Points $P$ and $Q$ are chosen on diagonal $AC$ of square $ABCD$ such that $AB=AP=CQ=1$. What is the measure of $\angle{PBQ}$ in degrees? [i]2019 CCA Math Bonanza Lightning Round #1.3[/i]

Find all $n \in \mathbb N$ such that $n = \varphi (n) + 402$, where $\varphi$ denotes the Euler phi function.

Prove that there exist two natural numbers $a,b$ such that $|a-m|+|b-n|>1000$ for any relatively prime natural numbers $m,n$.

A quadrilateral pyramid and a regular tetrahedron have edges that are all equal in length. They are glued together so that they have in common $1$ equilateral triangle . Prove that the resulting body has exactly $5$ sides.

In a country, there are towns, some of which are connected by roads. There is a route (not necessarily direct) between every two towns. The Minister of Education has ensured that every town without a school is connected via a direct road to a town that has a school. The Minister of State Optimization wants to ensure that there is a unique path between any two towns (without repeating traveled segments), which may require removing some roads. Is it always possible to achieve this without constructing additional schools while preserving what the Minister of Education has accomplished?

Given a natural number $n$, for what maximal value $k$ it is possible to construct a matrix of size $k \times n$ consisting only of elements $\pm 1$ in such a way that for any interchange of a $+1$ with a $-1$ or vice versa, its rank is equal to $k$?

Let $S_1, S_2,S_3$ be spheres of radii $a, b, c$ respectively whose centers lie on a line $l$. Sphere $S_2$ is externally tangent to $S_1$ and $S_3$, whereas $S_1$ and $S_3$ have no common points. A straight line t touches each of the spheres, Find the sine of the angle between $l$ and $t$

On the board a 20-digit number which have 10 ones and 10 twos in its decimal form is written. It is allowed to choose two different digits and to reverse the order of digits in the interval between them. Is it always possible to get a number divisible by 11 using such operations?

Find all the functions $f:[0,1]\rightarrow \mathbb{R}$ for which we have: \[|x-y|^2\le |f(x)-f(y)|\le |x-y|,\] for all $x,y\in [0,1]$.

Let $ABCD$ be a cyclic quadrilateral. The point $P$ is chosen on the line $AB$ such that the circle passing through $C,D$ and $P$ touches the line $AB$. Similarly, the point $Q$ is chosen on the line $CD$ such that the circle passing through $A,B$ and $Q$ touches the line $CD$. Prove that the distance between $P$ and the line $CD$ equals the distance between $Q$ and $AB$.

$21$ distinct numbers are chosen from the set $\{1,2,3,\ldots,2046\}.$ Prove that we can choose three distinct numbers $a,b,c$ among those $21$ numbers such that \[bc<2a^2<4bc\]

Let $P$ be a point outside a circle $\Gamma$, and let the two tangent lines through $P$ touch $\Gamma$ at $A$ and $B$. Let $C$ be on the minor arc $AB$, and let ray $PC$ intersect $\Gamma$ again at $D$. Let $\ell$ be the line through $B$ and parallel to $PA$. $\ell$ intersects $AC$ and $AD$ at $E$ and $F$, respectively. Prove that $B$ is the midpoint of $EF$.

The diagram below shows a circle with center $F$. The angles are related with $\angle BFC = 2\angle AFB$, $\angle CFD = 3\angle AFB$, $\angle DFE = 4\angle AFB$, and $\angle EFA = 5\angle AFB$. Find the degree measure of $\angle BFC$. [asy] size(4cm); pen dps = fontsize(10); defaultpen(dps); dotfactor=4; draw(unitcircle); pair A,B,C,D,E,F; A=dir(90); B=dir(66); C=dir(18); D=dir(282); E=dir(210); F=origin; dot("$F$",F,NW); dot("$A$",A,dir(90)); dot("$B$",B,dir(66)); dot("$C$",C,dir(18)); dot("$D$",D,dir(306)); dot("$E$",E,dir(210)); draw(F--E^^F--D^^F--C^^F--B^^F--A); [/asy]

Let $n$ and $k$ be positive integers. An $n$-tuple $(a_1, a_2,\ldots , a_n)$ is called a permutation if every number from the set $\{1, 2, . . . , n\}$ occurs in it exactly once. For a permutation $(p_1, p_2, . . . , p_n)$, we define its $k$-mutation to be the $n$-tuple $$(p_1 + p_{1+k}, p_2 + p_{2+k}, . . . , p_n + p_{n+k}),$$ where indices are taken modulo $n$. Find all pairs $(n, k)$ such that every two distinct permutations have distinct $k$-mutations. [i]Remark[/i]: For example, when $(n, k) = (4, 2)$, the $2$-mutation of $(1, 2, 4, 3)$ is $(1 + 4, 2 + 3, 4 + 1, 3 + 2) = (5, 5, 5, 5)$. [i]Proposed by Borna Šimić[/i]

Do there exist a sequence $a_1, a_2, \ldots , a_n, \ldots$ of positive integers such that for any positive integers $i, j$: $$d(a_i + a_j ) = i + j?$$ Here $d(n)$ is the number of positive divisors of a positive integer

Describe in geometric terms the set of points $(x,y)$ in the plane such that $x$ and $y$ satisfy the condition $t^2+yt+x\ge 0$ for all $t$ with $-1\le t\le 1$.

In the figure shown, the small circles have radius $1$. Calculate the area of the gray part of the figure. [img]https://1.bp.blogspot.com/-oy-WirJ6u9o/XzcFc3roVDI/AAAAAAAAMX8/qxNy5I_0RWUOxl-ZE52fnrwo0v0T7If9QCLcBGAsYHQ/s0/1998%2BMohr%2Bp1.png[/img]

An ant decides to walk on the perimeter of an $ABC$ triangle. The ant can start at any vertex. Whenever the ant is in a vertex, it chooses one of the adjacent vertices and walks directly (in a straight line) to the chosen vertex. a) In how many ways can the ant walk around each vertex exactly twice? b) In how many ways can the ant walk around each vertex exactly three times? Note: For each item, consider that the starting vertex is visited.

A string consisting of letters $A, C, G,$ and $U$ is [i]untranslatable[/i] if and only if it has no $\text{AUG}$ as a consecutive substring. For example, $\text{ACUGG}$ is untranslatable. Let $a_n$ denote the number of untranslatable strings of length $n.$ It is given that there exists a unique triple of real numbers $(x,y,z)$ such that $a_n = xa_{n-1} + ya_{n-2} +za_{n-3}$ for all integers $n \ge 100.$ Compute $(x, y,z)$

The numbers $A_1 , A_2 ,... , A_n$ are given. Prove, without calculating derivatives, that the value of $X$ that minimizes the sum $(X - A_1)^2 + (X -A_2)^2 + ...+ (X - A_n)^2$ is precisely the arithmetic mean of the given numbers.

A pack of $2n$ cards contains $n$ different pairs of cards. Each pair consists of two identical cards, either of which is called the twin of the other. A game is played between two players $A$ and $B$. A third person called the [i]dealer[/i] shuffles the pack and deals the cards one by one face upward onto the table. One of the players, called the [i]receiver[/i], takes the card dealt, provided he does not have already its twin. If he does already have the twin, his opponent takes the dealt card and becomes the receiver. $A$ is initially the receiver and takes the first card dealt. The player who first obtains a complete set of $n$ different cards wins the game. What fraction of all possible arrangements of the pack lead to $A$ winning? Prove the correctness of your answer.

Let $ABC$ be a triangle and $I$ the center of its incircle. $P$ is a point inside $ABC$ such that $\angle PBA +\angle PCA = \angle PBC + \angle PCB$. Prove that $AP\geq AI$ with equality iff $P=I$.

Find all poltnomials $ P(x)$ with real coefficients satisfying: For all $ a>1995$, the number of real roots of $ P(x)\equal{}a$ (incuding multiplicity of each root) is greater than 1995, and every roots are greater than 1995.

Let $\triangle ABC$ be an acute-angled triangle with $AB<AC$. Let $M$ and $N$ be the midpoints of $AB$ and $AC$, respectively; let $AD$ be an altitude in this triangle. A point $K$ is chosen on the segment $MN$ so that $BK=CK$. The ray $KD$ meets the circumcircle $\Omega$ of $ABC$ at $Q$. Prove that $C, N, K, Q$ are concyclic.

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.