A right triangle is drawn on the plane. How to use only a compass to mark two points, such that the distance between them is equal to the diameter of the circle inscribed in this triangle?

Let $P_1,P_2,P_3,P_4$ be four distinct points in the plane. Suppose $\ell_1,\ell_2, … , \ell_6$ are closed segments in that plane with the following property: Every straight line passing through at least one of the points $P_i$ meets the union $\ell_1 \cup \ell_2\cup … \cup\ell_6$ in exactly two points. Prove or disprove that the segments $\ell_i$ necessarily form a hexagon.

Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively. [i]Proposed by United Kingdom.[/i]

The common tangents to the circumcircle and an excircle of triangle $ABC$ meet $BC, CA,AB$ at points $A_1, B_1, C_1$ and $A_2, B_2, C_2$ respectively. The triangle $\Delta_1$ is formed by the lines $AA_1, BB_1$, and $CC_1$, the triangle $\Delta_2$ is formed by the lines $AA_2, BB_2,$ and $CC_2$. Prove that the circumradii of these triangles are equal.

How many numbers $\overline{abc}$ with $a,b,c>0$ there exists such that $$\overline{cba}\mid \overline{abc}$$ $\textbf{1.1.}$ The vertical line denotes that $\overline{cba}$ divides $\overline{abc}.$ [i]Proposed by Roger Carranza, Choluteca[/i]

In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$. [i]Proposed by Iman Maghsoudi[/i]

The vertices of a square sheet of paper are $ A$, $ B$, $ C$, $ D$. The sheet is folded in a way that the point $ D$ is mapped to the point $ D'$ on the side $ BC$. Let $ A'$ be the image of $ A$ after the folding, and let $ E$ be the intersection point of $ AB$ and $ A'D'$. Let $ r$ be the inradius of the triangle $ EBD'$. Prove that $ r\equal{}A'E$.

$162$ pluses and $144$ minuses are placed in a $30\times 30$ table in such a way that each row and each column contains at most $17$ signs. (No cell contains more than one sign.) For every plus we count the number of minuses in its row and for every minus we count the number of pluses in its column. Find the maximum of the sum of these numbers.

Find the sum of the prime factors of $777.$

Positive integers $a$, $b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation $ax^2-bx+c=0$ has exactly one real solution $r$, and this value $r$ is less than $100$. Compute the maximum possible value of $r$.

Let $BD$ be the external bisector of a triangle $ABC$ with $AB > BC$; $K$ and $K_1$ be the touching points of side $AC$ with the incircle and the excircle centered at $I$ and $I_1$ respectively. The lines $BK$ and $DI_1$ meet at point $X$, and the lines $BK_1$ and $DI$ meet at point $Y$. Prove that $XY \perp AC$.

Call a set of integers [i]unpredictable[/i] if no four elements in the set form an arithmetic sequence. How many unordered [i]unpredictable[/i] sets of five distinct positive integers $\{a, b, c, d, e\}$ exist such that all elements are strictly less than $12$? [i]Proposed by Anthony Yang[/i]

Prove that there exist infinitely many positive integers which cannot be written in form $a^{d(a)}+b^{d(b)}$ for some positive integers $a$ and $b$ For positive integer $d(a)$ denotes number of positive divisors of $a$ [i]Proposed by Borna Vukorepa[/i]

Circles \(\mathcal{C}_1\) and \(\mathcal{C}_2\) with centers at \(C_1\) and \(C_2\) respectively, intersect at two points \(A\) and \(B\). Points \(P\) and \(Q\) are varying points on \(\mathcal{C}_1\) and \(\mathcal{C}_2\), respectively, such that \(P\), \(Q\) and \(B\) are collinear and \(B\) is always between \(P\) and \(Q\). Let lines \(PC_1\) and \(QC_2\) intersect at \(R\), let \(I\) be the incenter of \(\Delta PQR\), and let \(S\) be the circumcenter of \(\Delta PIQ\). Show that as \(P\) and \(Q\) vary, \(S\) traces the arc of a circle whose center is concyclic with \(A\), \(C_1\) and \(C_2\).

Suppose that $f(x)=x^2-10x+21$. Find all distinct real roots of $f(f(x)+7)$.

Let $a_1,a_2,\ldots,a_n$ be non-negative real numbers ,$S_k= \sum\limits_{i=1}^{k}a_i $ $(1\le k\le n)$.Prove that$$\sum\limits_{i=1}^{n}\left(a_iS_i\sum\limits_{j=i}^{n}a^2_j\right)\le \sum\limits_{i=1}^{n}\left(a_iS_i\right)^2$$

For a positive integer $n$ determine all $n\times n$ real matrices $A$ which have only real eigenvalues and such that there exists an integer $k\geq n$ with $A + A^k = A^T$.

A page of an exercise book is painted with $23$ colours, arranged in squares. A pair of colours is called [i]good [/i] if there are neighbouring squares painted with these colours. What is the minimum number of good pairs?

Let $a_1 , a_2 , ..., a_n$ be non-zero integers that satisfy the equation $$a_1 +\dfrac{1}{a_2+\dfrac{1}{a_3+ ... \dfrac{1}{a_n+\dfrac{1}{x}} } } = x$$ for all values of $x$ for which the lefthand side of the equation makes sense. (a) Prove that $n$ is even. (b) What is the smallest n for which such numbers $a_1 , a_2 , ..., a_n$ exist? (M Skopenko)

A ball of mass $M$ and radius $R$ has a moment of inertia of $I=\frac{2}{5}MR^2$. The ball is released from rest and rolls down the ramp with no frictional loss of energy. The ball is projected vertically upward off a ramp as shown in the diagram, reaching a maximum height $y_{max}$ above the point where it leaves the ramp. Determine the maximum height of the projectile $y_{max}$ in terms of $h$. [asy] size(250); import roundedpath; path A=(0,0)--(5,-12)--(20,-12)--(20,-10); draw(roundedpath(A,1),linewidth(1.5)); draw((25,-10)--(12,-10),dashed+linewidth(0.5)); filldraw(circle((1.7,-1),1),lightgray); draw((25,-1)--(-1.5,-1),dashed+linewidth(0.5)); draw((23,-9.5)--(23,-1.5),Arrows(size=5)); label(scale(1.1)*"$h$",(23,-6.5),2*E); [/asy] (A) $h$ (B) $\frac{25}{49}h$ (C) $\frac{2}{5}h$ (D) $\frac{5}{7}h$ (E) $\frac{7}{5}h$

For every positive integer $n$ we define $a_{n}$ as the last digit of the sum $1+2+\cdots+n$. Compute $a_{1}+a_{2}+\cdots+a_{1992}$.

Suppose $p''(x) = 4x^2 + 4x + 2$ where $$p(x) = a_0 + a_1(x - 1) + a_2(x -2)^2 + a_3(x- 3)^4 + a_4(x-4)^4.$$ We have $p'(-3) = -24$ and $p(x)$ has the unique property that the sum of the third powers of the roots of $p(x)$ is equal to the sum of the fourth powers of the roots of $p(x)$ . Find $a_0$.

The $\textbf{symmetric difference}$, $\triangle$, of a pair of sets is the set of elements in exactly one set. For example, \[\{1,2,3\}\triangle\{2,3,4\}=\{1,4\}.\] There are fifteen nonempty subsets of $\{1,2,3,4\}$. Assign each subset to exactly one of the squares in the grid to the right so that the following conditions are satisfied. (i) If $A$ and $B$ are in squares connected by a solid line then $A\triangle B$ has exactly one element. (ii) If $A$ and $B$ are in squares connected by a dashed line then the largest element of $A$ is equal to the largest element of $B$. You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] size(150); defaultpen(linewidth(0.8)); draw((0,1)--(0,3)--(3,3)^^(2,3)--(2,2)--(3,2)--(3,1)--(1,1)--(1,2)--(0,2)^^(2,1)--(2,0)--(0,0)); draw(origin--(0,1)^^(1,0)--(3,2)^^(1,1)--(0,2)^^(1,2)--(0,3)^^(1,3)--(2,2),linetype("4 4")); real r=1/4; path square=(r,r)--(r,-r)--(-r,-r)--(-r,r)--cycle; int limit; for(int i=0;i<=3;i=i+1) { if (i==0) limit=2; else limit=3; for(int j=0;j<=limit;j=j+1) filldraw(shift(j,i)*square,white); } [/asy]

Each cell of the checkered plane is colored in one of $n^2$ colors so that in any square of $n \times n$ cells all colors occur. It is known that in some line all the colors occur. Prove that there exists a column colored in exactly $n$ colors.

Let $\vartriangle ABC$ be a triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\omega$. $AA'$, $BB'$ and $CC'$ are altitudes of $\vartriangle ABC$ with $A'$ in $BC$, $B'$ in $AC$ and $C'$ in $AB$. $P$ is a point on the segment $AA'$. The perpenicular line to $B'C'$ from $P$ intersects $BC$ at $K$. $AA'$ intersects $\omega$ at $M \ne A$. The lines $MK$ and $AO$ intersect at $Q$. Prove that $\angle CBQ = \angle PBA$.