$ AA_{1}$, $ BB_{1}$, $ CC_{1}$ are altitudes of an acute triangle $ ABC$. A circle passing through $ A_{1}$ and $ B_{1}$ touches the arc $ AB$ of its circumcircle at $ C_{2}$. The points $ A_{2}$, $ B_{2}$ are defined similarly. Prove that the lines $ AA_{2}$, $ BB_{2}$, $ CC_{2}$ are concurrent.

Given a circle $(O)$ and two fixed points $B,C$ on $(O),$ and an arbitrary point $A$ on $(O)$ such that the triangle $ABC$ is acute. $M$ lies on ray $AB,$ $N$ lies on ray $AC$ such that $MA=MC$ and $NA=NB.$ Let $P$ be the intersection of $(AMN)$ and $(ABC),$ $P\ne A.$ $MN$ intersects $BC$ at $Q.$ a) Prove that $A,P,Q$ are collinear. b) $D$ is the midpoint of $BC.$ Let $K$ be the intersection of $(M,MA)$ and $(N,NA),$ $K\ne A.$ $d$ is the line passing through $A$ and perpendicular to $AK.$ $E$ is the intersection of $d$ and $BC.$ $(ADE)$ intersects $(O)$ at $F,$ $F\ne A.$ Prove that $AF$ passes through a fixed point.

Let $ ABCD$ be a cyclic quadrilater. Denote $ P\equal{}AD\cap BC$ and $ Q\equal{}AB \cap CD$. Let $ E$ be the fourth vertex of the parallelogram $ ABCE$ and $ F\equal{}CE\cap PQ$. Prove that $ D,E,F$ and $ Q$ lie on the same circle.

Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$

Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$

The floor of a rectangular room is covered with square tiles. The room is 10 tiles long and 5 tiles wide. The number of tiles that touch the walls of the room is $\textbf{(A)}\ 26 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 34 \qquad \textbf{(D)}\ 46 \qquad \textbf{(E)}\ 50$

A rectangular field is 80 m long and 60 m wide. If fence posts are placed at the corners and are 10 m apart along the 4 sides of the field, how many posts are needed to completely fence the field? $\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 26 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 32$

A group of $6$ friends sit in the back row of an otherwise empty movie theater. Each row in the theater contains $8$ seats. Euler and Gauss are best friends, so they must sit next to each other, with no empty seat between them. However, Lagrange called them names at lunch, so he cannot sit in an adjacent seat to either Euler or Gauss. In how many different ways can the $6$ friends be seated in the back row? $\text{(A) }2520\qquad\text{(B) }3600\qquad\text{(C) }4080\qquad\text{(D) }5040\qquad\text{(E) }7200$

Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between $12^2$ and $13^2$? $\textbf{(A)}\ 105 \qquad \textbf{(B)}\ 147 \qquad \textbf{(C)}\ 156 \qquad \textbf{(D)}\ 165 \qquad \textbf{(E)}\ 175$

The diagram shows a magic square in which the sums of the numbers in any row, column or diagonal are equal. What is the value of $n$? $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 11$

Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.

Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$

Jean writes five tests and achieves the marks shown on the graph. What is her average mark on these five tests? [asy] draw(origin -- (0, 10.1)); for(int i = 0; i < 11; ++i) { draw((0, i) -- (10.5, i)); label(string(10*i), (0, i), W); } filldraw((1, 0) -- (1, 8) -- (2, 8) -- (2, 0) -- cycle, black); filldraw((3, 0) -- (3, 7) -- (4, 7) -- (4, 0) -- cycle, black); filldraw((5, 0) -- (5, 6) -- (6, 6) -- (6, 0) -- cycle, black); filldraw((7, 0) -- (7, 9) -- (8, 9) -- (8, 0) -- cycle, black); filldraw((9, 0) -- (9, 8) -- (10, 8) -- (10, 0) -- cycle, black); label("Test Marks", (5, 0), S); label(rotate(90)*"Marks out of 100", (-2, 5), W); [/asy] $\textbf{(A)}\ 74 \qquad \textbf{(B)}\ 76 \qquad \textbf{(C)}\ 70 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 79$

suppose this equation: x ² +y ² +z ² =w ² . show that the solution of this equation ( if w,z have same parity) are in this form: x=2d(XZ-YW), y=2d(XW+YZ),z=d(X ² +Y ² -Z ² -W ² ),w=d(X ² +Y ² +Z ² +W ² )

Prove that if the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are all distinct, then the polynomial \[(x-a_{1})^{2}(x-a_{2})^{2}\cdots (x-a_{n})^{2}+1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

(Wolstenholme's Theorem) Prove that if \[1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\] is expressed as a fraction, where $p \ge 5$ is a prime, then $p^{2}$ divides the numerator.

Claire takes a square piece of paper and folds it in half four times without unfolding, making an isosceles right triangle each time. After unfolding the paper to form a square again, the creases on the paper would look like

Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.

Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$ If $CN\parallel IM$ find $\frac{CN}{IM}$.

Find the number of positive integer n, which follows the following $ \bigstar $ $ n=[\frac{m^3}{2024}] $ $n$ has a positive integer $m$ that follows this equation ($ m \le 1000$)

We define an operation $\oplus$ on the set $\{0, 1\}$ by \[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\] For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$. For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.

Each of the 12 edges of a cube is coloured either red or green. Every face of the cube has at least one red edge. What is the smallest number of red edges? $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

Ten points are spaced equally around a circle. How many different chords can be formed by joining any 2 of these points? (A chord is a straight line joining two points on the circumference of a circle.) $\textbf{(A)}\ 9 \qquad \textbf{(B)}\ 45 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 55$

Two numbers have a sum of $32$. If one of the numbers is $ – 36$, what is the other number? $\textbf{(A)}\ 68 \qquad \textbf{(B)}\ -4 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 72 \qquad \textbf{(E)}\ -68$

A cube measures $10 \text{cm} \times 10 \text{cm} \times10 \text{cm}$ . Three cuts are made parallel to the faces of the cube as shown creating eight separate solids which are then separated. What is the increase in the total surface area? $\textbf{(A)}\ 300 \text{cm}^2 \qquad \textbf{(B)}\ 800 \text{cm}^2 \qquad \textbf{(C)}\ 1200 \text{cm}^2 \qquad \textbf{(D)}\ 600 \text{cm}^2 \qquad \textbf{(E)}\ 0 \text{cm}^2$