Cut the $9 \times 10$ grid rectangle along the grid lines into several squares so that there are exactly two of them with odd sidelengths.

In $ xyz$ space, find the volume of the solid expressed by the sytem of inequality: $ 0\leqq x\leqq 1,\ 0\leqq y\leqq 1,\ 0\leqq z\leqq 1$ $ x^2 \plus{} y^2 \plus{} z^2 \minus{} 2xy \minus{} 1\geqq 0$

At the junction of some countably infinite number of roads sits a greyhound. On one of the roads a hare runs, away from the junction. The only thing known is that the (maximal) speed of the hare is strictly less than the (maximal) speed of the greyhound (but not their precise ratio). Does the greyhound have a strategy for catching the hare in a finite amount of time? ([i]Dan Schwarz[/i])

$BE$ is the altitude of acute triangle $ABC$. The line $\ell$ touches the circumscribed circle of the triangle $ABC$ at point $B$. A perpendicular $CF$ is drawn from $C$ on line $\ell$. Prove that the lines $EF$ and $AB$ are parallel.

In a grid of dimensions $n \times n$, a part of the squares is marked with crosses such that in each at least half of the $4 \times 4$ squares are marked. Find the least possible the total number of marked squares in the grid.

For an ordered pair $(m,n)$ of distinct positive integers, suppose, for some nonempty subset $S$ of $\mathbb R$, that a function $f:S \rightarrow S$ satisfies the property that $f^m(x) + f^n(y) = x+y$ for all $x,y\in S$. (Here $f^k(z)$ means the result when $f$ is applied $k$ times to $z$; for example, $f^1(z)=f(z)$ and $f^3(z)=f(f(f(z)))$.) Then $f$ is called \emph{$(m,n)$-splendid}. Furthermore, $f$ is called \emph{$(m,n)$-primitive} if $f$ is $(m,n)$-splendid and there do not exist positive integers $a\le m$ and $b\le n$ with $(a,b)\neq (m,n)$ and $a \neq b$ such that $f$ is also $(a,b)$-splendid. Compute the number of ordered pairs $(m,n)$ of distinct positive integers less than $10000$ such that there exists a nonempty subset $S$ of $\mathbb R$ such that there exists an $(m,n)$-primitive function $f: S \rightarrow S$. [i]Proposed by Vincent Huang[/i]

Two runners start running along a circular track of unit length from the same starting point and int he same sense, with constant speeds $v_1$ and $v_2$ respectively, where $v_1$ and $v_2$ are two distinct relatively prime natural numbers. They continue running till they simultneously reach the starting point. Prove that (a) at any given time $t$, at least one of the runners is at a distance not more than $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units from the starting point. (b) there is a time $t$ such that both the runners are at least $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units away from the starting point. (All disstances are measured along the track). $[x]$ is the greatest integer function.

Let $N$ be the set of positive integers. If $A,B,C \ne \emptyset$, $A \cap B = B \cap C = C \cap A = \emptyset$ and $A \cup B \cup C = N$, we say that $A,B,C$ are partitions of $N$. Prove that there are no partitions of $N, A,B,C$, that satisfy the following: (i) $\forall a \in A, b \in B$, we have $a + b + 1 \in C$ (ii) $\forall b \in B, c \in C$, we have $b + c + 1 \in A$ (iii) $\forall c \in C, a \in A$, we have $c + a + 1 \in B$

Santa Claus wants to wrap presents. These are available in $n$ sizes $A_1 <A_2 <...<A_n$, and analogously, there are $n$ packaging sizes $B_1 <B_2 <...<B_n$, where $B_i$ is enough to all gift sizes $A_j$ can be grouped with $j\le i$, but too small for those with $j> i$. On the shelf to the right of Santa Claus are the gifts sorted by size, where the smallest are on the right, of course there can be several gifts of the same size, or none of a size at all. To his left is a shelf with packaging, and also these are sorted from small to large in the same direction. He's brooding in what way he should wrap the gifts and sees two methods for doing this, which depend on his thinking and laziness of movement have been optimized: a) He takes the present closest to him and puts it in the closest packaging, in which it fits in. b) He takes the packaging closest to him and packs in it the closest thing to him gift. In both cases he then does the same again, although of course the one he was using the gift and its packaging are missing, and so on. Once it is not large enough if the packaging or the present is not small enough, he / she will provide the present or the packaging back to its place on the shelf and takes the next-closest. Prove that both methods lead to the same result in the end, they are considered to be exactly the same gifts packed in the same packaging.

Define $f(x,y,z)=\frac{(xy+yz+zx)(x+y+z)}{(x+y)(y+z)(z+x)}$. Determine the set of real numbers $r$ for which there exists a triplet of positive real numbers satisfying $f(x,y,z)=r$.

Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.

Let $a(n)$ be the number of representations of the positive integer $n$ as an ordered sum of $1$'s and $2$'s. Let $b(n)$ be the number of representations of the positive integer $n$ as an ordered sum of integers greater than $1.$ Show that $a(n)=b(n+2)$ for each $n$.

Evaluate $ \int_{0}^{1}e^{\sqrt{e^{x}}}\ dx\plus{}2\int_{e}^{e^{\sqrt{e}}}\ln (\ln x)\ dx$.

One side of a square in on line $y=2x-17$, and two other points are on parabola $y=x^2$, then the minumum value of the area of the square is________.

Given are sheets and the numbers $00, 01, \ldots, 99$ are written on them. We must put them in boxes $000, 001, \ldots, 999$ so that the number on the sheet is the number on the box with one digit erased. What is the minimum number of boxes we need in order to put all the sheets?

Powers of $2$ in base $10$ start with $3$ or $4$ more frequently? What is their state in base $3$? First write down an exact form of the question.

Integers $n$ and $k$ satisfy $n > 2023k^3$. Kingdom Kitty has $n$ cities, with at most one road between each pair of cities. It is known that the total number of roads in the kingdom is at least $2n^{3/2}$. Prove that we can choose $3k + 1$ cities such that the total number of roads with both ends being a chosen city is at least $4k$.

There are $60$ friends who want to visit each others home during summer vacation. Everyday, they decide to either stay home or visit the home of everyone who stayed home that day. Find the minimum number of days required for everyone to have visited their friends’ homes.

The lines $L$ and $K$ are symmetric to each other with respect to the line $y=x$. If the equation of the line $L$ is $y=ax+b$ with $a\neq 0$ and $b \neq 0$, then the equation of $K$ is $y=$ $\text{(A)}\ \frac 1ax+b \qquad \text{(B)}\ -\frac 1ax+b \qquad \text{(C)}\ \frac 1ax - \frac ba \qquad \text{(D)}\ \frac 1ax+\frac ba \qquad \text{(E)}\ \frac 1ax -\frac ba$

The diameter $[AC]$ of a circle is divided into four equal segments by points $P, M$ and $Q$. Consider a segment $[BD]$ that passes through $P$ and cuts the circle at $B$ and $D$, such that $PD =\frac{3}{2} AP$. Knowing that the area of the triangle $[ABP]$ has measure $1$ cm$^2$ , calculate the area of $[ABCD]$? [img]https://1.bp.blogspot.com/-ibre0taeRo8/X4KiWWSROEI/AAAAAAAAMl4/xFNfpQBxmMMVLngp5OWOXRLMuaxf3nolQCLcBGAsYHQ/s154/1995%2Bportugal%2Bp5.png[/img]

Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$.

Determine the smallest radius a circle passing through EXACTLY three lattice points may have.

The locus of the centers of all circles of given radius $a$, in the same plane, passing through a fixed point, is: $ \textbf{(A) }\text{a point}\qquad\textbf{(B) }\text{ a straight line} \qquad\textbf{(C) }\text{two straight lines}\qquad\textbf{(D) }\text{a circle}\qquad$ $\textbf{(E) }\text{two circles} $

Show that [list=a][*] infinitely many perfect squares are a sum of a perfect square and a prime number, [*] infinitely many perfect squares are not a sum of a perfect square and a prime number. [/list]

Josanna's test scores to date are 90, 80, 70, 60, and 85. Her goal is to raise her test average at least 3 points with her next test. What is the minimum test score she would need to accomplish this goal? $ \textbf{(A)}\ 80 \qquad \textbf{(B)}\ 82 \qquad \textbf{(C)}\ 85 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 95 $