle of the triangle $ABC$. Prove that:
a) $AA_2 = BB_2 = OI$ where $O$ is the center of the circumscribed circle of the triangle $ABC$,
b) lines $AA_2$ and $BB_2$ intersect on the circumcircle of the triangle $ABC$.
2006 Oral Moscow Geometry Olympiad, 1
The diagonals of the inscribed quadrangle $ABCD$ intersect at point $K$. Prove that the tangent at point $K$ to the circle circumscribed around the triangle $ABK$ is parallel to $CD$.
(A Zaslavsky)
2008 Estonia Team Selection Test, 6
A [i]string of parentheses[/i] is any word that can be composed by the following rules.
1) () is a string of parentheses.
2) If $s$ is a string of parentheses then $(s)$ is a string of parentheses.
3) If $s$ and t are strings of parentheses then $st$ is a string of parentheses.
The [i]midcode [/i] of a string of parentheses is the tuple of natural numbers obtained by finding, for all pairs of opening and its corresponding closing parenthesis, the number of characters remaining to the left from the medium position between these parentheses, and writing all these numbers in non-decreasing order. For example, the midcode of $(())$ is $(2,2)$ and the midcode of ()() is $(1,3)$. Prove that midcodes of arbitrary two different strings of parentheses are different.
2012 European Mathematical Cup, 3
Prove that the following inequality holds for all positive real numbers $a$, $b$, $c$, $d$, $e$ and $f$
\[\sqrt[3]{\frac{abc}{a+b+d}}+\sqrt[3]{\frac{def}{c+e+f}} < \sqrt[3]{(a+b+d)(c+e+f)} \text{.}\]
[i]Proposed by Dimitar Trenevski.[/i]
2011 IberoAmerican, 3
Let $ABC$ be a triangle and $X,Y,Z$ be the tangency points of its inscribed circle with the sides $BC, CA, AB$, respectively. Suppose that $C_1, C_2, C_3$ are circle with chords $YZ, ZX, XY$, respectively, such that $C_1$ and $C_2$ intersect on the line $CZ$ and that $C_1$ and $C_3$ intersect on the line $BY$. Suppose that $C_1$ intersects the chords $XY$ and $ZX$ at $J$ and $M$, respectively; that $C_2$ intersects the chords $YZ$ and $XY$ at $L$ and $I$, respectively; and that $C_3$ intersects the chords $YZ$ and $ZX$ at $K$ and $N$, respectively. Show that $I, J, K, L, M, N$ lie on the same circle.