Olympiad problems

2006 Moldova MO 11-12, 4

Let $ABCDE$ be a right quadrangular pyramid with vertex $E$ and height $EO$. Point $S$ divides this height in the ratio $ES: SO=m$. In which ratio does the plane $(ABC)$ divide the lateral area of the pyramid.

2022 Assam Mathematical Olympiad, 14

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The following sum of three four digits numbers is divisible by $75$, $7a71 + 73b7 + c232$, where $a, b, c$ are decimal digits. Find the necessary conditions in $a, b, c$.

PEN G Problems, 19

Tags: induction , irrational number , geometry

Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with a rational area.

Kyiv City MO Juniors 2003+ geometry, 2015.8.3

Tags: geometry , angle bisector , isosceles , angle chasing , angle

In the isosceles triangle $ABC$, $ (AB = BC)$ the bisector $AD$ was drawn, and in the triangle $ABD$ the bisector $DE$ was drawn. Find the values of the angles of the triangle $ABC$, if it is known that the bisectors of the angles $ABD$ and $AED$ intersect on the line $AD$. (Fedak Ivan)

2024 Indonesia TST, G

Tags: geometry , incenter

Given an acute triangle $ABC$. The incircle with center $I$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M,N$ be the midpoint of the minor arc of $AB$ and $AC$ respectively. Prove that $M,F,E,N$ are collinear if and only if $\angle BAC =90$$^{\circ}$

1998 Bundeswettbewerb Mathematik, 2

Tags: sequence , number theory , consecutive , coprime , perfect square

Prove that there exists an infinite sequence of perfect squares with the following properties: (i) The arithmetic mean of any two consecutive terms is a perfect square, (ii) Every two consecutive terms are coprime, (iii) The sequence is strictly increasing.

2009 F = Ma, 8

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Determine the angular acceleration of the disk when $t=\text{2.0 s}$. (A) $\text{-12 rad/s}^2$. (B) $\text{-8 rad/s}^2$. (C) $\text{-4 rad/s}^2$. (D) $\text{-2 rad/s}^2$. (E) $\text{0 rad/s}^2$.

2024 HMNT, 8

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For all positive integers $r$ and $s,$ let $\text{Top}(r,s)$ denote the top number (i.e., numerator) when $r$ $s$ is written in simplified form. For instance, $\text{Top}(20,24) = 5.$ Compute the number of ordered pairs of positive integers $(a,z)$ such that $200 \le a \le 300$ and $\text{Top}(a,z) = \text{Top}(z,a-1).$

2014 Contests, 1

Tags: analytic geometry , trigonometry , trapezoid , perpendicular bisector , cyclic quadrilateral , trig identities , geometry

In a triangle $ABC$, let $D$ be the point on the segment $BC$ such that $AB+BD=AC+CD$. Suppose that the points $B$, $C$ and the centroids of triangles $ABD$ and $ACD$ lie on a circle. Prove that $AB=AC$.

1990 Kurschak Competition, 1

Tags: modular arithmetic , relatively prime , number theory

Let $p>2$ be a prime number and $n$ a positive integer. Prove that $pn^2$ has at most one positive divisor $d$ for which $n^2+d$ is a square number.

PEN S Problems, 28

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Let $A$ be the set of the $16$ first positive integers. Find the least positive integer $k$ satisfying the condition: In every $k$-subset of $A$, there exist two distinct $a, b \in A$ such that $a^2 + b^2$ is prime.

2019 Auckland Mathematical Olympiad, 1

Tags: trigonometry , algebra

Function $f$ satisfies the equation $f(\cos x) = \cos (17x)$. Prove that it also satisfies the equation $f(\sin x) = \sin (17x)$.

2019 Germany Team Selection Test, 1

Tags: algebra , functional equation

Let $\mathbb{Q}^+$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^+\to \mathbb{Q}^+$ satisfying$$f(x^2f(y)^2)=f(x^2)f(y)$$for all $x,y\in\mathbb{Q}^+$

1972 AMC 12/AHSME, 12

Tags: geometry , 3d geometry

The number of cubic feet in the volume of a cube is the same as the number of square inches in its surface area. The length of the edge expressed as a number of feet is $\textbf{(A) }6\qquad\textbf{(B) }864\qquad\textbf{(C) }1728\qquad\textbf{(D) }6\times 1728\qquad \textbf{(E) }2304$

2017 European Mathematical Cup, 3

Tags: geometry

Let $ABC$ be an acute triangle. Denote by $H$ and $M$ the orthocenter of $ABC$ and the midpoint of side $BC,$ respectively. Let $Y$ be a point on $AC$ such that $YH$ is perpendicular to $MH$ and let $Q$ be a point on $BH$ such that $QA$ is perpendicular to $AM.$ Let $J$ be the second point of intersection of $MQ$ and the circle with diameter $MY.$ Prove that $HJ$ is perpendicular to $AM.$ (Steve Dinh)

2013 IMC, 3

Tags: vector , induction , linear algebra , matrix , analytic geometry , geometry , 3d geometry

Suppose that $\displaystyle{{v_1},{v_2},...,{v_d}}$ are unit vectors in $\displaystyle{{{\Bbb R}^d}}$. Prove that there exists a unitary vector $\displaystyle{u}$ such that $\displaystyle{\left| {u \cdot {v_i}} \right| \leq \frac{1}{{\sqrt d }}}$ for $\displaystyle{i = 1,2,...,d}$. [b]Note.[/b] Here $\displaystyle{ \cdot }$ denotes the usual scalar product on $\displaystyle{{{\Bbb R}^d}}$. [i]Proposed by Tomasz Tkocz, University of Warwick.[/i]

2002 China Team Selection Test, 1

Tags: polynomial , inequalities , algebra

Let $P_n(x)=a_0 + a_1x + \cdots + a_nx^n$, with $n \geq 2$, be a real-coefficient polynomial. Prove that if there exists $a > 0$ such that \begin{align*} P_n(x) = (x + a)^2 \left( \sum_{i=0}^{n-2} b_i x^i \right), \end{align*} where $b_i$ are positive real numbers, then there exists some $i$, with $1 \leq i \leq n-1$, such that \[a_i^2 - 4a_{i-1}a_{i+1} \leq 0.\]

1945 Moscow Mathematical Olympiad, 093

Tags: 2-digit , number theory , perfect square

Find all two-digit numbers $\overline {ab}$ such that $\overline {ab} + \overline {ba}$ is a perfect square.

2018 HMNT, 3

Tags: geometry

$HOW,BOW,$ and $DAH$ are equilateral triangles in a plane such that $WO=7$ and $AH=2$. Given that $D,A,B$ are collinear in that order, find the length of $BA$.

1989 IMO Longlists, 99

Tags: geometry

Let $ ABC$ be an equilateral triangle and $ \Gamma$ the semicircle drawn exteriorly to the triangle, having $ BC$ as diameter. Show that if a line passing through $ A$ trisects $ BC,$ it also trisects the arc $ \Gamma.$

2006 Oral Moscow Geometry Olympiad, 3

Tags: geometry , locus , centroid

Two non-rolling circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ and radii $2R$ and $R$, respectively, are given on the plane. Find the locus of the centers of gravity of triangles in which one vertex lies on $C_1$ and the other two lie on $C_2$. (B. Frenkin)

2007 May Olympiad, 3

Tags: combinatorics

Eight children, all of different heights, must form an orderly line from smallest to largest. We will say that the row has exactly one error if there is a child that is immediately behind another taller than it, and everyone else (except the first in line) is immediately behind a shorter one. of how many ways the eight children can line up with exactly one mistake?

2023 International Zhautykov Olympiad, 5

Tags: number theory

We call a positive integer $n$ is $good$ , if there exist integers $a,b,c,x,y$ such that $n=ax^2+bxy+cy^2$ and $b^2-4ac=-20$. Prove that the product of any two good numbers is also a good number.

2022 CMIMC, 2.4

Tags: geometry

Circle $\Gamma$ has diameter $\overline{AB}$ with $AB = 6$. Point $C$ is constructed on line $AB$ so that $AB = BC$ and $A \neq C$. Let $D$ be on $\Gamma$ so that $\overleftrightarrow{CD}$ is tangent to $\Gamma$. Compute the distance from line $\overleftrightarrow{AD}$ to the circumcenter of $\triangle ADC$. [i]Proposed by Justin Hsieh[/i]

1990 IMO Longlists, 25

Tags: geometry , incenter , trigonometry , angle , midpoint

The incenter of the triangle $ ABC$ is $ K.$ The midpoint of $ AB$ is $ C_1$ and that of $ AC$ is $ B_1.$ The lines $ C_1K$ and $ AC$ meet at $ B_2,$ the lines $ B_1K$ and $ AB$ at $ C_2.$ If the areas of the triangles $ AB_2C_2$ and $ ABC$ are equal, what is the measure of angle $ \angle CAB?$