Olympiad problems

2017 NIMO Problems, 2

Tags:

Find the smallest positive integer $N$ for which $N$ is divisible by $19$, and when the digits of $N$ are read in reverse order, the result (after removing any leading zeroes) is divisible by $36$. [i]Proposed by Michael Tang[/i]

2010 China Northern MO, 5

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Let $a,b,c$ be positive real numbers such that $(a+2b)(b+2c)=9$. Prove that\[\sqrt{\frac{a^2+b^2}{2}}+2\sqrt[3]{\frac{b^3+c^3}{2}}\geq 3.\]

2012 Grigore Moisil Intercounty, 1

Tags: inequalities , geometry

For $ x\in\mathbb{R} , $ determine the minimum of $ \sqrt{(x-1)^2+\left( x^2-5\right)^2} +\sqrt{(x+2)^2+\left( x^2+1 \right)^2} $ and the maximum of $ \sqrt{(x-1)^2+\left( x^2-5\right)^2} -\sqrt{(x+2)^2+\left( x^2+1 \right)^2} . $ [i]Vasile Pop[/i]

Geometry Mathley 2011-12, 15.3

Tags: geometry , circumcircle , symmedian , perpendicular bisector

Triangle $ABC$ has circumcircle $(O,R)$, and orthocenter $H$. The symmedians through $A,B,C$ meet the perpendicular bisectors of $BC,CA,AB$ at $D,E, F$ respectively. Let $M,N, P$ be the perpendicular projections of H on the line $OD,OE,OF.$ Prove that $$\frac{OH^2}{R^2} =\frac{\overline{OM}}{\overline{OD}}+\frac{\overline{ON}}{\overline{OE}} +\frac{\overline{OP}}{\overline{OF}}$$ Đỗ Thanh Sơn

2016 AMC 10, 17

Tags: probability

Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$? $\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) }16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20$

1990 Tournament Of Towns, (277) 2

Tags: equilateral , geometry

A point $M$ is chosen on the arc $AC$ of the circumcircle of the equilateral triangle $ABC$. $P$ is the midpoint of this arc, $N$ is the midpoint of the chord $BM$ and $K$ is the foot of the perpendicular drawn from $P$ to $MC$. Prove that the triangle $ANK$ is equilateral. (I Nagel, Yevpatoria)

2014 NZMOC Camp Selection Problems, 4

Tags: combinatorial geometry , combinatorics , point

Given $2014$ points in the plane, no three of which are collinear, what is the minimum number of line segments that can be drawn connecting pairs of points in such a way that adding a single additional line segment of the same sort will always produce a triangle of three connected points?

1969 Miklós Schweitzer, 2

Tags: superior algebra

Let $ p\geq 7$ be a prime number, $ \zeta$ a primitive $ p$th root of unity, $ c$ a rational number. Prove that in the additive group generated by the numbers $ 1,\zeta,\zeta^2,\zeta^3\plus{}\zeta^{\minus{}3}$ there are only finitely many elements whose norm is equal to $ c$. (The norm is in the $ p$th cyclotomic field.) [i]K. Gyory[/i]

2024 AMC 12/AHSME, 14

Tags: remainder

How many different remainders can result when the $100$th power of an integer is divided by $125$? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }5 \qquad \textbf{(D) }25 \qquad \textbf{(E) }125 \qquad $

1953 AMC 12/AHSME, 13

Tags: trapezoid , geometry

A triangle and a trapezoid are equal in area. They also have the same altitude. If the base of the triangle is $ 18$ inches, the median of the trapezoid is: $ \textbf{(A)}\ 36\text{ inches} \qquad\textbf{(B)}\ 9\text{ inches} \qquad\textbf{(C)}\ 18\text{ inches}\\ \textbf{(D)}\ \text{not obtainable from these data} \qquad\textbf{(E)}\ \text{none of these}$