Olympiad problems

2013 China Team Selection Test, 1

For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define \[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\] Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.

2011 NIMO Problems, 7

Tags: greatest common divisor , algebra , polynomial , number theory

Let $P(x) = x^2 - 20x - 11$. If $a$ and $b$ are natural numbers such that $a$ is composite, $\gcd(a, b) = 1$, and $P(a) = P(b)$, compute $ab$. Note: $\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$. [i]Proposed by Aaron Lin [/i]

1978 IMO Longlists, 8

Tags: geometry , inequalities

For two given triangles $A_1A_2A_3$ and $B_1B_2B_3$ with areas $\Delta_A$ and $\Delta_B$, respectively, $A_iA_k \ge B_iB_k, i, k = 1, 2, 3$. Prove that $\Delta_A \ge \Delta_B$ if the triangle $A_1A_2A_3$ is not obtuse-angled.

1997 Chile National Olympiad, 6

Tags: combinatorics , combinatorial geometry

For each set $C$ of points in space, we designate by $P_C$ the set of planes containing at least three points of $C$. $\bullet$ Prove that there exists $C$ such that $\phi (P_C) = 1997$, where $\phi$ corresponds to the cardinality. $\bullet$ Determine the least number of points that $C$ must have so that the previous property can be fulfilled.

IV Soros Olympiad 1997 - 98 (Russia), 10.3

Tags: geometry , circumradius

What can angle $B$ of triangle $ABC$ be equal to if it is known that the distance between the feet of the altitudes drawn from vertices $A$ and $C$ is equal to half the radius of the circle circumscribed around this triangle?

2025 District Olympiad, P3

Tags: equation

Determine all positive real numbers $a,b,c,d$ such that $a+b+c+d=80$ and $$a+\frac{b}{1+a}+\frac{c}{1+a+b}+\frac{d}{1+a+b+c}=8.$$

2002 AMC 12/AHSME, 6

Tags:

Participation in the local soccer league this year is $10\%$ higher than last year. The number of males increased by $5\%$ and the number of females increased by $20\%$. What fraction of the soccer league is now female? $\textbf{(A) }\dfrac13\qquad\textbf{(B) }\dfrac4{11}\qquad\textbf{(C) }\dfrac25\qquad\textbf{(D) }\dfrac49\qquad\textbf{(E) }\dfrac12$

1992 Swedish Mathematical Competition, 6

Tags: collinear , analytic geometry

$(x_1, y_1), (x_2, y_2), (x_3, y_3)$ lie on a straight line and on the curve $y^2 = x^3$. Show that $\frac{x_1}{y_1} + \frac{x_2}{y_2}+\frac{x_3}{y_3} = 0$.

2012 South africa National Olympiad, 5

Tags: euler , geometry , circumcircle , ratio , homothety , geometric transformation

Let $ABC$ be a triangle such that $AB\neq AC$. We denote its orthocentre by $H$, its circumcentre by $O$ and the midpoint of $BC$ by $D$. The extensions of $HD$ and $AO$ meet in $P$. Prove that triangles $AHP$ and $ABC$ have the same centroid.

2007 Iran MO (3rd Round), 2

Tags: floor function , number theory

We call the mapping $ \Delta:\mathbb Z\backslash\{0\}\longrightarrow\mathbb N$, a degree mapping if and only if for each $ a,b\in\mathbb Z$ such that $ b\neq0$ and $ b\not|a$ there exist integers $ r,s$ such that $ a \equal{} br\plus{}s$, and $ \Delta(s) <\Delta(b)$. a) Prove that the following mapping is a degree mapping: \[ \delta(n)\equal{}\mbox{Number of digits in the binary representation of }n\] b) Prove that there exist a degree mapping $ \Delta_{0}$ such that for each degree mapping $ \Delta$ and for each $ n\neq0$, $ \Delta_{0}(n)\leq\Delta(n)$. c) Prove that $ \delta \equal{}\Delta_{0}$ [img]http://i16.tinypic.com/4qntmd0.png[/img]

Mathley 2014-15, 4

Tags: common tangent , mixtilinear , geometry

Let $(O)$ be the circumcircle of triangle $ABC$, and $P$ a point on the arc $BC$ not containing $A$. $(Q)$ is the $A$-mixtilinear circle of triangle $ABC$, and $(K), (L)$ are the $P$-mixtilinear circles of triangle $PAB, PAC$ respectively. Prove that there is a line tangent to all the three circles $(Q), (K)$ and $(L)$. Nguyen Van Linh, a student at Hanoi Foreign Trade University Cabinet

2001 China National Olympiad, 3

Tags: limit , modular arithmetic , number theory

Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\neq0$ such that (i) $m<2a$; (ii) $2n|(2am-m^2+n^2)$; (iii) $n^2-m^2+2mn\leq2a(n-m)$. For $(m, n)\in A$, let \[f(m,n)=\frac{2am-m^2-mn}{n}.\] Determine the maximum and minimum values of $f$.

2018 CHKMO, 4

Tags: geometry , combinatorial geometry , combinatorics

Suppose 2017 points in a plane are given such that no three points are collinear. Among the triangles formed by any three of these 2017 points, those triangles having the largest area are said to be [i]good[/i]. Prove that there cannot be more than 2017 good triangles.

2010 Tuymaada Olympiad, 1

Tags: limit , algebra , integration , calculus

We have a set $M$ of real numbers with $|M|>1$ such that for any $x\in M$ we have either $3x-2\in M$ or $-4x+5\in M$. Show that $M$ is infinite.

2020 Romanian Master of Mathematics, 5

Tags: analytic geometry , combinatorics

A [i]lattice point[/i] in the Cartesian plane is a point whose coordinates are both integers. A [i]lattice polygon[/i] is a polygon all of whose vertices are lattice points. Let $\Gamma$ be a convex lattice polygon. Prove that $\Gamma$ is contained in a convex lattice polygon $\Omega$ such that the vertices of $\Gamma$ all lie on the boundary of $\Omega$, and exactly one vertex of $\Omega$ is not a vertex of $\Gamma$.

2003 Federal Math Competition of S&M, Problem 2

Tags: combinatorics , geometry , combinatorial geometry

Given a segment $AB$ of length $2003$ in a coordinate plane, determine the maximal number of unit squares with vertices in the lattice points whose intersection with the given segment is non-empty.

2016 Iran MO (3rd Round), 3

Tags: angle bisector , geometry

Given triangle $\triangle ABC$ and let $D,E,F$ be the foot of angle bisectors of $A,B,C$ ,respectively. $M,N$ lie on $EF$ such that $AM=AN$. Let $H$ be the foot of $A$-altitude on $BC$. Points $K,L$ lie on $EF$ such that triangles $\triangle AKL, \triangle HMN$ are correspondingly similiar (with the given order of vertices) such that $AK \not\parallel HM$ and $AK \not\parallel HN$. Show that: $DK=DL$

2004 Iran Team Selection Test, 4

Tags: geometry

Let $ M,M'$ be two conjugates point in triangle $ ABC$ (in the sense that $ \angle MAB\equal{}\angle M'AC,\dots$). Let $ P,Q,R,P',Q',R'$ be foots of perpendiculars from $ M$ and $ M'$ to $ BC,CA,AB$. Let $ E\equal{}QR\cap Q'R'$, $ F\equal{}RP\cap R'P'$ and $ G\equal{}PQ\cap P'Q'$. Prove that the lines $ AG, BF, CE$ are parallel.

Today's calculation of integrals, 899

Tags: calculus computations , limit , integration , calculus

Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

2014 China Team Selection Test, 3

Tags: combinatorics

$A$ is the set of points of a convex $n$-gon on a plane. The distinct pairwise distances between any $2$ points in $A$ arranged in descending order is $d_1>d_2>...>d_m>0$. Let the number of unordered pairs of points in $A$ such that their distance is $d_i$ be exactly $\mu _i$, for $i=1, 2,..., m$. Prove: For any positive integer $k\leq m$, $\mu _1+\mu _2+...+\mu _k\leq (3k-1)n$.

1978 AMC 12/AHSME, 6

Tags:

The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations: \begin{align*}x&=x^2+y^2, \\ y&=2xy\end{align*} is $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4$

2006 Mathematics for Its Sake, 3

Tags: group theory , abstract algebra

Let be a group with $ 10 $ elements for which there exist two non-identity elements, $ a,b, $ having the property that $ a^2 $ and $ b^2 $ are the identity. Show that this group is not commutative.

1937 Moscow Mathematical Olympiad, 036

Tags: dodecahedron , plane , 3d geometry , regular hexagon , combinatorics , geometry , combinatorial geometry

* Given a regular dodecahedron. Find how many ways are there to draw a plane through it so that its section of the dodecahedron is a regular hexagon?

1998 AIME Problems, 8

Tags: limit

Except for the first two terms, each term of the sequence $1000, x, 1000-x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer $x$ produces a sequence of maximum length?

2012 Online Math Open Problems, 24

Tags: modular arithmetic , function

Find the number of ordered pairs of positive integers $(a,b)$ with $a+b$ prime, $1\leq a, b \leq 100$, and $\frac{ab+1}{a+b}$ is an integer. [i]Author: Alex Zhu[/i]