Olympiad problems

Ukrainian TYM Qualifying - geometry, 2019.10

At the altitude $AH_1$ of an acute non-isosceles triangle $ABC$ chose a point $X$ , from which draw the perpendiculars $XN$ and $XM$ on the sides $AB$ and $AC$ respectively. It turned out that $H_1A$ is the angle bisector $MH_1N$. Prove that $X$ is the point of intersection of the altitudes of the triangle $ABC$.

2009 IberoAmerican, 3

Tags: geometry , parallelogram , symmetry , circumcircle

Let $ C_1$ and $ C_2$ be two congruent circles centered at $ O_1$ and $ O_2$, which intersect at $ A$ and $ B$. Take a point $ P$ on the arc $ AB$ of $ C_2$ which is contained in $ C_1$. $ AP$ meets $ C_1$ at $ C$, $ CB$ meets $ C_2$ at $ D$ and the bisector of $ \angle CAD$ intersects $ C_1$ and $ C_2$ at $ E$ and $ L$, respectively. Let $ F$ be the symmetric point of $ D$ with respect to the midpoint of $ PE$. Prove that there exists a point $ X$ satisfying $ \angle XFL \equal{} \angle XDC \equal{} 30^\circ$ and $ CX \equal{} O_1O_2$. [i] Author: Arnoldo Aguilar (El Salvador)[/i]

2011 Estonia Team Selection Test, 4

Tags: inequalities , algebra

Let $a,b,c$ be positive real numbers such that $2a^2 +b^2=9c^2$.Prove that $\displaystyle \frac{2c}{a}+\frac cb \ge\sqrt 3$.

2009 Today's Calculation Of Integral, 408

Tags: calculus , integration , calculus computations , logarithm

Evaluate $ \int_1^e \{(1 \plus{} x)e^x \plus{} (1 \minus{} x)e^{ \minus{} x}\}\ln x\ dx$.

1963 IMO Shortlist, 1

Tags: algebra , equation , parametric equation

Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.

2011 Saudi Arabia BMO TST, 3

Tags: equal area , geometry , circumcircle , area

In an acute triangle $ABC$ the angle bisector $AL$, $L \in BC$, intersects its circumcircle at $N$. Let $K$ and $M$ be the projections of $L$ onto sides $AB$ and $AC$. Prove that triangle $ABC$ and quadrilateral $A K N M$ have equal areas.

2011 National Olympiad First Round, 20

Tags:

$100$ students participate in an exam with $5$ questions. Every question is answered by exactly $50$ students. What is the least possible value of number of students who answered at most $2$ questions? $\textbf{(A)}\ 21 \qquad\textbf{(B)}\ 18 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 16 \qquad\textbf{(E)}\ \text{None}$

2013 Taiwan TST Round 1, 1

Tags: geometry

Let P be a point in an acute triangle $ABC$, and $d_A, d_B, d_C$ be the distance from P to vertices of the triangle respectively. If the distance from P to the three edges are $d_1, d_2, d_3$ respectively, prove that \[d_A+d_B+d_C\geq 2(d_1+d_2+d_3)\]

1967 IMO Shortlist, 5

Tags: trigonometry , algebra , triangle , trigonometric identities

Show that a triangle whose angles $A$, $B$, $C$ satisfy the equality \[ \frac{\sin^2 A + \sin^2 B + \sin^2 C}{\cos^2 A + \cos^2 B + \cos^2 C} = 2 \] is a rectangular triangle.

2018 IFYM, Sozopol, 8

Tags: algebra , recurrence relation , recursion

The row $x_1, x_2,…$ is defined by the following recursion $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n}$ Prove that $\sum_{n=1}^{2018}{\frac{1}{x_n}}<3$.

2014 JHMMC 7 Contest, 15

Tags: purpleredblue

Rita the painter rolls a fair $6\text{-sided die}$that has $3$ red sides, $2$ yellow sides, and $1$ blue side. Rita rolls the die twice and mixes the colors that the die rolled. What is the probability that she has mixed the color purple?

1994 Poland - Second Round, 5

Tags: angle bisector , right angle , incircle , geometry

The incircle $\omega$ of a triangle $ABC$ is tangent to the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The angle bisector at $A$ meets $PQ$ at point $S$. Prove $\angle ASC = 90^o$ .

2008 Princeton University Math Competition, 8

Tags: algebra

Suppose that the roots of the quadratic $x^2 + ax + b$ are $\alpha$ and $\beta$. Then $\alpha^3$ and $\beta^3$ are the roots of some quadratic $x^2 + cx + d$. Find $c$ in terms of $a$ and $b$.

2006 Iran MO (3rd Round), 3

Tags: inequalities , trigonometry , geometry , incenter , geometric transformation , reflection , circumcircle

In triangle $ABC$, if $L,M,N$ are midpoints of $AB,AC,BC$. And $H$ is orthogonal center of triangle $ABC$, then prove that \[LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})\]

2022 USAMTS Problems, 2

Tags: function , functional equation , algebra

Let $Z^+$ denote the set of positive integers. Determine , with proof, if there exists a function $f:\mathbb{Z^+}\rightarrow\mathbb {Z^+}$ such that $f(f(f(f(f(n)))))$ = $2022n$ for all positive integers $n$.

2007 Greece JBMO TST, 1

Tags: circumcircle , geometry , angle

Let $ABC$ be a triangle with $\angle A=105^o$ and $\angle C=\frac{1}{4} \angle B$. a) Find the angles $\angle B$ and $\angle C$ b) Let $O$ be the center of the circumscribed circle of the triangle $ABC$ and let $BD$ be a diameter of that circle. Prove that the distance of point $C$ from the line $BD$ is equal to $\frac{BD}{4}$.

2020 MBMT, 6

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Given that $\sqrt{10} \approx 3.16227766$, find the largest integer $n$ such that $n^2 \leq 10,000,000$. [i]Proposed by Jacob Stavrianos[/i]

2001 IMO Shortlist, 5

Tags: geometry , circumcircle , trigonometry , triangle

Let $ABC$ be an acute triangle. Let $DAC,EAB$, and $FBC$ be isosceles triangles exterior to $ABC$, with $DA=DC, EA=EB$, and $FB=FC$, such that \[ \angle ADC = 2\angle BAC, \quad \angle BEA= 2 \angle ABC, \quad \angle CFB = 2 \angle ACB. \] Let $D'$ be the intersection of lines $DB$ and $EF$, let $E'$ be the intersection of $EC$ and $DF$, and let $F'$ be the intersection of $FA$ and $DE$. Find, with proof, the value of the sum \[ \frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}. \]

2018 AMC 12/AHSME, 22

Tags: algebra , polynomial

Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$? $\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286 $

2019 Thailand Mathematical Olympiad, 6

Tags: algebra , functional inequality

Determine all function $f:\mathbb{R}\to\mathbb{R}$ such that $xf(y)+yf(x)\leqslant xy$ for all $x,y\in\mathbb{R}$.

2019 Kurschak Competition, 1

Tags: geometry , geometric transformation , reflection , circumcircle

In an acute triangle $\bigtriangleup ABC$, $AB<AC<BC$, and $A_1,B_1,C_1$ are the projections of $A,B,C$ to the corresponding sides. Let the reflection of $B_1$ wrt $CC_1$ be $Q$, and the reflection of $C_1$ wrt $BB_1$ be $P$. Prove that the circumcirle of $A_1PQ$ passes through the midpoint of $BC$.

2009 District Olympiad, 3

Tags: inequalities

[b]a)[/b] For $ a,b\ge 0 $ and $ x,y>0, $ show that: $$ \frac{a^3}{x^2} +\frac{b^3}{y^2}\ge \frac{(a+b)^3}{(x+y)^2} . $$ [b]b)[/b] For $ a,b,c\ge 0 $ and $ x,y,z>0 $ under the condition $ a+b+c=x+y+z, $ prove that: $$ \frac{a^3}{x^2} +\frac{b^3}{y^2} +\frac{c^3}{z^2} \ge a+b+c. $$

2023 HMNT, 1

Tags: geometry

Let $ABC$ be an equilateral triangle with side length $2$ that is inscribed in a circle $\omega$. A chord of $\omega$ passes through the midpoints of sides $AB$ and $AC$. Compute the length of this chord.

1997 ITAMO, 2

Tags: odd , function , periodic , periodic function , algebra

Let a real function $f$ defined on the real numbers satisfy the following conditions: (i) $f(10+x) = f(10- x)$ (ii) $f(20+x) = - f(20- x)$ for all $x$. Prove that f is odd and periodic.

2000 IMO Shortlist, 5

Tags: geometry , rectangle , combinatorics , intersection , graph theory , double counting

In the plane we have $n$ rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is [i]nice[/i] if it has at least one of the vertices of the $n$ rectangles on the boundary. Prove that the sum of the numbers of the vertices of all nice regions is less than $40n$. (There can be nonconvex regions as well as regions with more than one boundary curve.)