Found problems: 85335
2014 PUMaC Geometry B, 2
Consider the pyramid $OABC$. Let the equilateral triangle $ABC$ with side length $6$ be the base. Also $9=OA=OB=OC$. Let $M$ be the midpoint of $AB$. Find the square of the distance from $M$ to $OC$.
2018 Tuymaada Olympiad, 1
Do there exist three different quadratic trinomials $f(x), g(x), h(x)$ such that the roots of the equation $f(x)=g(x)$ are $1$ and $4$, the roots of the equation $g(x)=h(x)$ are $2$ and $5$, and the roots of the equation $h(x)=f(x)$ are $3$ and $6$?
[i]Proposed by A. Golovanov[/i]
2009 Abels Math Contest (Norwegian MO) Final, 3a
In the triangle $ABC$ the edge $BC$ has length $a$, the edge $AC$ length $b$, and the edge $AB$ length $c$. Extend all the edges at both ends – by the length $a$ from the vertex $A, b$ from $B$, and $c$ from $C$. Show that the six endpoints of the extended edges all lie on a common circle.
[img]https://cdn.artofproblemsolving.com/attachments/8/7/14c8c6a4090d4fade28893729a510d263e7abb.png[/img]
2005 AMC 8, 23
Isosceles right triangle $ ABC$ encloses a semicircle of area $ 2\pi$. The circle has its center $ O$ on hypotenuse $ \overline{AB}$ and is tangent to sides $ \overline{AC}$ and $ \overline{BC}$. What is the area of triangle $ ABC$?
[asy]defaultpen(linewidth(0.8));pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2);
draw(circle(o, 2));
clip(a--b--c--cycle);
draw(a--b--c--cycle);
dot(o);
label("$C$", c, NW);
label("$A$", a, NE);
label("$O$", o, SE);
label("$B$", b, SW);[/asy]
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 3\pi\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 4\pi $
2024 Abelkonkurransen Finale, 3b
A $2024$-[i]table [/i]is a table with two rows and $2024$ columns containg all the numbers $1,2,\dots,4048$. Such a table is [i]evenly coloured[/i] if exactly half of the numbers in each row, and one number in each column, is coloured red. The [i]red sum[/i] in an evenly coloured $2024$-table is the sum of all the red numbers in the table.
Let $N$ be the largest number such that every $2024$-table has an even colouring with red sum $\ge N$. Determine $N$, and find the number of $2024$-tables such that every even colouring of the table has red sum $\le N$.
1999 AMC 12/AHSME, 20
The sequence $ a_1$, $ a_2$, $ a_3$, $ \dots$ satisfies $ a_1 \equal{} 19$, $ a_9 \equal{} 99$, and, for all $ n \ge 3$, $ a_n$ is the arithmetic mean of the first $ n \minus{} 1$ terms. Find $ a_2$.
$ \textbf{(A)}\ 29\qquad
\textbf{(B)}\ 59\qquad
\textbf{(C)}\ 79\qquad
\textbf{(D)}\ 99\qquad
\textbf{(E)}\ 179$
2005 Rioplatense Mathematical Olympiad, Level 3, 1
Find all numbers $n$ that can be expressed in the form $n=k+2\lfloor\sqrt{k}\rfloor+2$ for some nonnegative integer $k$.
2011 Tournament of Towns, 2
Peter buys a lottery ticket on which he enters an $n$-digit number, none of the digits being $0$. On the draw date, the lottery administrators will reveal an $n\times n$ table, each cell containing one of the digits from $1$ to $9$. A ticket wins a prize if it does not match any row or column of this table, read in either direction. Peter wants to bribe the administrators to reveal the digits on some cells chosen by Peter, so that Peter can guarantee to have a winning ticket. What is the minimum number of digits Peter has to know?
1985 Traian Lălescu, 1.3
Let $ a,b,c $ denote the lengths of a right triangle ($ a $ being the hypothenuse) that satisfy the equality $ a=2\sqrt{bc} . $
Find the angles of this triangle.
2018 Brazil National Olympiad, 5
Consider the sequence in which $a_1 = 1$ and $a_n$ is obtained by juxtaposing the decimal representation of $n$ at the end of the decimal representation of $a_{n-1}$. That is, $a_1 = 1$, $a_2 = 12$, $a_3 = 123$, $\dots$ , $a_9 = 123456789$, $a_{10} = 12345678910$ and so on. Prove that infinitely many numbers of this sequence are multiples of $7$.
2018-IMOC, C6
In a deck of cards, there are $kn$ cards numbered from $1$ to $n$ and there are $k$ cards of each number. Now, divide this deck into $k$ sub-decks with equal sizes. Prove that if $\gcd(k,n)=1$, then one could always pick $n$ cards, one from each sub-deck, such that the sum of those cards is divisible by $n$.
2000 Greece Junior Math Olympiad, 4
Four pupils decided to buy some mathematical books so that
(a) everybody buys exactly three different books , and
(b) every two of the pupils buy exactly one book in common.
What are the greatest and smallest number of different books they can buy?
2019 Math Prize for Girls Problems, 6
For each integer from 1 through 2019, Tala calculated the product of its digits. Compute the sum of all 2019 of Tala's products.
1999 AMC 12/AHSME, 8
At the end of $ 1994$, Walter was half as old as his grandmother. The sum of the years in which they were born was $ 3838$. How old will Walter be at the end of $ 1999$?
$ \textbf{(A)}\ 48 \qquad \textbf{(B)}\ 49\qquad \textbf{(C)}\ 53\qquad \textbf{(D)}\ 55\qquad \textbf{(E)}\ 101$
2019 Costa Rica - Final Round, 1
In a faraway place in the Universe, a villain has a medal with special powers and wants to hide it so that no one else can use it. For this, the villain hides it in a vertex of a regular polygon with $2019$ sides. Olcoman, the savior of the Olcomita people, wants to get the medal to restore peace in the Universe, for which you have to pay $1000$ olcolones for each time he makes the following move: on each turn he chooses a vertex of the polygon, which turns green if the medal is on it or in one of the four vertices closest to it, or otherwise red. Find the fewest olcolones Olcoman needs to determine with certainty the position of the medal.
2016 Kurschak Competition, 2
Prove that for any finite set $A$ of positive integers, there exists a subset $B$ of $A$ satisfying the following conditions:
[list][*]if $b_1,b_2\in B$ are distinct, then neither $b_1$ and $b_2$ nor $b_1+1$ and $b_2+1$ are multiples of each other, and
[*] for any $a\in A$, we can find a $b\in B$ such that $a$ divides $b$ or $b+1$ divides $a+1$.[/list]
2021 Princeton University Math Competition, B1
A nonempty word is called pronounceable if it alternates in vowels (A, E, I, O, U) and consonants (all other letters) and it has at least one vowel. How many pronounceable words can be formed using the letters P, U, M, A, C at most once each? Words of length shorter than $5$ are allowed.
2017 Greece Team Selection Test, 1
Let $ABC$ be an acute-angled triangle inscribed in circle $c(O,R)$ with $AB<AC<BC$,
and $c_1$ be the inscribed circle of $ABC$ which intersects $AB, AC, BC$ at
$F, E, D$ respectivelly. Let $A', B', C'$ be points which lie on $c$ such that the quadrilaterals
$AEFA', BDFB', CDEC'$ are inscribable.
(1) Prove that $DEA'B'$ is inscribable.
(2) Prove that $DA', EB', FC'$ are concurrent.
2017 South East Mathematical Olympiad, 6
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $O$, where $AC\perp BD$. $M$ be the midpoint of arc $ADC$. Circle $(DOM)$ intersect $DA,DC$ at $E,F$. Prove that $BE=BF$.
2007 Indonesia TST, 4
Let $ n$ and $ k$ be positive integers. Please, find an explicit formula for
\[ \sum y_1y_2 \dots y_k,\]
where the summation runs through all $ k\minus{}$tuples positive integers $ (y_1,y_2,\dots,y_k)$ satisfying $ y_1\plus{}y_2\plus{}\dots\plus{}y_k\equal{}n$.
2008 Kurschak Competition, 2
Let $n\ge 1$ and $a_1<a_2<\dots<a_n$ be integers. Let $S$ be the set of pairs $1\le i<j\le n$ for which $a_j-a_i$ is a power of $2$, and $T$ be the set of pairs $1\le i<j\le n$ with $j-i$ a power of $2$. (Here, the powers of $2$ are $1,2,4,\dots$.) Prove that $|S|\le |T|$.
1962 All Russian Mathematical Olympiad, 013
Given points $A' ,B' ,C' ,D',$ on the extension of the $[AB], [BC], [CD], [DA]$ sides of the convex quadrangle $ABCD$, such, that the following pairs of vectors are equal: $$[BB']=[AB], [CC']=[BC], [DD']=[CD], [AA']=[DA].$$ Prove that the quadrangle $A'B'C'D'$ area is five times more than the quadrangle $ABCD$ area.
1997 Pre-Preparation Course Examination, 1
Let $f: \mathbb R \to\mathbb R$ be a function such that $|f(x)| \leq 1$ for all $x \in \mathbb R$ and
\[f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac 17 \right) + f \left( x + \frac 16 \right), \quad \forall x \in \mathbb R.\]
Show that $f$ is a periodic function.
2006 Germany Team Selection Test, 1
A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off.
[i]Proposed by Australia[/i]
1995 Baltic Way, 19
The following construction is used for training astronauts:
A circle $C_2$ of radius $2R$ rolls along the inside of another, fixed circle $C_1$ of radius $nR$, where $n$ is an integer greater than $2$. The astronaut is fastened to a third circle $C_3$ of radius $R$ which rolls along the inside of circle $C_2$ in such a way that the touching point of the circles $C_2$ and $C_3$ remains at maximum distance from the touching point of the circles $C_1$ and $C_2$ at all times. How many revolutions (relative to the ground) does the astronaut perform together with the circle $C_3$ while the circle $C_2$ completes one full lap around the inside of circle $C_1$?