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Find (without calculus) a fifth degree polynomial $p(x)$ such that $p(x)+1$ is divisible by $(x-1)^3$ and $p(x)-1$ is divisible by $(x+1)^3$.

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Find the positive integer $n$ such that $133^5+110^5+84^5+27^5=n^5$.

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Assume that $x_1,\,x_2,\,\cdots,\,x_7$ are real numbers such that

$x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1\\4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12\\9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123$

Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$.

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Problem Of The Week #426 July 20th, 202]]>

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Solve the equation $\dfrac{7}{\sqrt{x^2-10x+26}+\sqrt{x^2-10x+29}+\sqrt{x^2-10x+41}}=x^4-9x^3+16x^2+15x+26$

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Given that $f(x)=ax^3+bx^2+cx+d$ and $|f'(x)|\le 1$ for $0\le x \le 1$. Find the maximum value of $a$.

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Determine $x^2+y^2+z^2+w^2$ if

$\dfrac{x^2}{2^2-1^2}+\dfrac{y^2}{2^2-3^2}+\dfrac{z^2}{2^2-5^2}+\dfrac{w^2}{2^2-7^2}=1,\\\dfrac{x^2}{4^2-1^2}+\dfrac{y^2}{4^2-3^2}+\dfrac{z^2}{4^2-5^2}+\dfrac{w^2}{4^2-7^2}=1,\\\dfrac{x^2}{6^2-1^2}+\dfrac{y^2}{6^2-3^2}+\dfrac{z^2}{6^2-5^2}+\dfrac{w^2}{6^2-7^2}=1,\\\dfrac{x^2}{8^2-1^2}+\dfrac{y^2}{8^2-3^2}+\dfrac{z^2}{8^2-5^2}+\dfrac{w^2}{8^2-7^2}=1$

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Remember to read the...

Problem Of The Week #423 June 29th, 2020]]>

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Let $a,\,b,\,c$ and $d$ be four non-negative real numbers satisfying the condition

$2(ab+ac+ad+bc+bd+cd)+abc+abd+acd+bcd=16$

Prove that

$a+b+c+d\ge \dfrac{2}{3}(ab+ac+ad+bc+bd+cd)$

and determine when equality occurs.

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Problem Of The Week #422 June 22nd, 2020]]>

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Solve for natural numbers for the identity below:

$\dfrac{1^4}{x}+\dfrac{2^4}{x+1}+\dfrac{3^4}{x+2}+\cdots+\dfrac{10^4}{x+9}=3025$

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For real numbers $a$ and $b$ that satisfy $a^3+12a^2+49a+69=0$ and $b^3-9b^2+28b-31=0$, find $a+b$.

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It takes 5 minutes to cross a certain bridge and 1000 people cross it in a day of 12 hours, all times of day being equally likely. Find the probability that there will be nobody on the bridge at noon.

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Find \(\displaystyle \int \left(x^{10}+\sqrt{1+x^{20}}\right)^{^{\Large\frac{21}{10}}}\,dx\) where $x\in \mathbb{R}$.

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Consider an acute triangle $ABC$ and let $P$ be an interior point of $ABC$. Suppose the lines $BP$ and $CP$, when produced, meet $AC$ and $AB$ in $E$ and $F$ respectively. Let $D$ be the point where $AP$ intersects the line segment $EF$ and $K$ be the foot of perpendicular from $D$ on to $BC$. Show that $DK$ bisects $\angle EKF$.

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Problem Of The Week #417 May 18th, 2020]]>

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Find all positive real solutions to the following system of solution:

$x^3+y^3+z^3=x+y+z$

$x^2+y^2+z^2=xyz$

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Show that for every real number $a$ the equation $8x^4-16x^3+16x^2-8x+a=0$ has at least one non-real root and find the sum of all the non-real roots of the equation.

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If $q>0$ and $p$ is a real number, prove that the polynomial $x^4-px^3+qx^2-\sqrt{q}x+1=0$ has no real roots.

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Let $a,\, b,\, c,\, d \in \mathbb{N} $ such that the equation $x^2-(a^2+b^2+c^2+d^2+1)x+ab+bc+cd+da=0$ has an integer solution. Prove that the other solution is integer too and that both solutions are perfect squares.

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Problem Of The Week #414 Apr 26th, 2020]]>

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Prove that in a triangle with sides $a, b$ and $c$ and opposite angles $A, B$ and $C$ (in radians), the following relation holds:

$\dfrac{aA+bB+cC}{a+b+c}\ge\dfrac{\pi}{3}$

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Suppose that the positive integers $x, y$ satisfy $2x^2+x=3y^2+y$. Show that $x-y, 2x+2y+1, 3x+3y+1$ are all perfect squares.

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Show that the set of real numbers $x$ which satisfy the inequality \(\displaystyle \sum_{k=1}^{70}\dfrac{k}{x-k}\ge \dfrac{5}{4}\) is a union of disjoint intervals, the sum of whose lengths is 1988.

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Problem Of The Week #409 Mar 20th, 2020]]>

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Find the minimum value of $(u-v)^2+\left(\sqrt{2-u^2}-\dfrac{9}{v}\right)^2$ for $0<u<\sqrt{2}$ and $v>0$.

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How many perfect squares are divisors of the product $1!\cdot 2! \cdot 3! \cdots 9!$?

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Consider the sequence of numbers 4, 7, 1, 8, 9, 7, 6, ... For $n>2$, the $n$th term of the sequence is the units digit of the sum of the two previous terms. Let $S_n$ denote the sum of the first $n$ terms of this sequence. What is the smallest value of $n$ for which $S_n>10000$?

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Problem Of The Week #407 Mar 4th, 2020]]>

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The parabola with equation $y=ax^2+bx+c$ and vertex $(h, k)$ is reflected about the line $y=k$. This results in the parabola with equation $y=dx^2+ex+f$. Find $a+b+c+d+e+f$.

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Let $n$ be the smallest positive integer such that $n$ is divisible by 20, $n^2$ is a perfect cube, and $n^3$ is a perfect square. What is the number of digits of $n$?

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Let $ABCD$ be a cyclic quadrilateral. The side lengths of $ABCD$ are distinct integers less than 15 such that $AB\cdot DA=BC \cdot CD$. What is the largest possible value of $BD$?

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Monic quadratic polynomials $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23,\,-21,\,-17$ and $-15$ and $Q(P(x))$ has zeros at $x=-59,\,-57,\,-51$ and $-49$. What is the sum of the minimum values of $P(x)$ and $Q(x)$?

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Problem Of The Week #403 Feb 2nd, 2020]]>

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Prove $\ln 2>\left(\dfrac{2}{5}\right)^{\frac{2}{5}}$.

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The set of real numbers $x$ for which

$\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge 1$

is the union of intervals of the form $a<x\le b$.

Find the sum of the lengths of these two intervals.

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Problem Of The Week #402 Jan 20th, 2020]]>

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Let $a>0$ and $P(x)$ be a polynomial with integer coefficients such that

$P(1)=P(3)=P(5)=P(7)=a$ and

$P(2)=P(4)=P(6)=P(8)=-a$.

What is the smallest possible value of $a$?

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The sides $a,\,b,\,c$ and $u,\,v,\,w$ of two triangles $ABC$ and $UVW$ are related by the equations

$u(v+w-u)=a^2\\v(w+u-v)=b^2\\w(u+v-w)=c^2\\$

Prove that $ABC$ are acute, and express the angles $U,\,V$ and $W$ in terms of $A,\,B$ and $C$.

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Evaluate \(\displaystyle \left\lfloor{\frac{1}{3}}\right\rfloor+\left\lfloor{\frac{2}{3}}\right\rfloor+\left\lfloor{\frac{2^2}{3}}\right\rfloor+\left\lfloor{\frac{2^3}{3}}\right\rfloor+\cdots+\left\lfloor{\frac{2^{1000}}{3}}\right\rfloor\), without the help of a calculator.

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Problem Of The Week #398 Dec 19th, 2019]]>

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A number $n$ has sum of digits 100, while $44n$ has sum of digits 800. Find the sum of the digits of $3n$.

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A polynomial $P(x)$ of the $n$-th degree satisfies $P(k)=2^k$ for $k=0,\,1,\,2,\,\cdots,\,n$. Find the value of $P(n+1)$.

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Find $d$ when $\sin^7 x=a\sin 7x+b\sin 5x+c\sin 3x+d\sin x$.

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$x^3+ax^2+bx+c$ has three distinct real roots, but $(x^2+x+2001)^3+a(x^2+x+2001)^2+b(x^2+x+2001)+c$ has no real roots. Show that $2001^3+a(2001^2)+b(2001)+c>\dfrac{1}{64}$.

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Prove that $\sin^n 2x+(\sin^n x - \cos^n x)^2\le 1$.

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The equations $x^2+ax+1=0$ and $x^2+bx+c=0$ have a common real root and the equations $x^2+x+a=0$ and $x^2+cx+b=0$ have a common real root. Find $a+b+c$.

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The positive reals $x,\,y$ satisfy $x^2+y^3\ge x^3+y^4$. Show that $x^3+y^3\le 2$.

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Given that $f(x)=x^2+12x+30$. Solve for the equation $f(f(f(f(f(x)))))=0$.

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Find the number of real roots of the equation $1+\dfrac{x}{1!}+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+\dfrac{x^5}{5!}+\dfrac{x^6}{6!}=0$.

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Factorize $(x+1)(x+2)(x+3)(x+6)-3x^2$.

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If $x$ and $y$ are positive real numbers that satisfy the equation $x+4\sqrt{xy}-2\sqrt{x}-4\sqrt{y}+4y=3$, evaluate $\dfrac{\sqrt{x}+2\sqrt{y}+2014}{4-\sqrt{x}-2\sqrt{y}}$.

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Given $a,\,b,\,c,\,d$ are roots of the equation $x^4-7x^3+3x^2-21x+1=0$.

Evaluate $(a+b+c)(b+c+d)(c+d+a)(d+a+b)$.

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Determine all triples $(a,\,b,\,c)$ of positive integers with $a^{(b^c)}=(a^b)^{c}$.

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Solve the equation $x^3(x+1)=2(x+a)(x+2a)$ where $a$ is a real parameter.

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Let \(\displaystyle \prod_{n=1}^{1996} (1+nx^{3n})=1+a_1x^{k_1}+a_2x^{k_2}+\cdots+a_mx^{k_m}\) where $a_1,\,a_2,\,\cdots a_m$ are non-zero and $k_1<k_2<\cdots<k_m$.

Find $a_{1996}$.

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Let $ABCD$ be an inscribed quadrilateral. Let $P$, $Q$ and $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$ and $AB$ respectively. Show that $PQ = QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ meet on $AC.$

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Problem Of The Week #382 Sep 5th, 2019]]>

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Find all non-negative integers $x,\,y$ satisfying $(xy-7)^2=x^2+y^2$.

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Show that there is a rational number $k$ such that $\sin 1^\circ \sin 2^\circ \cdots \sin 89^\circ \sin 90^\circ =k\sqrt{10}$.

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Find (without calculus) a fifth degree polynomial $P(x)$ such that $P(x)+1$ is divisible by $(x-1)^3$ and $P(x)-1$ is divisible by $(x+1)^3$.

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Let $a,\,b,\,c$ and $d$ be real numbers such that

$a+\sin b > c+ \sin d$ and

$b+\sin a > d + \sin c$.

Prove that $a+b>c+d$.

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Factor $x^8+4x^2+4$ into two non-constant polynomials with integer coefficients.

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Let $P(x)$ be any polynomial with integer coefficients such that $P(21)=17,\,P(32)=-247$ and $P(37)=33$.

Prove that if $P(N)=N+51$ for some integer $N$, then $N=26$.

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Let $a,\,b$ and $c$ be distinct real numbers such that

$a^3=3(b^2+c^2)-25\\b^3=3(c^2+a^2)-25\\c^3=3(a^2+b^2)-25$

Evaluate $abc$.

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Let $A=1+10+10^2+\cdots+10^{1997}$. Determine the 1000th digit after the decimal point of the square root of $A$ in base 10.

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Let $a, \, b,\,c$ and $d$ be integers with $a>b>c>d>0$.

Suppose that $ac+bd=(b+d+a-c)(b+d-a+c)$.

Prove that $ab+cd$ is not prime.

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Solve the equation $4x^6-6x^2+2\sqrt{2}=0$ without the help of a calculator.

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$ABCD$ is a trapezium with $AD$ is parallel to $BC$. Given that $BC=BD=1,\, AB=AC$, $CD<1$ and $\angle BAC+\angle BDC=180^\circ$, find $CD$.

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Solve the equation $x^3-3x=\sqrt{x+2}$.

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If the equation $ax^2+(c-b)x+(e-d)=0$ has real roots greater than 1, show that the equation $ax^4+bx^3+cx^2+dx+e=0$ has at least one real root.

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The equation $(x+a) (x+b) = 9$ has a root $a+b$.

Prove that $ab\le 1$.

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Without using a calculator, simplify \(\displaystyle \frac{\displaystyle\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}{\displaystyle\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}\).

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Given $a,b,c,d$ are real numbers such that

$ab+cd=4$

$ac+bd=8$

Find the maximum value of $abcd$.

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Prove that $\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+\cdots}}} - \sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8+\cdots}}}}}} = 1$.

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Prove $\sqrt[3]{\sqrt{3}\cos10^{\circ}+1}+\sqrt[3]{\sqrt{3}\cos110^{\circ}+1}+\sqrt[3]{\sqrt{3}\cos130^{\circ}+1}=\sqrt[3]{\dfrac{9}{2}}$.

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Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$.

Find the sum of the minimum values of $P(x)$ and $Q(x)$.

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Problem Of The Week #363 Apr 23rd, 2019]]>

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Let $a,\,b,\,c$ and $d$ be the real numbers which satisfy the system of equations below:

$a+b+2ab=3$

$b+c+2bc=4$

$c+d+2cd=5$

Evaluate $d+a+2da$.

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Let $a$, $b$, $c$ and $d$ be real numbers with $a^2 + b^2 + c^2 + d^2 = 4$.

Prove that the inequality $(a+2)(b+2) \ge cd$ holds.

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If $\{x,y,z\}⊂\Bbb{R}^+$ and

$x^2+xy+y^2=3\\y^2+yz+z^2=1\\x^2+xz+z^2=4$,

find the value of $xy+yz+zx$.

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Prove that $a^2+b^2\ge 8$ if $x^4+ax^3+2x^2+bx+1=0$ has at least one real root.

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Compute the least possible non-zero value of $A^2+B^2+C^2$ such that $A,\,B$, and $C$ are integers satisfying $A\log 16 +B\log 18 +C\log 24 = 0$.

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Compute \(\displaystyle \frac{\left\lfloor{\sqrt[4]{1}}\right\rfloor \cdot \left\lfloor{\sqrt[4]{3}}\right\rfloor \cdot\left\lfloor{\sqrt[4]{5}}\right\rfloor \cdots \left\lfloor{\sqrt[4]{2015}}\right\rfloor}{\left\lfloor{\sqrt[4]{2}}\right\rfloor \cdot \left\lfloor{\sqrt[4]{4}}\right\rfloor \cdot\left\lfloor{\sqrt[4]{6}}\right\rfloor \cdots \left\lfloor{\sqrt[4]{2016}}\right\rfloor}\).

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Remember to read the...

Problem Of The Week #357 Mar 12th, 2019]]>

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Evaluate \(\displaystyle \int_{0}^{\frac{\pi}{2}} \frac{\cos^4x+\sin x\cos^3 x+\sin^2 x \cos^2 x+\sin^3 x\cos x}{\sin^4 x+\cos^4 x+2\sin x\cos^3 x+2\sin^2 x \cos^2 x+2\sin^3 x\cos x}\,dx\).

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If $a,b,c$ are roots of the equation $x^3+3x^2-24x+1=0$, prove that $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}=0$.

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If $x^2-x-1$ divides $ax^{17}+bx^{16}+1$ for integer $a$ and $b$, find the possible value of $a-b$.

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Solve for non-negative integers $x$ and $y$ of $\sqrt{xy}=\sqrt{x+y}+\sqrt{x}+\sqrt{y}$.

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I won't be posting anything starting tomorrow onwards until February 12th, therefore, the result for High School POTW #351 will only be released on February 12th and I will post the POTW#352 (which is supposed to be posted next Tuesday@ February 5th) today too.

I hope you will enjoy solving these two weeks of high school problems I posted in a row.

Here is POTW #352:

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Without using a calculator, evaluate \(\displaystyle \frac{(4\times 7+2)(6\times...\)

Problem Of The Week #352 Jan 29th, 2019]]>

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Evaluate \(\displaystyle \int_{-1}^{1} \dfrac{x^2+1}{e^x+1}\,dx\).

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Let $ABC$ be a right triangle with right angle at $B$. Let $ACDE$ be a square drawn exterior to triangle $ABC$. If $M$ is the center of this square, evaluate $\angle MBC$.

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This week POTW is a follow-up question from last week problem

Prove that 2 is persistent.

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Problem Of The Week #349 Jan 15th, 2019]]>

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Call a number $T$ persistent if the following holds:

Whenever $a,\,b,\,c,\,d$ are real numbers different from $0$ and $1$ such that

$a+b+c+d=T$ and $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}=T$

we also have

$\dfrac{1}{1-a}+\dfrac{1}{1-b}+\dfrac{1}{1-c}+\dfrac{1}{1-d}=T$

Prove that $T$ must be equal to $2$ if $T$ is persistent.

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Remember to read the...

Problem Of The Week #348 Jan 8th, 2019]]>

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Let $ABCD$ be a square, and let $E$ be an internal point on side $AD$. Let $F$ be the foot of the perpendicular from $B$ to $CE$. Suppose $G$ is a point such that $BG = FG$, and the line through $G$ parallel to $BC$ passes through the midpoint of $EF$. Prove that $AC<2\cdot FG$.

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Problem Of The Week #347 Jan 1st, 2019]]>

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Solve the equation $\left\{\dfrac{1}{\sin^2{x}}\right\}-\left\{ \dfrac{1}{ \cos^2{x}}\right\}=\left\lfloor{\dfrac{1}{\tan^2{x}}}\right\rfloor -\left\lfloor{\dfrac{1}{\cot^2{x}}}\right\rfloor$, where $[ x ]$ denotes the integer part and $\{ x\}$ denotes the fractional part.

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Problem Of The Week #346 Dec 25th, 2018]]>

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Let $a$ and $b$ are positive real numbers such that $\dfrac{1}{a+1}+\dfrac{1}{b+1}=1.$

Prove that $\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}=1$`

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Let $a,\,b$ and $c$ be positive reals such that $a^2+b^2+c^2=\sqrt{ab+bc+ca}-\dfrac{1}{4}.$

Determine the values of $a,\,b $ and $c$.

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Prove $\cos (\cos 1) > \sin (\sin (\sin 1))$.

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Solve for the rational solution of the equation \(\displaystyle x+\sqrt{(x+1)(x+2)}+\sqrt{(x+2)(x+3)}+\sqrt{(x+3)(x+1)}=4\).

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Evaluate \(\displaystyle \sum_{k=0}^{89} \frac{1}{1+\tan^{3} (k^{\circ})}\).

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Given that $a,\,b$ and $c$ are roots for the equation $x^3-7x+7=0$.

Evaluate $\dfrac{1}{a^4}+\dfrac{1}{b^4}+\dfrac{1}{c^4}$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

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If $a,\,b,\,c,\,x,\,y,\,z \in \Bbb{R}$ and $a\ne x,\,b\ne y$ and $c \ne z$, solve the following system of equations:

$-a=b+y\\-b=c+z\\-c=a+x\\x=by\\y=cz\\z=ax$

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Evaluate $\log_{12}18\log_{24}54+5(\log_{12}18-\log_{24}54)$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

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Given that $A+B=C+D=E+F=\dfrac{\pi}{3}$ and $\dfrac{\sin A}{\sin B}\times \dfrac{\sin C}{\sin D} \times \dfrac{\sin E}{\sin F}=1 $.

Prove that $\dfrac{\sin (2A+F)}{\sin (2F+A)}\times \dfrac{\sin (2E+D)}{\sin (2D+E)} \times \dfrac{\sin (2C+B)}{\sin (2B+C)}=1$

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Remember to read the POTW submission guidelines to find out how to...

Problem Of The Week #337 Oct 25th, 2018]]>

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Given $a,\,b$ and $c$ are sides of $\triangle ABC$ such that $9a^{2}+9b^{2}=19c^{2}$.

Evaluate $\dfrac{\cot C}{\cot A+\cot B}$.

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Let $x$ and $y$ be real numbers satisfying both equations below:

\[x^4 + 8y = 4(x^3 - 1) - 16 \sqrt{3}\] \[y^4 + 8x = 4(y^3 - 1) + 16 \sqrt{3}.\]

Find $x^4 + y^4$.

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If $a,\,b$ and $c$ are roots of the polynomial $P(x) = x^3 - 2007x + 2002$, evaluate \(\displaystyle \prod_{\text{cyclic}}\frac{a-1}{a+1}\).

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duty while I was taking a break.

Here is this week's POTW:

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Suppose $a,\,b$ and $c$ are real numbers such that $abc \ne 0$.

Find $x,\,y$ and $z$ in terms of $a,\,b$ and $c$ such that

$a=bz+cy\\b=cx+az\\c=ay+bx$

Prove also that $\dfrac{1 - x^2}{a^2} = \dfrac{1 - y^2}{b^2} = \dfrac{1 - z^2}{c^2}$.

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Remember to read the...

Problem Of The Week #333 Sep 26th, 2018]]>

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Given that $n$ is an integer greater than $0$, when is $n^4+4$ prime?

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Find $\displaystyle\int\arcsin^2(x)\,dx.$

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Prove the identity $\sin(2x)+\sin(2y)+\sin(2z)=4\sin(x)\sin(y)\sin(z)$ where $x+y+z=\pi$.

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Here is this week's POTW:

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Consider the following triangle:

Show that the segments $AX$, $BY$, and $CZ$ are concurrent at $P$ if and only if $\frac{AZ}{BZ}\frac{BX}{CX}\frac{CY}{AY}=1$.

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Problem Of The Week #329 Aug 29th, 2018]]>