For a positive integer $n$, let $f(n)$ be the number of (not necessarily distinct) primes in the prime factorization of $k$. For example, $f(1) = 0, f(2) = 1, $ and $f(4) = f(6) = 2$. let $g(n)$ be the number of positive integers $k \leq n$ such that $f(k) \geq f(j)$ for all $j \leq n$. Find $g(1) + g(2) + \ldots + g(100)$.

A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$. [i]Proposed by Mongolia[/i]

Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$. Prove that line $AO$ passes through the midpoint of segment $BE$.

In the lower left corner of the square $7 \times 7$ board there is a king. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different ways can the king get to the upper right corner of the board if he is prohibited from visiting the central square?

Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive integer that is the product of three distinct prime numbers!" Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, you could start with any one of them, add a second, subtract a third, and get the fourth!" Cat says, "That's cool! My favorite number is not among those four numbers, though." Claire says, "Now I know your favorite number!" What is Cat's favorite number? [i]Proposed by Andrew Wu and Andrew Milas[/i]

We call natural numbers [i]similar[/i] if they are written with the same (decimal) digits. For example, 112, 121, 211 are similar numbers having the digits 1, 1, 2. Show that there exist three similar 1995-digit numbers with no zero digits, such that the sum of two of them equals the third. [i]S. Dvoryaninov[/i]

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

The diagram shows two congruent isosceles triangles in a $20\times20$ square which has been partitioned into four $10\times10$ squares. Find the area of the shaded region. [asy] import graph; size(4.4cm); real labelscalefactor = 0.5; pen dotstyle = black; fill((-2,5)--(0,1)--(1,3)--(1,5)--cycle,gray); draw((-3,5)--(1,5), linewidth(2.2)); draw((1,5)--(1,1), linewidth(2.2)); draw((1,1)--(-3,1), linewidth(2.2)); draw((-3,1)--(-3,5), linewidth(2.2)); draw((-1,5)--(-1,1), linewidth(2.2)); draw((-3,3)--(1,3), linewidth(2.2)); draw((-2,5)--(-3,3), linewidth(1.4)); draw((-2,5)--(0,1), linewidth(1.4)); draw((0,1)--(1,3), linewidth(1.4)); draw((-2,5)--(0,1)); draw((0,1)--(1,3)); draw((1,3)--(1,5)); draw((1,5)--(-2,5));[/asy]

Urn A contains $4$ white balls and $2$ red balls. Urn B contains $3$ red balls and $3$ black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What is the probability that the first ball drawn was red, given that the second ball drawn was black?

Let $n \ge 3$ be an integer. Find the minimal value of the real number $k_n$ such that for all positive numbers $x_1, x_2, ..., x_n$ with product $1$, we have $$\frac{1}{\sqrt{1 + k_nx_1}}+\frac{1}{\sqrt{1 + k_nx_2}}+ ... + \frac{1}{\sqrt{1 + k_nx_n}} \le n - 1.$$

Let $ABC$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $BC$ such that $\angle{CAX} = \angle{YAB}$. Suppose that: $1)$ $K$ and $S$ are the feet of the perpendiculars from from $B$ to the lines $AX$ and $AY$ respectively. $2)$ $T$ and $L$ are the feet of the perpendiculars from $C$ to the lines $AX$ and $AY$ respectively. Prove that $KL$ and $ST$ intersect on the line $BC$.

Determine all positive integers $n > 2$ , such that \[ \frac{1}{2} \varphi(n) \equiv 1 ( \bmod 6) \]

. In the isosceles triangle $ABC$ with $AC = BC$ we denote by $D$ the foot of the altitude through $C$. The midpoint of $CD$ is denoted by $M$. The line $BM$ intersects $AC$ in $E$. Prove that the length of $AC$ is three times that of $CE$.

In the expression $(x + 100) (x + 99) ... (x-99) (x-100)$, the brackets were expanded and similar terms were given. The expression $x^{201} + ...+ ax^2 + bx + c$ turned out. Find the numbers $a$ and $c$.

The figure shows an arbitrary (green) triangle in the center. White squares were built on its sides to the outside. Some of their vertices were connected by segments, white squares were built on them again to the outside, and so on. In the spaces between the squares, triangles and quadrilaterals were formed, which were painted in different colors. Prove that [list=a] [*]all colored quadrilaterals are trapezoids; [*]the areas of all polygons of the same color are equal; [*]the ratios of the bases of one-color trapezoids are equal; [*]if $S_0=1$ is the area of the original triangle, and $S_i$ is the area of the colored polygons at the $i^{\text{th}}$ step, then $S_1=1$, $S_2=5$ and for $n\geqslant 3$ the equality $S_n=5S_{n-1}-S_{n-2}$ is satisfied. [/list] [i]Proposed by F. Nilov[/i] [center][img width="40"]https://i.ibb.co/n8gt0pV/Screenshot-2023-03-09-174624.png[/img][/center]

Among $100$ points in the plane, no three collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points whose labels differ by at least $50$. [i]Proposed by Evan Chen[/i]

Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$.

Sarah and Darah play the following game. Sarah puts $ n$ coins numbered with $ 1,\dots,n$ on a table (Each coin is in HEAD or TAIL position.) At each step Darah gives a coin to Sarah and she (Sarah) let him (Dara) to change the position of all coins with number multiple of a desired number $ k$. At the end, all of the coins that are in TAIL position will be given to Sarah and all of the coins with HEAD position will be given to Darah. Prove that Sarah can put the coins in a position at the beginning of the game such that she gains at least $ \Omega(n)$ coins. [hide="Hint:"]Chernov inequality![/hide]

For how many permutations $(a_1, a_2, \cdots, a_{2007})$ of the integers from $1$ to $2007$ is there exactly one $i$ between $1$ and $2006$ such that $a_i > a_{i+1}$? Express your answer as $a \** b^c + d \** e^f$ for integers $a$, $b$, $c$, $d$, $e$, and $f$ with $a \nmid b$ and $d \nmid e$.

Let $a,b,c$ be a real numbers such that this equations: $a^2x + b^2y + c^2z = 1$ $xy + yz + xz = 1$ have only one solution $(x, y, z)$ in real numbers. Prove that $a, b, c$ are sides of the triangle

A circle and an ellipse with foci $F_{1}$, $F_{2}$ lying inside it are given. Construct a chord $AB$ of the circle touching the ellipse and such that $AF_{1}F_{2}B$ is a cyclic quadrilateral. Proposed by: A.Zaslavsky

In the plane we are given the circles $\Gamma$ and $\Delta$ tangent to each other and $\Gamma$ contains $\Delta$. The radius of $\Gamma$ is $R$ and of $\Delta$ is $\frac{R}{2}$. Prove that for each positive integer $n \geq 3$, the equation: \[ (p(1) - p(n))^2 = (n-1)^2 \cdot (2 \cdot (p(1) + p(n)) - (n-1)^2 - 8) \] is the necessary and sufficient condition for $n$ to exist $n$ distinct circles $\Upsilon_1, \Upsilon_2, \ldots, \Upsilon_n$ such that all these circles are tangent to $\Gamma$ and $\Delta$ and $\Upsilon_i$ is tangent to $\Upsilon_{i+1}$, and $\Upsilon_1$ has radius $\frac{R}{p(1)}$ and $\Upsilon_n$ has radius $\frac{R}{p(n)}$.

Given an isosceles triangle $ABC$ ($AB = AC$), the inscribed circle $\omega$ touches its sides $AB$ and $AC$ at points $K$ and $L$, respectively. On the extension of the side of the base $BC$, towards $B$, an arbitrary point $M$. is chosen. Line $M$ intersects $\omega$ at the point $N$ for the second time, line $BN$ intersects the second point $\omega$ at the point $P$. On the line $PK$, there is a point $X$ such that $K$ lies between $P$ and $X$ and $KX = KM$. Determine the locus of the point $X$.

Perimeter of triangle $ABC$ is $1$. Circle $\omega$ touches side $BC$, continuation of side $AB$ at $P$ and continuation of side $AC$ in $Q$. Line through midpoints $AB$ and $AC$ intersects circumcircle of $APQ$ at $X$ and $Y$. Find length of $XY$.

A simple pendulum experiment is constructed from a point mass $m$ attached to a pivot by a massless rod of length $L$ in a constant gravitational field. The rod is released from an angle $\theta_0 < \frac{\pi}{2}$ at rest and the period of motion is found to be $T_0$. Ignore air resistance and friction. At what angle $\theta_g$ during the swing is the tension in the rod the greatest? $\textbf{(A) } \text{The tension is greatest at } \theta_g = \theta_0.\\ \textbf{(B) } \text{The tension is greatest at }\theta_g = 0.\\ \textbf{(C) } \text{The tension is greatest at an angle } 0 < \theta_g < \theta_0.\\ \textbf{(D) } \text{The tension is constant.}\\ \textbf{(E) } \text{None of the above is true for all values of } \theta_0 \text{ with } 0 < \theta_{0} < \frac{\pi}{2}$