Found problems: 85335
2001 AMC 8, 13
Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie?
$ \text{(A)}\ 10\qquad\text{(B)}\ 20\qquad\text{(C)}\ 30\qquad\text{(D)}\ 50\qquad\text{(E)}\ 72 $
2006 Bulgaria Team Selection Test, 1
[b]Problem 4.[/b] Let $k$ be the circumcircle of $\triangle ABC$, and $D$ the point on the arc $\overarc{AB},$ which do not pass through $C$. $I_A$ and $I_B$ are the centers of incircles of $\triangle ADC$ and $\triangle BDC$, respectively. Proove that the circumcircle of $\triangle I_AI_BC$ touches $k$ iff \[ \frac{AD}{BD}=\frac{AC+CD}{BC+CD}. \]
[i] Stoyan Atanasov[/i]
2013 Stanford Mathematics Tournament, 23
Let $a$ and $b$ be the solutions to $x^2-7x+17=0$. Compute $a^4+b^4$.
2010 Tournament Of Towns, 1
In a multiplication table, the entry in the $i$-th row and the $j$-th column is the product $ij$ From an $m\times n$ subtable with both $m$ and $n$ odd, the interior $(m-2) (n-2)$ rectangle is removed, leaving behind a frame of width $1$. The squares of the frame are painted alternately black and white. Prove that the sum of the numbers in the black squares is equal to the sum of the numbers in the white squares.
2018 Serbia JBMO TST, 1
Let $AD$ be an internal angle bisector in triangle $\Delta ABC$.
An arbitrary point $M$ is chosen on the closed segment $AD$. A parallel to $BC$ through $M$ cuts $AB$ at $N$. Let $AD, CM$ cut circumcircle of $\Delta ABC$ at $K, L$, respectively. Prove that $K,N,L$ are collinear.
1968 Poland - Second Round, 1
Prove that if a polynomial with integer coefficients takes a value equal to $1$ in absolute value at three different integer points, then it has no integer zeros.
1971 IMO Longlists, 54
A set $M$ is formed of $\binom{2n}{n}$ men, $n=1,2,\ldots$. Prove that we can choose a subset $P$ of the set $M$ consisting of $n+1$ men such that one of the following conditions is satisfied:
$(1)$ every member of the set $P$ knows every other member of the set $P$;
$(2)$ no member of the set $P$ knows any other member of the set $P$.
1984 IMO Longlists, 40
Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.
2014 IberoAmerican, 3
Given a set $X$ and a function $f: X \rightarrow X$, for each $x \in X$ we define $f^1(x)=f(x)$ and, for each $j \ge 1$, $f^{j+1}(x)=f(f^j(x))$. We say that $a \in X$ is a fixed point of $f$ if $f(a)=a$. For each $x \in \mathbb{R}$, let $\pi (x)$ be the quantity of positive primes lesser or equal to $x$.
Given an positive integer $n$, we say that $f: \{1,2, \dots, n\} \rightarrow \{1,2, \dots, n\}$ is [i]catracha[/i] if $f^{f(k)}(k)=k$, for every $k=1, 2, \dots n$. Prove that:
(a) If $f$ is catracha, $f$ has at least $\pi (n) -\pi (\sqrt{n}) +1$ fixed points.
(b) If $n \ge 36$, there exists a catracha function $f$ with exactly $ \pi (n) -\pi (\sqrt{n}) + 1$ fixed points.
2007 Tournament Of Towns, 1
The sides of a convex pentagon are extended on both sides to form five triangles. If these triangles are congruent to one another, does it follow that the pentagon is regular?
1948 Putnam, B4
For what $\lambda$ does the equation
$$ \int_{0}^{1} \min(x,y) f(y)\; dy =\lambda f(x)$$
have continuous solutions which do not vanish identically in $(0,1)?$ What are these solutions?
2012 India National Olympiad, 4
Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be [i]good [/i]if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ such that the triangle is divided by these rays into $27$ [i]smaller triangles of equal area.[/i] Determine the number of good points for a given triangle $ABC$.
2011 IMO Shortlist, 1
Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.
[i]Proposed by Morteza Saghafian, Iran[/i]
2011 Tournament of Towns, 4
Does there exist a convex $N$-gon such that all its sides are equal and all vertices belong to the parabola $y = x^2$ for
a) $N = 2011$
b) $N = 2012$ ?
2008 All-Russian Olympiad, 3
Given a finite set $ P$ of prime numbers, prove that there exists a positive integer $ x$ such that it can be written in the form $ a^p \plus{} b^p$ ($ a,b$ are positive integers), for each $ p\in P$, and cannot be written in that form for each $ p$ not in $ P$.
2014 IPhOO, 6
A square plate has side length $L$ and negligible thickness. It is laid down horizontally on a table and is then rotating about the axis $\overline{MN}$ where $M$ and $N$ are the midpoints of two adjacent sides of the square. The moment of inertia of the plate about this axis is $kmL^2$, where $m$ is the mass of the plate and $k$ is a real constant. Find $k$.
[color=red]Diagram will be added to this post very soon. If you want to look at it temporarily, see the PDF.[/color]
[i]Problem proposed by Ahaan Rungta[/i]
2002 Pan African, 1
Find all functions $f: N_0 \to N_0$, (where $N_0$ is the set of all non-negative integers) such that $f(f(n))=f(n)+1$ for all $n \in N_0$ and the minimum of the set $\{ f(0), f(1), f(2) \cdots \}$ is $1$.
2012 Pre-Preparation Course Examination, 2
Prove that if a vector space is the union of some of it's proper subspaces, then number of these subspaces can not be less than the number of elements of the field of that vector space.
1998 National Olympiad First Round, 23
Ahmet and Betül play a game on $ n\times n$ $ \left(n\ge 7\right)$ board. Ahmet places his only piece on one of the $ n^{2}$ squares. Then Betül places her two pieces on two of the squares at the border of the board. If two squares have a common edge, we call them adjacent squares. When it is Ahmet's turn, Ahmet moves his piece either to one of the empty adjacent squares or to the out of the board if it is on one of the squares at the border of the board. When it is Betül's turn, she moves all her two pieces to the adjacent squares. If Ahmet's piece is already on one of the two squares that Betül has just moved to, Betül attacks to his piece and wins the game. If Ahmet manages to go out of the board, he wins the game. If Ahmet begins to move, he guarantees to win the game putting his piece on one of the $\dots$ squares at the beginning of the game.
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ n^{2} \qquad\textbf{(C)}\ \left(n\minus{}2\right)^{2} \qquad\textbf{(D)}\ 4\left(n\minus{}1\right) \qquad\textbf{(E)}\ 2n\minus{}1$
2016 Vietnam National Olympiad, 2
Given a triangle $ABC$ inscribed by circumcircle $(O)$. The angles at $B,C$ are acute angle. Let $M$ on the arc $BC$ that doesn't contain $A$ such that $AM$ is not perpendicular to $BC$. $AM$ meets the perpendicular bisector of $BC$ at $T$. The circumcircle $(AOT)$ meets $(O)$ at $N$ ($N\ne A$).
a) Prove that $\angle{BAM}=\angle{CAN}$.
b) Let $I$ be the incenter and $G$ be the foor of the angle bisector of $\angle{BAC}$. $AI,MI,NI$ intersect $(O)$ at $D,E,F$ respectively. Let ${P}=DF\cap AM, {Q}=DE\cap AN$. The circle passes through $P$ and touches $AD$ at $I$ meets $DF$ at $H$ ($H\ne D$).The circle passes through $Q$ and touches $AD$ at $I$ meets $DE$ at $K$ ($K\ne D$). Prove that the circumcircle $(GHK)$ touches $BC$.
Ukrainian TYM Qualifying - geometry, II.2
Is it true that when all the faces of a tetrahedron have the same area, they are congruent triangles?
1992 Baltic Way, 7
Let $ a\equal{}\sqrt[1992]{1992}$. Which number is greater
\[ \underbrace{a^{a^{a^{\ldots^{a}}}}}_{1992}\quad\text{or}\quad 1992?
\]
2023 Flanders Math Olympiad, 2
In the plane, the point $M$ is the midpoint of a line segment $[AB]$ and $\ell$ is an arbitrary line that has no has a common point with the line segment $[AB]$ (and is also not perpendicular to $[AB]$). The points $X$ and $Y$ are the perpendicular projections of $A$ and $B$ onto $\ell$, respectively. Show that the circumscribed circles of triangle $\vartriangle AMX$ and triangle $\vartriangle BMY$ have the same radius.
2021 Stars of Mathematics, 1
For every integer $n\geq 3$, let $s_n$ be the sum of all primes (strictly) less than $n$. Are there infinitely many integers $n\geq 3$ such that $s_n$ is coprime to $n$?
[i]Russian Competition[/i]
2006 Petru Moroșan-Trident, 2
Find the twice-differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that have the property that
$$ f'(x)+F(x)=2f(x)+x^2/2, $$
for any real numbers $ x; $ where $ F $ is a primitive of $ f. $
[i]Carmen Botea[/i]