Let $p$ be a prime number greater than $5.$ Let $V$ be the collection of all positive integers $n$ that can be written in the form $n = kp + 1$ or $n = kp - 1 \ (k = 1, 2, \ldots).$ A number $n \in V$ is called [i]indecomposable[/i] in $V$ if it is impossible to find $k, l \in V$ such that $n = kl.$ Prove that there exists a number $N \in V$ that can be factorized into indecomposable factors in $V$ in more than one way.

$P$ and $Q$ are points on the longest side $AB$ of triangle $ABC$ such that $AQ = AC$ and $BP = BC$. Prove that the circumcentre of triangle $CPQ$ coincides with the incentre of triangle $ABC$.

Solve the equation $$\sqrt[3]{x^3+6x^2-6x-1}=\sqrt{x^2+4x+1}$$

For $i=1,2,\ldots,7$, Zadam Heng chooses a positive integer $m_i$ at random such that each positive integer $k$ is chosen with probability $\frac{2^i-1}{2^{ik}}$. If $m_1+2m_2+\ldots+7m_7\neq35$, Zadam keeps rechoosing the $m_i$ until this equality holds. Given that he eventually stops, what is the probability that $m_4=1$ when Zadam stops? [i]2019 CCA Math Bonanza Tiebreaker Round #3[/i]

A country has $n$ cities, labelled $1,2,3,\dots,n$. It wants to build exactly $n-1$ roads between certain pairs of cities so that every city is reachable from every other city via some sequence of roads. However, it is not permitted to put roads between pairs of cities that have labels differing by exactly $1$, and it is also not permitted to put a road between cities $1$ and $n$. Let $T_n$ be the total number of possible ways to build these roads. (a) For all odd $n$, prove that $T_n$ is divisible by $n$. (b) For all even $n$, prove that $T_n$ is divisible by $n/2$.

We are given a convex four-sided pyramid with apex $S$ and base face $ABCD$ such that the pyramid has an inscribed sphere (i.e., it contains a sphere which is tangent to each race). By making cuts along the edges $SA,SB,SC,SD$ and rotating the faces $SAB,SBC,SCD,SDA$ outwards into the plane $ABCD$, we unfold the pyramid into the polygon $AKBLCMDN$ as shown in the figure. Prove that $K,L,M,N$ are concyclic. [i] Tibor Bakos and Géza Kós [/i]

The tangent line $l$ to the circumcircle of an acute triangle $ABC$ intersects the lines $AB, BC$, and $CA$ at points $C', A'$ and $B'$, respectively. Let $H$ be the orthocenter of a triangle $ABC$. On the straight lines A'H, B′H and C'H, respectively, points $A_1, B_1$ and $C_1$ (other than $H$) are marked such that $AH = AA_1, BH = BB_1$ and $CH = CC_1$. Prove that the circumcircles of triangles $ABC$ and $A_1B_1C_1$ are tangent.

Two circles $C_1(O_1)$ and $C_2(O_2)$ with distinct radii meet at points $A$ and $B$. The tangent from $A$ to $C_1$ intersects the tangent from $B$ to $C_2$ at point $M$. Show that both circles are seen from $M$ under the same angle.

Let $a$ be a natural number, $a>3$. Prove there is an infinity of numbers n, for which $a+n|a^{n}+1$

Find all positive integers of form abcd such that $$\overline{abcd} = a^{a+b+c+d} - a^{-a+b-c+d} + a$$

Determine all nonnegative integers $n$ having two distinct positive divisors with the same distance from $\tfrac{n}{3}$. (Richard Henner)

Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other

Let $ABC$ be a triangle and $K$ be its circumcircle. Let $P$ be the point of intersection of $BC$ with tangent in $A$ to $K$. Let $D$ and $E$ be the symmetrical points of $B$ and $A$, respectively, from $P$. Let $K_1$ be the circumcircle of triangle $DAC$ and let $K_2$ the circumscribed circle of triangle $APB$. We denote with $F$ the second intersection point of the circles $K_1$ and $K_2$ Then denote with $G$ the second intersection point of the circle $K_1$ with $BF$. Show that the lines $BC$ and $EG$ are parallel.

Let $P(x)$ be a polynomial with real coefficients and form the polynomial $$Q(x) = ( x^2 +1) P(x)P'(x) + x(P(x)^2 + P'(x)^2 ).$$ Given that the equation $P(x) = 0$ has $n$ distinct real roots exceeding $1$, prove or disprove that the equation $Q(x)=0$ has at least $2n - 1$ distinct real roots.

Given positive integers $m$ and $n$. Let $a_{i,j} ( 1 \le i \le m, 1 \le j \le n)$ be non-negative real numbers, such that $$ a_{i,1} \ge a_{i,2} \ge \cdots \ge a_{i,n} \text{ and } a_{1,j} \ge a_{2,j} \ge \cdots \ge a_{m,j} $$ holds for all $1 \le i \le m$ and $1 \le j \le n$. Denote $$ X_{i,j}=a_{1,j}+\cdots+a_{i-1,j}+a_{i,j}+a_{i,j-1}+\cdots+a_{i,1},$$ $$ Y_{i,j}=a_{m,j}+\cdots+a_{i+1,j}+a_{i,j}+a_{i,j+1}+\cdots+a_{i,n}.$$ Prove that $$ \prod_{i=1}^{m} \prod_{j=1}^{n} X_{i,j} \ge \prod_{i=1}^{m} \prod_{j=1}^{n} Y_{i,j}.$$

Suppose $a _ { n } > 0$ and $\sum _ { n = 1 } ^ { \infty } a _ { n }$ diverges. Determine whether $\sum _ { n = 1 } ^ { \infty } a _ { n } / S _ { n } ^ { 2 }$ converges, where $S _ { n } = a _ { 1 } + a _ { 2 } + \dots + a _ { n } .$

Let $P$ be a convex polyhedron. Jaroslav writes a non-negative real number to every vertex of $P$ in such a way that the sum of these numbers is $1$. Afterwards, to every edge he writes the product of the numbers at the two endpoints of that edge. Prove that the sum of the numbers at the edges is at most $\frac{3}{8}$.

[b]p1.[/b] What is the value of the sum $2 + 20 + 202 + 2022$? [b]p2.[/b] Find the smallest integer greater than $10000$ that is divisible by $12$. [b]p3.[/b] Valencia chooses a positive integer factor of $6^{10}$ at random. The probability that it is odd can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m + n$. [b]p4.[/b] How many three digit positive integers are multiples of $4$ but not $8$? [b]p5.[/b] At the Jane Street store, Andy accidentally buys $5$ dollars more worth of shirts than he had planned. Originally, including the tip to the cashier, he planned to spend all of the remaining $90$ dollars on his giftcard. To compensate for his gluttony, Andy instead gives the cashier a smaller, $12.5\%$ tip so that he still spends $90$ dollars total. How much percent tip was Andy originally planning on giving? [b]p6.[/b] Let $A,B,C,D$ be four coplanar points satisfying the conditions $AB = 16$, $AC = BC =10$, and $AD = BD = 17$. What is the minimum possible area of quadrilateral $ADBC$? [b]p7.[/b] How many ways are there to select a set of three distinct points from the vertices of a regular hexagon so that the triangle they form has its smallest angle(s) equal to $30^o$? [b]p8.[/b] Jaeyong rolls five fair $6$-sided die. The probability that the sum of some three rolls is exactly $8$ times the sum of the other two rolls can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p9.[/b] Find the least positive integer n for there exists some positive integer $k > 1$ for which $k$ and $k + 2$ both divide $\underbrace{11...1}_{n\,\,\,1's}$. [b]p10.[/b] For some real constant $k$, line $y = k$ intersects the curve $y = |x^4-1|$ four times: points $A$,$B$,$C$ and $D$, labeled from left to right. If $BC = 2AB = 2CD$, then the value of $k$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p11.[/b] Let a be a positive real number and $P(x) = x^2 -8x+a$ and $Q(x) = x^2 -8x+a+1$ be quadratics with real roots such that the positive difference of the roots of $P(x)$ is exactly one more than the positive difference of the roots of $Q(x)$. The value of a can be written as a common fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [b]p12.[/b] Let $ABCD$ be a trapezoid satisfying $AB \parallel CD$, $AB = 3$, $CD = 4$, with area $35$. Given $AC$ and $BD$ intersect at $E$, and $M$, $N$, $P$, $Q$ are the midpoints of segments $AE$,$BE$,$CE$,$DE$, respectively, the area of the intersection of quadrilaterals $ABPQ$ and $CDMN$ can be expressed as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]p13.[/b] There are $8$ distinct points $P_1, P_2, ... , P_8$ on a circle. How many ways are there to choose a set of three distinct chords such that every chord has to touch at least one other chord, and if any two chosen chords touch, they must touch at a shared endpoint? [b]p14.[/b] For every positive integer $k$, let $f(k) > 1$ be defined as the smallest positive integer for which $f(k)$ and $f(k)^2$ leave the same remainder when divided by $k$. The minimum possible value of $\frac{1}{x}f(x)$ across all positive integers $x \le 1000$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p15.[/b] In triangle $ABC$, let $I$ be the incenter and $O$ be the circumcenter. If $AO$ bisects $\angle IAC$, $AB + AC = 21$, and $BC = 7$, then the length of segment $AI$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.

Find the largest positive real number $p$ (if it exists) such that the inequality \[x^2_1+ x_2^2+ \cdots + x^2_n\ge p(x_1x_2 + x_2x_3 + \cdots + x_{n-1}x_n)\] is satisfied for all real numbers $x_i$, and $(a) n = 2; (b) n = 5.$ Find the largest positive real number $p$ (if it exists) such that the inequality holds for all real numbers $x_i$ and all natural numbers $n, n \ge 2.$

[b]R[/b]thea, a distant planet, is home to creatures whose DNA consists of two (distinguishable) strands of bases with a fixed orientation. Each base is one of the letters H, M, N, T, and each strand consists of a sequence of five bases, thus forming five pairs. Due to the chemical properties of the bases, each pair must consist of distinct bases. Also, the bases H and M cannot appear next to each other on the same strand; the same is true for N and T. How many possible DNA sequences are there on Rthea?

Prove that $ \left( \left.\left\{\begin{pmatrix} a & b & c \\ 3c & a & b \\ 3b & 3c & a\end{pmatrix} \right| a,b,c\in\mathbb{Q}\right\} ,+,\cdot\right) $ is a field.

A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. But during the first week the quarry only sells $60\%$ of the gravel at full price. The following week the quarry drops the price by $10\%$, and, again, it sells $60\%$ of the remaining gravel. Each week, thereafter, the quarry reduces the price by another $10\%$ and sells $60\%$ of the remaining gravel. This continues until there is only a handful of gravel left. How many dollars does the quarry collect for the sale of all its gravel?

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

The circles $\omega_1$ and $\omega_2$ intersect at $K{}$ and $L{}$. The line $\ell$ touches the circles $\omega_1$ and $\omega_2$ at the points $X{}$ and $Y{}$, respectively. The point $K{}$ lies inside the triangle $XYL$. The line $XK$ intersects $\omega_2$ a second time at the point $Z{}$. Prove that $LY$ is the bisector of the angle $XLZ$.