Prove that for every odd integer $n > 1$, there exist integers $a, b > 0$ such that, if we let $Q(x) = (x + a)^ 2 + b$, then the following conditions hold: $\bullet$ we have $\gcd(a, n) = gcd(b, n) = 1$; $\bullet$ the number $Q(0)$ is divisible by $n$; and $\bullet$ the numbers $Q(1), Q(2), Q(3), \dots$ each have a prime factor not dividing $n$.

The parallelogram $ABCD$ has $AB=a,AD=1,$ $\angle BAD=A$, and the triangle $ABD$ has all angles acute. Prove that circles radius $1$ and center $A,B,C,D$ cover the parallelogram if and only \[a\le\cos A+\sqrt3\sin A.\]

Suppose $(u,v)$ is an inner product on $\mathbb R^{n}$ and $f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is an isometry, that $f(0)=0$. 1) Prove that for each $u,v$ we have $(u,v)=(f(u),f(v)$ 2) Prove that $f$ is linear.

If the coordinate origin is inside the ellipse $k^2x^2+y^2-4kx+2ky+k^2-1=0$, then the range value of $k$ is $\text{(A)}|k|>1\qquad\text{(B)}|k|\neq1\qquad\text{(C)}-1<k<1\qquad\text{(D)}0<|k|<1$

Vitya wrote down $n{}$ different natural numbers in his notebook. For each pair of numbers from the notebook, he wrote out their smallest common multiple on the board. Could it happen for some $n>100$ that $n(n-1)/2$ numbers on the board are (in some order) consecutive terms of a non-constant arithmetic progression? [i]Proposed by S. Berlov[/i]

Let $ABCD$ be a square face of a cube with edge length $2$. A plane $P$ that contains $A$ and the midpoint of $\overline{BC}$ splits the cube into two pieces of the same volume. What is the square of the area of the intersection of $P$ and the cube?

Prove that $a^9 - a$ is divisible by $6$ for all integers $a$.

Determine all the quadruples of real numbers that satisfy the following: [i]The product of any three of these numbers plus the fourth is constant.[/i]

Let points $B$ and $C$ lie on the circle with diameter $AD$ and center $O$ on the same side of $AD$. The circumcircles of triangles $ABO$ and $CDO$ meet $BC$ at points $F$ and $E$ respectively. Prove that $R^2 = AF.DE$, where $R$ is the radius of the given circle. [i]Proposed by N.Moskvitin[/i]

There are $36$ cards in a deck arranged in the sequence spades, clubs, hearts, diamonds, spades, clubs, hearts, diamonds, etc. Somebody took part of this deck off the top, turned it upside down, and cut this part into the remaining part of the deck (i.e. inserted it between two consecutive cards). Then four cards were taken off the top, then another four, etc. Prove that in any of these sets of four cards, all the cards are of different suits. (A Merkov, Moscow)

Let $t$ be positive number. Draw two tangent lines to the palabola $y=x^{2}$ from the point $(t,-1).$ Denote the area of the region bounded by these tangent lines and the parabola by $S(t).$ Find the minimum value of $\frac{S(t)}{\sqrt{t}}.$

$f(x) = [x] + [2x] + [3x] + [4x] + [5x] + [6x]$. What values does $f$ take?

Does there exist a positive integer, $x$, such that $(x+2)^{2023}-x^{2023}$ has exactly $2023^{2023}$ factors? [i]Proposed by Wong Jer Ren[/i]

Let $n$ be the answer to this problem. $a$ and $b$ are positive integers satisfying $$3a + 5b \equiv 19 \,\,\, (mod \,\,\, n + 1)$$ $$4a + 2b \equiv 25 \,\,\, (mod \,\,\, n + 1)$$ Find $ 2a + 6b$.

A right circular cone has a height equal to three times its base radius and has volume 1. The cone is inscribed inside a sphere as shown. The volume of the sphere is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [center][img]https://snag.gy/92ikv3.jpg[/img][/center]

Consider two lines $\ell$ and $\ell ' $ and a fixed point $P$ equidistant from these lines. What is the locus of projections $M$ of $P$ on $AB$, where $A$ is on $\ell $, $B$ on $\ell ' $, and angle $\angle APB$ is right?

Let $ABC$ be a triangle such that $AB > BC$ and let $D$ be a variable point on the line segment $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$, lying on the opposite side of $BC$ from $A$ such that $\angle BAE = \angle DAC$. Let $I$ be the incenter of triangle $ABD$ and let $J$ be the incenter of triangle $ACE$. Prove that the line $IJ$ passes through a fixed point, that is independent of $D$. [i]Proposed by Merlijn Staps[/i]

Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$, compute the smallest possible value of $|P(0)|.$

Let $r_1,r_2,\dots ,r_n$ be real numbers. Given $n$ reals $a_1,a_2,\dots ,a_n$ that are not all equal to $0$, suppose that inequality \[r_1(x_1-a_1)+ r_2(x_2-a_2)+\dots + r_n(x_n-a_n)\leq\sqrt{x_1^2+ x_2^2+\dots + x_n^2}-\sqrt{a_1^2+a_2^2+\dots +a_n^2}\] holds for arbitrary reals $x_1,x_2,\dots ,x_n$. Find the values of $r_1,r_2,\dots ,r_n$.

A cube $ABCDA'B'C'D',AA'\parallel BB'\parallel CC'\parallel DD'$ is given. Denote $S$ the center of square $ABCD.$ Determine all points $X$ lying on some edge such that the volumes of tetrahedrons $ABDX$ and $CB'SX$ are the same.

On the Island of Camelot live $13$ grey, $15$ brown and $17$ crimson chameleons . If two chameleons of different colours meet , they both simultaneously change colour to the third colour (e .g . if a grey and brown chameleon meet each other they both change to crimson) . Is it possible that they will eventually all be the same colour? (V . G . Ilichev)

Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that: \[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0 \] and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that: \[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2} \] for all $ k \equal{} 1,2, \ldots$.

The periods of two periodic sequences are coprime (i.e. relatively prime) numbers $m$ and $n$.. What is the maximal length of initial sections of the two sequences which can coincide? (The period $p$ of a sequence $a_1$,$a_2$, $...$ is the minimal $p$ such that $a_n = a_{n+p}$ for all $n$.) (AY Belov)

Find the minimum value of $x^2+2x+ \frac{24}{x}$ for $x > 0$.

A [i]regular octahedron[/i] has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of $Q$? [asy] // Note: This diagram was not made by me. import graph; // The Solid // To save processing time, do not use three (dimensions) // Project (roughly) to two size(15cm); pair Fr, Lf, Rt, Tp, Bt, Bk; Lf=(0,0); Rt=(12,1); Fr=(7,-1); Bk=(5,2); Tp=(6,6.7); Bt=(6,-5.2); draw(Lf--Fr--Rt); draw(Lf--Tp--Rt); draw(Lf--Bt--Rt); draw(Tp--Fr--Bt); draw(Lf--Bk--Rt,dashed); draw(Tp--Bk--Bt,dashed); label(rotate(-8.13010235)*slant(0.1)*"$Q$", (4.2,1.6)); label(rotate(21.8014095)*slant(-0.2)*"$?$", (8.5,2.05)); pair g = (-8,0); // Define Gap transform real a = 8; draw(g+(-a/2,1)--g+(a/2,1), Arrow()); // Make arrow // Time for the NET pair DA,DB,DC,CD,O; DA = (6.92820323028,0); DB = (3.46410161514,6); DC = (DA+DB)/3; CD = conj(DC); O=(0,0); transform trf=shift(3g+(0,3)); path NET = O--(-2*DA)--(-2DB)--(-DB)--(2DA-DB)--DB--O--DA--(DA-DB)--O--(-DB)--(-DA)--(-DA-DB)--(-DB); draw(trf*NET); label("$7$",trf*DC); label("$Q$",trf*DC+DA-DB); label("$5$",trf*DC-DB); label("$3$",trf*DC-DA-DB); label("$6$",trf*CD); label("$4$",trf*CD-DA); label("$2$",trf*CD-DA-DB); label("$1$",trf*CD-2DA); [/asy] $\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}~5\qquad$