Olympiad problems

2008 Postal Coaching, 3

Let $ABC$ be a triangle. For any point $X$ on $BC$, let $AX$ meet the circumcircle of $ABC$ in $X'$. Prove or disprove: $XX'$ has maximum length if and only if $AX$ lies between the median and the internal angle bisector from $A$.

2010 Today's Calculation Of Integral, 613

Tags: calculus , integration , geometry , calculus computations , logarithm

Find the area of the part, in the $x$-$y$ plane, enclosed by the curve $|ye^{2x}-6e^{x}-8|=-(e^{x}-2)(e^{x}-4).$ [i]2010 Tokyo University of Agriculture and Technology entrance exam[/i]

2013 Kosovo National Mathematical Olympiad, 2

Tags: trigonometry

Solve equation $27\cdot3^{3\sin x}=9^{\cos^2x}$ where $x\in [0,2\pi )$

2004 Harvard-MIT Mathematics Tournament, 7

Tags: calculus , geometry

Find the area of the region in the $xy$-plane satisfying $x^6-x^2+y^2 \le 0$.

1990 Tournament Of Towns, (273) 1

Tags: combinatorics

The positive integers from $1$ to $n^2$ are placed arbitrarily on the squares of a chess board with dimensions $n\times n$. Prove that there are two adjacent squares (having a common vertex or a common side) such that the difference between the numbers placed on them is not less than $n + 1$. (N Sedrakyan, Yerevan)

2021 Azerbaijan IMO TST, 1

Tags: algebra , polynomial , linear combination

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

2002 AMC 10, 6

Tags: perimeter , rectangle , geometry

The perimeter of a rectangle is $100$ and its diagonal has length $x$. What is the area of this rectangle? $\textbf{(A) }625-x^2\qquad\textbf{(B) }625-\dfrac{x^2}2\qquad\textbf{(C) }1250-x^2\qquad\textbf{(D) }1250-\dfrac{x^2}2\qquad\textbf{(E) }2500-\dfrac{x^2}2$

2019 CCA Math Bonanza, L5.4

Tags:

Submit an integer between $0$ and $100$ inclusive as your answer to this problem. Suppose that $Q_1$ and $Q_3$ are the medians of the smallest $50\%$ and largest $50\%$ of submissions for this question. Your goal is to have your submission close to $D=Q_3-Q_1$. If you submit $N$, your score will be $2-2\sqrt{\frac{\left|N-D\right|}{\max\left\{D,100-D\right\}}}$. [i]2019 CCA Math Bonanza Lightning Round #5.4[/i]

2018 Sharygin Geometry Olympiad, 6

Tags: geometry

Let $CH$ be the altitude of a right-angled triangle $ABC$ ($\angle C = 90^{\circ}$) with $BC = 2AC$. Let $O_1$, $O_2$ and $O$ be the incenters of triangles $ACH$, $BCH$ and $ABC$ respectively, and $H_1$, $H_2$, $H_0$ be the projections of $O_1$, $O_2$, $O$ respectively to $AB$. Prove that $H_1H = HH_0 = H_0H_2$.

2020 Malaysia IMONST 1, 3

Tags: geometry , circles , square

Given a square with area $A$. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area $B$ lies inside the circle, such that all its vertices lie on the circle. Find the value of $\frac{A}{B}.$

2022 Purple Comet Problems, 10

Tags:

Find the positive integer $n$ such that a convex polygon with $3n + 2$ sides has $61.5$ percent fewer diagonals than a convex polygon with $5n - 2$ sides.

1990 AMC 12/AHSME, 11

Tags:

How man y positive integers less than $50$ have an odd number of positive integer divisors? $\text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 11$

2023 District Olympiad, P3

Tags: algebra , inequalities

Let $x,y{}$ and $z{}$ be positive real numbers satisfying $x+y+z=1$. Prove that [list=a] [*]\[1-\frac{x^2-yz}{x^2+x}=\frac{(1-y)(1-z)}{x^2+x};\] [*]\[\frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leqslant 0.\] [/list]

2007 AMC 12/AHSME, 14

Tags: vieta

Let $ a,$ $ b,$ $ c,$ $ d,$ and $ e$ be distinct integers such that \[ (6 \minus{} a)(6 \minus{} b)(6 \minus{} c)(6 \minus{} d)(6 \minus{} e) \equal{} 45. \]What is $ a \plus{} b \plus{} c \plus{} d \plus{} e?$ $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 17 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 30$

2008 Bulgarian Autumn Math Competition, Problem 12.3

Tags: function , algebra , functional equation in r , functional equation

Find all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that \[(f(x)f(y)-1)f(x+y)=2f(x)f(y)-f(x)-f(y)\quad \forall x,y\in \mathbb{R}\]

1935 Moscow Mathematical Olympiad, 013

Tags: geometry , geometric construction , triangle construction

The median, bisector, and height, all originate at the same vertex of a triangle. Given the intersection points of the median, bisector, and height with the circumscribed circle, construct the triangle.

2002 HKIMO Preliminary Selection Contest, 17

Tags: algebra

Let $a_0=2$ and for $n\geq 1$, $a_n=\frac{\sqrt3 a_{n-1}+1}{\sqrt3-a_{n-1}}$. Find the value of $a_{2002}$ in the form $p+q\sqrt3$ where $p$ and $q$ are rational numbers

2022 Purple Comet Problems, 4

Tags:

Of $450$ students assembled for a concert, $40$ percent were boys. After a bus containing an equal number of boys and girls brought more students to the concert, $41$ percent of the students at the concert were boys. Find the number of students on the bus.

2001 India National Olympiad, 2

Tags: modular arithmetic , polynomial , arithmetic sequence , number theory , algebra

Show that the equation $x^2 + y^2 + z^2 = ( x-y)(y-z)(z-x)$ has infintely many solutions in integers $x,y,z$.

2018 Hanoi Open Mathematics Competitions, 15

Tags: diophantine equation , number theory

Find all pairs of prime numbers $(p,q)$ such that for each pair $(p,q)$, there is a positive integer m satisfying $\frac{pq}{p + q}=\frac{m^2 + 6}{m + 1}$.

2013 All-Russian Olympiad, 2

Tags: rotation , geometric transformation , invariant , combinatorics , geometry

Circle is divided into $n$ arcs by $n$ marked points on the circle. After that circle rotate an angle $ 2\pi k/n $ (for some positive integer $ k $), marked points moved to $n$ [i] new points [/i], dividing the circle into $ n $ [i] new arcs[/i]. Prove that there is a new arc that lies entirely in the one of the old arсs. (It is believed that the endpoints of arcs belong to it.) [i]I. Mitrophanov[/i]

2011 N.N. Mihăileanu Individual, 4

Tags: calculus , integration , real analysis

Let be the sequence $ \left( I_n \right)_{n\ge 1} $ defined as $ I_n=\int_0^1 \frac{x^n}{\sqrt{x^{2n} +1}} dx . $ [b]a)[/b] Show that $ \left( I_n \right)_{n\ge 1} $ converges to $ 0. $ [b]b)[/b] Calculate $ \lim_{m\to\infty } m\cdot I_m. $ [b]c)[/b] Prove that the sequence $ \left( n\left( -n\cdot I_n +\lim_{m\to\infty } m\cdot I_m \right) \right)_{n\ge 1} $ is convergent.

2004 China Team Selection Test, 1

Tags: prime factorization , number theory

Let $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (may not distinct) and $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (may not distinct) be two groups of positive integers such that for any positive integer $ d$ larger than $ 1$, the numbers of which can be divided by $ d$ in group $ m_1$, $ m_2$, $ \cdots$, $ m_r$ (including repeated numbers) are no less than that in group $ n_1$, $ n_2$ $ \cdots$, $ n_s$ (including repeated numbers). Prove that $ \displaystyle \frac{m_1 \cdot m_2 \cdots m_r}{n_1 \cdot n_2 \cdots n_s}$ is integer.

2019 Kazakhstan National Olympiad, 3

Tags: number theory

Let $p$ be a prime number of the form $4k+1$ and $\frac{m}{n}$ is an irreducible fraction such that $$\sum_{a=2}^{p-2} \frac{1}{a^{(p-1)/2}+a^{(p+1)/2}}=\frac{m}{n}.$$ Prove that $p|m+n$. (Fixed, thanks Pavel)

2008 iTest Tournament of Champions, 5

Tags: geometry , 3d geometry

Two squares of side length $2$ are glued together along their boundary so that the four vertices of the first square are glued to the midpoints of the four sides of the other square, and vice versa. This gluing results in a convex polyhedron. If the square of the volume of this polyhedron is written in simplest form as $\tfrac{a+b\sqrt c}d$, what is the value of $a+b+c+d$?