Find the last two digits of each of the numbers $3^{1974}$ and $7^{1974}$.

Determine all triples $(a, b, c)$ of positive integers such that $$a! +b! = 2^{c!} .$$

An $h \times k$ minor of an $n \times n$ table is the $hk$ cells which lie in $h$ rows and $k$ columns. The semiperimeter of the minor is $h + k$. A number of minors each with semiperimeter at least $n$ together include all the cells on the main diagonal. Show that they include at least half the cells in the table.

Is there an infinite sequence $ a_0, a_1, a_2, \cdots $ of nonzero real numbers such that for $ n = 1, 2, 3, \cdots $ the polynomial \[ p_n(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \] has exactly $n$ distinct real roots?

Let $A_0A_1A_2$ be a scalene triangle. Find the locus of the centres of the equilateral triangles $X_0X_1X_2$ , such that $A_k$ lies on the line $X_{k+1}X_{k+2}$ for each $k=0,1,2$ (with indices taken modulo $3$).

The squares of a $3\times3$ grid are filled with positive integers such that $1$ is the label of the upper- leftmost square, $2009$ is the label of the lower-rightmost square, and the label of each square divides the ne directly to the right of it and the one directly below it. How many such labelings are possible?

We number a hundred blank cards on both sides with the numbers $1$ to $100$. The cards are then stacked in order, with the card with the number $1$ on top. The order of the cards is changed step by step as follows: at the $1$st step the top card is turned around, and is put back on top of the stack (nothing changes, of course), at the $2$nd step the topmost $2$ cards are turned around, and put back on top of the stack, up to the $100$th step, in which the entire stack of $100$ cards is turned around. At the $101$st step, again only the top card is turned around, at the $102$nd step, the top most $2$ cards are turned around, and so on. Show that after a finite number of steps, the cards return to their original positions.

A block $Z$ is formed by gluing one face of a solid cube with side length 6 onto one of the circular faces of a right circular cylinder with radius $10$ and height $3$ so that the centers of the square and circle coincide. If $V$ is the smallest convex region that contains Z, calculate $\lfloor\operatorname{vol}V\rfloor$ (the greatest integer less than or equal to the volume of $V$).

Let $ABC$ be a triangle with $BA=BC$ and $\angle ABC=90^{\circ}$. Let $D$ and $E$ be the midpoints of $CA$ and $BA$ respectively. The point $F$ is inside of $\triangle ABC$ such that $\triangle DEF$ is equilateral. Let $X=BF\cap AC$ and $Y=AF\cap DB$. Prove that $DX=YD$.

In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$. [i]Proposed by Andy Xu[/i]

A teacher and her 30 students play a game on an infinite cell grid. The teacher starts first, then each of the 30 students makes a move, then the teacher and so on. On one move the person can color one unit segment on the grid. A segment cannot be colored twice. The teacher wins if, after the move of one of the 31 players, there is a $1\times 2$ or $2\times 1$ rectangle , such that each segment from it's border is colored, but the segment between the two adjacent squares isn't colored. Prove that the teacher can win.

Define a sequence of integers $a_0=m, a_1=n$ and $a_{k+1}=4a_k-5a_{k-1}$ for all $k \ge 1$. Suppose $p>5$ is a prime with $p \equiv 1 \pmod{4}$. Prove that it is possible to choose $m,n$ such that $p \nmid a_k$ for any $k \ge 0$.

A triangle $ABC$ is inscribed in a circle with center $U$ and radius $r$. A tangent $c'$ to a larger circle $K(U, 2r)$ is drawn so that C lies between the lines $c = AB$ and $C'$. Lines $a'$ and $b'$ are analogously defined. The triangle formed by $a', b', c'$ is denoted $A'B'C'$. Prove that the three lines, joining the midpoints of pairs of parallel sides of the two triangles, have a common point.

The diagram shows a regular pentagon $ABCDE$ and a square $ABFG$. Find the degree measure of $\angle FAD$. [img]https://cdn.artofproblemsolving.com/attachments/7/5/d90827028f426f2d2772f7d7b875eea4909211.png[/img]

Let $a,b,c>0$ be real numbers with sum 1. Prove that \[ \frac{a^2}b + \frac{b^2}c + \frac{c^2} a \geq 3(a^2+b^2+c^2) . \]

Let $f_n$ be the Fibonacci numbers, defined by $f_0 = 1$, $f_1 = 1$, and $f_n = f_{n-1}+f_{n-2}$. For each $i$, $1 \le i \le 200$, we calculate the greatest common divisor $g_i$ of $f_i$ and $f_{2007}$. What is the sum of the distinct values of $g_i$?

What is the smallest positive integer than can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?

Incircle $\omega$ of triangle $ABC$ is tangent to sides $CB$ and $CA$ at $D$ and $E$, respectively. Point $X$ is the reflection of $D$ with respect to $B$. Suppose that the line $DE$ is tangent to the $A$-excircle at $Z$. Let the circumcircle of triangle $XZE$ intersect $\omega$ for the second time at $K$. Prove that the intersection of $BK$ and $AZ$ lies on $\omega$. Proposed by Mahdi Etesamifard and Alireza Dadgarnia

Triangle $ABC$ is said to be perpendicular to triangle $DEF$ if the perpendiculars from $A$ to $EF$,from $B$ to $FD$ and from $C$ to $DE$ are concurrent.Prove that if $ABC$ is perpendicular to $DEF$,then $DEF$ is perpendicular to $ABC$

For those real numbers $x_1 , x_2 , \ldots , x_{2011}$ where each of which satisfies $0 \le x_1 \le 1$ ($i = 1 , 2 , \ldots , 2011$), find the maximum of \[ x_1^3+x_2^3+ \cdots + x_{2011}^3 - \left( x_1x_2x_3 + x_2x_3x_4 + \cdots + x_{2011}x_1x_2 \right) \]

Let $\Gamma_1$ and $\Gamma_2$ be two circles touching each other externally at $R.$ Let $O_1$ and $O_2$ be the centres of $\Gamma_1$ and $\Gamma_2,$ respectively. Let $\ell_1$ be a line which is tangent to $\Gamma_2$ at $P$ and passing through $O_1,$ and let $\ell_2$ be the line tangent to $\Gamma_1$ at $Q$ and passing through $O_2.$ Let $K=\ell_1\cap \ell_2.$ If $KP=KQ$ then prove that the triangle $PQR$ is equilateral.

Find all pairs of positive integers \((a,b)\) with the following property: there exists an integer \(N\) such that for any integers \(m\ge N\) and \(n\ge N\), every \(m\times n\) grid of unit squares may be partitioned into \(a\times b\) rectangles and fewer than \(ab\) unit squares. [i]Proposed by Holden Mui[/i]

Let $\alpha$ and $\beta$ be positive integers such that $$ \frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16} . $$ Find the smallest possible value of $\beta$.

Determine the largest $3$ digit prime number that is a factor of ${2000 \choose 1000}$.

How many solutions does the equation $ \frac{[x]}{\{ x\}} =\frac{2007x}{2008} $ have? [i]Mihail Bălună[/i]