In $\Delta ABC$ points $A_1$, $B_1$, and $C_1$ are the tangential points of the excircles of $ABC$ with its sides. a) Prove that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $N$. b) If $AC+BC=3AB$, prove that the center of the inscribed circle of $ABC$, its tangential point with $AB$, and the point $N$ are collinear.

In an acute triangle $ABC$, let $D$, $E$, $F$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $BC$, $CA$, $AB$, respectively, and let $P$, $Q$, $R$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $EF$, $FD$, $DE$, respectively. Prove that $p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}$, where $p\left(T\right)$ denotes the perimeter of triangle $T$ . [i]Proposed by Hojoo Lee, Korea[/i]

Let $D,E,F$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$ such that $BD=CE=AF$ and $\angle BDF=\angle CED=\angle AFE.$ Show that $\triangle ABC$ is equilateral.

Suppose $A$, $B$, and $C$ are points in the plane with $AB=40$ and $AC=42$, and let $x$ be the length of the line segment from $A$ to the midpoint of $\overline{BC}$. Define a function $f$ by letting $f(x)$ be the area of $\triangle ABC$. Then the domain of $f$ is an open interval $(p,q)$, and the maximum value $r$ of $f(x)$ occurs at $x=s$. What is $p+q+r+s$? $ \textbf{(A) }909\qquad \textbf{(B) }910\qquad \textbf{(C) }911\qquad \textbf{(D) }912\qquad \textbf{(E) }913\qquad $

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that \[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\] If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

Let $ ABCDA’B’C’D’ $ a cube. $ M,P $ are the midpoints of $ AB, $ respectively, $ DD’. $ [b]a)[/b] Show that $ MP, A’C $ are perpendicular, but not coplanar. [b]b)[/b] Calculate the distance between the lines above.

Let $ABC$ be a triangle with $AB=AC$. Also, let $D\in[BC]$ be a point such that $BC>BD>DC>0$, and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$. Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$). [i]Greece[/i]

A triangle has lattice points as vertices and contains no other lattice points. Prove that its area is $\frac{1}{2}$.

Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime integers. Find $m + n$

A trapezium is given with parallel bases having lengths $1$ and $4$. Split it into two trapeziums by a cut, parallel to the bases, of length $3$. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in $m$ and $n$ trapeziums, respectively, so that all the $m + n$ trapezoids obtained have the same area. Determine the minimum possible value for $m + n$ and the lengths of the cuts to be made to achieve this minimum value.

This is a very classical problem. Let the $ABCD$ be a cyclic quadrilateral with circumcircle $\omega_1$.Lines $AC$ and $BD$ intersect at point $E$,and lines $AD$,$BC$ intersect at point $F$.Circle $\omega_2$ is tangent to segments $EB,EC$ at points $M,N$ respectively,and intersects with circle $\omega_1$ at points $Q,R$.Lines $BC,AD$ intersect line $MN$ at $S,T$ respectively.Show that $Q,R,S,T$ are concyclic.

$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD$. The crease is $EF$, where $E$ is on $AB$ and $F$is on $CD$. The dimensions $AE=8$, $BE=17$, and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C; draw(F--D--A--E); draw(E--B--C--F, linetype("4 4")); filldraw(E--F--Cp--Bp--cycle, white, black); pair point=( 12.5, 35/3 ); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$B^\prime$", Bp, dir(point--Bp)); label("$C^\prime$", Cp, dir(point--Cp));[/asy]

$ABC$ is triangle with $AB=BC$. $X,Y$ are midpoints of $AC$ and $AB$. $Z$ is base of perpendicular from $B$ to $CY$. Prove, that circumcenter of $XYZ$ lies on $AC$

Let $ABC$ be a right triangle with $\angle A= 90^{\circ}$. A circle $\omega$ centered on $BC$ is tangent to $AB$ at $D$ and $AC$ at $E$. Let $F$ and $G$ be the intersections of $\omega$ and $BC$ so that $F$ lies between $B$ and $G$. If lines $DG$ and $EF$ intersect at $X$, show that $AX=AD.$

Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$. (Nigeria)

Let $ABC$ be an equilateral triangle. On the small arc $AB{}$ of its circumcircle $\Omega$, consider the point $N{}$ such that the small arc $NB$ measures $30^\circ{}$. The perpendiculars from $N{}$ onto $AC$ and $AB$ intersect $\Omega$ again at $P{}$ and $Q{}$ respectively. Let $H_1,H_2$ and $H_3$ be the orthocenters of the triangles $NAB, QBC$ and $CAP$ respectively. [list=a] [*]Prove that the triangle $NPQ$ is equilateral. [*]Prove that the triangle $H_1H_2H_3$ is equilateral. [/list]

Prove that for any points $A,B,C,D$ in the plane, the following inequality holds \[\frac{AB}{DA+DB}+\frac{BC}{DB+DC}\geqslant\frac{AC}{DA+DC}.\]

A Yule log is shaped like a right cylinder with height $10$ and diameter $5$. Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\pi$. Compute $a$.

Points $D,E,F$ are on the sides $BC, CA$ and $AB$, respectively which satisfy $EF || BC$, $D_1$ is a point on $BC,$ Make $D_1E_1 || D_E, D_1F_1 || DF$ which intersect $AC$ and $AB$ at $E_1$ and $F_1$, respectively. Make $\bigtriangleup PBC \sim \bigtriangleup DEF$ such that $P$ and $A$ are on the same side of $BC.$ Prove that $E, E_1F_1, PD_1$ are concurrent. [color=red][Edit by Darij: See my post #4 below for a [b]possible correction[/b] of this problem. However, I am not sure that it is in fact the problem given at the TST... Does anyone have a reliable translation?][/color]

$\textbf{Bake Sale}$ Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. $\circ$ Art's cookies are trapezoids: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label("5 in", (2.5,0), S); label("3 in", (5,1.5), E); label("3 in", (3.5,3), N);[/asy] $\circ$ Roger's cookies are rectangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label("4 in", (2,0), S); label("2 in", (4,1), E);[/asy] $\circ$ Paul's cookies are parallelograms: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label("3 in", (1.5,0), S); label("2 in", (2.5,1), W);[/asy] $\circ$ Trisha's cookies are triangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label("3 in", (1.5,0), S); label("4 in", (3,2), E);[/asy] Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Who gets the fewest cookies from one batch of cookie dough? $ \textbf{(A)}\ \text{Art}\qquad\textbf{(B)}\ \text{Roger}\qquad\textbf{(C)}\ \text{Paul}\qquad\textbf{(D)}\ \text{Trisha}\qquad\textbf{(E)}\ \text{There is a tie for fewest.}$

A rhombus is given with one diagonal twice the length of the other diagonal. Express the side of the rhombus is terms of $K$, where $K$ is the area of the rhombus in square inches. ${{ \textbf{(A)}\ \sqrt{K} \qquad\textbf{(B)}\ \frac{1}{2}\sqrt{2K} \qquad\textbf{(C)}\ \frac{1}{3}\sqrt{3K} \qquad\textbf{(D)}\ \frac{1}{4}\sqrt{4K} }\qquad\textbf{(E)}\ \text{None of these are correct} } $

Let $a_{n}$ be the area surrounded by the curves $y=e^{-x}$ and the part of $y=e^{-x}|\cos x|,\ (n-1)\pi \leq x\leq n\pi \ (n=1,\ 2,\ 3,\ \cdots).$ Evaluate $\lim_{n\to\infty}(a_{1}+a_{2}+\cdots+a_{n}).$

[b]p1.[/b] Find the largest $4$ digit integer that is divisible by $2$ and $5$, but not $3$. [b]p2.[/b] The diagram below shows the eight vertices of a regular octagon of side length $2$. These vertices are connected to form a path consisting of four crossing line segments and four arcs of degree measure $270^o$. Compute the area of the shaded region. [center][img]https://cdn.artofproblemsolving.com/attachments/0/0/eec34d8d2439b48bb5cca583462c289287f7d0.png[/img][/center] [b]p3.[/b] Consider the numbers formed by writing full copies of $2023$ next to each other, like so: $$2023202320232023...$$ How many copies of $2023$ are next to each other in the smallest multiple of $11$ that can be written in this way? [b]p4.[/b] A positive integer $n$ with base-$10$ representation $n = a_1a_2 ...a_k$ is called [i]powerful [/i] if the digits $a_i$ are nonzero for all $1 \le i \le k$ and $$n = a^{a_1}_1 + a^{a_2}_2 +...+ a^{a_k}_k .$$ What is the unique four-digit positive integer that is [i]powerful[/i]? [b]p5.[/b] Six $(6)$ chess players, whose names are Alice, Bob, Crystal, Daniel, Esmeralda, and Felix, are sitting in a circle to discuss future content pieces for a show. However, due to fights they’ve had, Bob can’t sit beside Alice or Crystal, and Esmeralda can’t sit beside Felix. Determine the amount of arrangements the chess players can sit in. Two arrangements are the same if they only differ by a rotation. [b]p6.[/b] Given that the infinite sum $\frac{1}{1^4} +\frac{1}{2^4} +\frac{1}{3^4} +...$ is equal to $\frac{\pi^4}{90}$, compute the value of $$\dfrac{\dfrac{1}{1^4} +\dfrac{1}{2^4} +\dfrac{1}{3^4} +...}{\dfrac{1}{1^4} +\dfrac{1}{3^4} +\dfrac{1}{5^4} +...}$$ [b]p7.[/b] Triangle $ABC$ is equilateral. There are $3$ distinct points, $X$, $Y$ , $Z$ inside $\vartriangle ABC$ that each satisfy the property that the distances from the point to the three sides of the triangle are in ratio $1 : 1 : 2$ in some order. Find the ratio of the area of $\vartriangle ABC$ to that of $\vartriangle XY Z$. [b]p8.[/b] For a fixed prime $p$, a finite non-empty set $S = \{s_1,..., s_k\}$ of integers is $p$-[i]admissible [/i] if there exists an integer $n$ for which the product $$(s_1 + n)(s_2 + n) ... (s_k + n)$$ is not divisible by $p$. For example, $\{4, 6, 8\}$ is $2$-[i]admissible[/i] since $(4+1)(6+1)(8+1) = 315$ is not divisible by $2$. Find the size of the largest subset of $\{1, 2,... , 360\}$ that is two-,three-, and five-[i]admissible[/i]. [b]p9.[/b] Kwu keeps score while repeatedly rolling a fair $6$-sided die. On his first roll he records the number on the top of the die. For each roll, if the number was prime, the following roll is tripled and added to the score, and if the number was composite, the following roll is doubled and added to the score. Once Kwu rolls a $1$, he stops rolling. For example, if the first roll is $1$, he gets a score of $1$, and if he rolls the sequence $(3, 4, 1)$, he gets a score of $3 + 3 \cdot 4 + 2 \cdot 1 = 17$. What is his expected score? [b]p10.[/b] Let $\{a_1, a_2, a_3, ...\}$ be a geometric sequence with $a_1 = 4$ and $a_{2023} = \frac14$ . Let $f(x) = \frac{1}{7(1+x^2)}$. Find $$f(a_1) + f(a_2) + ... + f(a_{2023}).$$ [b]p11.[/b] Let $S$ be the set of quadratics $x^2 + ax + b$, with $a$ and $b$ real, that are factors of $x^{14} - 1$. Let $f(x)$ be the sum of the quadratics in $S$. Find $f(11)$. [b]p12.[/b] Find the largest integer $0 < n < 100$ such that $n^2 + 2n$ divides $4(n- 1)! + n + 4$. [b]p13.[/b] Let $\omega$ be a unit circle with center $O$ and radius $OQ$. Suppose $P$ is a point on the radius $OQ$ distinct from $Q$ such that there exists a unique chord of $\omega$ through $P$ whose midpoint when rotated $120^o$ counterclockwise about $Q$ lies on $\omega$. Find $OP$. [b]p14.[/b] A sequence of real numbers $\{a_i\}$ satisfies $$n \cdot a_1 + (n - 1) \cdot a_2 + (n - 2) \cdot a_3 + ... + 2 \cdot a_{n-1} + 1 \cdot a_n = 2023^n$$ for each integer $n \ge 1$. Find the value of $a_{2023}$. [b]p15.[/b] In $\vartriangle ABC$, let $\angle ABC = 90^o$ and let $I$ be its incenter. Let line $BI$ intersect $AC$ at point $D$, and let line $CI$ intersect $AB$ at point $E$. If $ID = IE = 1$, find $BI$. [b]p16.[/b] For a positive integer $n$, let $S_n$ be the set of permutations of the first $n$ positive integers. If $p = (a_1, ..., a_n) \in S_n$, then define the bijective function $\sigma_p : \{1,..., n\} \to \{1, ..., n\}$ such that $\sigma_p (i) = a_i$ for all integers $1 \le i \le n$. For any two permutations $p, q \in S_n$, we say $p$ and $q$ are friends if there exists a third permutation $r \in S_n$ such that for all integers $1 \le i \le n$, $$\sigma_p(\sigma_r (i)) = \sigma_r(\sigma_q(i)).$$ Find the number of friends, including itself, that the permutation $(4, 5, 6, 7, 8, 9, 10, 2, 3, 1)$ has in $S_{10}$. PS. You had better use hide for answers.

Triangle $ABC$ has orthocentre $H$. The feet of the perpendiculars from $H$ to the internal and external bisectors of $\hat{A}$ are $P$ and $Q$ respectively. Prove that $P$ is on the line that passes through $Q$ and the midpoint of $BC$. (Note: The ortohcentre of a triangle is the point where the three altitudes intersect.)