There is point $D$ on edge $AC$ isosceles triangle $ABC$ with base $BC$. There is point $K$ on the smallest arc $CD$ of circumcircle of triangle $BCD$. Ray $CK$ intersects line parallel to line $BC$ through $A$ at point $T$. Let $M$ be midpoint of segment $DT$. Prove that $\angle AKT=\angle CAM$.

Given is a triangle $ABC$ and the points $M, P$ lie on the segments $AB, BC$, respectively, such that $AM=BC$ and $CP=BM$. If $AP$ and $CM$ meet at $O$ and $2\angle AOM=\angle ABC$, find the measure of $\angle ABC$.

Let $a,b,c$ be non-negative real numbers such that $a+b\leq c+1, b+c\leq a+1$ and $c+a\leq b+1.$ Show that \[a^2+b^2+c^2\leq 2abc+1.\]

A positive integer is called [i]piola [/i] if the $9$ is the remainder obtained by dividing it by $2, 3, 4, 5, 6, 7, 8, 9$ and $10$ and it's digits are all different and nonzero. How many [i]piolas[/i] are there between $ 1$ and $100000$?

A person is locked in a room with a password-protected computer. If they enter the correct password, the door opens and they are freed. However, the password changes every time it is entered incorrectly. The person knows that the password is always a 10-digit number, and they also know that the password change follows a fixed pattern. This means that if the current password is \( b \) and \( a \) is entered, the new password is \( c \), which is determined by \( b \) and \( a \) (naturally, the person does not know \( c \) or \( b \)). Prove that regardless of the characteristics of this computer, the prisoner can free themselves. Proposed by Reza Tahernejad Karizi

Find all positive integers $n$ for which there exist real numbers $x_1, x_2,. . . , x_n$ satisfying all of the following conditions: (i) $-1 <x_i <1,$ for all $1\leq i \leq n.$ (ii) $ x_1 + x_2 + ... + x_n = 0.$ (iii) $\sqrt{1 - x_1^2} +\sqrt{1 - x^2_2} + ... +\sqrt{1 - x^2_n} = 1.$

On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz? ${ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}}\ 4\qquad\textbf{(E)}\ 5$

Each of the $100$ students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are $42$ students who cannot sing, $65$ students who cannot dance, and $29$ students who cannot act. How many students have two of these talents? $\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64$

Transform the equation $$ab^2 \left(\frac{1}{(a + c)^2} +\frac{1}{(a- c)^2} \right) = (a -b)$$ into a simpler form, given that $a > c \ge 0$, $b > 0$.

Find all functions $f : \mathbb{Z} \to\mathbb{ Z}$ such that \[ n^2+4f(n)=f(f(n))^2 \] for all $n\in \mathbb{Z}$. [i]Proposed by Sahl Khan, UK[/i]

$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.

Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$. Show that there are $p^2+1$ triples of integers $(x,y,z)$ with $0 \le x,y,z < p$ and such that $(x+y+z)^2 \equiv axyz \pmod p$

Consider the $4\times4$ array of $16$ dots, shown below. [asy] size(2cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); dot((0,3)); dot((1,3)); dot((2,3)); dot((3,3)); [/asy] Counting the number of squares whose vertices are among the $16$ dots and whose sides are parallel to the sides of the grid, we find that there are nine $1\times1$ squares, four $2\times2$ squares, and one $3\times3$ square, for a total of $14$ squares. We delete a number of these dots. What is the minimum number of dots that must be deleted so that each of the $14$ squares is missing at least one vertex?

Let $x,y,z$ be complex numbers satisfying \begin{align*} z^2 + 5x &= 10z \\ y^2 + 5z &= 10y \\ x^2 + 5y &= 10x \end{align*} Find the sum of all possible values of $z$. [i]Proposed by Aaron Lin[/i]

$\vartriangle ABC$ has $AB = 5$, $BC = 12$, and $AC = 13$. A circle is inscribed in $\vartriangle ABC$, and $MN$ tangent to the circle is drawn such that $M$ is on $\overline{AC}$, $N$ is on $\overline{BC}$, and $\overline{MN} \parallel \overline{AB}$. The area of $\vartriangle MNC$ is $m/n$ , where $m$ and $n $are relatively prime positive integers. Find $m + n$.

How many different solutions on the interval $[0, \pi]$ does the equation $$6\sqrt2 \sin x \cdot tgx - 2\sqrt2 tgx +3\sin x -1=0$$ have?

Given a sequence $s$ consisting of digits $0$ and $1$. For any positive integer $k$, define $v_k$ the maximum number of ways in any sequence of length $k$ that several consecutive digits can be identified, forming the sequence $s$. (For example, if $s=0110$, then $v_7=v_8=2$, because in sequences $0110110$ and $01101100$ one can find consecutive digits $0110$ in two places, and three pairs of $0110$ cannot meet in a sequence of length $7$ or $8$.) It is known that $v_n<v_{n+1}<v_{n+2}$ for some positive integer $n$. Prove that in the sequence $s$, all the numbers are the same. [i]A. Golovanov[/i]

In triangle $ABC$, $AB = AC$. Point $D$ is the midpoint of side $BC$. Point $E$ lies outside the triangle $ABC$ such that $CE \perp AB$ and $BE = BD$. Let $M$ be the midpoint of segment $BE$. Point $F$ lies on the minor arc $\widehat{AD}$ of the circumcircle of triangle $ABD$ such that $MF \perp BE$. Prove that $ED \perp FD.$ [asy] defaultpen(fontsize(10)); size(6cm); pair A = (3,10), B = (0,0), C = (6,0), D = (3,0), E = intersectionpoints( Circle(B, 3), C--(C+100*dir(B--A)*dir(90)) )[1], M = midpoint(B--E), F = intersectionpoints(M--(M+50*dir(E--B)*dir(90)), circumcircle(A,B,D))[0]; dot(A^^B^^C^^D^^E^^M^^F); draw(B--C--A--B--E--D--F--M^^circumcircle(A,B,D)); pair point = extension(M,F,A,D); pair[] p={A,B,C,D,E,F,M}; string s = "A,B,C,D,E,F,M"; int size = p.length; real[] d; real[] mult; for(int i = 0; i<size; ++i) { d[i] = 0; mult[i] = 1;} d[4] = -50; string[] k= split(s,","); for(int i = 0;i<p.length;++i) { label("$"+k[i]+"$",p[i],mult[i]*dir(point--p[i])*dir(d[i])); }[/asy]

Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or $b$. Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers such that \[x^{a}+y^{b}= z^{c}.\]

How many ordered pairs $(a, b)$ of positive integers satisfy the equation $$a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),$$ where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple? $\textbf{(A)}\ 0\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8$

Let $P(x)$ and $Q(x)$ be monic polynomials. Prove that the sum of the squares of the coeficients of the polynomial $P(x)Q(x)$ is not smaller than the sum of the squares of the free coefficients of $P(x)$ and $Q(x)$. [i]A. Galochkin, O. Ljashko[/i]

$ABCD$ is a parallelogram satisfying $AB = 7$, $BC = 2$, and $\angle DAB = 120^o$. Parallelogram $ECFA$ is contained in $ABCD$ and is similar to it. Find the ratio of the area of $ECFA$ to the area of $ABCD$.

Source: 2018 Canadian Open Math Challenge Part A Problem 4 ----- In the sequence of positive integers, starting with $2018, 121, 16, ...$ each term is the square of the sum of digits of the previous term. What is the $2018^{\text{th}}$ term of the sequence?

For $a, b, c>0,$ and $k\geq1,$ prove that \[\frac{a^{k+1}}{b^k+c^k}+\frac{b^{k+1}}{c^k+a^k}+\frac{c^{k+1}}{a^k+b^k}\geq\frac{3}{2}\sqrt{\frac{a^{k+1}+b^{k+1}+c^{k+1}}{{a^{k-1}+b^{k-1}+c^{k-1}}}}\] Author: MIHALY BENCZE

In a party of $1000$ people, the number of people who have shaken hands with at most $962$ people is less than or equal to $37$. Show that one can find $27$ people in the party who have all shaken hands with each other.