Olympiad problems

2010 Stanford Mathematics Tournament, 9

Tags: geometry , circumcircle , incenter , trigonometry , system of equations , algebra

For an acute triangle $ABC$ and a point $X$ satisfying $\angle{ABX}+\angle{ACX}=\angle{CBX}+\angle{BCX}$. Find the minimum length of $AX$ if $AB=13$, $BC=14$, and $CA=15$.

2017 QEDMO 15th, 5

Tags: number theory

Let $F$ be a finite subset of the integer numbers. We define a new subset $s(F)$ in that $a\in Z$ lies in $s (F)$ if and only if exactly one of the numbers $a$ and $a -1$ in $F$. In the same way one gets from $s (F)$ the set $s^2(F) = s (s (F))$ and by $n$-fold application of $s$ then iteratively further subsets $s^n (F)$. Prove there are infinitely many natural numbers $n$ for which $s^n (F) = F\cup \{a + n|a \in F\}$.

2011 AMC 10, 24

Tags: geometry , 3d geometry , tetrahedron , ratio , octahedron , rhombus

Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra? $ \textbf{(A)}\ \frac{1}{12}\qquad\textbf{(B)}\ \frac{\sqrt{2}}{12}\qquad\textbf{(C)}\ \frac{\sqrt{3}}{12}\qquad\textbf{(D)}\ \frac{1}{6}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{6} $

LMT Guts Rounds, 13

Tags:

A circle with center $O$ has radius $5,$ and has two points $A,B$ on the circle such that $\angle AOB = 90^{\circ}.$ Rays $OA$ and $OB$ are extended to points $C$ and $D,$ respectively, such that $AB$ is parallel to $CD,$ and the length of $CD$ is $200\%$ more than the radius of circle $O.$ Determine the length of $AC.$

2006 Kazakhstan National Olympiad, 3

Tags: combinatorics

The racing tournament has $12$ stages and $ n $ participants. After each stage, all participants, depending on the occupied place $ k $, receive points $ a_k $ (the numbers $ a_k $ are natural and $ a_1> a_2> \dots> a_n $). For what is the smallest $ n $ the tournament organizer can choose the numbers $ a_1 $, $ \dots $, $ a_n $ so that after the penultimate stage for any possible distribution of places at least two participants had a chance to take first place.

2020 Serbia National Math Olympiad, 3

Tags: circumcircle , triangle , geometry , angle

We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.

2005 MOP Homework, 6

Tags: combinatorics

A computer network is formed by connecting $2004$ computers by cables. A set $S$ of these computers is said to be independent if no pair of computers of $S$ is connected by a cable. Suppose that the number of cables used is the minimum number possible such that the size of any independent set is at most $50$. Let $c(L)$ be the number of cables connected to computer $L$. Show that for any distinct computers $A$ and $B$, $c(A)=c(B)$ if they are connected by a cable and $|c(A)-c(B)| \le 1$ otherwise. Also, find the number of cables used in the network.

2013 Polish MO Finals, 4

Tags: 3d geometry , tetrahedron , rotation , trapezoid , geometry

Given is a tetrahedron $ABCD$ in which $AB=CD$ and the sum of measures of the angles $BAD$ and $BCD$ equals $180$ degrees. Prove that the measure of the angle $BAD$ is larger than the measure of the angle $ADC$.

2019 Romania National Olympiad, 2

Tags: group theory , abstract algebra , superior algebra

Let $n \geq 4$ be an even natural number and $G$ be a subgroup of $GL_2(\mathbb{C})$ with $|G| = n.$ Prove that there exists $H \leq G$ such that $\{ I_2 \} \neq H$ and $H \neq G$ such that $XYX^{-1} \in H, \: \forall X \in G$ and $\forall Y \in H$

2019 Paraguay Mathematical Olympiad, 1

Tags: algebra , trinomial , quadratic

Elías and Juanca solve the same problem by posing a quadratic equation. Elijah is wrong when writing the independent term and gets as results of the problem $-1$ and $-3$. Juanca is wrong only when writing the coefficient of the first degree term and gets as results of the problem $16$ and $-2$. What are the correct results of the problem?

2011 NIMO Problems, 2

Tags:

The sum of three consecutive integers is $15$. Determine their product.

2015 Iran Geometry Olympiad, 2

Tags: midpoint , reflection , circumcircle , geometry

In acute-angled triangle $ABC$, $BH$ is the altitude of the vertex $B$. The points $D$ and $E$ are midpoints of $AB$ and $AC$ respectively. Suppose that $F$ be the reflection of $H$ with respect to $ED$. Prove that the line $BF$ passes through circumcenter of $ABC$. by Davood Vakili

1963 Poland - Second Round, 6

Tags: geometry , 3d geometry

From the point $ S $ of space arise $ 3 $ half-lines: $ SA $, $ SB $ and $ SC $, none of which is perpendicular to both others. Through each of these rays, a plane is drawn perpendicular to the plane containing the other two rays. Prove that the drawn planes intersect along one line $ d $.

2014 India IMO Training Camp, 2

Tags: number theory

Find all positive integers $x$ and $y$ such that $x^{x+y}=y^{3x}$.

2009 AMC 8, 11

Tags:

The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $ \$1.43$. Some of the $ 30$ sixth graders each bought a pencil, and they paid a total of $ \$1.95$. How many more sixth graders than seventh graders bought a pencil? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

2005 Finnish National High School Mathematics Competition, 3

Tags: absolute value , algebra

Solve the group of equations: \[\begin{cases} (x + y)^3 = z \\ (y + z)^3 = x \\ (z + x)^3 = y \end{cases}\]

2023 CMIMC Integration Bee, 13

Tags: calculus , integration

\[\int_0^1 2^{\sqrt x}\log^2(2)+\log^2(1+x)\,\mathrm dx\] [i]Proposed by Thomas Lam[/i]

2010 AMC 8, 9

Tags: percent

Ryan got $80\%$ of the problems on a $25$-problem test, $90\%$ on a $40$-problem test, and $70\%$ on a $10$-problem test. What percent of all problems did Ryan answer correctly? $ \textbf{(A)}\ 64 \qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 80\qquad\textbf{(D)}\ 84\qquad\textbf{(E)}\ 86 $

2013 India National Olympiad, 3

Tags: calculus , integration , inequalities , function , algebra , polynomial

Let $a,b,c,d \in \mathbb{N}$ such that $a \ge b \ge c \ge d $. Show that the equation $x^4 - ax^3 - bx^2 - cx -d = 0$ has no integer solution.

2017 CMIMC Computer Science, 6

Tags: computer science

Define a self-balanced tree to be a tree such that for any node, the size of the left subtree is within 1 of the size of the right subtree. How many balanced trees are there of size 2046?

1985 Traian Lălescu, 2.1

Tags: equation , algebra , floor

Solve $ \quad 5\lfloor x^2\rfloor -2\lfloor x\rfloor +2=0. $

1992 National High School Mathematics League, 7

Tags: arithmetic sequence , geometric series

For real numbers $x,y,z$, $3x,4y,5z$ are geometric series, $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are arithmetic sequence. Then $\frac{x}{z}+\frac{z}{x}=$________.

1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 3

Tags:

A bowling contest consists of several series. Mary got 185 points in her previous series and thereby increased her average score per series from 176 to 177 points. How many points would Mary need in her next series to increase her average to 178? $ \text{(A)}\ 184 \qquad \text{(B)}\ 185 \qquad \text{(C)}\ 186 \qquad \text{(D)}\ 187 \qquad \text{(E)}\ 188$

2023 Regional Competition For Advanced Students, 2

Tags: geometry , concurrency , rhombus

Let $ABCD$ be a rhombus with $\angle BAD < 90^o$. The circle passing through $D$ with center $A$ intersects the line $CD$ a second time in point $E$. Let $S$ be the intersection of the lines $BE$ and $AC$. Prove that the points $A$, $S$, $D$ and $E$ lie on a circle. [i](Karl Czakler)[/i]

2021 China Team Selection Test, 3

Tags: number theory , additive number theory

Given positive integer $n$. Prove that for any integers $a_1,a_2,\cdots,a_n,$ at least $\lceil \tfrac{n(n-6)}{19} \rceil$ numbers from the set $\{ 1,2, \cdots, \tfrac{n(n-1)}{2} \}$ cannot be represented as $a_i-a_j (1 \le i, j \le n)$.