Olympiad problems

1949-56 Chisinau City MO, 51

Determine graphically the number of roots of the equation $\sin x = \lg x$.

2019 Jozsef Wildt International Math Competition, W. 38

Tags: inequalities , triangle

Let $a$, $b$, $c$ be the sides of an acute triangle $\triangle ABC$ , then for any $x, y, z \geq 0$, such that $xy+yz+zx=1$ holds inequality:$$a^2x + b^2y + c^2z \geq 4F$$ where $F$ is the area of the triangle $\triangle ABC$

2012 AMC 12/AHSME, 5

Tags:

A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of $280$ pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad? $ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 96 $

1997 Austrian-Polish Competition, 6

Tags: function , induction , algebra

Show that there is no integer-valued function on the integers such that $f(m+f(n))=f(m)-n$ for all $m,n$.

2003 AMC 8, 12

Tags: probability , modular arithmetic

When a fair six-sided dice is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the $5$ faces than can be seen is divisible by $6$? $\textbf{(A)}\ 1/3 \qquad \textbf{(B)}\ 1/2 \qquad \textbf{(C)}\ 2/3 \qquad \textbf{(D)}\ 5/6 \qquad \textbf{(E)}\ 1$

TNO 2008 Junior, 6

Tags: algebra

(a) Given $2 + 4 + 6 + \dots + p = 6480$, find $p$. (b) Given $7 + 11 + 15 + \dots + q = 5250$, find $q$. (c) Given $2^2 + 4^2 + 6^2 + \dots + r^2 - 1^2 - 3^2 - 5^2 - \dots - (r-1)^2 = 2485$, compute $r$.

2002 Portugal MO, 3

Tags: combinatorial geometry , geometry , coloring , combinatorics

Daniel painted a rectangular painting measuring $2$ meters by $4$ meters with four colors. Knowing that he used more than two colors to paint the four corners of the painting, prove that he painted of the same color two points that are at least $\sqrt5$ meters

2018 Azerbaijan JBMO TST, 1

Tags: algebra , inequality

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

2016 Czech-Polish-Slovak Junior Match, 1

Tags: midpoint , angle chasing , angle , geometry

Let $ABC$ be a right-angled triangle with hypotenuse $AB$. Denote by $D$ the foot of the altitude from $C$. Let $Q, R$, and $P$ be the midpoints of the segments $AD, BD$, and $CD$, respectively. Prove that $\angle AP B + \angle QCR = 180^o$. Czech Republic

1999 Junior Balkan Team Selection Tests - Moldova, 3

Tags: number theory

On the board is written a number with nine non-zero and distinct digits. Prove that we can delete at most seven digits so that the number formed by the digits left to be a perfect square.

1995 Tournament Of Towns, (478) 2

Tags: irrational number , number theory , algebra

Let $p$ be the product of $n$ real numbers $x_1$, $x_2$,$...$, $x_n$. Prove that if $p - x_k$ is an odd integer for $k = 1, 2,..., n$, then each of the numbers $x_1$, $x_2$,$...$, $x_n$is irrational. (G Galperin)

1991 IMO Shortlist, 23

Tags: function , algebra , polynomial , functional equation

Let $ f$ and $ g$ be two integer-valued functions defined on the set of all integers such that [i](a)[/i] $ f(m \plus{} f(f(n))) \equal{} \minus{}f(f(m\plus{} 1) \minus{} n$ for all integers $ m$ and $ n;$ [i](b)[/i] $ g$ is a polynomial function with integer coefficients and g(n) = $ g(f(n))$ $ \forall n \in \mathbb{Z}.$

2022 IFYM, Sozopol, 2

Tags: algebra

We say that a rectangle and a triangle are [i]similar[/i], if they have the same area and the same perimeter. Let $P$ be a rectangle for which the ratio of the longer to the shorter side is at least $\lambda -1+\sqrt{\lambda (\lambda -2)}$ where $\lambda =\frac{3\sqrt{3}}{2}$. Prove that there exists a tringle that is [i]similar[/i] to $P$.

2006 Germany Team Selection Test, 3

Tags: induction , inequalities

Let $n$ be a positive integer, and let $b_{1}$, $b_{2}$, ..., $b_{n}$ be $n$ positive reals. Set $a_{1}=\frac{b_{1}}{b_{1}+b_{2}+...+b_{n}}$ and $a_{k}=\frac{b_{1}+b_{2}+...+b_{k}}{b_{1}+b_{2}+...+b_{k-1}}$ for every $k>1$. Prove the inequality $a_{1}+a_{2}+...+a_{n}\leq\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}}$.

2024 Tuymaada Olympiad, 1

Tags: number theory , nt construction

Prove that a positive integer of the form $n^4 +1$ can have more than $1000$ divisors of the form $a^4 +1$ with integral $a$.

2013 Online Math Open Problems, 25

Tags: geometry , perimeter , function , calculus , inequalities , geometric transformation

Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$. [i]Proposed by Evan Chen[/i]

2000 Romania Team Selection Test, 2

Tags: geometry , circumcircle , perimeter , geometric inequality , triangle

Let ABC be a triangle and $M$ be an interior point. Prove that \[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]

2020 MIG, 22

Tags:

Jane's uncle gives her a "$4$-balance." The $4$-balance acts like a normal balance scale, but it compares four masses instead of two, tilting towards the weight that is heaviest (if all four are equal, it stays balanced). He then gives her $25$ coins, one of which is a counterfeit heavier than the rest. What is the minimum number of uses of the $4$-balance needed to ensure she identifies the counterfeit? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$

1994 AIME Problems, 8

Tags: rotation , geometry , geometric transformation , linear algebra , matrix

The points $(0,0),$ $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.

2022 Kyiv City MO Round 1, Problem 3

Tags: geometry

In triangle $ABC$ $\angle B > 90^\circ$. Tangents to this circle in points $A$ and $B$ meet at point $P$, and the line passing through $B$ perpendicular to $BC$ meets the line $AC$ at point $K$. Prove that $PA = PK$. [i](Proposed by Danylo Khilko)[/i]

1949-56 Chisinau City MO, 2

Tags: number theory , last digit

What is the last digit of $777^{777}$?

2011 Kosovo National Mathematical Olympiad, 3

Tags: function , trigonometry , calculus , algebra , system of equations , inequalities

Find maximal value of the function $f(x)=8-3\sin^2 (3x)+6 \sin (6x)$

2016 Iran MO (2nd Round), 1

Tags: inequalities , algebra

If $0<a\leq b\leq c$ prove that $$\frac{(c-a)^2}{6c}\leq \frac{a+b+c}{3}-\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$$

2017 International Zhautykov Olympiad, 3

Tags: combinatorics , tiling , geometry , rectangle

Rectangle on a checked paper with length of a unit square side being $1$ Is divided into domino figures( two unit square sharing a common edge). Prove that you colour all corners of squares on the edge of rectangle and inside rectangle with $3$ colours such that for any two corners with distance $1$ the following conditions hold: they are coloured in different colour if the line connecting the two corners is on the border of two domino figures and coloured in same colour if the line connecting the two corners is inside a domino figure.

2015 Canadian Mathematical Olympiad Qualification, 4

Tags: geometry , cyclic quadrilateral

Given an acute-angled triangle $ABC$ whose altitudes from $B$ and $C$ intersect at $H$, let $P$ be any point on side $BC$ and $X, Y$ be points on $AB, AC$, respectively, such that $PB = PX$ and $PC = PY$. Prove that the points $A, H, X, Y$ lie on a common circle.