# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 1. Find prime numbers$a, b, c, d$so that$a>3 b>6 c>12 d$and $$a^{2}-b^{2}+c^{2}-d^{2}=1749.$$ 2. Given an isosceles triangle$A B C$with the vertex angle$A(\hat{A}=20^{\circ})$. Let$D$,$E$be the points on$A C$so that$D$is in between$A$and$E$, and$A D=C E=B C$. Find the measurement of the angle$\widehat{D B E}$. 3. Given a polynomial$f(x)=x^{2}+a x+b$where$a$,$b$are integers. Prove that there always exist integers$m$so that$f(m)=f(2021) \cdot f(2022)$. 4. Let$M$be a point lying inside the triangle$A B C$so that$\widehat{M B A}=\widehat{M C A}$. Draw the parallelogram$B M C D$. Show that the angle bisector of$\widehat{B A C}$and the angle bisector of$\widehat{M C D}$are perpendicular to each other. 5. Solve the system of equations $$\begin{cases} x y z+x+y+z &=x y+y z+z x+2 \\ \dfrac{1}{x^{2}-x+1}+\dfrac{1}{y^{2}-y+1}+\dfrac{1}{z^{2}-z+1} &=1\end{cases}$$ 6. Given numbers$x, y, z$which are greater than 1 and satisfy$x+y+z+2=x y z$. Prove that $$\sqrt{x^{2}-1}+\sqrt{y^{2}-1}+\sqrt{z^{2}-1} \geq 3 \sqrt{3}.$$ When does the equality happen? 7. Find all triangles$A B C$whose lengths of the sides are positive integers and the length of$A C$is equal to the length of the internal angle bisector of the angle$A$. 8. Two circles$(O)$and$\left(O^{\prime}\right)$intersect at two points$A$and$B$. Through a point$C$lying on the opposite ray of the ray$B A$draw the tangents$C D$and$C E$with$(O)$. The line segment$D E$intersects$\left(O^{\prime}\right)$at$F$. The tangent of$\left(O^{\prime}\right)$at$F$intersects$C D$and$C E$respectively at$M$and$N$. Show that$A B M N$is a cyclic quadrilateral. 9. Show that, for every positive integer$n$, we have $$\frac{1}{1} C_{8 n-1}^{1}-\frac{1}{2} C_{8 n-1}^{3}+\frac{1}{3} C_{8 n-1}^{\delta}-\ldots-\frac{1}{2 n-1} C_{8 n-1}^{4 n-5}+\frac{1}{2 n-1} C_{8 n-1}^{4 n-3} \leq \frac{(8 n-1) !}{((4 n) !)^{2}}-\frac{7}{4}.$$ 10. The sequence$\left(a_{n}\right)$is determined as follows $$a_{1}=1,\, a_{2}=3,\quad \log _{2} a_{n+2}=\log _{3}\left(a_{n}+1\right),\, \forall n=1,2, \ldots.$$ Show that the sequence$\left(a_{n}\right)$converges and find its limit. 11. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$so that $$f\left(x+y^{4}\right)=f(x)+y f\left(y^{3}\right),\, \forall x, y \in \mathbb{R}$$ 12. Given a quadrilateral$A B C D$and a point$E$is on$A B$. A point$F$varies on$C D$. The points$M$and$N$respectively are the perpendicular projections of$C$and$D$on$E F$. Assume that$P$is the intersection between the line passing through$M$and perpendicular to$A D$and the line passing through$N$and perpendicular to$B C$. Show that the incenter of the triangle$M N P$belongs to a fixed circle. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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Mathematics & Youth: 2021 Issue 525
2021 Issue 525
Mathematics & Youth