Compute the smallest positive integer that gives a remainder of $1$ when divided by $11$, a remainder of $2$ when divided by $21$, and a remainder of $5$ when divided by $51$. [i]2021 CCA Math Bonanza Lightning Round #3.3[/i]

Let $\triangle ABC$ be a triangle in a plane such that $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point in three-dimensional space such that $\angle{BDC}=\angle{CDA}=\angle{ADB}=90^\circ$. Let $d$ be the distance from $D$ to the plane containing $\triangle ABC$. The value $d^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by William Yue[/i]

Find the sum of the squares of the values $x$ that satisfy $\frac{1}{x} + \frac{2}{x+3}+\frac{3}{x+6} = 1$.

For which real numbers $a$, there exist pairwise different real numbers $x, y, z$ satisfying $$\frac{x^3+a}{y+z}=\frac{y^3+a}{x+z}=\frac{z^3+a}{x+y}= -3.$$

A triangle $ABC$ with $AC=20$ is inscribed in a circle $\omega$. A tangent $t$ to $\omega$ is drawn through $B$. The distance $t$ from $A$ is $25$ and that from $C$ is $16$.If $S$ denotes the area of the triangle $ABC$, find the largest integer not exceeding $\frac{S}{20}$

Suppose $\triangle ABC$ has circumcircle $\Gamma$, circumcentre $O$ and orthocentre $H$. Parallel lines $\alpha, \beta, \gamma$ are drawn through the vertices $A, B, C$, respectively. Let $\alpha ', \beta ', \gamma '$ be the reflections of $\alpha, \beta, \gamma$ in the sides $BC, CA, AB$, respectively. $(a)$ Show that $\alpha ', \beta ', \gamma '$ are concurrent if and only if $\alpha, \beta, \gamma$ are parallel to the Euler line $OH$. $(b)$ Suppose that $\alpha ', \beta ' , \gamma '$ are concurrent at the point $P$ . Show that $\Gamma$ bisects $OP$ .

Let $a,b$ and $c$ be the length of sides of a triangle.Then prove that $S\le\frac{a^2+b^2+c^2}{6}$ where $S$ is the area of triangle.

$p(n) $ is a product of all digits of n.Calculate: $ p(1001) + p(1002) + ... + p(2011) $

Prove that the system \begin{align*} x^6+x^3+x^3y+y & = 147^{157} \\ x^3+x^3y+y^2+y+z^9 & = 157^{147} \end{align*} has no solutions in integers $x$, $y$, and $z$.

Let $ABCDEF$ be a regular hexagon with sidelength $6$, and construct squares $ABGH$, $BCIJ$, $CDKL$, $DEMN$, $EFOP$, and $FAQR$ outside the hexagon. Find the perimeter of dodecagon $HGJILKNMPORQ$. [i]Proposed by Andrew Wu[/i]

On a $5 \times 5$ grid we randomly place two \emph{cars}, which each occupy a single cell and randomly face in one of the four cardinal directions. It is given that the two cars do not start in the same cell. In a \emph{move}, one chooses a car and shifts it one cell forward. The probability that there exists a sequence of moves such that, afterward, both cars occupy the same cell is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$. [i]Proposed by Sean Li[/i]

Find the smallest positive integer $n\neq 2004$ for which there exists a polynomial $f\in\mathbb{Z}[x]$ such that the equation $f(x)=2004$ has at least one, and the equation $f(x)=n$ has at least $2004$ different integer solutions.

A point $ P$ is selected at random from the interior of the pentagon with vertices $ A \equal{} (0,2)$, $B \equal{} (4,0)$, $C \equal{} (2 \pi \plus{} 1, 0)$, $D \equal{} (2 \pi \plus{} 1,4)$, and $ E \equal{} (0,4)$. What is the probability that $ \angle APB$ is obtuse? [asy] size(150); pair A, B, C, D, E; A = (0,1.5); B = (3,0); C = (2 *pi + 1, 0); D = (2 * pi + 1,4); E = (0,4); draw(A--B--C--D--E--cycle); label("$A$", A, dir(180)); label("$B$", B, dir(270)); label("$C$", C, dir(0)); label("$D$", D, dir(0)); label("$E$", E, dir(180)); [/asy] $ \displaystyle \textbf{(A)} \ \frac {1}{5} \qquad \textbf{(B)} \ \frac {1}{4} \qquad \textbf{(C)} \ \frac {5}{16} \qquad \textbf{(D)} \ \frac {3}{8} \qquad \textbf{(E)} \ \frac {1}{2}$

Let $ I$ be the incenter of triangle $ ABC$, and let $ A_1$, $ B_1$, $ C_1$ be arbitrary points on the segments $ (AI)$, $ (BI)$, $ (CI)$, respectively. The perpendicular bisectors of $ AA_1$, $ BB_1$, $ CC_1$ intersect each other at $ A_2$, $ B_2$, and $ C_2$. Prove that the circumcenter of the triangle $ A_2B_2C_2$ coincides with the circumcenter of the triangle $ ABC$ if and only if $ I$ is the orthocenter of triangle $ A_1B_1C_1$.

Let $ABC$ and $PQR$ be two triangles such that [list] [b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$. [b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$ [/list] Prove that $AB+AC=PQ+PR$.

Square with side 100 was cut by 99 horizontal and 99 vertical lines into 10000 rectangles (not necessarily with integer sides). How many rectangles in this square with area not exceeding 1 at least can be?

We reflected each vertex of a triangle on the opposite side. Prove that the area of the triangle formed by these three reflection points is smaller than the area of the initial triangle multiplied by five.

Find all positive integers $n$, for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.

Let $ABCD$ be a rectangle. We consider the points $E\in CA,F\in AB,G\in BC$ such that $DC\perp CA,EF\perp AB$ and $EG\perp BC$. Solve in the set of rational numbers the equation $AC^x=EF^x+EG^x$.

Let $AD$ be the altitude, $AE$ be the median, and $O$ be the circumcenter of a triangle $ABC.$ Points $X$ and $Y$ are selected inside the triangle such that $\angle BAX=\angle CAY,$ $OX\perp AX,$ and $OY\perp AY.$ Prove that points $D,E,X,Y$ are concyclic. (M. Kurskiy)

A set $S=\{ s_1,s_2,...,s_k\}$ of positive real numbers is "polygonal" if $k\geq 3$ and there is a non-degenerate planar $k-$gon whose side lengths are exactly $s_1,s_2,...,s_k$; the set $S$ is multipolygonal if in every partition of $S$ into two subsets,each of which has at least three elements, exactly one of these two subsets in polygonal. Fix an integer $n\geq 7$. (a) Does there exist an $n-$element multipolygonal set, removal of whose maximal element leaves a multipolygonal set? (b) Is it possible that every $(n-1)-$element subset of an $n-$element set of positive real numbers be multipolygonal?

Fix $2n$ distinct reals $x_1,y_1,...,x_n,y_n$ and dene the $n\times n$ matrix where its $(i, j)$-th element is $x_i + y_j$ for all $i, j = 1,..., n$. Show that if the products of the numbers in each column is always the same, then also the products of the numbers in each row is always the same. ( Alberto Alfarano)

Prove that among any five distinct rays $ Ox$, $ Oy$, $ Oz$, $ Ot$, $ Or$ in space there exist two which form an angle less than or equal to $ 90^{\circ}$.

In an acute triangle $ABC$ the angle $C$ is greater than the angle $A$. Let $AE$ be a diameter of the circumcircle of the triangle. Let the intersection point of the ray $AC$ and the tangent of the circumcircle through the vertex $B$ be $K$. The perpendicular to $AE$ through $K$ intersects the circumcircle of the triangle $BCK$ for the second time at point $D$. Prove that $CE$ bisects the angle $BCD$.

Take an arbitrary point $D$ on side $BC$ of triangle $ABC$ and draw a circle through point $D$ and the centers of the circles inscribed in triangles $ABD$ and $ACD$. Prove that all circles obtained for different points $D$ of side $BC$ have a common point.