Let $a,b,c,d,e,d,e,f$ be positive integers such that \(\dfrac a b < \dfrac c d < \dfrac e f\). Suppose \(af-be=-1\). Show that \(d \geq b+f\).

For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied: [list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$, [*] If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$[/list] Determine $N(n)$ for all $n\geq 2$. [i]Proposed by Dan Schwarz, Romania[/i]

A $3 \times 3 \times 4$ cuboid is constructed out of 36 white-coloured unit cubes. Then, all six of the cuboid's sides are coloured red. After that, the cuboid is dismantled into its constituent unit cubes. Then, randomly, all said unit cubes are constructed into the cuboid of its original size (and position). a) How many ways are there to position eight of its corner cubes so that the apparent sides of eight corner cubes are still red? (Cube rotations are still considered distinct configurations, and the position of the cuboid remains unchanged.) b) Determine the probability that after the reconstruction, all of its apparent sides are still red-coloured. (The cuboid is still upright, with the same dimensions as the original cuboid, without rotation.) [hide=Notes]The problem might have multiple interpretations. We agreed that this problem's wording was a bit ambiguous. Here's the original Indonesian version: Suatu balok berukuran $3 \times 3 \times 4$ tersusun dari 36 kubus satuan berwarna putih. Kemudian keenam permukaan balok diwarnai merah. Setelah itu, balok yang tersusun dari kubus-kubus satuan tersebut dibongkar. Kemudian, secara acak, semua kubus satuan disusun lagi menjadi balok seperti balok semula. a) Ada berapa cara menempatkan kedelapan kubus satuan yang berasal dari pojok sehingga kedelapan kubus di pojok yang tampak tetap berwarna merah? (Rotasi kubus dianggap konfigurasi yang berbeda, namun posisi balok tidak diubah.) b) Tentukan probabilitas balok yang tersusun lagi semua permukaannya berwarna merah. (Balok tegak tetap tegak dan balok tetap dalam suatu posisi.) [/hide]

Does there exist a two-variable polynomial $P(x, y)$ with real number coefficients such that $P(x, y)$ is positive exactly when $x$ and $y$ are both positive?

Positive reals $a,b,c$ satisfy $abc=1$. Prove that $\displaystyle 1+ \frac{3}{a+b+c} \ge \frac{6}{ab+bc+ca}$.

Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]

Let $ P$ be a polyhedron where every face is a regular polygon, and every edge has length $ 1$. Each vertex of $ P$ is incident to two regular hexagons and one square. Choose a vertex $ V$ of the polyhedron. Find the volume of the set of all points contained in $ P$ that are closer to $ V$ than to any other vertex.

Let $n\ge 3$ be an integer. Let $s(n)$ be the number of (ordered) pairs $(a;b)$ consisting of positive integers $a,b$ from the set $\{1,2,\dots ,n\}$ which satisfy $\gcd (a,b,n)=1$. Prove that $s(n)$ is divisible by $4$ for all $n\ge 3$. [i] (Vlad Robu) [/i]

In the sixth, seventh, eighth, and ninth basketball games of the season, a player scored $ 23$, $ 14$, $ 11$, and $ 20$ points, respectively. Her points-per-game average was higher after nine games than it was after the first five games. If her average after ten games was greater than $ 18$, what is the least number of points she could have scored in the tenth game? $ \textbf{(A)}\ 26\qquad \textbf{(B)}\ 27\qquad \textbf{(C)}\ 28\qquad \textbf{(D)}\ 29\qquad \textbf{(E)}\ 30$

In an $n$ by $n$ grid, each cell is filled with an integer between $1$ and $6$. The outmost cells all contain the number $1$, and any two cells that share a vertex has difference not equal to $3$. For any vertex $P$ inside the grid (not including the boundary), there are $4$ cells that have $P$ has a vertex. If these four cells have exactly three distinct numbers $i$, $j$, $k$ (two cells have the same number), and the two cells with the same number have a common side, we call $P$ an $ijk$-type vertex. Let there be $A_{ijk}$ vertices that are $ijk$-type. Prove that $A_{123}\equiv A_{246} \pmod 2$.

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

Show that there are infinitely many odd positive integers $n$ such that in binary $n$ has more $1$s than $n^2$.

The side lengths of a scalene triangle are roots of the polynomial $$x^3-20x^2+131x-281.3.$$ Find the square of the area of the triangle.

Of the following sets of data the only one that does not determine the shape of a triangle is: $ \textbf{(A)}\ \text{the ratio of two sides and the included angle} \\ \qquad\textbf{(B)}\ \text{the ratios of the three altitudes} \\ \qquad\textbf{(C)}\ \text{the ratios of the three medians} \\ \qquad\textbf{(D)}\ \text{the ratio of the altitude to the corresponding base} \\ \qquad\textbf{(E)}\ \text{two angles}$

Let $P_i(x) = x^2 + b_i x + c_i , i = 1,2, ..., n$ be pairwise distinct polynomials of degree $2$ with real coefficients so that for any $0 \le i < j \le n , i, j \in N$, the polynomial $Q_{i,j}(x) = P_i(x) + P_j(x)$ has only one real root. Find the greatest possible value of $n$.

From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$ a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$. It is found that $t_1$, the length of the tangent, is the mean proportional between $m$ and $n$. If $m$ and $t$ are integers, then $t$ may have the following number of values: $ \textbf{(A)}\ \text{zero} \qquad\textbf{(B)}\ \text{one} \qquad\textbf{(C)}\ \text{two} \qquad\textbf{(D)}\ \text{three} \qquad\textbf{(E)}\ \text{infinitely many} $

Show that given any 9 points inside a square of side 1 we can always find 3 which form a triangle with area less than $\frac 18$. [i]Bulgaria[/i]

An ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ passes point $(2,1)$, then all points $(x,y)$ on the ellipse that $|y|>1$ are (shown as shadow) [img]https://graph.baidu.com/resource/122481219e60931bb707101582696834.jpg[/img]

The price of $175$ Humpties is more than the price of $125$ Dumpties but less than that of $126$ Dumpties. Prove that you cannot buy three Humpties and one Dumpty for (a) $80$ cents. (b) $1$ dollar. (S Fomin, Leningrad) PS. (a) for Juniors , (a),(b) for Seniors

Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$. (Walther Janous)

A straight road consists of green and red segments in alternating colours, the first and last segment being green. Suppose that the lengths of all segments are more than a centimeter and less than a meter, and that the length of each subsequent segment is larger than the previous one. A grasshopper wants to jump forward along the road along these segments, stepping on each green segment at least once an without stepping on any red segment (or the border between neighboring segments). Prove that the grasshopper can do this in such a way that among the lengths of his jumps no more than $8$ different values occur. [i]Proposed by T. Korotchenko[/i]

Suppose that $ a,b,$ and $ \sqrt[3]{a}\plus{}\sqrt[3]{b}$ are rational numbers. Prove that $ \sqrt[3]{a}$ and $ \sqrt[3]{b}$ are also rational.

Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.

Let $ ABC $ be an obtuse triangle with $ \angle A > 90^{\circ} $. Let circle $ O $ be the circumcircle of $ ABC $. $ D $ is a point lying on segment $ AB $ such that $ AD = AC $. Let $ AK $ be the diameter of circle $ O $. Two lines $ AK $ and $ CD $ meet at $ L $. A circle passing through $ D, K, L $ meets with circle $ O $ at $ P ( \ne K ) $ . Given that $ AK = 2, \angle BCD = \angle BAP = 10^{\circ} $, prove that $ DP = \sin ( \frac{ \angle A}{2} )$.

Jody has $6$ distinguishable balls and $6$ distinguishable sticks, all of the same length. How many ways are there to use the sticks to connect the balls so that two disjoint non-interlocking triangles are formed? Consider rotations and reflections of the same arrangement to be indistinguishable.