A parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ is reflected about the $ x$-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of $ y \equal{} f(x)$ and $ y \equal{} g(x)$, respectively. Which of the following describes the graph of $ y \equal{} (f \plus{} g)(x)$? $ \textbf{(A)}\ \text{a parabola tangent to the }x\text{ \minus{} axis}$ $ \textbf{(B)}\ \text{a parabola not tangent to the }x\text{ \minus{} axis} \qquad \textbf{(C)}\ \text{a horizontal line}$ $ \textbf{(D)}\ \text{a non \minus{} horizontal line} \qquad \textbf{(E)}\ \text{the graph of a cubic function}$

As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ is between $C$ and $D$). If $CE = 14$, $AB > 14$, compute the minimal area of $\triangle AEG$. [asy] size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(real x, real y) { pair P = (x,y); dot(P,linewidth(3)); return P; } int big = 30, small = 14; filldraw((0,big)--(big+small,0)--(big,small)--cycle, rgb(0.9,0.5,0.5)); draw(scale(big)*unitsquare); draw(shift(big,0)*scale(small)*unitsquare); label("$A$",D2(0,big),NW); label("$B$",D2(0,0),SW); label("$C$",D2(big,0),SW); label("$D$",D2(big,big),N); label("$E$",D2(big+small,0),SE); label("$F$",D2(big+small,small),NE); label("$G$",D2(big,small),NE); [/asy]

If $r > s >0$ and $a > b > c$, prove that \[a^rb^s + b^rc^s + c^ra^s \ge a^sb^r + b^sc^r + c^sa^r.\]

Initially, there are $14$ numbers written on the board - zeros and ones. Every minute, Anton chooses half of the numbers on the board and adds $1$ to each of them, while Mykhailo multiplies all the other numbers by $8$. At some point (possibly initially), all the numbers on the board become equal. How many ones could have been on the board initially? [i]Proposed by Oleksii Masalitin[/i]

If $r$ is real number sampled at random with uniform probability, find the probability that $r$ is [i]strictly [/i] closer to a multiple of $58$ than it is to a multiple of $37$.

For $a\in \mathbb{Z}$ define \[ n_a=101a-100\cdot 2^a \] Show that, for $0\le a,b,c,d\le 99$ \[ n_a+n_b\equiv n_c+n_d\pmod{10100}\implies \{a,b\}=\{c,d\} \]

Let $\omega_1,\omega_2$ be two circles with centers $O_1$ and $O_2$, respectively. These two circles intersect each other at points $A$ and $B$. Line $O_1B$ intersects $\omega_2$ for the second time at point $C$, and line $O_2A$ intersects $\omega_1$ for the second time at point $D$ . Let $X$ be the second intersection of $AC$ and $\omega_1$. Also $Y$ is the second intersection point of $BD$ and $\omega_2$. Prove that $CX = DY$ . Proposed by Alireza Dadgarnia

Let $ f(x)$ be a continuous function defined in the interval $ 0\leq x\leq 1.$ Prove that $ \int_0^1 xf(x)f(1\minus{}x)\ dx\leq \frac{1}{4}\int_0^1 \{f(x)^2\plus{}f(1\minus{}x)^2\}\ dx.$

If the radius of a circle is increased by $ 1$ unit, the ratio of the new circumference to the new diameter is: $ \textbf{(A)}\ \pi \plus{} 2 \qquad \textbf{(B)}\ \frac{2 \pi \plus{} 1}{2} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{2 \pi \minus{} 1}{2} \qquad \textbf{(E)}\ \pi \minus{} 2$

Two straight lines perpendicular to each other meet each side of a triangle in points symmetric with respect to the midpoint of that side. Prove that these two lines intersect in a point on the nine-point circle.

Consider the orthogonal projections of the vertices $A$, $B$ and $C$ of triangle $ABC$ on external bisectors of $ \angle ACB$, $ \angle BAC$ and $ \angle ABC$, respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$, prove that $r^2+p^2=d^2$ [i]Proposed by Alexander Ivanov[/i]

The number of distinct positive integral divisors of $(30)^4$ excluding $1$ and $(30)^4$ is $ \textbf{(A)}\ 100 \qquad\textbf{(B)}\ 125 \qquad\textbf{(C)}\ 123 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ \text{none of these} $

Let $ABC$ be a triangle with $BC = a$, $CA = b$, and $AB = c$. The $A$-excircle is tangent to $\overline{BC}$ at $A_1$; points $B_1$ and $C_1$ are similarly defined. Determine the number of ways to select positive integers $a$, $b$, $c$ such that [list] [*] the numbers $-a+b+c$, $a-b+c$, and $a+b-c$ are even integers at most 100, and [*] the circle through the midpoints of $\overline{AA_1}$, $\overline{BB_1}$, and $\overline{CC_1}$ is tangent to the incircle of $\triangle ABC$. [/list]

Numbers $1,2,\ldots,50$ are written on the board. Anya does the following operation: removes the numbers $a$ and $b$ from the board and writes their sum - $a+b$, after which also notes down the number $ab(a+b)$. After $49$ of this operations only one number was left on the board. Anya summed up all the $49$ numbers in her notes and got $S$. a) Prove that $S$ does not depend on the order of Anya's actions. b) Calculate $S$.

Given a square with side $10$. Cut it into $100$ congruent quadrilaterals such that each of them is inscribed into a circle with diameter $\sqrt{3}$. [i](5 points)[/i] [i]Ilya Bogdanov[/i]

Let $x,y,z$ be positive reals for which: $\sum (xy)^{2}=6xyz$ Prove that: $\sum \sqrt{\frac{x}{x+yz}}\geq \sqrt{3}$.

A $ 9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $ C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $ \sqrt {13}$, ii) the closed broken line $ C_1C_2\ldots C_{96}C_1$ has a centre of symmetry? [i]Bulgaria[/i]

In a tetrahedron, all three pairs of opposite (skew) edges are mutually perpendicular. Prove that the midpoints of the six edges of the tetrahedron lie on one sphere.

Let $a$, $b$, $c$, $d$, $e$ be positive reals satisfying \begin{align*} a + b &= c \\ a + b + c &= d \\ a + b + c + d &= e.\end{align*} If $c=5$, compute $a+b+c+d+e$. [i]Proposed by Evan Chen[/i]

$ k$ black pieces are placed on $ k$ consecutive squares of top row starting from upper left of a $ 2\times 5$ board. We are placing white pieces on empty squares one by one in arbitrary order. Two squares is said to adjacent if they have common vertex. When a white piece is placed on a square, the pieces on adjacent squares change their color. For which $ k$, when all the squares are filled, it is possible that color of every piece is white? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

Let $P_0, P_1,\ldots, P_{2021}$ points on the unit circle of centre $O$ such that for each $n\in \{1,2,\ldots, 2021\}$ the length of the arc from $P_{n-1}$ to $P_n$ (in anti-clockwise direction) is in the interval $\left[\frac{\pi}2,\pi\right]$. Determine the maximum possible length of the vector: \[\overrightarrow{OP_0}+\overrightarrow{OP_1}+\ldots+\overrightarrow{OP_{2021}}.\] [i]Mihai Iancu[/i]

Convex quadrilateral $ABCD$ has $AB=20$, $BC=CD=26$, and $\angle{ABC}=90^\circ$. Point $P$ is on $DA$ such that $\angle{PBA}=\angle{ADB}$. If $PB=20$, compute the area of $ABCD$. [i]2019 CCA Math Bonanza Individual Round #13[/i]

A convex hexagon $ABCDEF$ with an interior point $P$ is given. Assume that $BCEF$ is a square and both $ABP$ and $PCD$ are right isosceles triangles with right angles at $B$ and $C$, respectively. Lines $AF$ and $DE$ intersect at $G$. Prove that $GP$ is perpendicular to $BC$. [i]Proposed by Patrik Bak - Slovakia[/i]

Let $a, b, c$ be real numbers such that $ab + bc + ca = 1$. Prove that $$\frac{(a + b)^2 + 1}{c^2+2}+\frac{(b + c)^2 + 1}{a^2+2}+ \frac{(c + a)^2 + 1}{b^2+2} \ge 3$$

Let $a_n$ be a recursively defined sequence with $a_0=2024$ and $a_{n+1}=a_n^3+5a_n^2+10a_n+6$ for $n\ge 0.$ Determine the value of $$\sum_{n=0}^{\infty} \frac{2^n(a_n+1)}{a_n^2+3a_n+4}.$$