Olympiad problems

2012 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: inequalities , algebra

For which real numbers $x$ and $\alpha$ inequality holds: $$\log _2 {x}+\log _x {2}+2\cos{\alpha} \leq 0$$

2025 Al-Khwarizmi IJMO, 8

Tags: combinatorics

There are $100$ cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from $1$ to $100$. In a move, Madina is allowed to choose any $3$ cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine? [i]Shubin Yakov, Russia[/i]

2010 Purple Comet Problems, 4

Tags: geometry

The grid below contains five rows with six points in each row. Points that are adjacent either horizontally or vertically are a distance one apart. Find the area of the pentagon shown. [asy] size(150); defaultpen(linewidth(0.9)); for(int i=0;i<=5;++i){ for(int j=0;j<=4;++j){ dot((i,j)); } } draw((3,0)--(0,1)--(1,4)--(4,4)--(5,2)--cycle); [/asy]

1994 Canada National Olympiad, 5

Tags: angle , triangle , geometry , trigonometry

Let $ABC$ be an acute triangle. Let $AD$ be the altitude on $BC$, and let $H$ be any interior point on $AD$. Lines $BH,CH$, when extended, intersect $AC,AB$ at $E,F$ respectively. Prove that $\angle EDH=\angle FDH$.

2013 China Team Selection Test, 3

Tags: inequalities , algebra

Let $n>1$ be an integer and let $a_0,a_1,\ldots,a_n$ be non-negative real numbers. Definite $S_k=\sum_{i\equal{}0}^k \binom{k}{i}a_i$ for $k=0,1,\ldots,n$. Prove that\[\frac{1}{n} \sum_{k\equal{}0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k\equal{}0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.\]

2015 Online Math Open Problems, 24

Tags:

Let $ABC$ be an acute triangle with incenter $I$; ray $AI$ meets the circumcircle $\Omega$ of $ABC$ at $M \neq A$. Suppose $T$ lies on line $BC$ such that $\angle MIT=90^{\circ}$. Let $K$ be the foot of the altitude from $I$ to $\overline{TM}$. Given that $\sin B = \frac{55}{73}$ and $\sin C = \frac{77}{85}$, and $\frac{BK}{CK} = \frac mn$ in lowest terms, compute $m+n$. [i]Proposed by Evan Chen[/i]

1975 IMO Shortlist, 14

Tags: approximation , inequality , sequence , recurrence relation , integration , algebra

Let $x_0 = 5$ and $x_{n+1} = x_n + \frac{1}{x_n} \ (n = 0, 1, 2, \ldots )$. Prove that \[45 < x_{1000} < 45. 1.\]

2004 Postal Coaching, 13

Tags: function , algebra

Find all functions $f,g : \mathbb{R} \times \mathbb{R} \mapsto \mathbb{R}^{+}$ such that \[ ( \sum_{j=1}^{n}a_{j}b_{j})^2 \leq (\sum_{j=1}^{n} f({a_{j},b_{j}))(\sum_{j=1}^{n} g({a_{j},b_{j})) \leq (\sum_{j=1}^{n} (a_j)^2 )(\sum_{j=1}^{n} (b_j)^2 ) }}\] for any two sets $a_j$ and $b_j$ of real numbers.

2001 India National Olympiad, 6

Tags: function , algebra

Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(x +y) = f(x) f(y) f(xy)$ for all $x, y \in \mathbb{R}.$

2021 AMC 12/AHSME Spring, 19

Tags:

Two fair dice, each with at least 6 faces, are rolled. On each face of each die is printed a distinct integer from 1 to the number of faces on that die, inclusive. The probability of rolling a sum of 7 is $\frac{3}{4}$ of the probability of rolling a sum of 10 and the probability of rolling a sum of 12 is $\frac{1}{12}$. What is the least possible number of faces on the two dice combined? $\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 19 \qquad\textbf{(E)}\ 20$

2002 Estonia National Olympiad, 1

Tags: divisible , gcd , lcm , divide , number theory

The greatest common divisor $d$ and the least common multiple $u$ of positive integers $m$ and $n$ satisfy the equality $3m + n = 3u + d$. Prove that $m$ is divisible by $n$.

2013 Stanford Mathematics Tournament, 8

Tags:

Farmer John owns 2013 cows. Some cows are enemies of each other, and Farmer John wishes to divide them into as few groups as possible such that each cow has at most 3 enemies in her group. Each cow has at most 61 enemies. Compute the smallest integer $G$ such that, no matter which enemies they have, the cows can always be divided into at most $G$ such groups?

1986 AMC 8, 10

Tags:

A picture $ 3$ feet across is hung in the center of a wall that is $ 19$ feet wide. How many feet from the end of the wall is the nearest edge of the picture? \[ \textbf{(A)}\ 1 \frac{1}{2} \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 9 \frac{1}{2} \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 22 \]

1961 Putnam, B2

Tags: line , probability

Let $a$ and $b$ be given positive real numbers, with $a<b.$ If two points are selected at random from a straight line segment of length $b,$ what is the probability that the distance between them is at least $a?$

2016 LMT, 25

Tags:

Let $ABCD$ be a trapezoid with $AB\parallel DC$. Let $M$ be the midpoint of $CD$. If $AD\perp CD, AC\perp BM,$ and $BC\perp BD$, find $\frac{AB}{CD}$. [i]Proposed by Nathan Ramesh

2014 Regional Olympiad of Mexico Center Zone, 4

Tags: geometry , ratio , reflection , square

Let $ABCD$ be a square and let $M$ be the midpoint of $BC$. Let $C ^ \prime$ be the reflection of $C$ wrt to $DM$. The parallel to $AB$ passing through $C ^ \prime$ intersects $AD$ at $R$ and $BC$ at $S$. Show that $$\frac {RC ^ \prime} {C ^\prime S} = \frac {3} {2}$$

2004 Thailand Mathematical Olympiad, 6

Tags: remainder , polynomial , algebra

Let $f(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$. Find the remainder when $f(x^7)$ is divided by $f(x)$.

1971 IMO Longlists, 50

Tags: geometry , polyhedron , 3d geometry , volume , combinatorial geometry

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

2021 Kosovo National Mathematical Olympiad, 3

Tags: algebra , inequalities

Let $a,b$ and $c$ be positive real numbers such that $a^5+b^5+c^5=ab^2+bc^2+ca^2$. Prove the inequality: $$\frac{a^2+b^2}{b}+\frac{b^2+c^2}{c}+\frac{c^2+a^2}{a}\geq 2(ab+bc+ca).$$

2003 Regional Competition For Advanced Students, 3

Tags: geometry

Given are two parallel lines $ g$ and $ h$ and a point $ P$, that lies outside of the corridor bounded by $ g$ and $ h$. Construct three lines $ g_1$, $ g_2$ and $ g_3$ through the point $ P$. These lines intersect $ g$ in $ A_1,A_2, A_3$ and $ h$ in $ B_1, B_2, B_3$ respectively. Let $ C_1$ be the intersection of the lines $ A_1B_2$ and $ A_2B_1$, $ C_2$ be the intersection of the lines $ A_1B_3$ and $ A_3B_1$ and let $ C_3$ be the intersection of the lines $ A_2B_3$ and $ A_3B_2$. Show that there exists exactly one line $ n$, that contains the points $ C_1,C_2,C_3$ and that $ n$ is parallel to $ g$ and $ h$.

Kyiv City MO Juniors 2003+ geometry, 2004.9.7

Tags: construction , geometry

The board depicts the triangle $ABC$, the altitude $AH$ and the angle bisector $AL$ which intersectthe inscribed circle in the triangle at the points $M$ and $N, P$ and $Q$, respectively. After that, the figure was erased, leaving only the points $H, M$ and $Q$. Restore the triangle $ABC$. (Bogdan Rublev)

2022 Taiwan TST Round 3, 4

Tags: ming , domain , function , functional equation , algebra

Let $\mathcal{X}$ be the collection of all non-empty subsets (not necessarily finite) of the positive integer set $\mathbb{N}$. Determine all functions $f: \mathcal{X} \to \mathbb{R}^+$ satisfying the following properties: (i) For all $S$, $T \in \mathcal{X}$ with $S\subseteq T$, there holds $f(T) \le f(S)$. (ii) For all $S$, $T \in \mathcal{X}$, there hold \[f(S) + f(T) \le f(S + T),\quad f(S)f(T) = f(S\cdot T), \] where $S + T = \{s + t\mid s\in S, t\in T\}$ and $S \cdot T = \{s\cdot t\mid s\in S, t\in T\}$. [i]Proposed by Li4, Untro368, and Ming Hsiao.[/i]

2011 Dutch IMO TST, 4

Tags: algebra , integer polynomial , polynomial

Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.

2005 Germany Team Selection Test, 3

Tags: inradius , euler , incenter , circumcircle , geometry , vector , inequalities

Let $ABC$ be a triangle with orthocenter $H$, incenter $I$ and centroid $S$, and let $d$ be the diameter of the circumcircle of triangle $ABC$. Prove the inequality \[9\cdot HS^2+4\left(AH\cdot AI+BH\cdot BI+CH\cdot CI\right)\geq 3d^2,\] and determine when equality holds.

2016 PUMaC Geometry B, 6

Tags: geometry

Let $D, E$, and $F$ respectively be the feet of the altitudes from $A, B$, and $C$ of acute triangle $\vartriangle ABC$ such that $AF = 28, FB = 35$ and $BD = 45$. Let $P$ be the point on segment $BE$ such that $AP = 42$. Find the length of $CP$.