The $27$ cells of a $3 \times 9$ grid are filled in using the numbers $1$ through $9$ so that each row contains $9$ different numbers, and each of the three $3 \times 3$ blocks heavily outlined in the example below contains $9$ different numbers, as in the first three rows of a Sudoku puzzle. [asy] unitsize(20); add(grid(9,3)); draw((0,0)--(9,0)--(9,3)--(0,3)--cycle, linewidth(2)); draw((3,0)--(3,3), linewidth(2)); draw((6,0)--(6,3), linewidth(2)); real a = 0.5; label("5",(a,a)); label("6",(1+a,a)); label("1",(2+a,a)); label("8",(3+a,a)); label("4",(4+a,a)); label("7",(5+a,a)); label("9",(6+a,a)); label("2",(7+a,a)); label("3",(8+a,a)); label("3",(a,1+a)); label("7",(1+a,1+a)); label("9",(2+a,1+a)); label("5",(3+a,1+a)); label("2",(4+a,1+a)); label("1",(5+a,1+a)); label("6",(6+a,1+a)); label("8",(7+a,1+a)); label("4",(8+a,1+a)); label("4",(a,2+a)); label("2",(1+a,2+a)); label("8",(2+a,2+a)); label("9",(3+a,2+a)); label("6",(4+a,2+a)); label("3",(5+a,2+a)); label("1",(6+a,2+a)); label("7",(7+a,2+a)); label("5",(8+a,2+a)); [/asy] The number of different ways to fill such a grid can be written as $p^a \cdot q^b \cdot r^c \cdot s^d$ where $p$, $q$, $r$, and $s$ are distinct prime numbers and $a$, $b$, $c$, $d$ are positive integers. Find $p \cdot a + q \cdot b + r \cdot c + s \cdot d$.

Find the area of the part bounded by two curves $y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $x$-axis. 1956 Tokyo Institute of Technology entrance exam

Each cell of an $m\times n$ board is filled with some nonnegative integer. Two numbers in the filling are said to be [i]adjacent[/i] if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a [i]garden[/i] if it satisfies the following two conditions: (i) The difference between any two adjacent numbers is either $0$ or $1$. (ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to $0$. Determine the number of distinct gardens in terms of $m$ and $n$.

The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image? [asy] import cse5;pathpen=black;pointpen=black; size(2cm); D((0,-2)--MP("y",(0,7),N)); D((-3,0)--MP("x",(5,0),E)); D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5)); [/asy][asy] import cse5;pathpen=black;pointpen=black; unitsize(0.2cm); D((0,-2)--MP("y",(0,7),N)); D(MP("\textbf{(A) }",(-3,0),W)--MP("x",(5,0),E)); D((1,0)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(3,4)--(3,5)--(0,5)); // D((18,-2)--MP("y",(18,7),N)); D(MP("\textbf{(B) }",(13,0),W)--MP("x",(21,0),E)); D((17,0)--(17,2)--(16,2)--(16,3)--(17,3)--(17,4)--(15,4)--(15,5)--(18,5)); // D((36,-2)--MP("y",(36,7),N)); D(MP("\textbf{(C) }",(29,0),W)--MP("x",(38,0),E)); D((31,0)--(31,1)--(33,1)--(33,2)--(34,2)--(34,1)--(35,1)--(35,3)--(36,3)); // D((0,-17)--MP("y",(0,-8),N)); D(MP("\textbf{(D) }",(-3,-15),W)--MP("x",(5,-15),E)); D((3,-15)--(3,-14)--(1,-14)--(1,-13)--(2,-13)--(2,-12)--(1,-12)--(1,-10)--(0,-10)); // D((15,-17)--MP("y",(15,-8),N)); D(MP("\textbf{(E) }",(13,-15),W)--MP("x",(22,-15),E)); D((15,-14)--(17,-14)--(17,-13)--(18,-13)--(18,-14)--(19,-14)--(19,-12)--(20,-12)--(20,-15)); [/asy]

Let $n \geq 2$ be an integer. Find the minimum $k$ for which there exists a partition of $\{1, 2, . . . , k\}$ into $n$ subsets $X_1,X_2, \cdots , X_n$ such that the following condition holds: for any $i, j, 1 \leq i < j \leq n$, there exist $x_i \in X_1, x_j \in X_2$ such that $|x_i - x_j | = 1.$

Let $a,b,c \ge 0$. Prove that $$\frac{(1+a^2)(1+b^2)(1+c^2)}{(1+a)(1+b)(1+c)}\ge \frac12 (1+abc)$$

For each $n\in\mathbb N$, determine the sign of $n^6+5n^5\sin n+1$. For which $n\in\mathbb N$ does it hold that $\frac{n^2+5n\cos n+1}{n^6+5n^5\sin n+1}\ge10^{-4}$.

Find all functions $f:[-1,1] \rightarrow \mathbb{R},$ which satisfy $$f(\sin{x})+f(\cos{x})=2020$$ for any real number $x.$

In the net drawn below, in how many ways can one reach the point $3n+1$ starting from the point $1$ so that the labels of the points on the way increase? [asy] import graph; size(12cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=12.32,ymin=-10.66,ymax=6.3; draw((1,2)--(xmax,0*xmax+2)); draw((1,0)--(xmax,0*xmax+0)); draw((0,1)--(1,2)); draw((1,0)--(0,1)); draw((1,2)--(3,0)); draw((1,0)--(3,2)); draw((3,2)--(5,0)); draw((3,0)--(5,2)); draw((5,2)--(7,0)); draw((5,0)--(7,2)); draw((7,2)--(9,0)); draw((7,0)--(9,2)); dot((1,0),linewidth(1pt)+ds); label("2",(0.96,-0.5),NE*lsf); dot((0,1),linewidth(1pt)+ds); label("1",(-0.42,0.9),NE*lsf); dot((1,2),linewidth(1pt)+ds); label("3",(0.98,2.2),NE*lsf); dot((2,1),linewidth(1pt)+ds); label("4",(1.92,1.32),NE*lsf); dot((3,2),linewidth(1pt)+ds); label("6",(2.94,2.2),NE*lsf); dot((4,1),linewidth(1pt)+ds); label("7",(3.94,1.32),NE*lsf); dot((6,1),linewidth(1pt)+ds); label("10",(5.84,1.32),NE*lsf); dot((3,0),linewidth(1pt)+ds); label("5",(2.98,-0.46),NE*lsf); dot((5,2),linewidth(1pt)+ds); label("9",(4.92,2.24),NE*lsf); dot((5,0),linewidth(1pt)+ds); label("8",(4.94,-0.42),NE*lsf); dot((8,1),linewidth(1pt)+ds); label("13",(7.88,1.34),NE*lsf); dot((7,2),linewidth(1pt)+ds); label("12",(6.8,2.26),NE*lsf); dot((7,0),linewidth(1pt)+ds); label("11",(6.88,-0.38),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

An $L$ is an arrangement of $3$ adjacent unit squares formed by deleting one unit square from a $2 \times 2$ square. a) How many $L$s can be placed on an $8 \times 8$ board (with no interior points overlapping)? b) Show that if any one square is deleted from a $1987 \times 1987$ board, then the remaining squares can be covered with $L$s (with no interior points overlapping).

For $a$ a positive real number, let $x_1$, $x_2$, $x_3$ be the roots of the equation $x^3-ax^2+ax-a=0$. Determine the smallest possible value of $x_1^3+x_2^3+x_3^3-3x_1x_2x_3$.

Let $ n,\ m$ be positive integers and $ \alpha ,\ \beta$ be real numbers. Prove the following equations. (1) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)(x \minus{} \beta)\ dx \equal{} \minus{} \frac 16 (\beta \minus{} \alpha)^3$ (2) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)\ dx \equal{} \minus{} \frac {n!}{(n \plus{} 2)!}(\beta \minus{} \alpha)^{n \plus{} 2}$ (3) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)^mdx \equal{} ( \minus{} 1)^{m}\frac {n!m!}{(n \plus{} m \plus{} 1)!}(\beta \minus{} \alpha)^{n \plus{} m \plus{} 1}$

On a board the following six vectors are written: $$(1, 0, 0), (-1, 0, 0), (0, 1, 0), (0, -1, 0), (0, 0, 1), (0, 0, -1).$$ Given two vectors $v$ and $w$ on the board, a move consists of erasing $v$ and $w$ and replacing them with $\frac{1}{\sqrt2} (v + w)$ and $\frac{1}{\sqrt2} (v - w)$. After some number of moves, the sum of the six vectors on the board is $u$. Find, with proof, the maximum possible length of $u$.

Let $ABC$ be a triangle with $\angle A = 90^o, \angle B = 75^o$ and $AB = 2$. The points $P$ and $Q$ on the sides $AC$ and $BC$ respectively are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$ . Calculate the measurement of the segment $QA $.

In the triangle $\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle.  Let $\alpha$ and $\beta$ be the angles at the vertices $A$ and $B$, respectively.  Suppose that the segment $IG$ is parallel to $AB$ and that $\beta = 2\tan^{-1}(1/3)$.  Find $\alpha$.

Let $n>1$ be an integer and $x_1,x_2,\ldots,x_n$ be $n{}$ arbitrary real numbers. Determine the minimum value of \[\sum_{i<j}|\cos(x_i-x_j)|.\]