Olympiad problems

1973 IMO Longlists, 2

Let $OX, OY$ and $OZ$ be three rays in the space, and $G$ a point "[i]between these rays[/i]" (i. e. in the interior of the part of the space bordered by the angles $Y OZ, ZOX$ and $XOY$). Consider a plane passing through $G$ and meeting the rays $OX, OY$ and $OZ$ in the points $A, B, C$, respectively. There are infinitely many such planes; construct the one which minimizes the volume of the tetrahedron $OABC$.

2017 F = ma, 16

Tags: acceleration , usapho

A rod moves freely between the horizontal floor and the slanted wall. When the end in contact with the floor is moving at v, what is the speed of the end in contact with the wall? $\textbf{(A)} v\frac{\sin{\theta}}{\cos(\alpha-\theta)}$ $\textbf{(B)}v\frac{\sin(\alpha - \theta)}{\cos(\alpha + \theta)} $ $\textbf{(C)}v\frac{\cos(\alpha - \theta)}{\sin(\alpha + \theta)}$ $\textbf{(D)}v\frac{\cos(\theta)}{\cos(\alpha - \theta)}$ $\textbf{(E)}v\frac{\sin(\theta)}{\cos(\alpha + \theta)}$

2020 BMT Fall, Tie 1

Tags: angle , geometry

An [i]exterior [/i] angle is the supplementary angle to an interior angle in a polygon. What is the sum of the exterior angles of a triangle and dodecagon ($12$-gon), in degrees?

2008 Cuba MO, 2

Tags: geometry , parallelogram

Consider the parallelogram $ABCD$. A circle is drawn that passes through $A$ and intersects side $AD$ at $N$, side $AB$ at $M$ and diagonal $AC$ in $P$ such that points $A, M, N, P$ are different. Prove that $$AP\cdot AC = AM \cdot AB + AN \cdot AD.$$

India EGMO 2023 TST, 6

Tags: angle chasing , isosceles triangle , geometry

Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose $P,Q,R$ are points on segments $AC, AB, BC$ respectively such that $AP = QB$, $\angle PBC = 90^\circ - \angle BAC$ and $RP = RQ$. Let $O_1, O_2$ be the circumcenters of $\triangle APQ$ and $\triangle CRP$. Prove that $BR = O_1O_2$. [i]Proposed by Atul Shatavart Nadig[/i]

2021 Saudi Arabia BMO TST, 1

Tags: algebra , polynomial

Do there exist two polynomials $P$ and $Q$ with integer coefficient such that i) both $P$ and $Q$ have a coefficient with absolute value bigger than $2021$, ii) all coefficients of $P \cdot Q$ by absolute value are at most $1$.

1989 AMC 8, 2

Tags: number theory , least common multiple

$\frac{2}{10}+\frac{4}{100}+\frac{6}{1000} =$ $\text{(A)}\ .012 \qquad \text{(B)}\ .0246 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .246 \qquad \text{(E)}\ 246$

2011 China Girls Math Olympiad, 5

Tags: complex number , algebra

A real number $\alpha \geq 0$ is given. Find the smallest $\lambda = \lambda (\alpha ) > 0$, such that for any complex numbers ${z_1},{z_2}$ and $0 \leq x \leq 1$, if $\left| {{z_1}} \right| \leq \alpha \left| {{z_1} - {z_2}} \right|$, then $\left| {{z_1} - x{z_2}} \right| \leq \lambda \left| {{z_1} - {z_2}} \right|$.

2008 Bulgaria Team Selection Test, 3

Tags: floor function , ceiling function , induction , algebra , polynomial , combinatorics

Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.

2000 Korea Junior Math Olympiad, 6

Tags: inequalities , algebra

$x, y, z$ are positive reals which their product is not smaller then their sum. Prove the inequality: $$\sqrt{2x^2+yz}+\sqrt{2y^2+zx}+\sqrt{2z^2+xy} \geq 9$$

1949 Putnam, B6

Tags: tangent , geometry

Let $C$ be a closed convex curve with a continuously turning tangent and let $O$ be a point inside $C.$ For each point $P$ on $C$ we define $T(P)$ as follows: Draw the tangent to $C$ at $P$ and from $O$ drop the perpendicular to that tangent. Then $T(P)$ is the point at which $C$ intersects this perpendicular. Starting now with a point $P_{0}$ on $C$, define points $P_n$ by $P_n =T(P_{n-1}).$ Prove that the points $P_{n}$ approach a limit and characterize all possible limit points. (You may assume that $T$ is continuous.)

1994 USAMO, 3

Tags:

A convex hexagon $ABCDEF$ is inscribed in a circle such that $AB = CD = EF$ and diagonals $AD$, $BE$, and $CF$ are concurrent. Let $P$ be the intersection of $AD$ and $CE$. Prove that $CP/PE = (AC/CE)^2$.

2014 Contests, 3

Tags: function , induction , algebra

Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying: i) $f(1)=f(2)=1$; ii)$f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$. For each integer $m\ge 2$, find the value of $f(2^m)$.

2006 District Olympiad, 2

Tags: inequalities , trigonometry

In triangle $ABC$ we have $\angle ABC = 2 \angle ACB$. Prove that a) $AC^2 = AB^2 + AB \cdot BC$; b) $AB+BC < 2 \cdot AC$.

2010 Sharygin Geometry Olympiad, 21

Tags: geometric transformation , reflection , trigonometry , geometry

A given convex quadrilateral $ABCD$ is such that $\angle ABD + \angle ACD > \angle BAC + \angle BDC.$ Prove that \[S_{ABD}+S_{ACD} > S_{BAC}+S_{BDC}.\]

1987 AMC 8, 1

Tags:

$.4+.02+.006=$ $\text{(A)}\ .012 \qquad \text{(B)}\ .066 \qquad \text{(C)}\ .12 \qquad \text{(D)}\ .24 \qquad \text{(E)} .426$

2023 Durer Math Competition (First Round), 4

Tags: concyclic , geometry

Let $k$ be a circle with diameter $AB$ and centre $O$. Let C be an arbitrary point on the circle different from $A$ and $B$. Let $D$ be the point for which $O$, $B$, $D$ and $C$ (in this order) are the four vertices of a parallelogram. Let $E$ be the intersection of the line $BD$ and the circle $k$, and let $F$ be the orthocenter of the triangle $OAC$. Prove that the points $O, D, E, C, F$ lie on a circle.

1983 AMC 12/AHSME, 21

Tags: function , inequalities , algebra , difference of squares , special factorizations

Find the smallest positive number from the numbers below $\text{(A)} \ 10-3\sqrt{11} \qquad \text{(B)} \ 3\sqrt{11}-10 \qquad \text{(C)} \ 18-5\sqrt{13} \qquad \text{(D)} \ 51-10\sqrt{26} \qquad \text{(E)} \ 10\sqrt{26}-51$

2014 NIMO Problems, 5

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We have a five-digit positive integer $N$. We select every pair of digits of $N$ (and keep them in order) to obtain the $\tbinom52 = 10$ numbers $33$, $37$, $37$, $37$, $38$, $73$, $77$, $78$, $83$, $87$. Find $N$. [i]Proposed by Lewis Chen[/i]

2011 Tournament of Towns, 7

Tags: matrix , combinatorics

In every cell of a square table is a number. The sum of the largest two numbers in each row is $a$ and the sum of the largest two numbers in each column is b. Prove that $a = b$.

2014 ASDAN Math Tournament, 8

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George and two of his friends go to a famous jiaozi restaurant, which serves only two kinds of jiaozi: pork jiaozi, and vegetable jiaozi. Each person orders exactly $15$ jiaozi. How many different ways could the three of them order? Two ways of ordering are different if one person orders a different number of pork jiaozi in both orders.

May Olympiad L2 - geometry, 2005.3

Tags: geometry

In a triangle $ABC$ with $AB = AC$, let $M$ be the midpoint of $CB$ and let $D$ be a point in $BC$ such that $\angle BAD = \frac{\angle BAC}{6}$. The perpendicular line to $AD$ by $C$ intersects $AD$ in $N$ where $DN = DM$. Find the angles of the triangle $BAC$.

2010 Dutch IMO TST, 5

Tags: system of equations , algebra

Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying $3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.

IMSC 2024, 1

Tags: divisibility , number theory

For a positive integer $n$ denote by $P_0(n)$ the product of all non-zero digits of $n$. Let $N_0$ be the set of all positive integers $n$ such that $P_0(n)|n$. Find the largest possible value of $\ell$ such that $N_0$ contains infinitely many strings of $\ell$ consecutive integers. [i]Proposed by Navid Safaei, Iran[/i]

1982 Spain Mathematical Olympiad, 5

Tags: construction , square , geometry

Construct a square knowing the sum of the diagonal and the side.