Four copies of an acute scalene triangle $\mathcal T$, one of whose sides has length $3$, are joined to form a tetrahedron with volume $4$ and surface area $24$. Compute the largest possible value for the circumradius of $\mathcal T$.

Bill’s age is one third larger than Tracy’s age. In $30$ years Bill’s age will be one eighth larger than Tracy’s age. How many years old is Bill?

$n$ pairs are formed from $n$ girls and $n$ boys at random. What is the probability of having at least one pair of girls? For which $n$ the probability is over $0,9?$

Let $n$ be a positive integer, and Megavan has a $(3n+1)\times (3n+1)$ board. All squares, except one, are tiled by non-overlapping $1\times 3$ triominoes. In each step, he can choose a triomino that is untouched in the step right before it, and then shift this triomino horizontally or vertically by one square, as long as the triominoes remain non-overlapping after this move. Show that there exist some $k$, such that after $k$ moves Megavan can no longer make any valid moves irregardless of the initial configuration, and find the smallest possible $k$ for each $n$. [i](Note: While he cannot undo a move immediately before the current step, he may still choose to move a triomino that has already been moved at least two steps before.)[/i] [i]Proposed by Ivan Chan Kai Chin[/i]

Let $a_1$ be any positive integer. For all $i$, write $5^{2020}$ times $a_i$ in base $10$, replace each digit with its remainder when divided by $2$, read off the result in binary, and call that $a_{i+1}$. Prove that $a_N = a_{N+2^{2020}}$ for all sufficiently large $N$.

Is there a sequence $a_{1}, . . . , a_{2016}$ of positive integers, such that every sum $$a_{r} + a_{r+1} + . . . + a_{s-1} + a_{s}$$ (with $1 \le r \le s \le 2016$) is a composite number, but: a) $GCD(a_{i}, a_{i+1}) = 1$ for all $i = 1, 2, . . . , 2015$; b) $GCD(a_{i}, a_{i+1}) = 1$ for all $i = 1, 2, . . . , 2015$ and $GCD(a_{i}, a_{i+2}) = 1$ for all $i = 1, 2, . . . , 2014$? $GCD(x, y)$ denotes the greatest common divisor of $x$, $y$. Proposed by Matija Bucić

Let $a, b, c$ be positive real numbers such that $a + b + c + ab + bc + ca + abc = 7$. Prove that $\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6$ .

Let $AB$ be a diameter of semicircle $h$. On this semicircle there is point $C$, distinct from points $A$ and $B$. Foot of perpendicular from point $C$ to side $AB$ is point $D$. Circle $k$ is outside the triangle $ADC$ and at the same time touches semicircle $h$ and sides $AB$ and $CD$. Touching point of $k$ with side $AB$ is point $E$, with semicircle $h$ is point $T$ and with side $CD$ is point $S$ $a)$ Prove that points $A$, $S$ and $T$ are collinear $b)$ Prove that $AC=AE$

Find the number of lattice points that the line $19x+20y=1909$ passes through in Quadrant I.

Let $A,B,C,O$ be points in the plane such that angles $\angle AOB,\angle BOC, \angle COA$ are obtuse. On $OA,OB,OC$ points $X,Y,Z$ respectively are chosen, such that $OX=OY=OZ$. On segments $OX,OY,OZ$ points $K,L,M$ respectively are chosen. The lines $AL$ and $BK$ intersect at point $R$, which isn't on $XY$. The segment $XY$ intersects $AL,BK$ at points $R_1,R_2$. The lines $BM$ and $CL$ intersect at point $P$, which isn't on $YZ$. The segment $YZ$ intersects $BM,CL$ at points $P_1,P_2$. The lines $CK$ and $AM$ intersect at point $Q$, which isn't on $ZX$. The segment $ZX$ intersects $CK,AM$ at points $Q_1,Q_2$. Suppose that $PP_1=PP_2$ and $QQ_1=QQ_2$. Prove that $RR_1=RR_2$.

The diagonals of a trapezoid $ABCD$ with $AD \parallel BC$ intersect at point $P$. Point $Q$ lies between the parallel lines $AD$ and $BC$ such that the line $CD$ separates points $P$ and $Q$, and $\angle AQD=\angle CQB$. Prove that $\angle BQP = \angle DAQ$.

A rectangular storage bin measures $10$ feet by $12$ feet, is $3$ feet tall, and sits on a flat plane. A pile of dirt is pushed up against the outside of the storage bin so that it slants down from the top of the storage bin to points on the ground $4$ feet away from the base of the storage bin as shown. The number of cubic feet of dirt needed to form the pile can be written as $m + n \pi$ where $m$ and $n$ are positive integers. Find $m + n.$

Triangle $\triangle PQR$, with $PQ=PR=5$ and $QR=6$, is inscribed in circle $\omega$. Compute the radius of the circle with center on $\overline{QR}$ which is tangent to both $\omega$ and $\overline{PQ}$.

Let $n$ be a positive integer. Consider the sum $x_1y_1 + x_2y_2 +\cdots + x_ny_n$, where that values of the variables $x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n$ are either 0 or 1. Let $I(n)$ be the number of 2$n$-tuples $(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n)$ such that the sum of the number is odd, and let $P(n)$ be the number of 2$n$-tuples $(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n)$ such that the sum is an even number. Show that: \[ \frac{P(n)}{I(n)}=\frac{2^n+1}{2^n-1} \]

Given a positive integer $n\ge 3$, Arándano and Banana play a game. Initially, numbers $1,2,3,\dots,n$ are written on the blackboard. Alternatingly and starting with Arándano, the players erase numbers from the board one at a time, until exactly three numbers remain on the board. Banana wins the game if the last three numbers on the board are the sides of a nondegenerate triangle, and Arándano wins otherwise. Determine, in terms of $n$, who has a winning strategy.

Let $A$ be a $n \times n$ matrix of real numbers such that $A^2+\mathbb{I}=A$, where $\mathbb{I}$ is the identity $n \times n$ matrix. Prove that the matrix $A^{3n}$ , where $\nu\in\mathbb{Z}$ takes only two values and find those values.

Positive integers $a, b$ and $c$ are all less than $2020$. We know that $a$ divides $b + c$, $b$ divides $a + c$ and $c$ divides $a + b$. How many such ordered triples $(a, b, c)$ are there? Note: In an ordered triple, the order of the numbers matters, so the ordered triple $(0, 1, 2)$ is not the same as the ordered triple $(2, 0, 1)$.

Nine circles are drawn around an arbitrary triangle as in the figure. All circles tangent to the same side of the triangle have equal radii. Three lines are drawn, each one connecting one of the triangle’s vertices to the center of one of the circles touching the opposite side, as in the figure. Show that the three lines are concurrent. (N. Beluhov)

Let $a$ and $b$ be two-digit positive integers. Find the greatest possible value of $a+b$, given that the greatest common factor of $a$ and $b$ is $6$. [i]Proposed by Jacob Xu[/i] [hide=Solution][i]Solution[/i]. $\boxed{186}$ We can write our two numbers as $6x$ and $6y$. Notice that $x$ and $y$ must be relatively prime. Since $6x$ and $6y$ are two digit numbers, we just need to check values of $x$ and $y$ from $2$ through $16$ such that $x$ and $y$ are relatively prime. We maximize the sum when $x = 15$ and $y = 16$, since consecutive numbers are always relatively prime. So the sum is $6 \cdot (15+16) = \boxed{186}$.[/hide]

Two triangles intersect to form seven finite disjoint regions, six of which are triangles with area 1. The last region is a hexagon with area \(A\). Compute the minimum possible value of \(A\). [i]Proposed by Karthik Vedula[/i]

Compute the number of dates in the year $2023$ such that when put in $\text{MM / DD / YY}$ form, the three numbers are in strictly increasing order. For example, $06/18/23$ is such a date since $6 < 18 < 23,$ while today, $11/11/23,$ is not.

Do there exist positive integers $a_1<a_2<\ldots<a_{100}$ such that for $2\le k\le100$, the least common multiple of $a_{k-1}$ and $a_k$ is greater than the least common multiple of $a_k$ and $a_{k+1}$?

Let $a_1,a_2,\dots,a_{22}$ be positive integers with sum $59$. Prove the inequality \[\frac{a_1}{a_1+1}+\frac{a_2}{a_2+1}+\dots+\frac{a_{22}}{a_{22}+1}<16.\]

In trapezoid $ ABCD$ with bases $ AB$ and $ CD$, we have $ AB\equal{}52$, $ BC\equal{}12$, $ CD\equal{}39$, and $ DA\equal{}5$. The area of $ ABCD$ is [asy] pair A,B,C,D; A=(0,0); B=(52,0); C=(38,20); D=(5,20); dot(A); dot(B); dot(C); dot(D); draw(A--B--C--D--cycle); label("$A$",A,S); label("$B$",B,S); label("$C$",C,N); label("$D$",D,N); label("52",(A+B)/2,S); label("39",(C+D)/2,N); label("12",(B+C)/2,E); label("5",(D+A)/2,W);[/asy] $ \text{(A)}\ 182 \qquad \text{(B)}\ 195 \qquad \text{(C)}\ 210 \qquad \text{(D)}\ 234 \qquad \text{(E)}\ 260$

Find all pairs of positive integers $(n, k)$ such that all sufficiently large odd positive integers $m$ are representable as $$m=a_1^{n^2}+a_2^{(n+1)^2}+\ldots+a_k^{(n+k-1)^2}+a_{k+1}^{(n+k)^2}$$ for some non-negative integers $a_1, a_2, \ldots, a_{k+1}$.