In Survev.io, Calvin observes that he has exactly twice as much blue ammo as red ammo. After firing one blue bullet and $9$ red bullets, he remarks that the amount of blue ammo he has is divisible by $5$ and the amount of red ammo he has is divisible by $7$. Find the least amount of red ammo he could have started with.

Which of the two numbers is bigger : $\sqrt{1997}+2\sqrt{1999} + 2\sqrt{2001} + \sqrt{2003}$ or $2\sqrt{1998} +2\sqrt{2000}+2\sqrt{2002}$ ?

Let triangle $ABC$ have $\angle BAC = 45^{\circ}$ and circumcircle $\Gamma$ and let $M$ be the intersection of the angle bisector of $\angle BAC$ with $\Gamma$. Let $\Omega$ be the circle tangent to segments $\overline{AB}$ and $\overline{AC}$ and internally tangent to $\Gamma$ at point $T$. Given that $\angle TMA = 45^{\circ}$ and that $TM = \sqrt{100 - 50 \sqrt{2}}$, the length of $BC$ can be written as $a \sqrt{b}$, where $b$ is not divisible by the square of any prime. Find $a + b$.

Niki has $15$ dollars more than twice as much money as her sister Amy. If Niki gives Amy $30$ dollars, then Niki will have hals as much money as her sister. How many dollars does Niki have?

From a point $P$ outside of a circle with centre $O$, tangent segments $\overline{PA}$ and $\overline{PB}$ are drawn. If $\frac1{\left|\overline{OA}\right|^2}+\frac1{\left|\overline{PA}\right|^2}=\frac1{16}$, then $\left|\overline{AB}\right|=$? $\textbf{(A)}~4$ $\textbf{(B)}~6$ $\textbf{(C)}~8$ $\textbf{(D)}~10$

From a square of side length $1$, four identical triangles are removed, one at each corner, leaving a regular octagon. What is the area of the octagon?

In a solid sphere of radius $R$ the density $\rho$ is a function of $r$, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $k r^2$, where $k$ is a constant, find $\rho$ as a function of $r.$ Find also the magnitude of the force of attraction at a point outside the sphere at a distance $r$ from the center.

After years at sail, you and your crew have found the island that houses the great treasure of Scottybeard, the greatest pirate to ever sail the high seas. The island takes the shape of a unit square, and the treasure (which we treat as a single point) could be buried under any point on the island. To assist you in finding his treasure, Scotty has left a peculiar instrument. To use this instrument, you may draw any directed line (possibly one that never hits the island!), and the instrument will tell you whether the treasure lies to the "left" or the "right" of the line.* [asy] draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((1,0.5)--(0.8,0.42),arrow=Arrow()); draw((0.8,0.42)--(0.6,0.34),arrow=Arrow()); draw((0.6,0.34)--(0.4,0.26),arrow=Arrow()); draw((0.4,0.26)--(0.2,0.18),arrow=Arrow()); draw((0.2,0.18)--(0,0.1)); label("``Right''", (0.5,0.55)); label("``Left''", (0.8,0.2)); [/asy] However, Scotty also left a trap! If the instrument ever reports ``left'' three times in a row or ``right'' three times in a row, the island will suddenly sink into the sea, submerging the treasure forever and drowning you and your crew! You want to avoid this at all costs. To minimize the amount of energy spent digging, you would like to narrow down the set of possible locations of the treasure to be as small as possible. However, Scotty left one last trick; you can only use the instrument 12 times before it breaks! Devise an algorithm to use the instrument no more than 12 times that can never result in the island sinking and narrows the worst-case space of possible locations of the treasure to have as small an area as possible. * [size=75]Where "left" or "right" is taken with respect to an observer walking along the line in its designated direction. There is also a probability zero chance the treasure is precisely on the line; this won't affect anything, but for the sake of clarity let's say the instrument reports "left" in this case.[/size] [b]Scoring:[/b] An algorithm that achieves a worst-case area of $K$ will be awarded: [list] [*] 1 point for any $K<1$ [*] 10 points for $K=\tfrac 14$ [*] 20 points for $\tfrac 1{128}<K<\tfrac 14$ [*] 30 points for $K=\tfrac 1{128}$ [*] 50 points for $K_{\text{min}}<K<\tfrac 1 {128}$ [*] 75 points for $K=K_{\text{min}}$ [*] 100 points for $K=K_{\text{min}}$, with a proof that this is optimal [/list] (where $K_{\text{min}}$ is the smallest possible worst-case area, which we are not disclosing to avoid giving anything away) [i]Proposed by Connor Gordon[/i]

There are $32$ students in the class with $10$ interesting group. Each group contains exactly $16$ students. For each couple of students, the square of the number of the groups which are only involved by just one of the two students is defined as their $interests-disparity$. Define $S$ as the sum of the $interests-disparity$ of all the couples, $\binom{32}{2}\left ( =\: 496 \right )$ ones in total. Determine the minimal possible value of $S$.

Find all the triples of integers $ (a, b,c)$ such that: \[ \begin{array}{ccc}a\plus{}b\plus{}c &\equal{}& 24\\ a^{2}\plus{}b^{2}\plus{}c^{2}&\equal{}& 210\\ abc &\equal{}& 440\end{array}\]

Outside the parallelogram $ABCD$ on its sides $AB$ and $BC$ are constructed equilateral triangles $ABK$, and $BCM$. Prove that the triangle $KMD$ is equilateral.

Show that if all lateral edges of a pentagonal pyramid are of equal length and all the angles between neighboring lateral faces are equal, then the pyramid is regular.

Cris has the equation $-2x^2 + bx + c = 0$, and Cristian increases the coefficients of the Cris equation by $1$, obtaining the equation $-x^2 + (b + 1) x + (c + 1) = 0$. Mariloli notices that the real solutions of the Cristian's equation are the squares of the real solutions of the Cris equation. Find all possible values that can take the coefficients $b$ and $c$.

Let $n\in\mathbb{Z}$, $n\geq 2$. Find all functions $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ such that $$f(x_1+\dots +x_n)^2=\sum_{i=1}^nf(x_i) ^2+ 2\sum_{i<j}f(x_ix_j),$$ for all $x_1,\dots ,x_n\in\mathbb{R}_{>0}$. [i]Proposed by Andrei Vila[/i]

Let $\alpha < 0 < \beta$ and consider the polynomial $f(x) = x(x-\alpha)(x-\beta)$. Let $S$ be the set of real numbers $s$ such that $f(x) - s$ has three different real roots. For $s\in S$, let $p(x)$ the product of the smallest and largest root of $f(x)-s$. Determine the smallest possible value that $p(s)$ for $s\in S$.

Find all triples $(x,y,z)$ of positive real numbers such that $$ \begin{cases} 2x^2+y^3=3 \\ 3y^2+z^3=4 \\ 4z^2+x^3=5 \\ \end{cases} $$ [i]M. Zorka[/i]

Is it possible to find a set of $100$ (or $200$) points on the boundary of a cube such that this set remains fixed under all rotations which leave the cube fixed ?

Let $S \subset [0, 1]$ be a set of 5 points with $\{0, 1\} \subset S$. The graph of a real function $f : [0, 1] \to [0, 1]$ is continuous and increasing, and it is linear on every subinterval $I$ in $[0, 1]$ such that the endpoints but no interior points of $I$ are in $S$. We want to compute, using a computer, the extreme values of $g(x, t) = \frac{f(x+t)-f(x)}{ f(x)-f(x-t)}$ for $x - t, x + t \in [0, 1]$. At how many points $(x, t)$ is it necessary to compute $g(x, t)$ with the computer?

Which is greater $$\sqrt{n}^{\sqrt{n+1}} \;\; \; \text{or}\;\;\; \sqrt{n+1}^{\sqrt{n}}$$ where $n>8?$

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 $X = Y$.

Let $BH$ be an altitude of a right-angled triangle $ABC$ ($\angle B = 90^o$). The incircle of triangle $ABH$ touches $AB,AH$ in points $H_1, B_1$, the incircle of triangle $CBH$ touches $CB,CH$ in points $H_2, B_2$, point $O$ is the circumcenter of triangle $H_1BH_2$. Prove that $OB_1 = OB_2$.

A rhombus with its incircle is given. At each vertex of the rhombus a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii $r_1,r_2$, while the incircle has radius $r$. Given that $r_1$ and $r_2$ are natural numbers and that $r_1r_2=r$, find $r_1,r_2,$ and $r$.

For some integers $u$,$ v$, and $w$, the equation $$26ab - 51bc + 74ca = 12(a^2 + b^2 + c^2)$$ holds for all real numbers a, b, and c that satisfy $$au + bv + cw = 0$$ Find the minimum possible value of $u^2 + v^2 + w^2$.

Maria and Bilyana play the following game. Maria has $2024$ fair coins and Bilyana has $2023$ fair coins. They toss every coin they have. Maria wins if she has strictly more heads than Bilyana, otherwise Bilyana wins. What is the probability of Maria winning this game?

Let $S$ be a set with an asymmetric relation $<$; this means that if $a,b\in S$ and $a<b$, then we do not have $b<a$. Prove that there exists a set $T$ containing $S$ with an asymmetric relation $\prec$ with the property that if $a,b\in S$, then $a<b$ if and only if $a\prec b$, and if $x,y\in T$ with $x\prec y$, then there exists $t\in T$ such that $x\prec t\prec y$.