Found problems: 85335
Let $ABC$ be a non-rectangle triangle with $M$ the middle of $BC$. Let $D$ be a point on the line $AB$ such that $CA=CD$ and let $E$ be a point on the line $BC$ such that $EB=ED$. The parallel to $ED$ passing through $A$ intersects the line $MD$ at the point $I$ and the line $AM$ intersects the line $ED$ at the point $J$. Show that the points $C, I$ and $J$ are aligned.
Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $ 20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $ 10$ vertical feet above the bottom?
$ \textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 7.5 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 15$
Consider acute $\triangle ABC$ with altitudes $AA_1, BB_1$ and $CC_1$ ($A_1 \in BC,B_1 \in AC,C_1 \in AB$). A point $C' $ on the extension of $B_1A_1$ beyond $A_1$ is such that $A_1C' = B_1C_1$. Analogously, a point $B'$ on the extension of A$_1C_1$ beyond $C_1$ is such that $C_1B' = A_1B_1$ and a point $A' $ on the extension of $C_1B_1$ beyond $B_1$ is such that $B_1A' = C_1A_1$. Denote by $A'', B'', C''$ the symmetric points of $A' , B' , C'$ with respect to $BC, CA$ and $AB$ respectively. Prove that if $R, R'$ and R'' are circumradiii of $\triangle ABC, \triangle A'B'C'$ and $\triangle A''B''C''$, then $R, R'$ and $R'' $ are sidelengths of a triangle with area equals one half of the area of $\triangle ABC$.
Define the function $f:\mathbb{R} \backslash \{-1,1\} \to \mathbb{R}$ to be
\[f(x) = \sum_{a,b=0}^{\infty} \frac{x^{2^a3^b}}{1-x^{2^{a+1}3^{b+1}}} .\]
Suppose that $f\left(y\right)-f\left(\tfrac{1}{y}\right)=2016$. Then $y$ can be written in simplest form as $\tfrac{p}{q}$. Find $p+q$. ($\mathbb{R} \backslash \{-1,1\}$ refers to the set of real numbers excluding $-1$ and $1$.)
Let $Z^+$ denote the set of positive integers. Determine , with proof, if there exists a function $f:\mathbb{Z^+}\rightarrow\mathbb {Z^+}$ such that $f(f(f(f(f(n)))))$ = $2022n$ for all positive integers $n$.
Let $G$ be a group, with operation $*$. Suppose that
(i) $G$ is a subset of $\mathbb{R}^3$ (but $*$ need not be related to addition of vectors);
(ii) For each $\mathbf{a},\mathbf{b}\in G,$ either $\mathbf{a}\times\mathbf{b}=\mathbf{a}*\mathbf{b}$ or $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ (or both), where $\times$ is the usual cross product in $\mathbb{R}^3.$
Prove that $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ for all $\mathbf{a},\mathbf{b}\in G.$
Let $ABCD$ be a square with $AB=6.$ A point $P$ in the interior is $2$ units away from side $BC$ and $3$ units away from side $CD.$ What is the distance from $P$ to $A?$
Let $\epsilon$ be a plane and $k_1, k_2, k_3$ be spheres on the same side of $\epsilon$. The spheres $k_1, k_2, k_3$ touch the plane at points $T_1, T_2, T_3$, respectively, and $k_2$ touches $k_1$ at $S_1$ and $k_3$ at $S_3$. Prove that the lines $S_1T_1$ and $S_3T_3$ intersect on the sphere $k_2$. Describe the locus of the intersection point.
Find all triples of natural numbers $ (x, y, z) $ that satisfy the condition $ (x + 1) ^ {y + 1} + 1 = (x + 2) ^ {z + 1}. $
For $n \ge 2$, consider $n$ boxes aligned from left to right. In each box, one puts a ball that can be red, blue or white such that the following condition is ful lled:
[i]Each box is neighboring at least one box containing a ball of the same color.[/i]
We denote by $I_n$ the number of such con gurations.
a) Determine $I_{11}$. Justify your answer.
b) Find, with proof, the general formula for $I_n$.
Find the least positive integer $n$ ($n\geq 3$), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.
Let $DIAL$, $FOR$, and $FRIEND$ be regular polygons in the plane. If $ID=1$, find the product of all possible areas of $OLA$.
[i]Proposed by Andrew Gu.[/i]
Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a three times differentiable function. Prove that there exists $w \in [-1,1]$ such that \[ \frac{f'''(w)}{6} = \frac{f(1)}{2}-\frac{f(-1)}{2}-f'(0). \]
Find the smallest positive integer, $n$, which can be expressed as the sum of distinct positive integers $a,b,c$ such that $a+b,a+c,b+c$ are perfect squares.
Let $N = 12!$ and denote by $X$ the set of positive divisors of $N$ other than $1$. A [i]pseudo-ultrafilter[/i] $U$ is a nonempty subset of $X$ such that for any $a,b \in X$:
\begin{itemize}
\item If $a$ divides $b$ and $a \in U$ then $b \in U$.
\item If $a,b \in U$ then $\gcd(a,b) \in U$.
\item If $a,b \notin U$ then $\operatorname{lcm} (a,b) \notin U$.
\end{itemize}
How many such pseudo-ultrafilters are there?
[i]Proposed by Evan Chen[/i]
How many distinct ordered pairs of integers $(b, m, t)$ satisfy the equation $b^8+m^4+t^2+1 = 2019$?
Let $\{x_1, x_2, x_3, ..., x_n\}$ be a set of $n$ distinct positive integers, such that the sum of any $3$ of them is a prime number. What is the maximum value of $n$?
a) One has a set of stones with weights $1, 2, \ldots, 20$ grams. Find all $k$ for which it is possible to place $k$ and the rest $20-k$ stones from the set respectively on the two pans of a balance so that equilibrium is achieved.
b) One has a set of stones with weights $1, 2, \ldots, 51$ grams. Find all $k$ for which it is possible to place $k$ and the rest $51-k$ stones from the set respectively on the two pans of a balance so that equilibrium is achieved.
c) One has a set of stones with weights $1, 2, \ldots, n$ grams ($n\in\mathbb{N}$). Find all $n$ and $k$ for which it is possible to place $k$ and the rest $n-k$ stones from the set respectively on the two pans of a balance so that equilibrium is achieved.
[size=75] a) and b) were proposed at the festival, c) is a generalization[/size]
Show that the inequality $(x^2 + y^2z^2) (y^2 + x^2z^2) (z^2 + x^2y^2) \ge 8xy^2z^3$ is valid for all integers $x, y$ and $z$.When does equality apply?
Let $M$ be the midpoint of $AC$ in an acute triangle $ABC$. Let $K$ be a point on the minor arc $AC$, such that $\angle AKM=90^{o}$. Let $BK \cap AM=X$ and the $A$-altitude meets $BM$ at $Y$. Show that $XY \parallel AB$.
Let $ABC$ be an isosceles triangle with apex $A$ and altitude $AD$. On $AB$, choose a point $F$ distinct from $B$ such that $CF$ is tangent to the incircle of $ABD$. Suppose that $\vartriangle BCF$ is isosceles. Show that those conditions uniquely determine:
a) which vertex of $BCF$ is its apex,
b) the size of $\angle BAC$
Three circles $\mathcal{K}_1$, $\mathcal{K}_2$, $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$. The point $O$ lies inside the triangle $ABC$.
Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$. Let $ \alpha = \frac {OA_1}{AA_1} $, $ \beta= \frac {OB_1}{BB_1} $ and $ \gamma = \frac {OC_1}{CC_1} $ and let $R$ be the circumradius of the triangle $ABC$. Prove that
\[ \alpha R_1 + \beta R_2 + \gamma R_3 \geq R. \]
In an acute triangle $ ABC $ with circumcircle $ \Omega $ and circumcenter $ O $, the circle $ \Gamma $ is drawn, passing through the points $ A $, $ O $ and $ C $ together with its diameter $ OQ $, then the points $ M $ and $ N $ are chosen on the lines $ AQ $ and $ AC $, respectively, in such a way that the quadrilateral $ AMBN $ is a parallelogram.
Prove that the point of intersection of the lines $ MN $ and $ BQ $ lies on the circle $ \Gamma $.
A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equal the third, what is the probability that at least one $2$ is tossed?
$\displaystyle \textbf{(A)} \ \frac{1}{6} \qquad \textbf{(B)} \ \frac{91}{216} \qquad \textbf{(C)} \ \frac{1}{2} \qquad \textbf{(D)} \ \frac{8}{15} \qquad \textbf{(E)} \ \frac{7}{12}$