Olympiad problems

2007 Alexandru Myller, 1

Solve $ x^3-y^3=2xy+7 $ in integers.

2011 Armenian Republican Olympiads, Problem 1

Tags: function , algebra

Does there exist a function $f\colon \mathbb{R}\to\mathbb{R}$ such that for any $x>y,$ it satisfies $f(x)-f(y)>\sqrt{x-y}.$

The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is $75^\circ$, then the largest angle is $\textbf{(A) }95^\circ\qquad \textbf{(B) }100^\circ\qquad \textbf{(C) }105^\circ\qquad \textbf{(D) }110^\circ\qquad \textbf{(E) }115^\circ$

2010 Contests, 3

Tags: geometry , 3D geometry , trigonometry , vector , linear algebra , matrix , inequalities

What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak? Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.

2010 Stanford Mathematics Tournament, 2

Tags:

Write $0.2010\overline{228}$ as a fraction.

2004 Bulgaria Team Selection Test, 1

Tags: function , algebra proposed , algebra

Find all $k>0$ such that there exists a function $f : [0,1]\times[0,1] \to [0,1]$ satisfying the following conditions: $f(f(x,y),z)=f(x,f(y,z))$; $f(x,y) = f(y,x)$; $f(x,1)=x$; $f(zx,zy) = z^{k}f(x,y)$, for any $x,y,z \in [0,1]$

2000 Regional Competition For Advanced Students, 2

Tags: algebra

For any real number $a$, find all real numbers $x$ that satisfy the following equation. $$(2x + 1)^4 + ax(x + 1) - \frac{x}{2}= 0$$

2003 Turkey Team Selection Test, 6

Tags: combinatorics proposed , combinatorics

For all positive integers $n$, let $p(n)$ be the number of non-decreasing sequences of positive integers such that for each sequence, the sum of all terms of the sequence is equal to $n$. Prove that \[\dfrac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.\]

Geometry Mathley 2011-12, 14.2

Tags: Euler Circle , tangent circles , angle bisector , geometry , Euler

The nine-point Euler circle of triangle $ABC$ is tangent to the excircles in the angle $A,B,C$ at $Fa, Fb, Fc$ respectively. Prove that $AF_a$ bisects the angle $\angle CAB$ if and only if $AFa$ bisects the angle $\angle F_bAF_c$. Đỗ Thanh Sơn

2016 HMNT, 22-24

Tags: HMMT

22. Let the function $f : \mathbb{Z} \to \mathbb{Z}$ take only integer inputs and have integer outputs. For any integers $x$ and $y$, f satisfies $$f(x) + f(y) = f(x + 1) + f(y - 1)$$ If $f(2016) = 6102$ and $f(6102) = 2016$, what is $f(1)$? 23. Let $d$ be a randomly chosen divisor of $2016$. Find the expected value of $$\frac{d^2}{d^2 + 2016}$$ 24. Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjecent vertices consists of edges with at least two different colors. What is the smallest possible value of $N$?

2018-2019 SDML (High School), 13

Tags: NEEDS DIAGRAM , SDML High School , 2018-2019 Round 2 , geometry , 3D geometry

A steel cube has edges of length $3$ cm, and a cone has a diameter of $8$ cm and a height of $24$ cm. The cube is placed in the cone so that one of its interior diagonals coincides with the axis of the cone. What is the distance, in cm, between the vertex of the cone and the closest vertex of the cube? [NEEDS DIAGRAM] $ \mathrm{(A) \ } \frac{12\sqrt6-3\sqrt3}{4} \qquad \mathrm{(B) \ } \frac{9\sqrt6-3\sqrt3}{2} \qquad \mathrm {(C) \ } 5\sqrt3 \qquad \mathrm{(D) \ } 6\sqrt6 - \sqrt3 \qquad \mathrm{(E) \ } 6\sqrt6$

2021 Federal Competition For Advanced Students, P2, 5

Tags: geometry , reflection , cyclic quadrilateral , Concyclic , collinear

Let $ABCD$ be a convex cyclic quadrilateral with diagonals $AC$ and $BD$. Each of the four vertixes are reflected across the diagonal on which the do not lie. (a) Investigate when the four points thus obtained lie on a straight line and give as simple an equivalent condition as possible to the cyclic quadrilateral $ABCD$ for it. (b) Show that in all other cases the four points thus obtained lie on one circle. (Theresia Eisenkölbl)

Indonesia MO Shortlist - geometry, g6.7

Tags: geometry , parallelogram

Let $ABCD$ be a parallelogram. $E$ and $F$ are on $BC, CD$ respectively such that the triangles $ABE$ and $BCF$ have the same area. Let $BD$ intersect $AE, AF$ at $M, N$ respectively. Prove there exists a triangle whose side lengths are $BM, MN, ND$.

Croatia MO (HMO) - geometry, 2010.7

Tags: geometry , circumcircle , Circumcenter , collinear

Given a non- isosceles triangle $ABC$. Let the points $B'$ and $C'$ be symmetric to the points $B$ and $C$ wrt $AC$ and $AB$ respectively. If the circles circumscribed around triangles $ABB'$ and $ACC'$ intersect at point $P$, prove that the line $AP$ passes through the center of the circumcircle of the triangle $ABC$.

2019 Jozsef Wildt International Math Competition, W. 32

Tags: Summation , Sequences , inequalities

Let $u_k$, $v_k$, $a_k$ and $b_k$ be non-negative real sequences such as $u_k > a_k$ and $v_k > b_k$, where $k = 1, 2,\cdots , n$. If $0 < m_1 \leq u_k \leq M_1$ and $0 < m_2 \leq v_k \leq M_2$, then $$\sum \limits_{k=1}^n(lu_kv_k-a_kb_k)\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^2\right)\right)^\frac{1}{2}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^2\right)\right)^\frac{1}{2}$$where$$l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}$$

2005 Today's Calculation Of Integral, 9

Tags: calculus , integration , logarithms , trigonometry , absolute value , calculus computations

Calculate the following indefinite integrals. [1] $\int (x^2+4x-3)^2(x+2)dx$ [2] $\int \frac{\ln x}{x(\ln x+1)}dx$ [3] $\int \frac{\sin \ (\pi \log _2 x)}{x}dx$ [4] $\int \frac{dx}{\sin x\cos ^ 2 x}$ [5] $\int \sqrt{1-3x}\ dx$

1992 Brazil National Olympiad, 6

Tags: search , combinatorics proposed , combinatorics

Given a set of n elements, find the largest number of subsets such that no subset is contained in any other

2004 Harvard-MIT Mathematics Tournament, 3

Tags: number theory , algebra

How many ordered pairs of integers $(a, b)$ satisfy all of the following inequalities? $$a^2 + b^2 < 16$$ $$a^2 + b^2 < 8a$$ $$a^2 + b^2 < 8b$$

2023 Canadian Junior Mathematical Olympiad, 2

Tags: geometry , angle bisector , circumcircle

An acute triangle is a triangle that has all angles less that $90^{\circ}$ ($90^{\circ}$ is a Right Angle). Let $ABC$ be a right-angled triangle with $\angle ACB =90^{\circ}.$ Let $CD$ be the altitude from $C$ to $AB,$ and let $E$ be the intersection of the angle bisector of $\angle ACD$ with $AD.$ Let $EF$ be the altitude from $E$ to $BC.$ Prove that the circumcircle of $BEF$ passes through the midpoint of $CE.$

1978 AMC 12/AHSME, 5

Tags: AMC

Four boys bought a boat for $\textdollar 60$. The first boy paid one half of the sum of the amounts paid by the other boys; the second boy paid one third of the sum of the amounts paid by the other boys; and the third boy paid one fourth of the sum of the amounts paid by the other boys. How much did the fourth boy pay? $\textbf{(A) }\textdollar 10\qquad\textbf{(B) }\textdollar 12\qquad\textbf{(C) }\textdollar 13\qquad\textbf{(D) }\textdollar 14\qquad \textbf{(E) }\textdollar 15$