Olympiad problems

Denmark (Mohr) - geometry, 2013.5

The angle bisector of $A$ in triangle $ABC$ intersects $BC$ in the point $D$. The point $E$ lies on the side $AC$, and the lines $AD$ and $BE$ intersect in the point $F$. Furthermore, $\frac{|AF|}{|F D|}= 3$ and $\frac{|BF|}{|F E|}=\frac{5}{3}$. Prove that $|AB| = |AC|$. [img]https://1.bp.blogspot.com/-evofDCeJWPY/XzT9dmxXzVI/AAAAAAAAMVY/ZN87X3Cg8iMiULwvMhgFrXbdd_f1f-JWwCLcBGAsYHQ/s0/2013%2BMohr%2Bp5.png[/img]

OMMC POTM, 2022 9

Tags: least common multiple , inequalities , number theory

For positive integers $a_1 < a_2 < \dots < a_n$ prove that $$\frac{1}{\operatorname{lcm}(a_1, a_2)}+\frac{1}{\operatorname{lcm}(a_2, a_3)}+\dots+\frac{1}{\operatorname{lcm}(a_{n-1}, a_n)} \leq 1-\frac{1}{2^{n-1}}.$$ [i]Proposed by Evan Chang (squareman), USA[/i]

1980 IMO Longlists, 5

Tags: geometry , algebra , analytic geometry , polynomial

In a rectangular coordinate system we call a horizontal line parallel to the $x$ -axis triangular if it intersects the curve with equation \[y = x^4 + px^3 + qx^2 + rx + s\] in the points $A,B,C$ and $D$ (from left to right) such that the segments $AB, AC$ and $AD$ are the sides of a triangle. Prove that the lines parallel to the $x$ - axis intersecting the curve in four distinct points are all triangular or none of them is triangular.

1970 Canada National Olympiad, 7

Tags: number theory

Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by 3.

2021 South Africa National Olympiad, 3

Tags: number theory , prime , sum of squares , power of 2 , perfect square

Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.

2005 Indonesia MO, 7

Tags: geometry

Let $ ABCD$ be a convex quadrilateral. Square $ AB_1A_2B$ is constructed such that the two vertices $ A_2,B_1$ is located outside $ ABCD$. Similarly, we construct squares $ BC_1B_2C$, $ CD_1C_2D$, $ DA_1D_2A$. Let $ K$ be the intersection of $ AA_2$ and $ BB_1$, $ L$ be the intersection of $ BB_2$ and $ CC_1$, $ M$ be the intersection of $ CC_2$ and $ DD_1$, and $ N$ be the intersection of $ DD_2$ and $ AA_1$. Prove that $ KM$ is perpendicular to $ LN$.

2020-2021 OMMC, 12

Tags: polynomial

Let $P(x) = x^3 + 8x^2 - x + 3$ and let the roots of $P$ be $a, b,$ and $c.$ The roots of a monic polynomial $Q(x)$ are $ab - c^2, ac - b^2, bc - a^2.$ Find $Q(-1).$

2019 Saudi Arabia JBMO TST, 3

Tags: number theory , digit

Find all positive integers of form abcd such that $$\overline{abcd} = a^{a+b+c+d} - a^{-a+b-c+d} + a$$

2023 Harvard-MIT Mathematics Tournament, 5

Tags:

Suppose $E$, $I$, $L$, $V$ are (not necessarily distinct) nonzero digits in base ten for which [list] [*] the four-digit number $\underline{E}\ \underline{V}\ \underline{I}\ \underline{L}$ is divisible by $73$, and [*] the four-digit number $\underline{V}\ \underline{I}\ \underline{L}\ \underline{E}$ is divisible by $74$. [/list] Compute the four-digit number $\underline{L}\ \underline{I}\ \underline{V}\ \underline{E}$.

PEN G Problems, 2

Tags: inequalities , irrational number

Prove that for any positive integers $ a$ and $ b$ \[ \left\vert a\sqrt{2}\minus{}b\right\vert >\frac{1}{2(a\plus{}b)}.\]

1962 All Russian Mathematical Olympiad, 021

Tags: number theory

Given $1962$ -digit number. It is divisible by $9$. Let $x$ be the sum of its digits. Let the sum of the digits of $x$ be $y$. Let the sum of the digits of $y$ be $z$. Find $z$.

2010 F = Ma, 21

Tags:

The gravitational self potential energy of a solid ball of mass density $\rho$ and radius $R$ is $E$. What is the gravitational self potential energy of a ball of mass density $\rho$ and radius $2R$? (A) $2E$ (B) $4E$ (C) $8E$ (D) $16E$ (E) $32E$

2006 Korea Junior Math Olympiad, 4

Tags: analytic geometry , geometric transformation , algebra , transformation

In the coordinate plane, define $M = \{(a, b),a,b \in Z\}$. A transformation $S$, which is defined on $M$, sends $(a,b)$ to $(a + b, b)$. Transformation $T$, also defined on $M$, sends $(a, b)$ to $(-b, a)$. Prove that for all $(a, b) \in M$, we can use $S,T$ denitely to map it to $(g,0)$.

2009 AMC 12/AHSME, 12

Tags: factorial , geometric sequence , ratio

The fifth and eighth terms of a geometric sequence of real numbers are $ 7!$ and $ 8!$ respectively. What is the first term? $ \textbf{(A)}\ 60\qquad \textbf{(B)}\ 75\qquad \textbf{(C)}\ 120\qquad \textbf{(D)}\ 225\qquad \textbf{(E)}\ 315$

2018 ASDAN Math Tournament, 1

Tags:

Given that $x\ge0$, $y\ge0$, $x+2y\le6$, and $2x+y\le6$, compute the maximum possible value of $x+y$.

2020 Harvest Math Invitational Team Round Problems, HMI Team #10

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10. Let $p=47$ be a prime. Call a function $f$ defined on the integers [i]lit[/i] if $f(x)$ is an integer from 1 to $p$ inclusive and $f(x+p)=f(x)$ for all integers $x$. How many [i]lit[/i] functions $g$ are there such that for all integers $x$, $p$ divides $g(x^2)-g(x)-x^8+x$? [i]Proposed by Monkey_king1[/i]

1965 AMC 12/AHSME, 34

Tags: function , quadratic , calculus

For $ x \ge 0$ the smallest value of $ \frac {4x^2 \plus{} 8x \plus{} 13}{6(1 \plus{} x)}$ is: $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \frac {25}{12} \qquad \textbf{(D)}\ \frac {13}{6} \qquad \textbf{(E)}\ \frac {34}{5}$

2016 Postal Coaching, 1

Tags: algebra , polynomial , blackboard , combinatorics

If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x), f(x)g(x), f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3 - 3x^2 + 5$ and $x^2 - 4x$ are written on the blackboard. Can we write a nonzero polynomial of the form $x^n - 1$ after a finite number of steps? Justify your answer.

1977 AMC 12/AHSME, 2

Tags:

Which one of the following statements is false? All equilateral triangles are $\textbf{(A)} \ \text{ equiangular} \qquad \textbf{(B)} \ \text{isosceles} \qquad \textbf{(C)} \ \text{regular polygons } \qquad \textbf{(D)} \ \text{congruent to each other} \qquad \textbf{(E)} \ \text{similar to each other} $

2006 Germany Team Selection Test, 2

Tags: modular arithmetic , factorial , number theory , divisibility , exponential , chinese remainder theorem

Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

1989 All Soviet Union Mathematical Olympiad, 487

Tags: algebra , combinatorics

$7$ boys each went to a shop $3$ times. Each pair met at the shop. Show that $3$ must have been in the shop at the same time.

2000 Portugal MO, 1

Tags: algebra , combinatorics

Consider the following table where initially all squares contain zeros: $ \begin{tabular}{ | l | c | r| } \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline \end{tabular} $ To change the table, the following operation is allowed: a $2 \times 2$ square formed by adjacent squares is chosen, and a unit is added to all its numbers. Complete the following table, knowing that it was obtained by a sequence of permitted operations $ \begin{tabular}{ | l | c | r| } \hline 14 & & \\ \hline 19 & 36 & \\ \hline & 16 & \\ \hline \end{tabular} $

2005 Baltic Way, 14

Tags: parallelogram , geometry

Let the medians of the triangle $ABC$ meet at $G$. Let $D$ and $E$ be different points on the line $BC$ such that $DC=CE=AB$, and let $P$ and $Q$ be points on the segments $BD$ and $BE$, respectively, such that $2BP=PD$ and $2BQ=QE$. Determine $\angle PGQ$.

2016 Online Math Open Problems, 3

Tags: geometry

In a rectangle $ABCD$, let $M$ and $N$ be the midpoints of sides $BC$ and $CD$, respectively, such that $AM$ is perpendicular to $MN$. Given that the length of $AN$ is $60$, the area of rectangle $ABCD$ is $m \sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m+n$. [i]Proposed by Yannick Yao[/i]

Taiwan TST 2015 Round 1, 2

Tags: geometry

Given any triangle $ABC.$ Let $O_1$ be it's circumcircle, $O_2$ be it's nine point circle, $O_3$ is a circle with orthocenter of $ABC$, $H$, and centroid $G$, be it's diameter. Prove that: $O_1,O_2,O_3$ share axis. (i.e. chose any two of them, their axis will be the same one, if $ABC$ is an obtuse triangle, the three circle share two points.)