Source: 2017 Canadian Open Math Challenge, Problem C1 ----- For a positive integer $n$, we define function $P(n)$ to be the sum of the digits of $n$ plus the number of digits of $n$. For example, $P(45) = 4 + 5 + 2 = 11$. (Note that the first digit of $n$ reading from left to right, cannot be $0$). $\qquad$(a) Determine $P(2017)$. $\qquad$(b) Determine all numbers $n$ such that $P(n) = 4$. $\qquad$(c) Determine with an explanation whether there exists a number $n$ for which $P(n) - P(n + 1) > 50$.

There are forty weights: $1, 2, \cdots , 40$ grams. Ten weights with even masses were put on the left pan of a balance. Ten weights with odd masses were put on the right pan of the balance. The left and the right pans are balanced. Prove that one pan contains two weights whose masses differ by exactly $20$ grams. [i](4 points)[/i]

Let $n$ be a positive integer. Show that \[\sum^{n}_{k=1}\tan^{2}\frac{k \pi}{2n+1}\] is an odd integer.

Let $ K$ be a subset of a group $ G$ that is not a union of lift cosets of a proper subgroup. Prove that if $ G$ is a torsion group or if $ K$ is a finite set, then the subset \[ \bigcap _{k \in K} k^{-1}K\] consists of the identity alone. [i]L. Redei[/i]

In a one-round football tournament, three points were awarded for a victory. All the teams scored different numbers of points. If not three, but two points were given for a victory, then all teams would also have a different number of points, but each team's place would be different. What is the smallest number of teams for which this is possible? [i]Proposed by A. Zaslavsky[/i]

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

[b]2A. [/b] Two triangles $ABC$ and $ABD$ share a common side. $ABC$ is drawn such that its entire area lies inside the larger triangle $ABD$. If $AB = 20$, side $AD$ meets side $AB$ at a right angle, and point $C$ is between points $A$ and $D$, then find the area outside of triangle $ABC$ but within $ABD$, given that both triangles have integral side lengths and $AB$ is the smallest side of either triangle. $ABC$ and $ABD$ are both primitive right triangles. [i] (1 point)[/i] [b]2B.[/b] Find the sum of all positive integral factors of the correct answer to [b]2A[/b]. [i](2 points)[/i] [b]2C.[/b] Let $B$ be the sum of the digits of the correct answer to [b]2B[/b] above. If the solution to the functional equation $21*f(x) - 7*f(1/x) = Bx$ is of the form $(Ax^2 + C) / Dx$, find $C$, given that $A$, $C$, and $D$ are relatively prime (they don’t share a common prime factor). [i](3 points)[/i] [hide=ANSWER KEY]2A.780 2B. 2352 2C. 3[/hide]

$\triangle ABC$ has the property that $\angle ACB = 90^{\circ}$. Let $D$ and $E$ be points on $AB$ such that $D$ is on ray $BA$, $E$ is on segment $AB$, and $\angle DCA = \angle ACE$. Let the circumcircle of $\triangle CDE$ hit $BC$ at $F \ne C$, and $EF$ hit $AC$ and $DC$ at $P$ and $Q$, respectively. If $EP = FQ$, then the ratio $\frac{EF}{PQ}$ can be written as $a+\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$. [i]Proposed by Kevin Zhao[/i]

If $ p$ and $ q$ are primes and $ x^2 \minus{} px \plus{} q \equal{} 0$ has distinct positive integral roots, then which of the following statements are true? $ \text{I. The difference of the roots is odd.}$ $ \text{II. At least one root is prime.}$ $ \text{III. } p^2 \minus{} q \text{ is prime.}$ $ \text{IV. } p \plus{} q \text{ is prime.}$ $ \textbf{(A)}\ \text{I only} \qquad \textbf{(B)}\ \text{II only} \qquad \textbf{(C)}\ \text{II and III only} \qquad$ $ \textbf{(D)}\ \text{I, II and IV only} \qquad \textbf{(E)}\ \text{All are true.}$

Some club has 42 members. It’s known that among 31 arbitrary club members, we can find one pair of a boy and a girl that they know each other. Show that from club members we can choose 12 pairs of knowing each other boys and girls.

(a) $n$ is a natural number. $d_1,\dots,d_n,r_1,\dots ,r_n$ are natural numbers such that for each $i,j$ that $1\leq i < j \leq n$ we have $(d_i,d_j)=1$ and $d_i\geq 2$. Prove that there exist an $x$ such that (i) $1 \leq x \leq 3^n$ (ii)For each $1 \leq i \leq n$ \[x \overset{d_i}{\not{\equiv}} r_i\] (b) For each $\epsilon >0$ prove that there exists natural $N$ such that for each $n>N$ and each $d_1,\dots,d_n,r_1,\dots ,r_n$ satisfying the conditions above there exists an $x$ satisfying (ii) such that $1\leq x \leq (2+\epsilon )^n$. Time allowed for this exam was 75 minutes.

Given triangle $ABC$ in the plane such that $\angle CAB = a > \pi/2$. Let $PQ$ be a segment whose midpoint is the point $A$. Prove that $(BP+CQ) \tan a/2 \geq BC$.

Show that for any vectors $a, b$ in Euclidean space, \[|a \times b|^3 \leq \frac{3 \sqrt 3}{8} |a|^2 |b|^2 |a-b|^2\] Remark. Here $\times$ denotes the vector product.

In triangle $ABC$, side $BC$ is equal to $a$, and the angles of the triangle adjacent to it are equal to $\alpha$ and $\beta$. A circle passing through points $A $and $B$ intersects lines $CA$ and $CB$ for second time at points $ P$ and $M$. It is known that straight line $RM$ passes through the center of the circle circumscribed around $ABC$. Find the length of the segment $PM$

In a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ intersect at $X$. Let the circumcircles of triangles $AXD$ and $BXC$ intersect again at $Y$ . If $X$ is the incentre of triangle $ABY$ , show that $\angle CAD = 90^o$.

For odd positive integers $n$, define $f(n)$ to be the smallest odd integer greater than $n$ that is not relatively prime to $n$. Compute the smallest $n$ such that $f(f(n))$ is not divisible by $3$.

Let $X=(x_1,x_2,......,x_9)$ be a permutation of the set $\{1,2,\ldots,9\}$ and let $A$ be the set of all such $X$ . For any $X \in A$, denote $f(X)=x_1+2x_2+\cdots+9x_9$ and $ M=\{f(X)|X \in A \}$. Find $|M|$. ($|S|$ denotes number of members of the set $S$.)

Prove that if $n$ is a positive integer then $2 (n^2 + 1) - n$ is not a square of an integer.

Compute the sum of all positive integers $n$ such that $n^n$ has 325 positive integer divisors. (For example, $4^4=256$ has 9 positive integer divisors: 1, 2, 4, 8, 16, 32, 64, 128, 256.)

Parallel lines $\ell_1$ and $\ell_2$ are drawn in a plane. Points $A_1, A_2, \dots, A_n$ are chosen on $\ell_1$, and points $B_1, B_2, \dots, B_{n+1}$ are chosen on $\ell_2$. All segments $A_iB_j$ are drawn, such that $1 \le i \le n$ and $1 \le j \le n+1$. Let the number of total intersections between these segments (not including endpoints) be denoted by $Q$. Given that no three segments are concurrent, besides at endpoints, prove that $Q$ is divisible by 3. [i]Proposed by Lewis Chen[/i]

A bug travels from $A$ to $B$ along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there? [asy] size(10cm); draw((0.0,0.0)--(1.0,1.7320508075688772)--(3.0,1.7320508075688772)--(4.0,3.4641016151377544)--(6.0,3.4641016151377544)--(7.0,5.196152422706632)--(9.0,5.196152422706632)--(10.0,6.928203230275509)--(12.0,6.928203230275509)); draw((0.0,0.0)--(1.0,1.7320508075688772)--(3.0,1.7320508075688772)--(4.0,3.4641016151377544)--(6.0,3.4641016151377544)--(7.0,5.196152422706632)--(9.0,5.196152422706632)--(10.0,6.928203230275509)--(12.0,6.928203230275509)); draw((3.0,-1.7320508075688772)--(4.0,0.0)--(6.0,0.0)--(7.0,1.7320508075688772)--(9.0,1.7320508075688772)--(10.0,3.4641016151377544)--(12.0,3.464101615137755)--(13.0,5.196152422706632)--(15.0,5.196152422706632)); draw((6.0,-3.4641016151377544)--(7.0,-1.7320508075688772)--(9.0,-1.7320508075688772)--(10.0,0.0)--(12.0,0.0)--(13.0,1.7320508075688772)--(15.0,1.7320508075688776)--(16.0,3.464101615137755)--(18.0,3.4641016151377544)); draw((9.0,-5.196152422706632)--(10.0,-3.464101615137755)--(12.0,-3.464101615137755)--(13.0,-1.7320508075688776)--(15.0,-1.7320508075688776)--(16.0,0)--(18.0,0.0)--(19.0,1.7320508075688772)--(21.0,1.7320508075688767)); draw((12.0,-6.928203230275509)--(13.0,-5.196152422706632)--(15.0,-5.196152422706632)--(16.0,-3.464101615137755)--(18.0,-3.4641016151377544)--(19.0,-1.7320508075688772)--(21.0,-1.7320508075688767)--(22.0,0)); draw((0.0,-0.0)--(1.0,-1.7320508075688772)--(3.0,-1.7320508075688772)--(4.0,-3.4641016151377544)--(6.0,-3.4641016151377544)--(7.0,-5.196152422706632)--(9.0,-5.196152422706632)--(10.0,-6.928203230275509)--(12.0,-6.928203230275509)); draw((3.0,1.7320508075688772)--(4.0,-0.0)--(6.0,-0.0)--(7.0,-1.7320508075688772)--(9.0,-1.7320508075688772)--(10.0,-3.4641016151377544)--(12.0,-3.464101615137755)--(13.0,-5.196152422706632)--(15.0,-5.196152422706632)); draw((6.0,3.4641016151377544)--(7.0,1.7320508075688772)--(9.0,1.7320508075688772)--(10.0,-0.0)--(12.0,-0.0)--(13.0,-1.7320508075688772)--(15.0,-1.7320508075688776)--(16.0,-3.464101615137755)--(18.0,-3.4641016151377544)); draw((9.0,5.1961524)--(10.0,3.464101)--(12.0,3.46410)--(13.0,1.73205)--(15.0,1.732050)--(16.0,0)--(18.0,-0.0)--(19.0,-1.7320)--(21.0,-1.73205080)); draw((12.0,6.928203)--(13.0,5.1961524)--(15.0,5.1961524)--(16.0,3.464101615)--(18.0,3.4641016)--(19.0,1.7320508)--(21.0,1.732050)--(22.0,0)); dot((0,0)); dot((22,0)); label("$A$",(0,0),WNW); label("$B$",(22,0),E); filldraw((2.0,1.7320508075688772)--(1.6,1.2320508075688772)--(1.75,1.7320508075688772)--(1.6,2.232050807568877)--cycle,black); filldraw((5.0,3.4641016151377544)--(4.6,2.9641016151377544)--(4.75,3.4641016151377544)--(4.6,3.9641016151377544)--cycle,black); filldraw((8.0,5.196152422706632)--(7.6,4.696152422706632)--(7.75,5.196152422706632)--(7.6,5.696152422706632)--cycle,black); filldraw((11.0,6.928203230275509)--(10.6,6.428203230275509)--(10.75,6.928203230275509)--(10.6,7.428203230275509)--cycle,black); filldraw((4.6,0.0)--(5.0,-0.5)--(4.85,0.0)--(5.0,0.5)--cycle,white); filldraw((8.0,1.732050)--(7.6,1.2320)--(7.75,1.73205)--(7.6,2.2320)--cycle,black); filldraw((11.0,3.4641016)--(10.6,2.9641016)--(10.75,3.46410161)--(10.6,3.964101)--cycle,black); filldraw((14.0,5.196152422706632)--(13.6,4.696152422706632)--(13.75,5.196152422706632)--(13.6,5.696152422706632)--cycle,black); filldraw((8.0,-1.732050)--(7.6,-2.232050)--(7.75,-1.7320508)--(7.6,-1.2320)--cycle,black); filldraw((10.6,0.0)--(11,-0.5)--(10.85,0.0)--(11,0.5)--cycle,white); filldraw((14.0,1.7320508075688772)--(13.6,1.2320508075688772)--(13.75,1.7320508075688772)--(13.6,2.232050807568877)--cycle,black); filldraw((17.0,3.464101615137755)--(16.6,2.964101615137755)--(16.75,3.464101615137755)--(16.6,3.964101615137755)--cycle,black); filldraw((11.0,-3.464101615137755)--(10.6,-3.964101615137755)--(10.75,-3.464101615137755)--(10.6,-2.964101615137755)--cycle,black); filldraw((14.0,-1.7320508075688776)--(13.6,-2.2320508075688776)--(13.75,-1.7320508075688776)--(13.6,-1.2320508075688776)--cycle,black); filldraw((16.6,0)--(17,-0.5)--(16.85,0)--(17,0.5)--cycle,white); filldraw((20.0,1.7320508075688772)--(19.6,1.2320508075688772)--(19.75,1.7320508075688772)--(19.6,2.232050807568877)--cycle,black); filldraw((14.0,-5.196152422706632)--(13.6,-5.696152422706632)--(13.75,-5.196152422706632)--(13.6,-4.696152422706632)--cycle,black); filldraw((17.0,-3.464101615137755)--(16.6,-3.964101615137755)--(16.75,-3.464101615137755)--(16.6,-2.964101615137755)--cycle,black); filldraw((20.0,-1.7320508075688772)--(19.6,-2.232050807568877)--(19.75,-1.7320508075688772)--(19.6,-1.2320508075688772)--cycle,black); filldraw((2.0,-1.7320508075688772)--(1.6,-1.2320508075688772)--(1.75,-1.7320508075688772)--(1.6,-2.232050807568877)--cycle,black); filldraw((5.0,-3.4641016)--(4.6,-2.964101)--(4.75,-3.4641)--(4.6,-3.9641016)--cycle,black); filldraw((8.0,-5.1961524)--(7.6,-4.6961524)--(7.75,-5.19615242)--(7.6,-5.696152422)--cycle,black); filldraw((11.0,-6.9282032)--(10.6,-6.4282032)--(10.75,-6.928203)--(10.6,-7.428203)--cycle,black);[/asy] $ \textbf{(A)}\ 2112\qquad\textbf{(B)}\ 2304\qquad\textbf{(C)}\ 2368\qquad\textbf{(D)}\ 2384\qquad\textbf{(E)}\ 2400 $

The road that goes from the town to the mountain cottage is $76$ km long. A group of hikers finished it in $10$ days, never travelling more than $16$ km in two consecutive days, but travelling at least $23$ km in three consecutive days. Find the maximum ammount of kilometers that the hikers may have traveled in one day.

Points $A,B,C$ are on a plane such that $AB=BC=CA=6$. At any step, you may choose any three existing points and draw that triangle's circumcentre. Prove that you can draw a point such that its distance from an previously drawn point is: $(a)$ greater than 7 $(b)$ greater than 2019

Let $ ABCDE$ be a convex pentagon such that \[ \angle BAC \equal{} \angle CAD \equal{} \angle DAE\qquad \text{and}\qquad \angle ABC \equal{} \angle ACD \equal{} \angle ADE. \]The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$. [i]Proposed by Zuming Feng, USA[/i]

A cockroach is crawling along the walls inside a cubical room that has an edge length of $\text{3 m}$. If the cockroach starts from the back lower left hand corner of the cube and finishes at the front upper right hand corner, what is the magnitude of the displacement of the cockroach? (a) $\text{3}\sqrt{2} \ \text{m}$ (b) $\text{3}\sqrt[3]{2} \ \text{m}$ (c) $\text{3}\sqrt{3} \ \text{m}$ (d) $\text{3 m}$ (e) $\text{9 m}$