Let two circles be externally tangent at point $C$, with parallel diameters $A_1A_2, B_1B_2$ (i.e. the quadrilateral $A_1B_1B_2A_2$ is a trapezoid with bases $A_1A_2$ and $B_1B_2$ or parallelogram). Circle with the center on the common internal tangent to these two circles, passes through the intersection point of lines $A_1B_2$ and $A_2B_1$ as well intersects those lines at points $M, N$. Prove that the line $MN$ is perpendicular to the parallel diameters $A_1A_2, B_1B_2$. (Yuri Biletsky)

Suppose that $ \left(u_n\right)$ is a sequence of real numbers satisfying $ u_{n \plus{} 2} \equal{} 2u_{n \plus{} 1} \plus{} u_{n}$, and that $ u_3 \equal{} 9$ and $ u_6 \equal{} 128$. What is $ u_5$? $ \textbf{(A)}\ 40 \qquad \textbf{(B)}\ 53 \qquad \textbf{(C)}\ 68 \qquad \textbf{(D)}\ 88 \qquad \textbf{(E)}\ 104$

Find all functions $f: R \to R$ such that, for any $x, y \in R$: $f\left( f\left( x \right)-y \right)\cdot f\left( x+f\left( y \right) \right)={{x}^{2}}-{{y}^{2}}$

Let $T_n$ denotes the least natural such that $$n\mid 1+2+3+\cdots +T_n=\sum_{i=1}^{T_n} i$$ Find all naturals $m$ such that $m\ge T_m$. [i]Proposed by Nicolás De la Hoz [/i]

Find all prime numbers $p$ for which the number $p^2+11$ has less than $11$ divisors.

The sequence $a_{n,k} \ , k = 1, 2, 3,\ldots, 2^n \ , n = 0, 1, 2,\ldots,$ is defined by the following recurrence formula: \[a_1 = 2,\qquad a_{n,k} = 2a_{n-1,k}^3, \qquad , a_{n,k+2^{n-1}} =\frac 12 a_{n-1,k}^3\]\[\text{for} \quad k = 1, 2, 3,\ldots, 2^{n-1} \ , n = 0, 1, 2,\ldots\] Prove that the numbers $a_{n,k}$ are all different.

Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions: (i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$; (ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$; (iii) there exists a rational number $a>1$ such that $f(a)=a$. Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$. [i]Proposed by Bulgaria[/i]

Quadrilateral $ABCD$ is inscribed in a circle $O$. Let $AB\cap CD=E$ and $P\in BC, EP\perp BC$, $R\in AD, ER\perp AD$, $EP\cap AD=Q, ER\cap BC=S$ Let $K$ be the midpoint of $QS$ Prove that $E, K, O$ are collinear.

Is there a number $n$ such that one can write $n$ as the sum of $2017$ perfect squares and (with at least) $2017$ distinct ways?

For positive integer $n,$ let $s(n)$ be the sum of the digits of n when n is expressed in base ten. For example, $s(2022) = 2 + 0 + 2 + 2 = 6.$ Find the sum of the two solutions to the equation $n - 3s(n) = 2022.$

Let $ABCD$ be a convex quadrilateral with the property that $MA \cdot MC + MA \cdot CD = MB \cdot MD$, where $M$ is the intersection of the diagonals $AC$ and $BD$. The angle bisector of $\angle ACD$ is drawn intersecting ray $\overrightarrow{BA}$ at $K$. Prove that $BC = DK$ if and only if $AB \parallel CD$.

Let $f$ be a differentiable function such that $f'(0)=4$ and $f(0)=3$. Compute $$\lim_{x\rightarrow\infty}\left(\frac{f\left(\frac{1}{x}\right)}{f(0)}\right)^x.$$

Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon? [asy] // diagram by SirCalcsALot size(250); real side1 = 1.5; real side2 = 4.0; real side3 = 6.5; real pos = 2.5; pair s1 = (-10,-2.19); pair s2 = (15,2.19); pen grey1 = rgb(100/256, 100/256, 100/256); pen grey2 = rgb(183/256, 183/256, 183/256); fill(circle(origin + s1, 1), grey1); for (int i = 0; i < 6; ++i) { draw(side1*dir(60*i)+s1--side1*dir(60*i-60)+s1,linewidth(1.25)); } fill(circle(origin, 1), grey1); for (int i = 0; i < 6; ++i) { fill(circle(pos*dir(60*i),1), grey2); draw(side2*dir(60*i)--side2*dir(60*i-60),linewidth(1.25)); } fill(circle(origin+s2, 1), grey1); for (int i = 0; i < 6; ++i) { fill(circle(pos*dir(60*i)+s2,1), grey2); fill(circle(2*pos*dir(60*i)+s2,1), grey1); fill(circle(sqrt(3)*pos*dir(60*i+30)+s2,1), grey1); draw(side3*dir(60*i)+s2--side3*dir(60*i-60)+s2,linewidth(1.25)); } [/asy] $\textbf{(A)}\ 35 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 39 \qquad \textbf{(D)}\ 43 \qquad \textbf{(E)}\ 49$

It takes Mary $ 30$ minutes to walk uphill $ 1$ km from her home to school, but it takes her only $ 10$ minutes to walk from school to home along the same route. What is her average speed, in km/hr, for the round trip? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 3.125 \qquad \textbf{(C)}\ 3.5 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 4.5$

Suppose that $ K$ is a compact Hausdorff space and $ K\equal{} \cup_{n\equal{}0}^{\infty}A_n$, where $ A_n$ is metrizable and $ A_n \subset A_m$ for $ n<m$. Prove that $ K$ is metrizable. [i]Z. Balogh[/i]

Let $C$ be a cube. By connecting the centres of the faces of $C$ with lines we form an octahedron $O$. By connecting the centers of each face of $O$ with lines we get a smaller cube $C'$. What is the ratio between the side length of $C$ and the side length of $C'$?

There are three non-negative numbers $ a, b, c $ such that the sum of each two is not less than the remaining one. Prove that $$ \sqrt{a+b-c} + \sqrt{a-b+c} + \sqrt{-a+b+c} \leq \sqrt{a} + \sqrt{b} + \sqrt{c}.$$

Let $ABC$ be a triangle with $\angle BAC= 45^o$ . Altitudes $AD, BE, CF$ meet at $H$. $EF$ cuts $BC$ at $P$. $I$ is the midpoint of $BC$, $IF$ cuts $PH$ in $Q$. a) Prove that $\angle IQH = \angle AIE$. b) Let $(K)$ be the circumcircle of triangle $ABC$, $(J)$ be the circumcircle of triangle $KPD$. $CK$ cuts circle $(J)$ at $G$, $IG$ cuts $(J)$ at $M$, $JC$ cuts circle of diameter $BC$ at $N$. Prove that $G, N, M, C$ lie on the same circle.

The alphabet of the tribe AAB consists of the only letters $A$ and $B$. However, if you insert or delete the combination $AAA$ or $BBB$ for any words, the meaning of the word will not change. In addition, if $AB$ is replaced with $BBAA$, or vice versa, the meaning of the word doesn't change. The same holds for $BA$ and $AABB$. Is it true that $AB$ and $BA$ have the same meaning?

The circle $x^2 + y^2 = 25$ intersects the abscissa in points $A$ and $B$. Let $P$ be a point that lies on the line $x = 11$, $C$ is the intersection point of this line with the $Ox$ axis, and the point $Q$ is the intersection point of $AP$ with the given circle. It turned out that the area of the triangle $AQB$ is four times smaller than the area of the triangle $APC$. Find the coordinates of $Q$.

The roots of $x^3 + a x^2 + b x + c = 0$ are $\alpha, \beta$ and $\gamma.$ Find the cubic whose roots are $\alpha^3, \beta^3, \gamma^3.$

Let $n+1, n \geq 1$ positive integers be formed by taking the product of $n$ given prime numbers (a prime number can appear several times or also not appear at all in a product formed in this way.) Prove that among these $n+1$ one can find some numbers whose product is a perfect square.

Let $f = f(x,y,z)$ be a polynomial in three variables $x$, $y$, $z$ such that $f(w,w,w) = 0$ for all $w \in \mathbb{R}$. Show that there exist three polynomials $A$, $B$, $C$ in these same three variables such that $A + B + C = 0$ and \[ f(x,y,z) = A(x,y,z) \cdot (x-y) + B(x,y,z) \cdot (y-z) + C(x,y,z) \cdot (z-x). \] Is there any polynomial $f$ for which these $A$, $B$, $C$ are uniquely determined?

[b]p1.[/b] A matrix is a rectangular array of numbers. For example, $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$ are $2 \times 2$ matrices. A [i]saddle [/i] point in a matrix is an entry which is simultaneously the smallest number in its row and the largest number in its column. a. Write down a $2 \times 2$ matrix which has a saddle point, and indicate which entry is the saddle point. b. Write down a $2 \times 2$ matrix which has no saddle point c. Prove that a matrix of any size, all of whose entries are distinct, can have at most one saddle point. [b]p2.[/b] a. Find four different pairs of positive integers satisfying the equation $\frac{7}{m}+\frac{11}{n}=1$. b. Prove that the solutions you have found in part (a) are all possible pairs of positive integers satisfying the equation $\frac{7}{m}+\frac{11}{n}=1$. [b]p3.[/b] Let $ABCD$ be a quadrilateral, and let points $M, N, O, P$ be the respective midpoints of sides $AB$, $BC$, $CD$, $DA$. a. Show, by example, that it is possible that $ABCD$ is not a parallelogram, but $MNOP$ is a square. Be sure to prove that your construction satisfies all given conditions. b. Suppose that $MO$ is perpendicular to $NP$. Prove that $AC = BD$. [b]p4.[/b] A [i]Pythagorean triple[/i] is an ordered collection of three positive integers $(a, b, c)$ satisfying the relation $a^2 + b^2 = c^2$. We say that $(a, b, c)$ is a [i]primitive [/i] Pythagorean triple if $1$ is the only common factor of $a, b$, and $c$. a. Decide, with proof, if there are infinitely many Pythagorean triples. b. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form $(a, b, c)$ where $c = b + 2$. c. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form $(a, b, c)$ where $c = b + 3$. [b]p5.[/b] Let $x$ and $y$ be positive real numbers and let $s$ be the smallest among the numbers $\frac{3x}{2}$,$\frac{y}{x}+\frac{1}{x}$ and $\frac{3}{y}$. a. Find an example giving $s > 1$. b. Prove that for any positive $x$ and $y,s <2$. c. Find, with proof, the largest possible value of $s$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with $BC=13, CA=11, AB=10$. Let $A_1$ be the midpoint of $BC$. A variable line $\ell$ passes through $A_1$ and meets $AC,AB$ at $B_1,C_1$. Let $B_2,C_2$ be points with $B_2B=B_2C, B_2C_1\perp AB, C_2B=C_2C, C_2B_1 \perp AC$, and define $P=BB_2\cap CC_2$. Suppose the circles of diameters $BB_2, CC_2$ meet at a point $Q\neq A_1$. Given that $Q$ lies on the same side of line $BC$ as $A$, the minimum possible value of $\dfrac{PB}{PC}+\dfrac{QB}{QC}$ can be expressed in the form $\dfrac{a\sqrt{b}}{c}$ for positive integers $a,b,c$ with $\gcd (a,c)=1$ and $b$ squarefree. Determine $a+b+c$. [i]Proposed by Vincent Huang[/i]