Given is a chessboard 8x8. We have to place  $n$  black queens and  $n$  white queens, so that no two queens attack. Find the maximal possible  $n$ .
(Two queens attack each other when they have different colors. The queens of the same color don't attack each other)


Given that P be a polynomial with each root is real and each coefficient is either $1$ or $-1$, determine the maximum degree of P.


All the numbers  $1,2,...,9$  are written in the cells of a  $3\times 3$  table (exactly one number in a cell) . Per move it is allowed to choose an arbitrary  $2\times2$  square of the table and either decrease by  $1$  or increase by  $1$  all four numbers of the square. After some number of such moves all numbers of the table become equal to some number  $a$ . Find all possible values of  $a$ .

I.Voronovich


Determine the number of solutions of the simultaneous equations  $ x^2 \plus{} y^3 \equal{} 29$  and  $ \log_3 x \cdot \log_2 y \equal{} 1.$


$201$  positive integers are written on a line, such that both the first one and the last one are equal to  $19999$ . Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equal to a unique integer. Find the second-to-last term on the line.


Find minimum of  $x+y+z$  where  $x$ ,  $y$  and  $z$  are real numbers such that  $x \geq 4$ ,  $y \geq 5$ ,  $z \geq 6$  and  $x^2+y^2+z^2 \geq 90$


Determine all positive real  $M$  such that for any positive reals  $a,b,c$ , at least one of  $a + \dfrac{M}{ab}, b + \dfrac{M}{bc}, c + \dfrac{M}{ca}$  is greater than or equal to  $1+M$ .

Solution: \boxed{M = \frac{1}{2}}


Let  $x, y,z$  satisfy the following inequalities  $\begin{cases} | x + 2y - 3z| \le 6
| x - 2y + 3z|  \le  6
| x - 2y - 3z|  \le 6
| x + 2y + 3z|  \le  6 \end{cases}$ Determine the greatest value of  $M = |x| + |y| + |z|$ .


For all  $x,y,z$  positive real numbers, find the all  $c$  positive real numbers that providing  $$ \frac{x^3y+y^3z+z^3x}{x+y+z}+\frac{4c}{xyz}\ge2c+2 $$

Solution: \boxed{c = 1}


Find all odd prime numbers  $p$  for which there exists a natural number  $g$  for which the sets \[A=\left\{ \left( {{k}^{2}}+1 \right)\,\bmod p|\,k=1,2,\ldots ,\frac{p-1}{2} \right\}\] and \[B=\left\{ {{g}^{k}}\bmod \,p|\,k=1,2,...,\frac{p-1}{2} \right\}\] are equal.

Solution: \boxed{p = 3}


Find all triples of positive integers  $(a,b,c)$  such that the following equations are both true:
I-   $a^2+b^2=c^2$ II-  $a^3+b^3+1=(c-1)^3$

Solution: \boxed{(6, 8, 10)}


$N $  different numbers are written on blackboard and one of these numbers is equal to  $0$ .One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between  $-2016$  and  $2016$  were written on blackboard(and some other numbers maybe). Find  the smallest possible value of  $N $ .

Solution: \boxed{N = 2}


For a positive integer  $n>1$ , let  $g(n)$  denote the largest positive proper divisor of  $n$  and  $f(n)=n-g(n)$ . For example,  $g(10)=5, f(10)=5$  and  $g(13)=1,f(13)=12$ . Let  $N$  be the smallest positive integer such that  $f(f(f(N)))=97$ . Find the largest integer not exceeding  $\sqrt{N}$


Determine all pairs  $(f,g)$  of functions from the set of positive integers to itself that satisfy \[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\] for every positive integer  $n$ . Here,  $f^k(n)$  means  $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$ .

*Proposed by Bojan Bašić, Serbia*

Solution: \boxed{(f(n) = n, g(n) = 1)}


Let  $f(x) = x^2 + 6x + 7$ . Determine the smallest possible value of  $f(f(f(f(x))))$  over all real numbers  $x.$


Determine the smallest possible value of  $$ |2^m - 181^n|, $$  where  $m$  and  $n$  are positive integers.


Let  $ABCD$  be a cyclic quadrilateral, and  $E$  be the intersection of its diagonals. If  $m(\widehat{ADB}) = 22.5^\circ$ ,  $|BD|=6$ , and  $|AD|\cdot|CE|=|DC|\cdot|AE|$ , find the area of the quadrilateral  $ABCD$ .


Determine the smallest positive integer  $M$  with the following property:
For every choice of integers  $a,b,c$ , there exists a polynomial  $P(x)$  with integer coefficients  so that  $P(1)=aM$  and  $P(2)=bM$  and  $P(4)=cM$ .

*Proposed by Gerhard Woeginger, Austria*


In the plane a point  $O$  is and a sequence of points  $P_1, P_2, P_3, \ldots$  are given. The distances  $OP_1, OP_2, OP_3, \ldots$  are  $r_1, r_2, r_3, \ldots$  Let  $\alpha$  satisfies  $0 < \alpha < 1.$  Suppose that for every  $n$  the distance from the point  $P_n$  to any other point of the sequence is  $\geq r^{\alpha}_n.$  Determine the exponent  $\beta$ , as large as possible such that for some  $C$  independent of  $n$
\[r_n \geq Cn^{\beta}, n = 1,2, \ldots\]

Solution: \boxed{\beta = \frac{1}{2(1 - \alpha)}}


No matter how  $n$  real numbers on the interval  $[1,2013]$  are selected, find the least possible value of  $n$  for which it is possible to find a scalene polygon such that its sides are equal to some of the numbers selected.


What fraction of the Earth's volume lies above the  $45$  degrees north parallel? You may assume the Earth is a perfect sphere. The volume in question is the smaller piece that we would get if the sphere were sliced into two pieces by a plane.


Let  $\alpha$  and  $\beta$  be reals. Find the least possible value of   $$ (2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2. $$


Find all positive integers  $a$  such that for any positive integer  $n\ge 5$  we have   $2^n-n^2\mid a^n-n^a$ .

Solution: \boxed{a = 2 \text{ or } a = 4}


Suppose that  $x_1 < x_2 < \dots < x_n$  is a sequence of positive integers such that  $x_k$  divides  $x_{k+2}$  for each  $k = 1, 2, \dots, n-2$ .  Given that  $x_n = 1000$ , what is the largest possible value of  $n$ ?

*Proposed by Evan Chen*


Find all positive integers  $n$  for which all positive divisors of  $n$  can be put into the cells of a rectangular table under the following constraints:

- each cell contains a distinct divisor;
- the sums of all rows are equal; and
- the sums of all columns are equal.

Solution: \boxed{n = 1}


Find all integers  $k$  such that all roots of the following polynomial are also integers:   $$ f(x)=x^3-(k-3)x^2-11x+(4k-8). $$

Solution: \boxed{k = 5}


A paper equilateral triangle with area 2019 is folded over a line parallel to one of its sides.  What is the greatest possible area of the overlap of folded and unfolded parts of the triangle?


Determine all the pairs  $(a,b)$  of positive integers, such that all the following three conditions are satisfied:
1-  $b>a$  and  $b-a$  is a prime number
2- The last digit of the number  $a+b$  is  $3$ 3- The number  $ab$  is a square of an integer.

Solution: \boxed{(4, 9)}


Derek fills a square  $10$  by  $10$  grid with  $50$   $1$ s and  $50$   $2$ s. He takes the product of the numbers in each of the  $10$  rows. He takes the product of the numbers in each of the  $10$  columns. He then sums these  $20$  products up to get an integer  $N.$  Find the minimum possible value of  $N.$


For two quadratic trinomials  $P(x)$  and  $Q(x)$  there is a linear function  $\ell(x)$  such that  $P(x)=Q(\ell(x))$  for all real  $x$ . How many such linear functions  $\ell(x)$  can exist?

*(A. Golovanov)*


Find all positive integers  $ n$  such that there exists sequence consisting of  $ 1$  and  $ - 1: a_{1},a_{2},\cdots,a_{n}$  satisfying  $ a_{1}\cdot1^2 + a_{2}\cdot2^2 + \cdots + a_{n}\cdot n^2 = 0.$

Solution: \boxed{n \equiv 0 \text{ or } 3 \pmod{4}}


Find the largest real number  $\lambda$  with the following property: for any positive real numbers  $p,q,r,s$  there exists a complex number  $z=a+bi$ ( $a,b\in \mathbb{R})$   such that   $$  |b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0. $$


For a set  $ P$  of five points in the plane, no three of them being collinear, let  $ s(P)$  be the numbers of acute triangles formed by vertices in  $ P$ .
Find the maximum value of  $ s(P)$  over all such sets  $ P$ .


Find all positive integers  $n$  such that there exist non-constant polynomials with integer coefficients  $f_1(x),...,f_n(x)$  (not necessarily distinct) and  $g(x)$  such that   $$ 1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x) $$


Given the equation $x^3+y^3=x^2yz+xy^2z+2$, determine how many integer triples $(x,y,z)$ satisfy the equation.


Find the greatest integer  $n$ , such that there are  $n+4$  points  $A$ ,  $B$ ,  $C$ ,  $D$ ,  $X_1,\dots,~X_n$  in the plane with  $AB\ne CD$  that satisfy the following condition: for each  $i=1,2,\dots,n$  triangles  $ABX_i$  and  $CDX_i$  are equal.

Solution: \boxed{n = 4}


Let  $S$  be the set of all positive integers  $n$  such that each of the numbers  $n + 1$ ,  $n + 3$ ,  $n + 4$ ,  $n + 5$ ,  $n + 6$ , and  $n + 8$  is composite. Determine the largest integer  $k$  with the following property: For each  $n \in S$  there exist at least  $k$  consecutive composite integers in the set
{ $n, n +1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9$ }.


Find all prime numbers  $p$  such that there exists a unique  $a \in \mathbb{Z}_p$  for which  $a^3 - 3a + 1 = 0.$

Solution: \boxed{p = 3}


Alice and Bob want to play a game. In the beginning of the game, they are teleported to two random position on a train, whose length is  $ 1 $  km. This train is closed and dark. So they dont know where they are.

Fortunately, both of them have iPhone 133, it displays some information:
1. Your facing direction
2. Your total walking distance
3. whether you are at the front of the train
4. whether you are at the end of the train
Moreover, one may see the information of the other one. Once Alice and Bob meet, the game ends.

Alice and Bob can only discuss their strategy before the game starts. Find the least value  $ x $  so that they are guarantee to end the game with total walking distance  $ \le x $  km.

Solution: \boxed{x = 1.5}


A positive interger number  $k$  is called “ $t-m$ ”-property if forall positive interger number  $a$ , there exists a positive integer number  $n$  such that ${{1}^{k}}+{{2}^{k}}+{{3}^{k}}+...+{{n}^{k}} \equiv a (\bmod m).$ a) Find all positive integer numbers  $k$  which has  $t-20$ -property.
b) Find smallest positive integer number  $k$  which has  $t-{{20}^{15}}$ -property.

Solution: \boxed{k = 4}


Given the function $f(x) = |x| + \left|\frac{1 - 2013x}{2013 - x}\right|$, where $x \in [-1, 1]$, find the smallest value of $f(x)$.


Find the smallest positive integer  $j$  such that for every polynomial  $p(x)$  with integer coefficients and for every integer  $k,$  the integer
\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\]
(the  $j$ -th derivative of  $p(x)$  at  $k$ ) is divisible by  $2016.$


Find all pairs  $(a,b)$  of positive integers, such that for **every**  $n$  positive integer, the equality  $a^n+b^n=c_n^{n+1}$  is true, for some  $c_n$  positive integer.

Solution: \boxed{(2, 2)}


If  $a,b,c,d$  are Distinct Real no. such that  $a = \sqrt{4+\sqrt{5+a}}$  $b = \sqrt{4-\sqrt{5+b}}$  $c = \sqrt{4+\sqrt{5-c}}$  $d = \sqrt{4-\sqrt{5-d}}$ Then  $abcd = $


Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units.  Given that the volume of this set is  $(m + n \pi)/p$ , where  $m$ ,  $n$ , and  $p$  are positive integers, and  $n$  and  $p$  are relatively prime, find  $m + n + p$ .


$ABCD$  is a rectangle with  $AB = CD = 2$ . A circle centered at  $O$  is tangent to  $BC$ ,  $CD$ , and  $AD$  (and hence has radius  $1$ ). Another circle, centered at  $P$ , is tangent to circle  $O$  at point  $T$  and is also tangent to  $AB$  and  $BC$ . If line  $AT$  is tangent to both circles at  $T$ , find the radius of circle  $P$ .

Solution: \boxed{3 - 2\sqrt{2}}


$101$  numbers are written on a blackboard:  $1^2, 2^2, 3^2, \cdots, 101^2$ . Alex choses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number.


Let  $n\geq 2$  be an integer. Determine all sets of real numbers  $(a_1, a_2,\dots, a_n)$  such that  $a_1-2a_2, a_2-2a_3,\dots, a_{n-1}-2a_n, a_n-2a_1$  are rearrangements of  $a_1, a_2,\dots, a_n$ .

Solution: \boxed{a_1 = a_2 = \cdots = a_n = 0}


Determine the largest positive integer  $n$  for which there exists a set  $S$  with exactly  $n$  numbers such that
- each member in  $S$  is a positive integer not exceeding  $2002$ ,
- if  $a,b\in S$  (not necessarily different), then  $ab\not\in S$ .


In the game of *Winners Make Zeros*, a pair of positive integers  $(m,n)$  is written on a sheet of paper.  Then the game begins, as the players make the following legal moves:

- If  $m\geq n$ , the player choose a positive integer  $c$  such that  $m-cn\geq 0$ , and replaces  $(m,n)$  with  $(m-cn,n)$ .
- If  $m<n$ , the player choose a positive integer  $c$  such that  $n-cm\geq 0$ , and replaces  $(m,n)$  with  $(m,n-cm)$ .

When  $m$  or  $n$  becomes  $0$ , the game ends, and the last player to have moved is declared the winner.  If  $m$  and  $n$  are originally  $2007777$  and  $2007$ , find the largest choice the first player can make for  $c$  (on his first move) such that the first player has a winning strategy after that first move.


Let the number  $x$ . Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent  of the number  $x$  (for example,  $x\cdot x=x^{2}$ ,  $x^{2}\cdot x^{2}=x^{4}$ ,  $x^{4}: x=x^{3}$ , etc). Determine the minimal number of operations needed for calculating  $x^{2006}$ .


Let  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ ,  $f$  and  $g$  be seven distinct positive integers not bigger than  $7$ . Find all primes which can be expressed as  $abcd+efg$


Let  $ABC$  be a triangle with  $AC > AB$ .  Let  $P$  be the intersection point of the perpendicular bisector of  $BC$  and the internal angle bisector of  $\angle{A}$ .  Construct points  $X$  on  $AB$  (extended) and  $Y$  on  $AC$  such that  $PX$  is perpendicular to  $AB$  and  $PY$  is perpendicular to  $AC$ .  Let  $Z$  be the intersection point of  $XY$  and  $BC$ .
Determine the value of  $\frac{BZ}{ZC}$ .


Let  $S$  be the set of positive integers  $n$  such that
\[3\cdot
 \varphi (n)=n,\]
where  $\varphi (n)$  is the number of positive integers  $k\leq n$  such that  $\gcd (k, n)=1$ . Find
\[\sum_{n\in S} \, \frac{1}{n}.\]

[i]Proposed by Nathan Ramesh


Find all pairs  $(a,b)$  of prime positive integers  $a,b$  such that number  $A=3a^2b+16ab^2$  equals to a square of an integer.

Solution: \boxed{(19, 19) \text{ and } (2, 3)}


Triangle  $ABC$  is right angled such that  $\angle ACB=90^{\circ}$  and  $\frac {AC}{BC} = 2$ . Let the line parallel to side  $AC$  intersects line segments  $AB$  and  $BC$  in  $M$  and  $N$  such that  $\frac {CN}{BN} = 2$ . Let  $O$  be the intersection point of lines  $CM$  and  $AN$ . On segment  $ON$  lies point  $K$  such that  $OM+OK=KN$ . Let  $T$  be the intersection point of angle bisector of  $\angle ABC$  and line from  $K$  perpendicular to  $AN$ . Determine value of   $\angle MTB$ .


Graph  $G_1$  of a quadratic trinomial  $y = px^2 + qx + r$  with real coefficients intersects the graph  $G_2$  of a quadratic trinomial  $y = x^2$  in points  $A$ ,  $B$ . The intersection of tangents to  $G_2$  in points  $A$ ,  $B$  is point  $C$ . If  $C \in G_1$ , find all possible values of  $p$ .

Solution: \boxed{p = 2}


Consider all functions  $f: \mathbb{Z} \rightarrow \mathbb{Z}$  satisfying \[f(f(x) + 2x + 20) = 15. \]

Call an integer  $n$   $\textit{good}$  if  $f(n)$  can take any integer value. In other words, if we fix  $n$ , for any integer  $m$ , there exists a function  $f$  such that  $f(n) = m.$  Find the sum of all good integers  $x.$


For which real numbers  $k > 1$   does there exist a bounded set of positive real numbers  $S$   with at
least  $3$  elements such that  $$ k(a - b)\in S $$  for all  $a,b\in  S $  with  $a > b?$ Remark: A set of positive real numbers  $S$   is bounded if there exists a positive real number  $M$  such that $x < M$   for all  $x \in  S.$

Solution: \boxed{k = 2}


In a convex quadrilateral  $ABCD$ ,  $\angle ABC = 90^o$  ,  $\angle BAC = \angle  CAD$ ,  $AC = AD, DH$  is the alltitude of the triangle  $ACD$ . In what ratio does the line  $BH$  divide the segment  $CD$ ?


Call a sequence of positive integers  $(a_n)_{n \ge 1}$  a "CGMO sequence" if  $(a_n)_{n \ge 1}$  strictly increases, and for all integers  $n \ge 2022$ ,  $a_n$  is the smallest integer such that there exists a non-empty subset of  $\{a_{1}, a_{2}, \cdots, a_{n-1} \}$   $A_n$  where  $a_n \cdot \prod\limits_{a \in A_n} a$  is a perfect square.

Proof: there exists  $c_1, c_2 \in \mathbb{R}^{+}$  s.t. for any "CGMO sequence"  $(a_n)_{n \ge 1}$  , there is a positive integer  $N$  that satisfies any  $n \ge N$ ,  $c_1 \cdot n^2 \le a_n \le c_2 \cdot n^2$ .

Solution: \boxed{c_1 \cdot n^2 \le a_n \le c_2 \cdot n^2}


$ AA_{3}$  and  $ BB_{3}$  are altitudes of acute-angled  $ \triangle ABC$ . Points  $ A_{1}$  and  $ B_{1}$  are second points of intersection lines  $ AA_{3}$  and  $ BB_{3}$  with circumcircle of  $ \triangle ABC$  respectively.  $ A_{2}$  and  $ B_{2}$  are points on  $ BC$  and  $ AC$  respectively.  $ A_{1}A_{2}\parallel AC$ ,  $ B_{1}B_{2}\parallel BC$ . Point  $ M$  is midpoint of  $ A_{2}B_{2}$ .  $ \angle BCA \equal{} x$ . Find  $ \angle A_{3}MB_{3}$ .


Denote by  $P(n)$  the greatest prime divisor of  $n$ . Find all integers  $n\geq 2$  for which \[P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor\]

Solution: \boxed{n = 3}


Suppose  $\overline{AB}$  is a segment of unit length in the plane. Let  $f(X)$  and  $g(X)$  be functions of the plane such that  $f$  corresponds to rotation about  $A$   $60^\circ$  counterclockwise and  $g$  corresponds to rotation about  $B$   $90^\circ$  clockwise. Let  $P$  be a point with  $g(f(P))=P$ ; what is the sum of all possible distances from  $P$  to line  $AB$ ?

Solution: \boxed{\frac{1 + \sqrt{3}}{2}}


A positive integer  $n\geq 4$  is called *interesting* if there exists a complex number  $z$  such that  $|z|=1$  and \[1+z+z^2+z^{n-1}+z^n=0.\] Find how many interesting numbers are smaller than  $2022.$


A sequence of numbers  $ a_1, a_2, a_3, ...$  satisfies

(i)  $ a_1 \equal{} \frac{1}{2}$
(ii)  $ a_1\plus{}a_2 \plus{} \cdots \plus{} a_n \equal{} n^2 a_n \ (n \geq 1)$

Determine the value of  $ a_n \ (n \geq 1)$ .

Solution: \boxed{a_n = \frac{1}{n(n+1)}}


$\definecolor{A}{RGB}{190,0,60}\color{A}\fbox{A1.}$  Find all  $f:\mathbb{R}\rightarrow \mathbb{R}$  such that   $$ \definecolor{A}{RGB}{80,0,200}\color{A} x^4+y^4+z^4\ge f(xy)+f(yz)+f(zx)\ge xyz(x+y+z) $$  holds for all  $a,b,c\in\mathbb{R}$ .

*Proposed by***usjl**.
#1733

Solution: \boxed{f(x) = x^2}


Parallelogram  $ABCD$  is such that angle  $B < 90$  and  $AB<BC$ . Points E and F are on the circumference of  $\omega$  inscribing triangle ABC, such that tangents to  $\omega$  in those points pass through D. If  $\angle EDA= \angle{FDC}$ , find  $\angle{ABC}$ .


For a positive integer  $n$  consider all its divisors  $1 = d_1 < d_2 < \ldots < d_k = n$ . For  $2 \le i \le k-1$ , let's call divisor  $d_i$  good, if  $d_{i-1}d_{i+1}$  isn't divisible by  $d_i$ . Find all  $n$ , such that the number of their good divisors is smaller than the number of their prime distinct divisors.

*Proposed by Mykhailo Shtandenko*

Solution: \boxed{n = 1}


Michelle has a word with  $2^n$  letters, where a word can consist of letters from any alphabet. Michelle performs a swicheroo on the word as follows: for each  $k = 0, 1, \ldots, n-1$ , she switches the first  $2^k$  letters of the word with the next  $2^k$  letters of the word. For example, for  $n = 3$ , Michelle changes \[
ABCDEFGH \to BACDEFGH \to CDBAEFGH \to EFGHCDBA
\] in one switcheroo.

In terms of  $n$ , what is the minimum positive integer  $m$  such that after Michelle performs the switcheroo operation  $m$  times on any word of length  $2^n$ , she will receive her original word?


Let  $ABC$  be an equilateral triangle and a point M inside the triangle such that  $MA^2 = MB^2 +MC^2$ . Draw an equilateral triangle  $ACD$  where  $D \ne B$ . Let the point  $N$  inside  $\vartriangle ACD$  such that  $AMN$  is an equilateral triangle. Determine  $\angle BMC$ .


Given a positive integer  $n$ , consider a triangular array with entries  $a_{ij}$  where  $i$  ranges from  $1$  to  $n$  and  $j$  ranges from  $1$  to  $n-i+1$ . The entries of the array are all either  $0$  or  $1$ , and, for all  $i > 1$  and any associated  $j$  ,  $a_{ij}$  is  $0$  if  $a_{i-1,j} = a_{i-1,j+1}$ , and  $a_{ij}$  is  $1$  otherwise. Let  $S$  denote the set of binary sequences of length  $n$ , and define a map  $f \colon S \to S$  via  $f \colon (a_{11}, a_{12},\cdots ,a_{1n}) \to (a_{n1}, a_{n-1,2}, \cdots , a_{1n})$ . Determine the number of fixed points of  $f$ .

Solution: \boxed{2^{\left\lceil \frac{n}{2} \right\rceil}}


It is given an acute triangle   $ABC$  ,  $AB \neq AC$   where the feet of altitude from  $A$  its  $H$ .  In the extensions of the sides  $AB$  and  $AC$   (in the direction of  $B$  and  $C$ ) we take the points  $P$  and  $Q$  respectively such that  $HP=HQ$  and the points  $B,C,P,Q$  are concyclic.
Find the ratio  $\tfrac{HP}{HA}$ .


Find the greatest possible value of  $pq + r$ , where p, q, and r are (not necessarily distinct) prime numbers satisfying  $pq + qr + rp = 2016$ .


Given that  $a,b,c$  are integers with  $abc = 60$ , and that complex number  $\omega \neq 1$  satisfies  $\omega^3=1$ , find the minimum possible value of  $|a + b\omega + c\omega^2|$ .


Find all primes  $p$ , such that there exist positive integers  $x$ ,  $y$  which satisfy  $$ \begin{cases}
p + 49 = 2x^2
p^2 + 49 = 2y^2
\end{cases} $$

Solution: \boxed{p = 23}


Let  $ m, n \geq 1$  be two coprime integers and let also  $ s$  an arbitrary integer. Determine the number of subsets  $ A$  of   $ \{1, 2, ..., m \plus{} n \minus{} 1\}$  such that  $ |A| \equal{} m$  and  $ \sum_{x \in A} x \equiv s \pmod{n}$ .

Solution: \boxed{\frac{1}{n} \binom{m+n-1}{m}}


$(BEL 1)$  A parabola  $P_1$  with equation  $x^2 - 2py = 0$  and parabola  $P_2$  with equation  $x^2 + 2py = 0, p > 0$ , are given. A line  $t$  is tangent to  $P_2.$  Find the locus of pole  $M$  of the line  $t$  with respect to  $P_1.$

Solution: \boxed{x^2 + 2py = 0}


Let  $ABC$  be a triangle with  $\angle A = 120^\circ$ ,  $I$  be the incenter, and  $M$  be the midpoint of  $BC$ . The line passing through  $M$  and parallel to  $AI$  meets the circle with diameter  $BC$  at points  $E$  and  $F$  ( $A$  and  $E$  lie on the same semiplane with respect to  $BC$ ). The line passing through  $E$  and perpendicular to  $FI$  meets  $AB$  and  $AC$  at points  $P$  and  $Q$  respectively. Find the value of  $\angle PIQ$ .

Solution: \boxed{\angle PIQ = 90^\circ}


Find the largest positive integer  $n$  ( $n \ge 3$ ), so that there is a convex  $n$ -gon, the tangent of each interior angle is an integer.

Solution: \boxed{n_{\text{max}} = 8}


A configuration of  $4027$  points in the plane is called Colombian if it consists of  $2013$  red points and  $2014$  blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied:

i) No line passes through any point of the configuration.

ii) No region contains points of both colors.

Find the least value of  $k$  such that for any Colombian configuration of  $4027$  points, there is a good arrangement of  $k$  lines.

Proposed by *Ivan Guo* from *Australia.*


Aaron takes a square sheet of paper, with one corner labeled  $A$ . Point  $P$  is chosen at random inside of the square and Aaron folds the paper so that points  $A$  and  $P$  coincide. He cuts the sheet along the crease and discards the piece containing  $A$ . Let  $p$  be the probability that the remaining piece is a pentagon. Find the integer nearest to  $100p$ .

*Proposed by Aaron Lin*


Find all positive integer  $n$  and nonnegative integer  $a_1,a_2,\dots,a_n$  satisfying: $i$  divides exactly  $a_i$  numbers among  $a_1,a_2,\dots,a_n$ , for each  $i=1,2,\dots,n$ .
( $0$  is divisible by all integers.)

Solution: \boxed{n = 1}


The positive real numbers  $a, b, c$  satisfy the equality  $a + b + c = 1$ . For every natural number  $n$  find the minimal possible value of the expression   $$ E=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c} $$

Solution: \boxed{\frac{3^{n+2} + 3}{2}}


Given that the integers $0 \leq a < 2^{2008}$ and $0 \leq b < 8$ satisfy the equivalence $7(a+2^{2008}b) \equiv 1 \pmod{2^{2011}}$, determine the value of b.


Determine all polynomials  $P$  with real coeffients such that  $1 + P(x) = \frac12 (P(x -1) + P(x + 1))$  for all real  $x$ .

Solution: \boxed{P(x) = x^2 + bx + c}


An odd integer  $ n \ge 3$  is said to be nice if and only if there is at least one permutation  $ a_{1}, \cdots, a_{n}$  of  $ 1, \cdots, n$  such that the  $ n$  sums  $ a_{1} \minus{} a_{2} \plus{} a_{3} \minus{} \cdots \minus{} a_{n \minus{} 1} \plus{} a_{n}$ ,  $ a_{2} \minus{} a_{3} \plus{} a_{3} \minus{} \cdots \minus{} a_{n} \plus{} a_{1}$ ,  $ a_{3} \minus{} a_{4} \plus{} a_{5} \minus{} \cdots \minus{} a_{1} \plus{} a_{2}$ ,  $ \cdots$ ,  $ a_{n} \minus{} a_{1} \plus{} a_{2} \minus{} \cdots \minus{} a_{n \minus{} 2} \plus{} a_{n \minus{} 1}$  are all positive. Determine the set of all `nice' integers.

Solution: \boxed{n \equiv 1 \pmod{4}}


Find all integers  $n$  greater than or equal to  $4$  that satisfy the following conditions:

- Take an arbitrary convex  $n$ -gon  $P$  on the coordinate plane whose vertices are lattice points (points whose coordinates are both integers). There are a total of  $n$  triangles that share two sides with  $P$ . Let  $S_1, S_2, \ldots, S_n$  be their areas, and let  $S$  be the area of  $P$ . Then, the greatest common divisor of  $2S_1, 2S_2, \ldots, 2S_n$  divides the integer  $2S$ .

Solution: \boxed{n = 4}


The number of integer solutions $x$ for the equation $(12x -1)(6x - 1)(4x -1)(3x - 1) = 330$ must be determined.


Consider the domain  $D$  expressed by the following system inequality:
\[x^2+(y-1)^2\leq 1,\ x\geq \frac{\sqrt{2}}{3}.\]

Suppose that the line  $l$  passes through the origin and the common part of  $l$  and  $D$  is line segment.
Find the maximum length of the line segment  $L$ , then when  $L$  is maximized, find the value of  $\cos \theta$  for the angle  $\theta\ \left(0<\theta <\frac{\pi}{2}\right)$  between the  $x$ -axis and  $l$ .

Solution: \boxed{\cos \theta = \frac{\sqrt{3}}{3}}


Find all non-zero real numbers  $ x, y, z$  which satisfy the system of equations:
\[ (x^2 \plus{} xy \plus{} y^2)(y^2 \plus{} yz \plus{} z^2)(z^2 \plus{} zx \plus{} x^2) \equal{} xyz\]
\[ (x^4 \plus{} x^2y^2 \plus{} y^4)(y^4 \plus{} y^2z^2 \plus{} z^4)(z^4 \plus{} z^2x^2 \plus{} x^4) \equal{} x^3y^3z^3\]

Solution: \boxed{\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right)}


Radii of five concentric circles  $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$  form a geometric progression with common ratio  $q$  in this order. What is the maximal value of  $q$  for which it's possible to draw a broken line  $A_0A_1A_2A_3A_4$  consisting of four equal segments such that  $A_i$  lies on  $\omega_i$  for every  $i=\overline{0,4}$ ?

<details><summary>thanks </summary>Thanks to the user Vlados021 for translating the problem.</details>

Solution: \boxed{\frac{1 + \sqrt{5}}{2}}


Find all triplets of real numbers  $(x,y,z)$  such that the following equations are satisfied simultaneously:
\begin{align*}
 x^3+y=z^2
y^3+z=x^2
z^3+x =y^2
\end{align*}

Solution: \boxed{(0, 0, 0)}


In triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Find angle B.

Solution: \boxed{\angle B = 45^\circ}


What is the value of  $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $   $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $


Consider a sequence  $\{a_n\}_{n\geq 0}$  such that  $a_{n+1}=a_n-\lfloor{\sqrt{a_n}}\rfloor\ (n\geq 0),\ a_0\geq 0$ .

(1) If  $a_0=24$ , then find the smallest  $n$  such that  $a_n=0$ .

(2) If  $a_0=m^2\ (m=2,\ 3,\ \cdots)$ , then for  $j$  with  $1\leq j\leq m$ , express  $a_{2j-1},\ a_{2j}$  in terms of  $j,\ m$ .

(3) Let  $m\geq 2$  be integer and for integer  $p$  with  $1\leq p\leq m-1$ , let  $a\0=m^2-p$ . Find  $k$  such that  $a_k=(m-p)^2$ , then
find the smallest  $n$  such that  $a_n=0$ .

Solution: \boxed{2m - 1}


Graphistan has  $2011$  cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer  $k$  such that no matter how these flights are arranged it is possible to travel between any two cities in Graphistan riding only GA flights as long as the absolute values of the difference between the number of flights originating and terminating at any city is not more than  $k.$


How many primes $p$ are there such that $2p^4-7p^2+1$ is equal to the square of an integer?


Find the number of pairs  $(n,C)$  of positive integers such that  $C\leq 100$  and  $n^2+n+C$  is a perfect square.


Three positive reals  $x , y , z $  satisfy  $x^2 + y^2 = 3^2
y^2 + yz + z^2 = 4^2
x^2 + \sqrt{3}xz + z^2 = 5^2 .$
Find the value of  $2xy + xz + \sqrt{3}yz$


Every single point on the plane with integer coordinates is coloured either red, green or blue. Find the least possible positive integer  $n$  with the following property: no matter how the points are coloured, there is always a triangle with area  $n$  that has its  $3$  vertices with the same colour.


To any triangle with side lengths  $a,b,c$  and the corresponding angles  $\alpha, \beta, \gamma$  (measured in radians), the 6-tuple  $(a,b,c,\alpha, \beta, \gamma)$  is assigned. Find the minimum possible number  $n$  of distinct terms in the 6-tuple assigned to a scalene triangle.


Let  $p$  be a prime number and  $F=\left \{0,1,2,...,p-1 \right \}$ . Let  $A$  be a proper subset of  $F$  that satisfies the following property:  if  $a,b \in A$ , then  $ab+1$  (mod  $p$ )  $ \in A$ .
How many elements can  $A$  have? (Justify your answer.)

Solution: \boxed{|A| = 1}


Solve the equation for positive integers  $m, n$ :
\[\left \lfloor \frac{m^2}n \right \rfloor + \left \lfloor \frac{n^2}m \right \rfloor = \left \lfloor \frac mn + \frac nm \right \rfloor +mn\]

Solution: \boxed{(m, n) = (2, 1)}


An  $n \times n$  complex matrix  $A$  is called \emph{t-normal} if $AA^t = A^t A$  where  $A^t$  is the transpose of  $A$ .  For each  $n$ ,
determine the maximum dimension of a linear space of complex  $n
\times n$  matrices consisting of t-normal matrices.

Proposed by Shachar Carmeli, Weizmann Institute of Science


Let  $x_1,\ldots,x_n$  be a sequence with each term equal to  $0$  or  $1$ . Form a triangle as follows: its first row is  $x_1,\ldots,x_n$  and if a row if  $a_1, a_2, \ldots, a_m$ , then the next row is  $a_1 + a_2, a_2 + a_3, \ldots, a_{m-1} + a_m$  where the addition is performed modulo  $2$  (so  $1+1=0$ ). For example, starting from  $1$ ,  $0$ ,  $1$ ,  $0$ , the second row is  $1$ ,  $1$ ,  $1$ , the third one is  $0$ ,  $0$  and the fourth one is  $0$ .

A sequence is called good it is the same as the sequence formed by taking the last element of each row, starting from the last row (so in the above example, the sequence is  $1010$  and the corresponding sequence from last terms is  $0010$  and they are not equal in this case). How many possibilities are there for the sequence formed by taking the first element of each row, starting from the last row, which arise from a good sequence?

Solution: \boxed{2^{\lceil n/2 \rceil}}


$13$  fractions are corrected by using each of the numbers  $1,2,...,26$  once.**Example:** $\frac{12}{5},\frac{18}{26}.... $ What is the maximum number of fractions which are integers?


$n$  is a positive integer. Let  $a(n)$  be the smallest number for which  $n\mid a(n)!$ Find all solutions of:  $$ \frac{a(n)}{n}=\frac{2}{3} $$

Solution: \boxed{n = 9}


Below is a diagram showing a  $6 \times 8$  rectangle divided into four  $6 \times 2$  rectangles and one diagonal line. Find the total perimeter of the four shaded trapezoids.


We call the polynomial  $P (x)$  simple if the coefficient of each of its members belongs to the set  $\{-1, 0, 1\}$ .
Let  $n$  be a positive integer,  $n> 1$ . Find the smallest possible number of terms with a non-zero coefficient in a simple  $n$ -th degree polynomial with all values at integer places are divisible by  $n$ .


Find the greatest integer  $A$  for which in any permutation of the numbers  $1, 2, \ldots , 100$  there exist ten consecutive numbers whose sum is at least  $A$ .


Determine the maximum number of bishops that we can place in a  $8 \times 8$  chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop.

Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals the  $2$  main diagonals and the ones parallel to those ones.


There are  $2024$  people, which are knights and liars and some of them are friends. Every person is asked for the number of their friends and the answers were  $0,1, \ldots, 2023$ . Every knight answered truthfully, while every liar changed the real answer by exactly  $1$ . What is the minimal number of liars?


Write number 2013 in a sum of m composite numbers. What is the largest value of m?


If integers  $a$ ,  $b$ ,  $c$ , and  $d$  satisfy  $ bc + ad = ac + 2bd = 1 $ , find all possible values of  $ \frac {a^2 + c^2}{b^2 + d^2} $ .


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.


If the integers $1,2,\dots,n$ can be divided into two sets such that each of the two sets does not contain the arithmetic mean of its any two elements, find the largest possible value of $n$.


For positive integer  $n$  we define  $f(n)$  as sum of all of its positive integer divisors (including  $1$  and  $n$ ). Find all positive integers  $c$  such that there exists strictly increasing infinite sequence of positive integers  $n_1, n_2,n_3,...$  such that for all  $i \in \mathbb{N}$  holds  $f(n_i)-n_i=c$

Solution: \boxed{c = 1}


How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself?


Let  $n\ge 3$ . Suppose  $a_1, a_2, ... , a_n$  are  $n$   distinct in pairs real numbers.
In terms of  $n$ , find the smallest possible number of different assumed values by the following  $n$  numbers:  $$ a_1 + a_2, a_2 + a_3,..., a_{n- 1} + a_n, a_n + a_1 $$


Given an acute triangle  $ABC$  with altituties AD and BE. O circumcinter of  $ABC$ .If o lies on the segment DE then find the value of  $sinAsinBcosC$


Let  $ABC$  be an acute-angled triangle and  $P$  be a point in its interior. Let  $P_A,P_B$  and  $P_c$  be the images of  $P$  under reflection in the sides  $BC,CA$ , and  $AB$ , respectively. If  $P$  is the orthocentre of the triangle  $P_AP_BP_C$  and if the largest angle of the triangle that can be formed by the line segments $ PA, PB$ . and  $PC$  is  $x^o$ , determine the value of  $x$ .


Find all positive integers  $m$  for which  $2001\cdot S (m) = m$  where  $S(m)$  denotes the sum of the digits of  $m$ .


For some real number  $c,$  the graphs of the equation  $y=|x-20|+|x+18|$  and the line  $y=x+c$  intersect at exactly one point. What is  $c$ ?


Given that $ n$ is any whole number, determine the smallest positive integer that always divides $ n^2(n^2 \minus{} 1)$.


Let  $A$ ,  $B$ ,  $C$ ,  $D$ ,  $E$ ,  $F$ ,  $G$ ,  $H$ ,  $I$  be nine points in space such that  $ABCDE$ ,  $ABFGH$ , and  $GFCDI$  are each regular pentagons with side length  $1$ . Determine the lengths of the sides of triangle  $EHI$ .


Given the equation $2\cos(2x)\left(\cos(2x)-\cos\left(\frac{2014\pi^2}{x}\right)\right)=\cos(4x)-1$, find the sum of all positive real solutions $x$ to this equation.


Find all quadruples  $(p, q, r, n)$  of prime numbers  $p, q, r$  and positive integer numbers  $n$ , such that   $$ p^2 = q^2 + r^n $$  (Walther Janous)

Solution: \boxed{(p, q, r, n) = (3, 2, 5, 1) \text{ and } (p, q, r, n) = (5, 3, 2, 4)}


For real numbers  $a$  and  $b$ , define  $$ f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}. $$  Find the smallest possible value of the expression   $$ f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b). $$

Solution: \boxed{4 \sqrt{2018}}


For how many natural numbers  $n$  between  $1$  and  $2014$  (both inclusive) is  $\frac{8n}{9999-n}$  an integer?


Let  $M$  be the intersection point of the medians  $AD$  and  $BE$  of a right triangle  $ABC$  ( $\angle C=90^\circ$ ). It is known that the circumcircles of triangles  $AEM$  and  $CDM$  are tangent. Find the angle  $\angle BMC.$


Find the largest  $n$  such that the last nonzero digit of  $n!$  is  $1$ .


Let  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$  be real strictly positive real numbers such that  $abcde = 1$ . Then is true the following inequality:  $$ \frac{de}{a(b+1)}+\frac{ea}{b(c+1)}+\frac{ab}{c(d+1)}+\frac{bc}{d(e+1)}+\frac{cd}{e(a+1)}\geq \frac{5}{2} $$


Find all integers  $n\geq 1$  such that there exists a permutation  $(a_1,a_2,...,a_n)$  of  $(1,2,...,n)$  such that  $a_1+a_2+...+a_k$  is divisible by  $k$  for  $k=1,2,...,n$

Solution: \boxed{n = 1}


Every cell of an $8\times8$ chessboard contains either $1$ or $-1$. Given that there are at least four rows such that the sum of numbers inside the cells of those rows is positive, determine the maximum number of columns such that the sum of numbers inside the cells of those columns is less than $-3$.


$a_0, a_1, \ldots, a_{100}$  and  $b_1, b_2,\ldots, b_{100}$  are sequences of real numbers, for which the property holds: for all  $n=0, 1, \ldots, 99$ , either  $$ a_{n+1}=\frac{a_n}{2} \quad \text{and} \quad b_{n+1}=\frac{1}{2}-a_n, $$  or  $$ a_{n+1}=2a_n^2 \quad \text{and} \quad b_{n+1}=a_n. $$  Given  $a_{100}\leq a_0$ , what is the maximal value of  $b_1+b_2+\cdots+b_{100}$ ?


In a pile you have 100 stones. A partition of the pile in  $ k$  piles is *good* if:

1) the small piles have different numbers of stones;

2) for any partition of one of the small piles in 2 smaller piles, among the  $ k \plus{} 1$  piles you get 2 with the same number of stones (any pile has at least 1 stone).

Find the maximum and minimal values of  $ k$  for which this is possible.

Solution: \boxed{k_{\text{max}} = 13}


Let  $a,b,c$  be distinct real numbers such that  $a+b+c>0$ . Let  $M$  be the set of  $3\times 3$  matrices with the property that each line and each column contain all given numbers  $a,b,c$ . Find  $\{\max \{ \det A \mid A \in M \}$  and the number of matrices which realise the maximum value.

*Mircea Becheanu*


Let  $ABC$  be an equilateral triangle. Denote by  $D$  the midpoint of  $\overline{BC}$ , and denote the circle with diameter  $\overline{AD}$  by  $\Omega$ . If the region inside  $\Omega$  and outside  $\triangle ABC$  has area  $800\pi-600\sqrt3$ , find the length of  $AB$ .

*Proposed by Eugene Chen*


What is the largest positive integer  $n < 1000$  for which there is a positive integer  $m$  satisfying \[\text{lcm}(m,n) = 3m \times \gcd(m,n)?\]


Find the number of ordered  $64$ -tuples  $\{x_0,x_1,\dots,x_{63}\}$  such that  $x_0,x_1,\dots,x_{63}$  are distinct elements of  $\{1,2,\dots,2017\}$  and
\[x_0+x_1+2x_2+3x_3+\cdots+63x_{63}\]
is divisible by  $2017.$


Find all integers  $ n \ge 3$  such that there are  $ n$  points in space, with no three on a line and no four on a circle, such that all the circles pass through three points between them are congruent.

Solution: \boxed{n = 4}


$p(m)$  is the number of distinct prime divisors of a positive integer  $m>1$  and  $f(m)$  is the  $\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor$  th smallest prime divisor of  $m$ . Find all positive integers  $n$  satisfying the equation:   $$ f(n^2+2) + f(n^2+5) = 2n-4 $$

Solution: \boxed{n = 5}


The towns in one country are connected with bidirectional airlines, which are paid in at least one of the two directions. In a trip from town A to town B there are exactly 22 routes that are free. Find the least possible number of towns in the country.


$2021$  points are given on a circle. Each point is colored by one of the  $1,2, \cdots ,k$  colors. For all points and colors  $1\leq r \leq k$ , there exist an arc such that at least half of the points on it are colored with  $r$ . Find the maximum possible value of  $k$ .

Solution: \boxed{k = 2}


Let  $x, y$  be two positive integers, with  $x> y$ , such that  $2n = x + y$ , where n is a number two-digit integer. If  $\sqrt{xy}$  is an integer with the digits of  $n$  but in reverse order, determine the value of  $x - y$


Let  $\triangle ABC$  be a triangle with  $AB = 7$ ,  $AC = 8$ , and  $BC = 3$ . Let  $P_1$  and  $P_2$  be two distinct points on line  $AC$  ( $A, P_1, C, P_2$  appear in that order on the line) and  $Q_1$  and  $Q_2$  be two distinct points on line  $AB$  ( $A, Q_1, B, Q_2$  appear in that order on the line) such that  $BQ_1 = P_1Q_1 = P_1C$  and  $BQ_2 = P_2Q_2 = P_2C$ . Find the distance between the circumcenters of  $BP_1P_2$  and  $CQ_1Q_2$ .


Let  $T_1$  and  $T_2$  be the points of tangency of the excircles of a triangle  $ABC$  with its sides  $BC$  and  $AC$  respectively. It is known that the reflection of the incenter of  $ABC$  across the midpoint of  $AB$  lies on the circumcircle of triangle  $CT_1T_2$ . Find  $\angle BCA$ .


Given positive integers $a$, $b$, and $c$ such that $a<b<c$, and the system of equations $2x+y=2003$ and $y=|x-a|+|x-b|+|x-c|$ has exactly one solution, calculate the minimum value of $c$.


Triangle  $\vartriangle ABC$  has circumcenter  $O$  and orthocenter  $H$ . Let  $D$  be the foot of the altitude from  $A$  to   $BC$ , and suppose  $AD = 12$ . If  $BD = \frac14 BC$  and   $OH  \parallel  BC$ , compute  $AB^2$ .
.


We say that a polygon  $P$  is inscribed in another polygon  $Q$  when all the vertices of  $P$  belong to the perimeter of  $Q$ . We also say in this case that  $Q$  is circumscribed to  $P$ . Given a triangle  $T$ , let  $\ell$  be the largest side of a square inscribed in  $T$   and  $L$  is the shortest side of a square circumscribed to  $T$  . Find the smallest possible value of the ratio  $L/\ell$  .


Perimeter of triangle  $ABC$  is  $1$ . Circle  $\omega$  touches side  $BC$ , continuation of side  $AB$  at  $P$  and continuation of side  $AC$  in  $Q$ . Line through midpoints  $AB$  and  $AC$  intersects circumcircle of  $APQ$  at  $X$  and  $Y$ .
Find length of  $XY$ .


Let  $\{a_n\}$  be a sequence of integers satisfying the following conditions.

- $a_1=2021^{2021}$
- $0 \le a_k < k$  for all integers  $k \ge 2$
- $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$  is multiple of  $k$  for all positive integers  $k$ .

Determine the  $2021^{2022}$ th term of the sequence  $\{a_n\}$ .


An infinite sequence of digits  $1$  and  $2$  is determined by the following two properties:

i) The sequence is built by writing, in some order, blocks  $12$  and blocks  $112.$ ii) If each block  $12$  is replaced by  $1$  and each block  $112$  by  $2$ , the same sequence is again obtained.

In which position is the hundredth digit  $1$ ? What is the thousandth digit of the sequence?


$25.$  Let  $C$  be the answer to Problem  $27.$  What is the  $C$ -th smallest positive integer with exactly four positive factors? $26.$  Let  $A$  be the answer to Problem  $25.$  Determine the absolute value of the difference between the two positive integer roots of the quadratic equation  $x^2-Ax+48=0$   $27.$  Let  $B$  be the answer to Problem  $26.$  Compute the smallest integer greater than  $\frac{B}{\pi}$


Among all pairs of real numbers  $(x, y)$  such that  $\sin \sin x = \sin \sin y$  with  $-10 \pi \le x, y \le 10 \pi$ , Oleg randomly selected a pair  $(X, Y)$ .  Compute the probability that  $X = Y$ .


Find all positive integers  $n$  for which there exist real numbers  $x_1, x_2,. . . , x_n$  satisfying all of the following conditions:
(i)  $-1 <x_i <1,$  for all  $1\leq  i \leq  n.$ (ii)  $ x_1 + x_2 + ... + x_n = 0.$ (iii)  $\sqrt{1 - x_1^2} +\sqrt{1 - x^2_2} + ... +\sqrt{1 - x^2_n} = 1.$

Solution: \boxed{n = 1}


A  $5\times5$  grid of squares is filled with integers. Call a rectangle *corner-odd* if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?

Note: A rectangles must have four distinct corners to be considered *corner-odd*; i.e. no  $1\times k$  rectangle can be *corner-odd* for any positive integer  $k$ .


Let  $k$ ,  $m$  and  $n$  be integers such that  $1<n \leq m-1 \leq k$ . Find maximum size of subset  $S$  of set  $\{1,2,...,k\}$  such that sum of any  $n$  different elements from  $S$  is not: $a)$  equal to  $m$ , $b)$  exceeding  $m$

Solution: \boxed{\left\lfloor \frac{m}{n} + \frac{n-1}{2} \right\rfloor}


Suppose  $a$  and  $b$  are positive integers with a curious property:  $(a^3 - 3ab +\tfrac{1}{2})^n + (b^3 +\tfrac{1}{2})^n$  is an integer for at least  $3$ , but at most finitely many different choices of positive integers  $n$ . What is the least possible value of  $a+b$ ?


Find the maximum value of  $$ \int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx $$  over all continuously differentiable functions  $f:[0,1]\to\mathbb R$  with  $f(0)=0$  and  $$ \int^1_0|f'(x)|^2dx\le1. $$


For each vertex of triangle  $ABC$ , the angle between the altitude and the bisectrix from this vertex was found. It occurred that these angle in vertices  $A$  and  $B$   were equal. Furthermore the angle in vertex  $C$  is greater than two remaining angles. Find angle  $C$  of the triangle.

Solution: \boxed{\gamma = 120^\circ}


Let  $S$  be the set of all ordered triples  $(p,q,r)$  of prime numbers for which at least one rational number  $x$  satisfies  $px^2+qx+r=0.$  Which primes appear in seven or more elements of  $S?$


Simone begins working on three tasks at 8:00 AM. The first two tasks each take 45 minutes, and after completing these, she takes a 15-minute break. The third task takes twice as long as each of the first two tasks. Calculate the time at which she finishes the third task.

Solution: \boxed{11:15 \text{ AM}}


A rectangle with dimensions 10 units by 5 units is cut in half across its length, creating two congruent squares. Determine the side length of one of these squares.


Alex paid $250 for a set of four tires with a deal that included a fourth tire for $10. What was the regular price of one tire?


Luis's flight took off from Chicago at 3:15 PM and landed in Denver at 4:42 PM. Chicago is in the Central Time Zone, while Denver is in the Mountain Time Zone, which is 1 hour behind Chicago. If his flight took $h$ hours and $m$ minutes, with $0 < m < 60$, calculate the value of $h + m$.


Given that the score of Elena's video was $150$ and that $60\%$ of the votes were likes, calculate the total number of votes cast on Elena's video at that point.


Given that the car ran exclusively on its battery for the first $60$ miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of $0.03$ gallons per mile, and the car averaged $75$ miles per gallon on the whole trip, determine the total length of the trip in miles.


A right circular cone and a sphere have the same base radius, denoted as \( r \). The volume of the cone is one-third that of the sphere. Find the ratio of the altitude of the cone (\( h \)) to the radius of its base (\( r \)).


Given that the first vessel was $\frac{3}{4}$ full of oil, and after transferring all the oil to the second vessel, the second vessel was $\frac{5}{8}$ full, determine the ratio of the volume of the first vessel to the volume of the second vessel.


Tom's age $T$ years is equal to the sum of the ages of his four children. His age $M$ years ago was three times the sum of their ages then. Find the value of $T/M$.


The cost of the shirts was $\frac{5}{6}$ of the price at which he actually sold them, and a discount of $\frac{1}{4}$ was advertised. Find the ratio of the cost to the marked price.


Suppose $M$ minutes of internet usage can be purchased from a service provider for $P$ pennies. Determine the number of minutes of internet usage that can be purchased for $E$ euros, where 1 euro is equivalent to 100 pennies.


Carlos took $60\%$ of a whole pie. Maria took one fourth of the remainder. What portion of the whole pie was left?


If one side of a triangle is $16$ inches and the opposite angle is $45^{\circ}$, calculate the diameter of the circumscribed circle.

Solution: \boxed{16\sqrt{2}\text{ inches}}


Given Tom's age is $T$ years, which is also the sum of the ages of his four children, and his age $N$ years ago was three times the sum of their ages then, calculate $T/N$.


Given Tom's age is $T$ years, and the combined age of his partner and his four children is $T$ years. If $N$ years ago, Tom's age was thrice the sum of his partner's and children's ages then, find the value of $T/N$.


Alice commutes to her office daily, taking normally 30 minutes in moderate traffic and 18 minutes in less traffic with an increased speed of 12 miles per hour. Determine the distance from Alice's home to her office in miles.


A $9 \times 9$ board consists of alternating light and dark squares. Calculate the difference between the number of dark squares and the number of light squares.


If the digit 2 is placed after a three-digit number whose hundreds' digit is $a$, tens' digit is $b$, and units' digit is $c$, calculate the resulting four-digit number.

Solution: \boxed{1000a + 100b + 10c + 2}


Given that Ella has two containers, the first is $\tfrac{3}{4}$ full of juice and the second is completely empty, she transfers all the juice from the first container into the second container, and then the second container is $\tfrac{5}{8}$ full. What is the ratio of the volume of the first container to the volume of the second container?


Given that $\frac{2}{3}$ of the marbles are blue and the rest are red, what fraction of the marbles will be red after the number of red marbles is doubled and the number of blue marbles is increased by 50%?


The point \( P(a, b) \) in the \( xy \)-plane is first rotated counterclockwise by \( 180^\circ \) around the point \( (2, 3) \) and then reflected about the line \( y = x \). The image of \( P \) after these two transformations is at \( (1, -4) \). Find the value of \( b - a \).


Given that a triangle and a trapezoid are on parallel bases and share the same height, the base of the triangle is 24 inches, and the area of the triangle is 192 square inches, find the length of the median of the trapezoid assuming it has the same area as the triangle.


Find the number of three-digit whole numbers where the sum of the digits is $27$.


Theresa's parents have decided to buy her a new phone if she spends an average of 12 hours per week helping around the house for 7 weeks. For the first 6 weeks, she helps around the house for 14, 10, 13, 9, 12, and 11 hours. Calculate the number of hours she must work in the final week to earn the phone.


Given a 6x6 formation consisting of black and white square tiles with 12 black tiles and 24 white tiles spaced out uniformly, the pattern is surrounded by a border of white tiles. What is the new ratio of black tiles to white tiles in the extended pattern?


Consider a regular 25-gon. It has a certain number of lines of symmetry, $L$, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. Calculate $L + \frac{R}{2}$.


Given that Luis wants to arrange his sticker collection in rows with exactly 4 stickers in each row, and he has 29 stickers initially, find the minimum number of additional stickers Luis must purchase so that the total number of stickers can be exactly split into 5 equal groups without any stickers left over.


A laptop is originally priced at $1200$ dollars and is put on sale for $30\%$ off. If a $12\%$ tax was added to the sale price, calculate the total selling price (in dollars) of the laptop.


Mr. Lee sold two speakers at $\textdollar{1.44}$ each. One speaker was sold at a 20% profit, whereas the other was sold at a 10% loss. Did Mr. Lee break even, or how much did he gain or lose in the overall transaction?
A) Lost 8 cents
B) Gained 8 cents
C) Broke even
D) Lost 16 cents
E) Gained 24 cents

Solution: \boxed{B) Gained 8 cents}


During a storm, Alex and Jamie are fishing 2 miles from the shore when their boat begins to leak. Water enters the boat at a rate of 15 gallons per minute due to the leak. Additionally, because of the heavy rain, an extra 5 gallons of water per minute accumulates in the boat. The boat will sink if it takes in more than 60 gallons of water. Alex rows towards the shore at a constant rate of 3 miles per hour while Jamie tries to bail out water. Determine the minimum rate at which Jamie must bail water, in gallons per minute, to ensure they reach the shore without the boat sinking.


The ratio of $a$ to $b$ is $5:4$, the ratio of $c$ to $d$ is $4:1$, and the ratio of $d$ to $b$ is $2:5$. Find the ratio of $a$ to $c$.


Two pitchers, one holding 800 mL and another 500 mL, are partially filled with apple juice. The first pitcher is 1/4 full and the second pitcher is 1/5 full. Water is added to fill each pitcher completely, then both pitchers are poured into a container. What fraction of the mixture in the container is apple juice?


In order to average 85 for five tests, determine the lowest score Shauna could earn on one of the other two tests, given that her scores on the first three tests are 82, 88, and 93.


Calculate the sum of the numbers on the top faces that occurs with the same probability as when the sum is $15$ when $8$ standard dice are rolled.


Given Mr. Green's garden measures $18$ steps by $25$ steps, and each of his steps is $2.5$ feet long, calculate the area of the garden in square feet, and then determine how many pounds of potatoes Mr. Green can expect from his garden given that he expects half a pound of potatoes per square foot.


Calculate the sum of the numbers from 80 to 100, excluding 90.


Given that five fish can be traded for three loaves of bread, one loaf of bread can be exchanged for six apples, and two apples can be traded for a bag of rice, determine the value of one fish in bags of rice.


The ratio of $p$ to $q$ is $5:4$, the ratio of $r$ to $s$ is $4:3$, and the ratio of $s$ to $q$ is $1:8$. Express the ratio of $p$ to $r$.


Given the expression $1-(-2)-3-(-4)-5-(-6)-7-(-8)$, evaluate the expression.


Jonathan has 1000 coins, consisting of pennies (1-cent coins) and dimes (10-cent coins), with at least one penny and one dime. Determine the difference in cents between the greatest possible and least amounts of money that Jonathan can have.


If $\frac{2}{3}$ of the marbles are blue and the rest are red, what fraction of the marbles will be blue after the number of blue marbles is doubled and the number of red marbles stays the same?


Given that $|x^2 - 14x + 44| = 4$, calculate the sum of all real numbers $x$.


Given that Alice's car averages 30 miles per gallon of gasoline, and Bob's car averages 20 miles per gallon of gasoline, and Alice drives 120 miles and Bob drives 180 miles, calculate the combined rate of miles per gallon of gasoline for both cars.


Given the operation $a @ b = \frac{a \times b}{a+b}$, and $(a, b)$ are positive integers, calculate $35 @ 77$.


Evaluate the expression $\frac{36 - 6 \times 3}{6 \div 3 \times 2}$.


Marie completes four tasks consecutively without taking any breaks. Each subsequent task takes 10 minutes longer than the previous one. The first task begins at 8:00 AM and takes 30 minutes. Determine the completion time of the fourth task.

Solution: \boxed{11:00 \text{ AM}}


For real numbers \( x \) and \( y \), define \( x \spadesuit y = (x+y)(x-y) \). Evaluate the expression \( 2 \spadesuit (3 \spadesuit (1 \spadesuit 4)) \).


Jenny bakes a 24-inch by 20-inch pan of banana bread. The banana bread is cut into pieces that measure 3 inches by 4 inches. How many pieces of banana bread does the pan contain?


Given the equation $\frac{1}{x^2} - \frac{1}{y^2} = \frac{1}{z}$, express the value of \( z \) in terms of \( x \) and \( y $.

Solution: \boxed{\frac{x^2y^2}{y^2 - x^2}}


The arithmetic mean of four numbers is 92. Given that the largest of these numbers is 108, calculate the mean of the remaining three numbers.


Given a right-angled triangle $I$ with base $(a+b)$ and height equal to $(a+b)$, find the area of a second triangle $II$ with double the area of triangle $I$.


Find the smallest positive integer $x$ for which $540x = M^2$, where $M$ is an integer.


Given that the number $3.17171717\ldots$ can be written as a fraction in lowest terms, find the sum of its numerator and denominator.


Given that in a math test, $15\%$ of the students scored $60$ points, $20\%$ scored $75$ points, $40\%$ scored $85$ points, and the rest scored $95$ points, calculate the difference between the mean and median score of the students' scores on this test.


Given that there are three mathematics teachers: Mrs. Germain with 13 students, Mr. Newton with 10 students, and Mrs. Young with 12 students, and 2 students are taking classes from both Mrs. Germain and Mr. Newton and 1 additional student is taking classes from both Mrs. Germain and Mrs. Young. Determine the number of distinct students participating in the competition from all three classes.


A bakery prepares a large $24$-inch by $30$-inch tray of brownies. Each brownie is cut into pieces that measure $3$ inches by $4$ inches. Calculate the number of brownie pieces obtained from the tray.


Ahn selects a two-digit integer \( n \), subtracts it from 300, then triples the result. What is the largest number Ahn can achieve?


Given the ratio of $a$ to $b$ is $5:4$, of $b$ to $c$ is $2:3$, and of $c$ to $d$ is $3:5$, find the ratio of $a$ to $d$.


A chip is drawn randomly from each of three boxes that contain chips numbered $1$, $2$, $4$. What is the probability that the product of the numbers on the three chips is even?


Suppose $P$ packets of chips can be purchased from a vending machine for $R$ dimes. How many packets of chips can be purchased for $M$ dollars, where $1$ dollar is worth $10$ dimes?


Define the operation $x \otimes y$ as $x^3 + y^3$. Evaluate the expression $h \otimes (h \otimes h)$.

Solution: \boxed{h^3 + 8h^9}


Given that Alice holds 2 green, 2 red, and 1 blue jelly bean and Carl holds 3 green, 2 yellow, and 1 red jelly bean, find the probability that the colors of the jelly beans they each pick match.


If there are a total of 10 children and they ride past a checkpoint with 26 wheels in total, how many tricycles are involved?


The number halfway between $\frac{1}{12}$ and $\frac{1}{15}$ is equal to what value?


Given Emilia's scaled model of a lighthouse has a cylindrical base that holds 0.15 liters of water, while the original lighthouse has a cylindrical base that holds 150,000 liters of water, and the original model is 60 meters tall, determine the height of Emilia's model in centimeters.

Solution: \boxed{60 \text{ cm}}


Ava holds 2 green and 2 red jelly beans in her hand, and Ben holds 2 green, 3 yellow, and 3 red jelly beans in his hand. Each randomly picks a jelly bean to show the other, find the probability that the colors match.


Given that Route X is 7 miles long and Mrs. Davis's average speed on this route is 35 miles per hour, calculate the time it takes to travel Route X. Given that Route Y is 8 miles long, but includes a 1-mile stretch through a busy downtown area where her average speed drops to 10 miles per hour, and the rest of the route she can travel at 50 miles per hour, calculate the time it takes to travel Route Y. By how many minutes is Route Y slower than Route X?


How many ordered pairs of positive integers $(M,N)$ satisfy the equation $\frac{M}{4} = \frac{12}{N}$


Given that Alice sells an item at $15 less than the list price and receives 15% of her selling price as her commission, and Bob sells the item at $25 less than the list price and receives 25% of his selling price as his commission, find the list price.


Given Barney's bicycle odometer initially reads $2552$. After riding $5$ hours one day and $4$ hours the next day, the odometer reads another palindrome, $2772$. Calculate the average speed in miles per hour.


What is the ratio of the least common multiple of 210 and 462 to the greatest common factor of 210 and 462?


Given Sarah and her two sisters ordered a 20-slice pizza, and Sarah ate 3 slices by herself and shared an additional slice equally with both her sisters. Determine the fraction of the pizza that Sarah consumed.


Given that all of David's telephone numbers have the form $555-abc-def$, where $a$, $b$, $c$, $d$, $e$, and $f$ are distinct digits and in increasing order, and none is either $0$ or $1$, calculate the total number of different telephone numbers that David can have.


A container holds a collection of triangular and square tiles. There are 32 tiles in total, with a sum of 110 edges. Find the number of square tiles.


Suppose that $\frac{1}{2}$ of $12$ apples are worth as much as $10$ pears. How many pears are worth as much as $\frac{3}{4}$ of $6$ apples?


Given that $60$ people are wearing sunglasses, $40$ people are wearing caps, and a randomly selected person wearing a cap has a $\frac{2}{5}$ probability of also wearing sunglasses, find the probability that a person wearing sunglasses is also wearing a cap.


Determine the sum of all values of $y$ for which $g(4y) = 10$, given that $g\left(\dfrac{x}{2}\right) = 2x^2 - x + 3$.


Given that a pan of brownies measures $24$ inches by $15$ inches and is cut into pieces that measure $3$ inches by $2$ inches, calculate the number of pieces of brownie the pan contains.


Suppose there is a calculator with a special key that replaces the number $x$ currently displayed with the number given by the formula $\frac{1}{1-x}$. Initially, the calculator displays the number 7. After the special key is pressed 101 times in a row, what number will the calculator display?


Given a soccer team with 18 players and a total salary cap of $600,000, each player must receive a minimum salary of $20,000. Determine the maximum possible salary for a single player.


A cone filled with water has a base radius of $10 \mathrm{cm}$ and a height of $15 \mathrm{cm}$. This water is then poured into a cylindrical container with a base radius of $20 \mathrm{cm}$. Calculate the height of the water in the cylindrical container.


Given a competition that starts at 3:00 p.m. and continues for 875 minutes, determine the exact ending time of the competition.

Solution: \boxed{5:35 \text{ a.m.}}


Given that the perimeter of an equilateral triangle is $3p$, find its area.


Consider a 9x9 chessboard-like layout where each row alternates between starting with a dark square and a light square, beginning with a dark square. Determine how many more dark squares there are than light squares on this chessboard.


Let $F = 0.\overline{726}$ be an infinite repeating decimal. When $F$ is written as a fraction in lowest terms, find the difference between the denominator and the numerator.


A store owner bought $2000$ pencils at $0.15$ each. If he sells them for $0.30$ each, calculate the number of pencils he must sell to make a profit of exactly $150.00$.


Handy Meg worked a total of 1 and 3/4 hours on Monday, 1 hour 15 minutes on Wednesday, from 9:15 to 11:30 on Thursday morning, and 45 minutes on Saturday. She is paid 4 dollars per hour. Calculate her total earnings for the week.


Joe walked one-third of the distance from home to school in 9 minutes, and he bikes 5 times as fast as he walks. If he biked the remaining two-thirds of the distance, calculate the total time it took Joe to get from home to school.


How many three-digit numbers are divisible by 17?


Given a camping group where $50\%$ of the members are girls, and after three boys leave and one girl joins, $60\%$ of the group consists of girls, calculate the initial number of girls in the group.


A merchant buys goods at a discount of $30\%$ off the list price. He wants to mark the goods so that he can offer a discount of $20\%$ on the marked price and still achieve a profit of $30\%$ on the selling price. What percent of the list price must he mark the goods.


A cart rolls down a hill, traveling 5 inches in the first second. It accelerates so that during each successive 1-second interval, it travels 9 inches more than during the previous 1-second interval. The cart takes 40 seconds to reach the bottom of the hill. Calculate the total distance traveled by the cart, in inches.


David's flight took off from Boston at 11:07 AM and landed in Toronto at 2:49 PM. Both cities are in the same time zone. If his flight took $h$ hours and $m$ minutes, with $0 < m < 60$, calculate the value of $h + m$.


Let $x = -2023$. Find the value of $\left| \left| |x| - x \right| - |x| \right| - x$.


Suppose that $m$ and $n$ are positive integers such that $98m = n^{3}$. Find the minimum possible value of $m + n$.

Solution: \boxed{m + n = 42}


Given a deck of cards consisting of four red cards numbered 1, 2, 3, 4; four green cards numbered 1, 2, 3, 4; and four yellow cards numbered 1, 2, 3, 4, calculate the probability of drawing a winning pair, where a winning pair consists of two cards of the same color or two cards with the same number.


Solution: \boxed{\frac{6}{29} < \frac{8}{25} < \frac{10}{31}}


What is the smallest whole number larger than the perimeter of any triangle with sides of length $7$ and $17$?


Given the equation $\frac{A}{10}=\frac{5}{B}$, how many ordered pairs of positive integers $(A, B)$ satisfy this equation.


A truck with six tires (five road tires and one spare) rotated so that each tire was used for an equal amount of distance during the first 36,000 miles the truck traveled. Determine the number of miles each tire was used.


At a conference of $40$ people, there are $25$ people who all know each other and $15$ people who know no one. People who know each other wave, and people who do not know each other shake hands. How many handshakes occur between people who do not know each other?


Given that every 7-digit whole number can be a telephone number except those that begin with 0, 1, or 2, calculate the fraction of telephone numbers that begin with 9 and end with an even number.


The Cougars and the Panthers played two halves, with the first half total of 38 points, and the Cougars leading by 16 points. Given that the overall total score of the two teams was 58 and the Cougars won by 22 points overall, calculate the total points scored by the Panthers in the second half.


Alice selects a real number uniformly at random from the interval $[0, 1500]$ and Bob chooses a real number uniformly at random from the interval $[0, 4500]$. Determine the probability that Bob's number is greater than Alice's number.


Let $G = 0.\overline{726}$, an infinite repeating decimal with the digits $7$, $2$, and $6$ repeating. When $G$ is written as a fraction in lowest terms, find the difference between the denominator and the numerator.


John's flight departed from Los Angeles at 9:15 AM and arrived in New York at 6:25 PM. Los Angeles is in the Pacific Time Zone (PT), and New York is in the Eastern Time Zone (ET), which is 3 hours ahead of PT. If the flight took $h$ hours and $m$ minutes, with $0 < m < 60$, calculate $h + m$.


If a pentagon is inscribed in a circle, find the sum of the angles inscribed in the five arcs cut off by the sides of the pentagon.


Given the quadratic equation $px^2 + qx + r = 0$, for one root to be triple the other, determine the relationship between the coefficients $p,\,q,\,r$.

Solution: \boxed{3q^2 = 16pr}


When the sum of the first fifteen terms of an arithmetic progression is three times the sum of the first ten terms, determine the ratio of the first term to the common difference.


Given a geometric sequence with first term $b$ and a common ratio $s$ where $-1 < s < 1$, calculate the limit of the sum of the cubes of the terms in this infinite series.

Solution: \boxed{\frac{b^3}{1 - s^3}}


A square and a circle share the same perimeter. Let \(A\) be the area of the square, and \(B\) the area of the circle. Find the value of \(\frac{A}{B}\).


Suppose the estimated cost to send a person to the planet Mars is now $25$ billion dollars. If this cost is shared equally by $200$ million people in the U.S., calculate the amount each person should pay.


Given that $\boxed{n}$ is the sum of the positive factors of $n$ and $p=17$, a prime number, find $\boxed{\boxed{p}}$.


Two jars contain the same number of marbles, where each marble is either blue or green. In Jar $1$, the ratio of blue to green marbles is $7:2$, and in Jar $2$, the ratio of blue to green marbles is $8:1$. If there are $135$ green marbles in all, calculate the difference in the number of blue marbles in Jar $1$ and Jar $2$.


What is the hundreds digit of $(12! - 8!)$.


Find the sum of the two smallest prime factors of 264.


Given that the smartphone's battery lasts for 36 hours when not being used and 4 hours when being used continuously, and that it has been on for 12 hours with 2 hours of usage, calculate the battery life when not being used after the described period.


Given an inverted cone with a base radius of $15 \mathrm{cm}$ and a height of $15 \mathrm{cm}$, and a cylinder with a horizontal base radius of $18 \mathrm{cm}$, determine the height in centimeters of the water in the cylinder after $10\%$ of the water is lost from the cone.


Suppose that f(x+5) = 5x^2 + 9x + 6 and f(x)=ax^2 + bx + c. Determine the value of a + b + c.


Keiko walks once around a track at a constant speed every day. The track has straight sides and semicircular ends. The track has a width of $10$ meters, and it takes her $60$ seconds longer to walk around the outside edge of the track than around the inside edge. Determine Keiko's speed in meters per second.


Given the numbers 2, 3, 0, 3, 1, 4, 0, 3, 5, 2, calculate the sum of the mean, median, and mode.

Solution: \boxed{2.3 + 2.5 + 3 = 7.8}


Find the value of $y$ that satisfies the equation $16^{-3} = \frac{4^{72/y}}{8^{37/y} \cdot 16^{28/y}}$.


What is $\heartsuit(1) + \heartsuit(2) + \heartsuit(3)$, where $\heartsuit(x) = \frac{x^2 - x + 3}{2}$?


Cagney and Lacey and Pat can frost a cupcake every 25 seconds, 35 seconds and 45 seconds, respectively. Calculate the total number of cupcakes that they can frost in 10 minutes.


Let \(a_1, a_2, \ldots\) and \(b_1, b_2, \ldots\) be arithmetic progressions where \(a_1 = 15\), \(b_1 = 45\), and \(a_{50} + b_{50} = 150\). Find the sum of the first fifty terms of the progression \(a_1 + b_1, a_2 + b_2, \ldots\) .


The ratio of 9-th-graders to 10-th-graders is 3:2, the ratio of 9-th-graders to 11-th-graders is 5:4, and the ratio of 9-th-graders to 12-th-graders is 7:6. Find the smallest number of students that could be participating in this science fair.


The sum of the first hundred positive odd integers subtracted from the sum of the first hundred positive even integers.


Given that Alice, Bob, and Carlos went on a camping trip and agreed to split the total costs of the trip. During the trip, Alice paid $130, Bob paid $150, and Carlos paid $200. Alice then gave Carlos $a dollars, and Bob gave Carlos $b dollars so that all three paid an equal share of the total costs, determine the value of $a-b$.


What is the value of $\dfrac{13! - 12!}{10!}$


The bakery owner turns on his bakery oven at $\text{9:00}\ {\small\text{AM}}$. At $\text{12:00}\ {\small\text{PM}}$, the oven has baked one-fourth of the day's bread. Calculate the time when the oven will complete baking all the bread.

Solution: \boxed{9:00 PM}


Eight posts are equally spaced along one side of a straight alley. The total distance from the first post to the fifth post is 100 meters. Calculate the distance from the first post to the last post.


Given Isabella's house has 4 bedrooms, each with a length of 15 feet, a width of 12 feet, a height of 9 feet, and doorways and windows occupying 80 square feet, calculate the total square feet of walls that must be painted.


Josh buys $5$ cookies and $3$ cupcakes, paying a total of $23$ dollars, and Emily purchases $4$ cookies and $4$ cupcakes and spends $21$ dollars total; assuming cookies and cupcakes have constant prices, determine the ratio of the price of a cupcake to the price of a cookie.


Given $r$ be the result of tripling the base and doubling the exponent of $a^b$, where $b$ does not equal $0$, determine the value of $x$ if $r$ equals the product of $a^b$ by $x^b$.


Given the quadratic equation $x^2 - (a+b)x + ab(1-k) = 0$ and that one of the roots is $1$, determine the other root.

Solution: \boxed{a + b - 1}


Given that $a * b$ means $4a - 2b$, determine the value of $x$ such that $3 * (6 * x) = 2$.


Let $\triangle ABC$ be a triangle where $B$ is twice the measure of $A$, i.e., $B = 2A$. After drawing an altitude from the vertex $C$ to the base $AB$ which divides $C$ into $C_1$ and $C_2$, where $C_2$ is adjacent to side $a$, find the relationship between $C_1$, $C_2$, and the angle $A$.
A) $C_1 - C_2 = A$
B) $C_1 - C_2 = 2A$
C) $C_1 + C_2 = 3A$
D) $C_1 - C_2 = 3A$
E) $C_1 + C_2 = A$

Solution: \boxed{C_1 - C_2 = A}


For how many positive integer values of \(N\) is the expression \(\dfrac{48}{N+3}\) an integer?


For how many integers \( x \) is the number \( x^4 - 63x^2 + 62 \) negative?


Alice and Bob play a game on a circle with 15 equally-spaced points, and they start on point 15. Alice moves 7 points clockwise and Bob moves 11 points counterclockwise in each turn. Find the number of turns it will take for them to meet again.


When Peetu was a boy, he could cycle 18 miles in 2 hours and 15 minutes. As an older adult, he can now walk 6 miles and rest for half an hour within the same 3 hours. Calculate the difference in time it takes for him to travel a mile now compared to when he was a boy.


A store prices an item such that when 7% sales tax is added, the final price is precisely $m$ dollars, where $m$ is a positive integer. Find the smallest value of $m$.


Let $x$ and $y$ be two-digit positive integers with a mean of $55$. Find the maximum value of the ratio $\frac{x}{y}$.


A box contains $30$ red balls, $23$ green balls, $21$ yellow balls, $17$ blue balls, $14$ white balls, and $12$ black balls. Calculate the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $18$ balls of a single color will be drawn.


Given $R$ varies directly as the square of $S$ and inversely as $T^2$, and when $R = 2$ and $T = 2$, $S = 1$, find $S$ when $R = 50$ and $T = 5$.


How many odd positive 4-digit integers are divisible by 5 but do not contain the digits 0 or 5?


A number \( y \) is \( 3 \) more than the product of its reciprocal and its additive inverse. Express the value of \( y \).


The ratio of 9th-graders to 7th-graders is 4:5, and the ratio of 9th-graders to 8th-graders is 7:6. Find the smallest number of students participating in the annual science fair.


How many primes less than 150 have 7 as the ones digit?


A movie marathon started at 3:00 p.m. and lasted for 1575 minutes.  Determine the time at which the movie marathon ended.

Solution: \boxed{5:15 \text{ p.m. the next day}}


Given the number $762$, convert it into its factorial base representation $762 = b_1 + b_2 \times 2! + b_3 \times 3! + \ldots + b_n \times n!$, where $0 \leq b_k \leq k$. Determine the value of $b_4$.


Given c_1 = 30, d_1 = 90, and c_{100} + d_{100} = 300, find the sum of the first hundred terms of the progression c_1 + d_1, c_2 + d_2, ...


Given that the sum $7^0 + 7^1 + 7^2 + \cdots + 7^{10}$ is divided by 5, find the remainder.


On a 9x9 checkerboard, the 81 whole numbers from 1 through 81 are written in order, starting with 1 in the top left corner and moving left to right, then top to bottom, and filling each row one by one. Calculate the sum of the numbers in the four corners of this 9x9 checkerboard.


Find the quadratic equation whose roots sum up to 12 and the absolute difference of whose roots is 4.

Solution: \boxed{x^2 - 12x + 32 = 0}


Given that the absolute value of $x$ is less than $4\pi$, calculate the number of integer values of $x$.


A rectangular floor that is $18$ feet wide and $24$ feet long is tiled with one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, calculate the number of tiles the bug visits.


Given five positive consecutive integers starting with $a$, find the average of seven consecutive integers that start with the average $b$.

Solution: \boxed{a + 5}


Let Lucas wrote down one integer three times and another integer four times. The sum of these seven numbers is 140, and one of the numbers is 20. What is the other number?


Given that the list contains the elements 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, determine the number that should be removed so that the average of the remaining numbers is $8.5$.


Mr. Green measures his rectangular garden by walking two of the sides and finds that it is $18$ steps by $25$ steps. Each of Mr. Green's steps is $2.5$ feet long. Mr. Green expects three-quarters of a pound of potatoes per square foot from his garden. Determine the total amount of potatoes in pounds that Mr. Green expects to harvest from his garden.


If Rosella has a 5 × 7 index card and shortening the length of one side by 2 inches results in an area of 21 square inches, determine the area of the card if instead she shortens the length of the other side by 2 inches.


Given that \(\frac{1}{5}\) of the team's total points were scored by Linda, \(\frac{3}{8}\) were scored by Maria, and Kelly scored 18 points, and the other 8 teammates scored no more than 2 points each, find the total number of points scored by these 8 teammates.


Given a deck consisting of five red cards and five green cards, each labeled from $A$ to $E$, find the probability of drawing a winning pair, where a winning pair is defined as two cards of the same color or two cards of the same letter.


Given the rate at which Suzanna rides her bike, with an update rate of 1.5 miles every 10 minutes, calculate the distance Suzanna would have ridden after riding for 40 minutes but taking a 10-minute break after the first 20 minutes.


Let m and n be positive integers such that 50m = n^3. Determine the minimum possible value of m + n.


The first four terms of an arithmetic sequence are $p$, $11$, $4p-r$, and $4p+r$. Find the 1005th term of this sequence.


How many four-digit numbers divisible by 17 are also even?


For every dollar Ben spent on pastries, David spent $50$ cents less. Ben paid $20$ more than David. Calculate the amount Ben and David spent together in the pastry shop.


Given the equation $3^{3x^3 - 9x^2 + 15x - 5} = 1$, determine the number of real solutions.


Given the length of a rectangular field is $5.2$ meters and the width is $2.7$ meters, calculate the area in square meters after converting all dimensions to centimeters.

Solution: \boxed{14.04 \text{ m}^2}


Find the sum of all prime numbers between $1$ and $100$ that are simultaneously $1$ less than a multiple of $6$ and $2$ less than a multiple of $7$.


Given $a \star b = \frac{a \times b - (a - b)}{a + b}$ for $a,b$ positive integers, calculate the value of $7 \star 5$.


Given the operation $a \star b = a^2 + b^2 - ab$, evaluate the expression $(x+2y) \star (y+3x)$.

Solution: \boxed{7x^2 + 3y^2 + 3xy}


Let $x=-2023$. Calculate the value of $\bigg|$ $||x|-x|-|x+7|$ $\bigg|$ $-x-7$.


In a right triangle where the ratio of the legs is 1:3, a perpendicular is dropped from the vertex of the right angle to the hypotenuse. Find the ratio of the segments created on the hypotenuse by this perpendicular.


Find the number of distinct ordered pairs $(x,y)$ where $x$ and $y$ are positive integers satisfying the equation $x^4y^4 - 14x^2y^2 + 49 = 0$.


Shelby drives her scooter at a speed of 35 miles per hour in sunny conditions and 25 miles per hour when it rains. On one particular day, she drove during sunny weather and rainy weather, covering a total of 22.5 miles in 50 minutes. Calculate the time she drove while it was raining.


A driver travels for 3 hours at 50 miles per hour. The car uses gasoline at the rate of 25 miles per gallon. He earns $0.60 per mile driven, and the only expense he has is gasoline, which costs $2.50 per gallon. Additionally, if he drives more than 100 miles, a maintenance cost of $10 is added to his overall expense. Calculate the driver's net rate of pay, in dollars per hour, including expenses.


There are twenty-four $4$-digit numbers that use each of the four digits $3$, $4$, $6$, and $8$ exactly once. Determine the numerical value of the number in the $17\text{th}$ position in the list when the numbers are listed in numerical order from smallest to largest.


Let $M = 47^{4} + 4\cdot47^{3} + 6\cdot47^{2} + 4\cdot47 + 1$. How many positive integers are factors of $M$?


Given that the students in third grade, fourth grade, and fifth grade run an average of $14$, $18$, and $11$ minutes per day, respectively, there are three times as many third graders as fourth graders, and four times as many fourth graders as fifth graders. Calculate the average number of minutes run per day by these students.


A triangle and a trapezoid have equal areas and share the same altitude. If the base of the triangle is 24 inches, and the base of the trapezoid is halved in length compared to the triangle, determine the length of the median of the trapezoid.


Four fair dice with eight faces each are tossed at random. Calculate the probability that the four numbers turned up can be arranged to form an arithmetic progression with a common difference of two.


A store owner bought 2000 pencils at 0.15 each. If he sells them for 0.30 each, determine the number of pencils that must be sold to make a profit of exactly 150.00.


Given that $\frac{3}{8}$ of the knights are red and the remainder are blue, $\frac{1}{4}$ of the knights are magical, and the fraction of red knights who are magical is $3$ times the fraction of blue knights who are magical, calculate the fraction of red knights who are magical.


Determine the number of terms with rational coefficients in the expansion of $\left(x \sqrt[4]{5} + y \sqrt{7}\right)^{500}$.


Suppose each vertex of a tetrahedron needs to be labeled with an integer from $1$ to $4$, each integer being used exactly once, so that the sum of the numbers on the vertices of each face is the same for all four faces. Arrangements that can be rotated into each other are considered identical. Determine the number of different such arrangements that are possible.


Count the number of positive even multiples of $5$ less than $3000$ that are perfect squares.


Evaluate the value of $\text{rem} \left(2\frac{1}{4}, \frac{5}{6}\right)$.


Given the numbers 2, 3, 4, 3, 1, 6, 3, 7, determine the sum of the mean, median, and mode of these numbers.


Given a deck of three red cards labeled $A$, $B$, $C$, three green cards labeled $A$, $B$, $C$, and three blue cards labeled $A$, $B$, $C$, calculate the probability of drawing a winning pair.


Given a worker's yearly salary of $48,000 with a 10% reduction, determine the percentage increase to the new salary required to restore the original salary, and the monetary difference in earnings for 6 months at the reduced salary before the increase takes effect.


A cylindrical tank has a radius $R = 10$ inches and height $H = 5$ inches. The volume of the tank is increased by the same fixed amount when the radius $R$ is multiplied by a factor $k$ as when the height $H$ is increased by $k$ inches. Determine the value of $k$ that satisfies this condition.

Solution: \boxed{\frac{1 + \sqrt{101}}{10}}


Given that $2^6 \cdot 3^9 = a^b$, where $a$ and $b$ are positive integers, find the smallest possible value for $a+b$.


What is the value of $\dfrac{13! - 12!}{10!}$?


Given that $20\%$ of the students scored $60$ points, $20\%$ scored $75$ points, $40\%$ scored $85$ points, and the rest scored $95$ points, calculate the difference between the median and mean score of the student's scores on this test.


The mean of the 9 data values 50, 85, x, 30, 70, 200, 75, 95, 55 are all equal to x.


Given that $x \neq 3$, $y \neq 5$, and $z \neq 7$, evaluate the expression $\frac{x-3}{7-z} \cdot \frac{y-5}{3-x} \cdot \frac{z-7}{5-y}$.


The graphs of $y = -|x-a|^2 + b$ and $y = |x-c|^2 + d$ intersect at points $(1,4)$ and $(7,2)$. Find $a+c$.


A triangle is inscribed in a circle. If an angle is inscribed into each of the three segments outside the triangle, determine the sum of these three angles, expressed in degrees.


Given that the arithmetic average of three positive integers added to a fourth integer results in the numbers $35, 27, 25$, and $19$, determine one of the original integers.


Johann has $80$ fair coins. Each coin is flipped. Any coin that lands on tails is flipped again up to three more times (for a total of four flips per coin). Find the expected number of coins that show heads after all the tosses are completed.


How many real numbers \( x \) satisfy the equation \( 3^{2x+1} - 3^{x+2} - 9 \cdot 3^x + 27 = 0 \)?


Suppose that $\tfrac{3}{4}$ of $16$ bananas are worth as much as $10$ oranges. How many oranges are worth as much as $\tfrac{3}{5}$ of $15$ bananas?


Given that Randy drove the first quarter of his trip on gravel road, the next 40 miles on pavement, then 10 miles through a city, and the remaining one-sixth on dirt road, determine the total length of his trip in miles.


The formula used to determine the number of individuals earning more than $x$ dollars is $N = 2 \times 10^9 \times x^{-2}$. Find the minimum income, in dollars, of the wealthiest $500$ individuals.


Evaluate the expression: $(3 \times 4 \times 5 \times 6) \left(\frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6}\right)$.


In an updated letter and numeral grid, you are trying to spell "AMC9" starting from a central 'A'. You can move from one character to an adjacent character (directly up, down, left, or right, but not diagonal). Assuming there are 4 'M's reachable from the 'A', 3 'C's reachable from each 'M', and now 3 '9's reachable from each 'C', how many different paths can spell "AMC9"?


Given that the odometer initially displays $12321$ miles, and after $4$ hours, the next higher palindrome is displayed, calculate the average speed in miles per hour during this $4$-hour period.


Two numbers have their difference, their sum, and their product corresponding to each other in the ratio 1:8:30. Find the product of the two numbers.


Two objects start from points $A$ and $B$, and move towards each other. If the total distance between $A$ and $B$ is $m$ and the product of the individual distances traveled before meeting is $n^2$, determine the polynomial whose roots are the distances traveled by each object from $A$ and $B$ before meeting.

Solution: \boxed{x^2 - mx + n^2}


Find the sum of all prime numbers between $1$ and $50$ that are simultaneously $1$ greater than a multiple of $4$ and $1$ less than a multiple of $3$.


Given Professor Lee has nine different language books on a bookshelf, consisting of two Arabic, three French, and four English books, arrange the nine books on the shelf keeping the Arabic books together and the English books together.


Determine the largest integer n for which 5^n is a factor of the sum 48!+49!+50!.


If the radius of a circle is increased by 2 units, calculate the ratio of the new circumference to the original circumference.

Solution: \boxed{1 + \frac{2}{r}}


Given that Emily received $\frac{1}{4}$ of all votes and Fiona received $\frac{1}{3}$ of all votes, and that together they received 77 votes, calculate the total number of votes cast in the election.


Charles has \(7q + 3\) quarters and Richard has \(3q + 7\) quarters. Calculate the difference in their money in nickels.


Let \(N\) be a two-digit positive integer with the property that the sum of \(N\) and the number obtained by reversing the order of the digits of \(N\) is a multiple of 13. How many integers \(N\) satisfy this property.


Given a triangle with an area of $24$, one side of length $8$, and the median to that side of length $7.5$, calculate the sine of the angle $\theta$ between that side and the median.


Let a sequence $\{v_n\}$ be defined by $v_1=7$ and the relationship $v_{n+1}-v_n=5+6(n-1), n=1,2,3\cdots.$ Express $v_n$ as a polynomial in $n$ and find the algebraic sum of its coefficients.


Simplify $\left(\sqrt[4]{81} + \sqrt{8 \frac{3}{4} }\right)^2$.

Solution: \boxed{\frac{71 + 12\sqrt{35}}{4}}


Given the weights of boxes in pounds with three different combinations, $135$, $139$, $142$, and $145$, determine the combined weight of the four boxes.


Given that the city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water, and Logan's miniature water tower now holds 0.2 liters, determine the height of Logan's tower in meters.


Samantha turned 14 years old in the year she took the seventh AMC 8, and she moved to the U.S. three years before she took her first AMC 8. Given the first AMC 8 was held in 1983, calculate the year Samantha was born.


What is the largest quotient that can be formed using two numbers chosen from the set $\{ -30, -5, -1, 0, 2, 10, 15 \}$?


Crystal decides to alter her running routine slightly. She heads due north for 2 miles, then goes northwest for 2 miles, followed by a southwest direction for 2 miles, and she finishes with a final segment directly back to her starting point. Calculate the distance of this final segment of her run.


Given positive integers $a$ and $b$ that are each less than $10$, find the smallest possible value for $2 \cdot a - a \cdot b$.


When $n$ standard 6-sided dice are rolled, find the smallest possible value of $S$ such that the probability of obtaining a sum of 2000 is greater than zero and is the same as the probability of obtaining a sum of $S$.


For real numbers $a$ and $b$, define $a \diamond b = \sqrt{a^2 + b^2}$. Evaluate the value of $(10 \diamond 24) \diamond ((-24) \diamond (-10))$.


If four times the larger of two numbers is six times the smaller, and the sum of these two numbers is 36, calculate the value of the larger of the two numbers.


Given two positive integers $a$ and $b$, where $a$ is a two-digit number that was incorrectly reversed in a multiplication with $b$, yielding the product $143$, calculate the correct value of the product of $a$ and $b$.


Given a rectangular floor measuring $a$ by $b$ feet, where $a$ and $b$ are positive integers and $b > a$, an artist paints a rectangle on the floor with the sides of the rectangle parallel to the floor. The unpainted part of the floor forms a border of width $2$ feet around the painted rectangle and occupies half the area of the whole floor. Determine the number of possibilities for the ordered pair $(a,b)$.


Given the expression $\left(x\sqrt[4]{2} + y\sqrt{5}\right)^{500}$, calculate the number of terms in the expansion with rational coefficients.


If $y$ cows give $y+2$ cans of milk in $y+4$ days, how many days will it take $y+4$ cows to give $y+7$ cans of milk?


The total surface area of a rectangular solid is $150\text{ cm}^2$, and the total length of all its edges is $60\text{ cm}$. Calculate the length in cm of any one of its interior diagonals.


Given the quadratic equation \( 3x^2 + 4x + 5 = 0 \) with roots \( u \) and \( v \), another quadratic equation \( x^2 + mx + n = 0 \) has roots \( u^2+1 \) and \( v^2+1 \). Find the value of \( m \).


Chloe chooses a real number uniformly at random from the interval $[0, 3000]$. Independently, Max chooses a real number uniformly at random from the interval $[0, 4500]$. Determine the probability that Max's number is at least twice as large as Chloe's number.


Given Alice has 30 apples, in how many ways can she share them with Becky and Chris so that each of the three people has at least three apples.


Given that 15% of the students scored 60 points, 20% scored 75 points, 40% scored 85 points, and the remainder scored 95 points, calculate the difference between the mean and median score of the students' scores on this test.


For a given digit $d$, determine the number of values of $d$ for which $2.0d05 > 2.010$.


Calculate the value of the product $\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\left(\frac{4\cdot6}{5\cdot5}\right)\cdots\left(\frac{98\cdot100}{99\cdot99}\right)\left(\frac{99\cdot101}{100\cdot100}\right)$.


If \( a \) and \( b \) are positive numbers such that \( a^b = b^a \) and \( b = 4a \), then find the value of \( a \).


Given that $15\%$ of the students scored $60$ points, $25\%$ scored $75$ points, $20\%$ scored $85$ points, and the rest scored $95$ points, calculate the difference between the mean and median score of the students' scores on this test.


A lemming sits at a corner of a square with side length $15$ meters. It runs $9.3$ meters along a diagonal toward the opposite corner. It stops, makes a $45^{\circ}$ right turn and runs $3$ meters. Calculate the average of the shortest distances from the lemming to each side of the square.


In a new linear mapping setup, segment $\overline{AB}$ of length 10 units is mapped to segment $\overline{A'B'}$ which is 5 units long. If $D$ and $D'$ are the respective midpoints of $\overline{AB}$ and $\overline{A'B'}$, and if $x$ is the distance from a point $P$ on $\overline{AB}$ to $D$, while $y$ is the distance from the associated point $P'$ on $\overline{A'B'}$ to $D'$, find the value of $x + y$ when $x = 3$ units.


Suppose $\tan \gamma$ and $\tan \delta$ are the roots of the quadratic equation $x^2 - mx + n = 0$, and $\cot \gamma$ and $\cot \delta$ are the roots of another quadratic equation $x^2 - ux + v = 0$. Determine the product $uv$.


Define a sequence of real numbers $a_1$, $a_2$, $a_3$, $\dots$ by $a_1 = 1$ and $a_{n + 1}^2 = 49a_n^2$ for all $n \geq 1$. Find the value of $a_{100}$.


Find the sum of the elements in the 30th set of consecutive integers starting from {1}, increasing by one additional element in each subsequent set.


The sums of three whole numbers taken in pairs are 14, 20, and 22. Calculate the value of the middle number.


Let M be the second smallest positive integer that is divisible by every positive integer less than 10. What is the sum of the digits of M?


Three counterfeit coins of equal weight are mixed with 12 identical genuine coins. The weight of each counterfeit coin is different from the weight of each genuine coin. Two pairs of coins are selected at random without replacement from the 15 coins, first a pair of two and then another pair of two from the remaining. What is the probability that all four selected coins are genuine?


Given Menkara has a $4 \times 6$ index card, if she shortens the length of one side of this card by $2$ inches, the card would have an area of $12$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?


Given equations of the form $x^2 + bx + c = 0$, determine the number of such equations that have real roots and have coefficients $b$ and $c$ selected from the set of integers $\{1, 2, 3, 4, 5, 7\}$.


Given that the acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ belong to the Fibonacci sequence (where each number is the sum of the two preceding ones, starting from 1 and 2), what is the least possible value of $b$?


Given the number $3033$, determine the number of different four-digit numbers that can be formed by rearranging its digits.


How many whole numbers are between $\sqrt{50}$ and $\sqrt{200}$?


Given the student multiplied $99$ by $2.cd$ instead of $2.\overline{cd}$, and his result was $0.75$ less than the correct result, find the 2-digit number $cd$.


Ella has two older twin sisters. The product of their three ages is 72. What is the sum of their three ages?


Given that x and y are distinct nonzero real numbers such that x + 3/x = y + 3/y, find the product xy.


Given that $3$ steps equals $4$ strides, $5$ leaps equals $2$ steps, and $7$ leaps equals $6$ yards, determine the value of one yard in terms of strides.


Circle $O$ has points $C$ and $D$ on the same side of diameter $\overline{AB}$, where $\angle AOC = 40^\circ$ and $\angle DOB = 60^\circ$. Find the ratio of the area of the smaller sector $COD$ to the area of the entire circle.


Evaluate the expression $(16)^{-2^{-3}}$.


The graph of $y = \log_2 x$ is rotated $90^{\circ}$ counter-clockwise about the origin. Find the equation of the resulting graph, $G'$.

Solution: \boxed{y = 2^{-x}}


A dealer bought n radios for $200 dollars. He donated three radios to a community bazaar at half their cost. He sold the rest of the radios at a profit of $10$ per radio. If the overall profit was $100$, find the least possible value of n.


Given that $8x + 15y = 120$ and the point (x, y) is in the first quadrant, determine the minimum value of $\sqrt{x^2 + y^2}$.


Given Randy drove the first one-fourth of his trip on a gravel road, the next 30 miles on pavement, and the remaining one-sixth on a dirt road, calculate the total length of Randy's trip in miles.


Three distinct vertices of a tetrahedron are chosen at random. Determine the probability that the plane determined by these three vertices contains points inside the tetrahedron.


If the digit 2 is placed after a three-digit number whose hundreds' digit is $h$, tens' digit is $t$, and units' digit is $u$, find the expression that represents the new number formed.

Solution: \boxed{1000h + 100t + 10u + 2}


Given a square grid of 121 uniformly spaced points, where one point $P$ is at the center, determine the probability that a randomly chosen point $Q$ among the remaining 120 points lies on a line of symmetry for the square.


Triangle $PQR$ is inscribed in a circle. The measure of the non-overlapping minor arcs $PQ$, $QR$, and $RP$ are, respectively, $y+60^{\circ}$, $2y+40^{\circ}$, and $3y-10^{\circ}$. Determine the measure of interior angle $Q$.


Evaluate the double summation: $\sum^{50}_{i=1} \sum^{50}_{j=1} 2(i+j)$.


The scout troop bought 1500 candy bars at a rate of three for $1 dollar and sold them at a price of two for $1.5 dollars, incurring a fixed cost of 50 dollars for setting up their sales booth. What was their profit, in dollars?