Found problems: 85335
2011 AMC 8, 19
How many rectangles are in this figure?
[asy]
pair A,B,C,D,E,F,G,H,I,J,K,L;
A=(0,0);
B=(20,0);
C=(20,20);
D=(0,20);
draw(A--B--C--D--cycle);
E=(-10,-5);
F=(13,-5);
G=(13,5);
H=(-10,5);
draw(E--F--G--H--cycle);
I=(10,-20);
J=(18,-20);
K=(18,13);
L=(10,13);
draw(I--J--K--L--cycle);[/asy]
$ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12 $
2002 Manhattan Mathematical Olympiad, 3
Let us consider all rectangles with sides of length $a,b$ both of which are whole numbers. Do more of these rectangles have perimeter $2000$ or perimeter $2002$?
2011 National Olympiad First Round, 1
Which one is true for a quadrilateral $ABCD$ such that perpendicular bisectors of $[AB]$ and $[CD]$ meet on the diagonal $[AC]$?
$\textbf{(A)}\ |BA| + |AD| \leq |BC| + |CD| \\
\textbf{(B)}\ |BD| \leq |AC| \\
\textbf{(C)}\ |AC| \leq |BD| \\
\textbf{(D)}\ |AD| + |DC| \leq |AB| + |BC| \\
\textbf{(E)}\ \text{None}$
2008 Bosnia and Herzegovina Junior BMO TST, 4
On circle are $ 2008$ blue and $ 1$ red point(s) given. Are there more polygons which have a red point or those which dont have it??
2025 Austrian MO National Competition, 2
Let $\triangle{ABC}$ be an acute triangle with $BC > AC$. Let $S$ be the centroid of triangle $ABC$ and let $F$ be the foot of the perpendicular from $C$ to side $AB$. The median $CS$ intersects the circumcircle $\gamma$ of triangle $\triangle{ABC}$ at a second point $P$. Let $M$ be the point where $CS$ intersects $AB$. The line $SF$ intersects the circle $\gamma$ at a point $Q$, such that $F$ lies between $S$ and $Q$. Prove that the points $M,P,Q$ and $F$ lie on a circle.
[i](Karl Czakler)[/i]
2021 Saudi Arabia Training Tests, 19
Let $ABC$ be a triangle with $AB < AC$ inscribed in $(O)$. Tangent line at $A$ of $(O)$ cuts $BC$ at $D$. Take $H$ as the projection of $A$ on $OD$ and $E,F$ as projections of $H$ on $AB,AC$.Suppose that $EF$ cuts $(O)$ at $R,S$. Prove that $(HRS)$ is tangent to $OD$
2007 IMC, 3
Call a polynomial $ P(x_{1}, \ldots, x_{k})$ [i]good[/i] if there exist $ 2\times 2$ real matrices $ A_{1}, \ldots, A_{k}$ such that
$ P(x_{1}, \ldots, x_{k}) = \det \left(\sum_{i=1}^{k}x_{i}A_{i}\right).$
Find all values of $ k$ for which all homogeneous polynomials with $ k$ variables of degree 2 are good. (A polynomial is homogeneous if each term has the same total degree.)
2002 China Team Selection Test, 3
Let \[ f(x_1,x_2,x_3) = -2 \cdot (x_1^3+x_2^3+x_3^3) + 3 \cdot (x_1^2(x_2+x_3) + x_2^2 \cdot (x_1+x_3) + x_3^2 \cdot ( x_1+x_2 ) - 12x_1x_2x_3. \] For any reals $r,s,t$, we denote \[ g(r,s,t)=\max_{t\leq x_3\leq t+2} |f(r,r+2,x_3)+s|. \] Find the minimum value of $g(r,s,t)$.
2021 CMIMC, 8
Determine the number of functions $f$ from the integers to $\{1,2,\cdots,15\}$ which satisfy $$f(x)=f(x+15)$$
and
$$f(x+f(y))=f(x-f(y))$$
for all $x,y$.
[i]Proposed by Vijay Srinivasan[/i]
2010 Czech And Slovak Olympiad III A, 5
On the board are written numbers $1, 2,. . . , 33$. In one step we select two numbers written on the product of which is the square of the natural number, we wipe off the two chosen numbers and write the square root of their product on the board. This way we continue to the board only the numbers remain so that the product of neither of them is a square. (In one we can also wipe out two identical numbers and replace them with the same number.) Prove that at least $16$ numbers remain on the board.
2017 CCA Math Bonanza, L3.4
A random walk is a process in which something moves from point to point, and where the direction of movement at each step is randomly chosen. Suppose that a person conducts a random walk on a line: he starts at $0$ and each minute randomly moves either $1$ unit in the positive direction or $1$ unit in the negative direction. What is his expected distance from the origin after $6$ moves?
[i]2017 CCA Math Bonanza Lightning Round #3.4[/i]
2013 Today's Calculation Of Integral, 894
Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$
1989 Bundeswettbewerb Mathematik, 1
For a given positive integer $n$, let $f(x) =x^{n}$. Is it possible for the decimal number
$$0.f(1)f(2)f(3)\ldots$$
to be rational? (Example: for $n=2$, we are considering $0.1491625\ldots$)
2015 Rioplatense Mathematical Olympiad, Level 3, 1
Let $ABC$ be a triangle and $P$ a point on the side $BC$. Let $S_1$ be the circumference with center $B$ and radius $BP$ that cuts the side $AB$ at $D$ such that $D$ lies between $A$ and $B$. Let $S_2$ be the circumference with center $C$ and radius $CP$ that cuts the side $AC$ at $E$ such that $E$ lies between $A$ and $C$. Line $AP$ cuts $S_1$ and $S_2$ at $X$ and $Y$ different from $P$, respectively. We call $T$ the point of intersection of $DX$ and $EY$. Prove that $\angle BAC+ 2 \angle DTE=180$
1990 Vietnam National Olympiad, 1
A triangle $ ABC$ is given in the plane. Let $ M$ be a point inside the triangle and $ A'$, $ B'$, $ C'$ be its projections on the sides $ BC$, $ CA$, $ AB$, respectively. Find the locus of $ M$ for which $ MA \cdot MA' \equal{} MB \cdot MB' \equal{} MC \cdot MC'$.
2010 Federal Competition For Advanced Students, Part 1, 3
Given is the set $M_n=\{0, 1, 2, \ldots, n\}$ of nonnegative integers less than or equal to $n$. A subset $S$ of $M_n$ is called [i]outstanding[/i] if it is non-empty and for every natural number $k\in S$, there exists a $k$-element subset $T_k$ of $S$.
Determine the number $a(n)$ of outstanding subsets of $M_n$.
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 3)[/i]
2024 Regional Olympiad of Mexico Southeast, 2
Let \(ABC\) be an acute triangle with circumradius \(R\). Let \(D\) be the midpoint of \(BC\) and \(F\) the midpoint of \(AB\). The perpendicular to \(AC\) through \(F\) and the perpendicular to \(BC\) through \(B\) intersect at \(N\). Prove that \(ND = R\).
2017 ELMO Problems, 6
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$:
(i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$
(ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$
[i]Proposed by Ashwin Sah[/i]
LMT Guts Rounds, 18
Congruent unit circles intersect in such a way that the center of each circle lies on the circumference of the other. Let $R$ be the region in which two circles overlap. Determine the perimeter of $R.$
2023 Princeton University Math Competition, B2
Let $f$ be a polynomial with degree at most $n-1$. Show that
$$
\sum_{k=0}^n\left(\begin{array}{l}
n \\
k
\end{array}\right)(-1)^k f(k)=0
$$
2024 Argentina National Olympiad Level 2, 1
Let $ABC$ be an equilateral triangle with side length $8$, and let $D$, $E$, and $F$ be points on the sides $BC$, $CA$, and $AB$ respectively. Suppose that $BD = 2$ and $\angle ADE = \angle DEF = 60^\circ$. Calculate the length of segment $AF$.
1989 AMC 12/AHSME, 14
$\cot 10 + \tan 5 =$
$\textbf{(A)}\ \csc 5 \qquad
\textbf{(B)}\ \csc 10 \qquad
\textbf{(C)}\ \sec 5 \qquad
\textbf{(D)}\ \sec 10 \qquad
\textbf{(E)}\ \sin 15$
2007 Tournament Of Towns, 1
Let $ABCD$ be a rhombus. Let $K$ be a point on the line $CD$, other than $C$ or $D$, such that $AD = BK$. Let $P$ be the point of intersection of $BD$ with the perpendicular bisector of $BC$. Prove that $A, K$ and $P$ are collinear.
1986 All Soviet Union Mathematical Olympiad, 420
The point $M$ belongs to the side $[AC]$ of the acute-angle triangle $ABC$. Two circles are circumscribed around triangles $ABM$ and $BCM$ . What $M$ position corresponds to the minimal area of those circles intersection?
2022 APMO, 1
Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.