Let $n$ be a non-negative integer. Find all non-negative integers $a,b,c,d$ such that \[a^2+b^2+c^2+d^2=7\cdot 4^n\]

Prove that there exists a set of infinitely many positive integers such that the elements of no finite subset of this set add up to a perfect square.

Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $BAC$ of $\Omega$. Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.

Prove that if $ a $ and $ b $ are real numbers and the polynomial $ ax^3 - ax^2 + 9bx - b $ has three positive roots, then they are equal.

An epidemic influenza broke out in the elves city. First day some of them were infected by the external source of infection and nobody later was infected by the external source. The elf is infected when visiting his ill friend. In spite of the situation every healthy elf visits all his ill friends every day. The elf is ill one day exactly, and has the immunity at least on the next day. There is no graftings in the city. Prove that a) If there were some elves immunised by the external source on the first day, the epidemic influenza can continue arbitrary long time. b) If nobody had the immunity on the first day, the epidemic influenza will stop some day.

Albrecht writes numbers on the points of the first quadrant with integer coordinates in the following way: If at least one of the coordinates of a point is 0, he writes 0; in all other cases the number written on point $(a, b)$ is one greater than the average of the numbers written on points $ (a+1 , b-1) $ and $ (a-1,b+1)$ . Which numbers could he write on point $(121, 212)$? Note: The elements of the first quadrant are points where both of the coordinates are non- negative.

Suppose $a_{1} < a_{2}< \cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences. [i]Ray Li[/i]

We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A [i]bicoloured triangulation[/i] is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which share an edge have different colours. We call a positive integer $n \ge 4$ [i]triangulable[/i] if every regular $n$-gon has a bicoloured triangulation such that for each vertex $A$ of the $n$-gon the number of black triangles of which $A$ is a vertex is greater than the number of white triangles of which $A$ is a vertex. Find all triangulable numbers.

Each different letter in the following addition represents a different decimal digit. The sum is a six-digit integer whose digits are all equal. $$\begin{tabular}{ccccccc} & P & U & R & P & L & E\\ + & & C & O & M & E & T \\ \hline \\ \end{tabular}$$ Find the greatest possible value that the five-digit number $COMET$ could represent.

Let $ABC$ be an acute triangle with $AC > AB$. Let $\Gamma$ be the circle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smaller arc $BC$ of this circle. Let $I$ be the incenter of $ABC$ and let $E$ and $F$ be points on sides $AB$ and $AC$, respectively, such that $AE = AF$ and $I$ lies on the segment $EF$. Let $P$ be the second intersection point of the circumcircle of the triangle $AEF$ with $\Gamma$ with $P \ne A$. Let $G$ and $H$ be the intersection points of the lines $PE$ and $PF$ with $\Gamma$ different from $P$, respectively. Let $J$ and $K$ be the intersection points of lines $DG$ and $DH$ with lines AB and $AC$, respectively. Show that the line $JK$ passes through the midpoint of $BC$.

Let $b=\log_53$. What is $\log_b\left(\log_35\right)$? $\text{(A) }-1\qquad\text{(B) }-\frac{3}{5}\qquad\text{(C) }0\qquad\text{(D) }\frac{3}{5}\qquad\text{(E) }1$

Let $ f:[-1,1]\longrightarrow\mathbb{R} $ a derivable function and a non-negative integer $ n. $ Show that there is a $ c\in [-1,1] $ so that: $$ \int_{-1}^1 x^{2n+1} f(x)dx =\frac{2}{2n+3}f'(c). $$

A rectangular prism with dimensions $20$ cm by $1$ cm by $7$ cm is made with blue $1$ cm unit cubes. The outside of the rectangular prism is coated in gold paint. If a cube is chosen at random and rolled, what is the probability that the side facing up is painted gold?

In a quadrilateral $ABCD$ with $\angle A = 60^o, \angle B = 90^o, \angle C = 120^o$, the point $M$ of intersection of the diagonals satisfies $BM = 1$ and $MD = 2$. (a) Prove that the vertices of $ABCD$ lie on a circle and find the radius of that circle. (b) Find the area of quadrilateral $ABCD$.

Find all functions $f:\mathbb N\to\mathbb N$ satisfying $$x+f^{f(x)}(y)\mid2(x+y)$$for all $x,y\in\mathbb N$.

Determine all the positive integers with more than one digit, all distinct, such that the sum of its digits is equal to the product of its digits.

Let \(ABC\) be a triangle and \(D\) be the mid-point of \(BC\). Suppose the angle bisector of \(\angle ADC\) is tangent to the circumcircle of triangle \(ABD\) at \(D\). Prove that \(\angle A=90^{\circ}\).

Let $ABC$ be an acute triangle with $AB<AC$ and $D,E,F$ be the contact points of the incircle $(I)$ with $BC,AC,AB$. Let $M,N$ be on $EF$ such that $MB \perp BC$ and $NC \perp BC$. $MD$ and $ND$ intersect the $(I)$ in $D$ and $Q$. Prove that $DP=DQ$.

The sequence $ (a_n)$ satisfies $ a_1 \equal{} 1$ and $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$ for $ n \geq 1$. Let $ k$ be the least integer greater than $ 1$ for which $ a_k$ is an integer. Find $ k$.

If the integer $k$ is added to each of the numbers $36$, $300$, and $596$, one obtains the squares of three consecutive terms of an arithmetic series. Find $k$.

$\triangle ABC$ has side lengths $AB=20$, $BC=15$, and $CA=7$. Let the altitudes of $\triangle ABC$ be $AD$, $BE$, and $CF$. What is the distance between the orthocenter (intersection of the altitudes) of $\triangle ABC$ and the incenter of $\triangle DEF$?

Let $ABC$ be a triangle inscribed in the circle $(O)$. Let $I$ be the center of the circle inscribed in the triangle and $D$ the point of contact of the circle inscribed with the side $BC$. Let $M$ be the second intersection point of the bisector $AI$ with the circle $(O)$ and let $P$ be the point where the line $DM$ intersects the circle $(O)$ . Show that $PA \perp PI$.

In the diagram, two circles, each with center D, have radii of $1$ and $2$. The total area of the shaded region is $\frac{5}{12}$ of the area of the larger circle. How many degrees are in the measure of $\angle ADC$? [asy] int angle = 100; path A = arc(0, 1, 0, angle); path B = arc(0, 1, angle, 360); path C = arc(0, 2, 0, angle); path D = arc(0, 2, angle, 360); filldraw(C -- origin -- cycle, gray); filldraw(D -- origin -- cycle, white); filldraw(A -- origin -- cycle, white); filldraw(B -- origin -- cycle, gray); label("$D$", origin, NE); label("$C$", (2, 0), E); label("$A$", (2, 0) * dir(angle), N); [/asy] $\mathrm{(A) \ } 100 \qquad \mathrm{(B) \ } 105 \qquad \mathrm {(C) \ } 110 \qquad \mathrm{(D) \ } 115 \qquad \mathrm{(E) \ } 120$

Suppose that $n\ge 8$ persons $P_1,P_2,\dots,P_n$ meet at a party. Assume that $P_k$ knows $k+3$ persons for $k=1,2,\dots,n-6$. Further assume that each of $P_{n-5},P_{n-4},P_{n-3}$ knows $n-2$ persons, and each of $P_{n-2},P_{n-1},P_n$ knows $n-1$ persons. Find all integers $n\ge 8$ for which this is possible. (It is understood that "to know" is a symmetric nonreflexive relation: if $P_i$ knows $P_j$ then $P_j$ knows $P_i$; to say that $P_i$ knows $p$ persons means: knows $p$ persons other than herself/himself.)

Is it possible to locate in a rectangle of $5$ cm by $ 8$ cm, $51$ circles of diameter $ 1$ cm, so that they don't overlap? Could it be possible for more than $40$ circles ?