Olympiad problems

2012 Gulf Math Olympiad, 2

Prove that if $a, b, c$ are positive real numbers, then the least possible value of \[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}\] is $6$. For which values of $a, b$ and $c$ is equality attained?

2004 Postal Coaching, 4

Tags: combinatorics , function

In how many ways can a $2\times n$ grid be covered by (a) 2 monominoes and $n-1$ dominoes (b) 4 monominoes and $n-2$ dominoes.

2008 National Chemistry Olympiad, 11

Tags: percent

For the reaction: $2X + 3Y \rightarrow 3Z$, the combination of $2.00$ moles of $X$ with $2.00$ moles of $Y$ produces $1.75 $ moles of $Z$. What is the percent yield of this reaction? $\textbf{(A)}\hspace{.05in}43.8\%\qquad\textbf{(B)}\hspace{.05in}58.3\%\qquad\textbf{(C)}\hspace{.05in}66.7\%\qquad\textbf{(D)}\hspace{.05in}87.5\%\qquad $

2001 IMO Shortlist, 5

Tags: composite number , number theory

Let $a > b > c > d$ be positive integers and suppose that \[ ac + bd = (b+d+a-c)(b+d-a+c). \] Prove that $ab + cd$ is not prime.

1950 Moscow Mathematical Olympiad, 184

Tags: point , combinatorics , combinatorial geometry

* On a circle, $20$ points are chosen. Ten non-intersecting chords without mutual endpoints connect some of the points chosen. How many distinct such arrangements are there?

2010 Dutch IMO TST, 3

Tags: number theory , perfect square

Let $n\ge 2$ be a positive integer and $p $ a prime such that $n|p-1$ and $p | n^3-1$. Show $ 4p-3$ is a square.

2022 Tuymaada Olympiad, 3

Tags: number theory , combinatorics

Is there a colouring of all positive integers in three colours so that for each positive integer the numbers of its divisors of any two colours differ at most by $2?$

2011 Czech-Polish-Slovak Match, 1

Tags: polynomial , algebra

A polynomial $P(x)$ with integer coefficients satisfies the following: if $F(x)$, $G(x)$, and $Q(x)$ are polynomials with integer coefficients satisfying $P\Big(Q(x)\Big)=F(x)\cdot G(x)$, then $F(x)$ or $G(x)$ is a constant polynomial. Prove that $P(x)$ is a constant polynomial.

Champions Tournament Seniors - geometry, 2008.4

Tags: geometry , 3d geometry , sphere , pyramid

Given a quadrangular pyramid $SABCD$, the basis of which is a convex quadrilateral $ABCD$. It is known that the pyramid can be tangent to a sphere. Let $P$ be the point of contact of this sphere with the base $ABCD$. Prove that $\angle APB + \angle CPD = 180^o$.

2001 Swedish Mathematical Competition, 1

Tags: algebra , product

Show that if we take any six numbers from the following array, one from each row and column, then the product is always the same: 4 6 10 14 22 26 6 9 15 21 33 39 10 15 25 35 55 65 16 24 40 56 88 104 18 27 45 63 99 117 20 30 50 70 110 130

2022 Yasinsky Geometry Olympiad, 6

Tags: geometry , incenter , fixed point , fixed , circumcircle

Let $s$ be an arbitrary straight line passing through the incenter $I$ of the triangle $ABC$ . Line $s$ intersects lines $AB$ and $BC$ at points $D$ and $E$, respectively. Points $P$ and $Q$ are the centers of the circumscribed circles of triangles $DAI$ and $CEI$, respectively, and point $F$ is the second intersection point of these circles. Prove that the circumcircle of the triangle $PQF$ is always passes through a fixed point on the plane regardless of the position of the straight line $s$. (Matvii Kurskyi)

2017 F = ma, 11

Tags: angular momentum , momentum , collision

A small hard solid sphere of mass m and negligible radius is connected to a thin rod of length L and mass 2m. A second small hard solid sphere, of mass M and negligible radius, is fired perpendicularly at the rod at a distance h above the sphere attached to the rod, and sticks to it. A: h = 0 B: h = L/3 C: h = L/2 D: h = L E: Any L will work

2007 AIME Problems, 12

Tags: rotation , geometry

In isosceles triangle $ABC$, $A$ is located at the origin and $B$ is located at $(20, 0)$. Point $C$ is in the first quadrant with $AC = BC$ and $\angle BAC = 75^\circ$. If $\triangle ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive y-axis, the area of the region common to the original and the rotated triangle is in the form $p\sqrt{2}+q\sqrt{3}+r\sqrt{6}+s$ where $p$, $q$, $r$, $s$ are integers. Find $(p-q+r-s)/2$.

2015 IFYM, Sozopol, 2

Tags: geometry

Let $ABCD$ be an inscribed quadrilateral and $P$ be an inner point for it so that $\angle PAB=\angle PBC=\angle PCD=\angle PDA$. The lines $AD$ and $BC$ intersect in point $Q$ and lines $AB$ and $CD$ – in point $R$. Prove that $\angle (PQ,PR)=\angle (AC,BD)$.

1991 China Team Selection Test, 1

Tags: geometry

We choose 5 points $A_1, A_2, \ldots, A_5$ on a circumference of radius 1 and centre $O.$ $P$ is a point inside the circle. Denote $Q_i$ as the intersection of $A_iA_{i+2}$ and $A_{i+1}P$, where $A_7 = A_2$ and $A_6 = A_1.$ Let $OQ_i = d_i, i = 1,2, \ldots, 5.$ Find the product $\prod^5_{i=1} A_iQ_i$ in terms of $d_i.$

1991 French Mathematical Olympiad, Problem 1

Tags: number theory

(a) Suppose that $x_n~(n\ge0)$ is a sequence of real numbers with the property that $x_0^3+x_1^3+\ldots+x_n^3=(x_0+x_1+\ldots+x_n)^2$ for each $n\in\mathbb N$. Prove that for each $n\in\mathbb N_0$ there exists $m\in\mathbb N_0$ such that $x_0+x_1+\ldots+x_n=\frac{m(m+1)}2$. (b) For natural numbers $n$ and $p$, we define $S_{n,p}=1^p+2^p+\ldots+n^p$. Find all natural numbers $p$ such that $S_{n,p}$ is a perfect square for each $n\in\mathbb N$.

2017 China National Olympiad, 6

Tags: prc , inequalities , convex-concave inequalities , sqing orz , inequality

Given an integer $n \geq2$ and real numbers $a,b$ such that $0<a<b$. Let $x_1,x_2,\ldots, x_n\in [a,b]$ be real numbers. Find the maximum value of $$\frac{\frac{x^2_1}{x_2}+\frac{x^2_2}{x_3}+\cdots+\frac{x^2_{n-1}}{x_n}+\frac{x^2_n}{x_1}}{x_1+x_2+\cdots +x_{n-1}+x_n}.$$

1984 Czech And Slovak Olympiad IIIA, 3

Tags: algebra , recurrence relation , floor function

Let the sequence $\{a_n\}$ , $n \ge 0$ satisfy the recurrence relation $$a_{n + 2} =4a_{n + 1}-3a_n, \ \ (1) $$ Let us define the sequence $\{b_n\}$ , $n \ge 1$ by the relation $$b_n= \left[ \frac{a_{n+1}}{a_{n-1}} \right]$$ where we put $b_n =1$ for $a_{n-1}=0$. Prove that, starting from a certain term, the sequence also satisfies the recurrence relation (1). Note: $[x]$ indicates the whole part of the number $x$.

2001 National Olympiad First Round, 10

Tags:

At each step, we are changing the places of exactly two numbers from the sequence $1$, $2$, $3$, $4$, $5$, $6$, $7$. How many different arrangements can be formed after two steps? $ \textbf{(A)}\ 88 \qquad\textbf{(B)}\ 100 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\ 176 \qquad\textbf{(E)}\ 441 $

2018 AMC 12/AHSME, 17

Tags:

Let $p$ and $q$ be positive integers such that \[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\] and $q$ is as small as possible. What is $q-p$? $\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19 $

1973 Spain Mathematical Olympiad, 2

Tags: inequalities , system of equations , system , algebra

Determine all solutions of the system $$\begin{cases} 2x - 5y + 11z - 6 = 0 \\ -x + 3y - 16z + 8 = 0 \\ 4x - 5y - 83z + 38 = 0 \\ 3x + 11y - z + 9 > 0 \end{cases}$$ in which the first three are equations and the last one is a linear inequality.

2015 Saint Petersburg Mathematical Olympiad, 3

Tags: combinatorics , number theory

There are weights with mass $1,3,5,....,2i+1,...$ Let $A(n)$ -is number of different sets with total mass equal $n$( For example $A(9)=2$, because we have two sets $9=9=1+3+5$). Prove that $A(n) \leq A(n+1)$ for $n>1$

2003 IMO Shortlist, 8

Tags: modular arithmetic , number theory , prime number , perfect power

Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions: (i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements; (ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power. What is the largest possible number of elements in $A$ ?

2023 Taiwan TST Round 1, A

Tags: cringe

Let $f:\mathbb{N}\to\mathbb{R}_{>0}$ be a given increasing function that takes positive values. For any pair $(m,n)$ of positive integers, we call it [i]disobedient[/i] if $f(mn)\neq f(m)f(n)$. For any positive integer $m$, we call it [i]ultra-disobedient[/i] if for any nonnegative integer $N$, there are always infinitely many positive integers $n$ satisfying that $(m,n), (m,n+1),\ldots,(m,n+N)$ are all disobedient pairs. Show that if there exists some disobedient pair, then there exists some ultra-disobedient positive integer. [i] Proposed by usjl[/i]

Russian TST 2019, P3

Tags: geometry

Let $H{}$ be the orthocenter of the acute-angled triangle $ABC$. In the triangle $BHC$, the median $HM$ and the symedian $HL$ are drawn. The point $K{}$ is marked on the line $LH$ so that $\angle AKL=90^\circ$. Prove that the circumcircles of the triangles $ABC$ and $KLM$ are tangent.