Question bank VMO 1963

Three students of class X, Dần, Mão and Thìn, while out walking see a driver violate traffic regulations. None remembers the entire registration number, but each remembers one feature. Dần remembers that its first two digits are equal. Mão remembers that its last two digits are equal. Thìn asserts that the four-digit number is a perfect square. Let us try to find that number.

Determine the following limits:
$$\lim_{x\to\frac12}\frac{8x^3-1}{6x^2-5x+1}\qquad\&\qquad\lim_{x\to a}\frac{\sqrt{x-b}-\sqrt{a-b}}{x^2-a^2}.$$

Solve $\cos^2x+\cos^22x+\cos^23x+\cos^24x=2$.

A parallelepiped $ABCD.A'B'C'D'$ is cut by the plane through $A,B',D'$. Prove that its diagonal $A'C$ meets the triangular section at that triangle's centroid.

The base of a right prism is a rhombus $ABCD$ of side $a$ and acute angle $60^\circ$. Join $B'$ to the midpoint $M$ of $AD$, and $D'$ to the midpoint $N$ of $AB$. The lines $B'M$ and $D'N$ intersect and form angle $B'OD'=\alpha$. Find the prism's volume.

A district youth delegates' conference on combating drought has 47 young men and women. Ms Lê recognizes 16 men, Ms Đào recognizes 17 men, Ms Mận recognizes 18 men, ..., and Ms Cam, the last woman, knows all the men present. How many men and how many women attend?

(a) Determine $m$ so that $x^2+(2m+6)x+4m+12=0$ has two real roots both greater than $-1$.

(b) Differentiate
$$f(u)=\frac{u+\sqrt{u^2+1}}{\sqrt{1+u^2}-u}+\frac{\sqrt{1+u^2}-u}{u+\sqrt{1+u^2}}.$$

Solve $\sin^3x\cos3x+\cos^3x\sin3x=3/8$.

A triangular pyramid $S.ABC$ has face $SCB$ perpendicular to the base, edges $SC=SB=1$, and all its face angles at the apex equal to $60^\circ$. Find its volume.

Find side $a$ and area $S$ of a triangle given angles $A,B$ and semiperimeter $p$. Determine $S$ for $p\approx23.6$, $A\approx52^\circ42'$, $B=46^\circ16'$.