Olympiad problems

2012-2013 SDML (Middle School), 1

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On planet Polyped, every creature has either $6$ legs or $10$ legs. In a room with $20$ creatures and $156$ legs, how many of the creatures have $6$ legs?

1998 VJIMC, Problem 4-M

A function $f:\mathbb R\to\mathbb R$ has the property that for every $x,y\in\mathbb R$ there exists a real number $t$ (depending on $x$ and $y$) such that $0<t<1$ and $$f(tx+(1-t)y)=tf(x)+(1-t)f(y).$$ Does it imply that $$f\left(\frac{x+y}2\right)=\frac{f(x)+f(y)}2$$ for every $x,y\in\mathbb R$?

1987 IMO Shortlist, 7

Tags: inequality system , algebra

Given five real numbers $u_0, u_1, u_2, u_3, u_4$, prove that it is always possible to find five real numbers $v0, v_1, v_2, v_3, v_4$ that satisfy the following conditions: $(i)$ $u_i-v_i \in \mathbb N, \quad 0 \leq i \leq 4$ $(ii)$ $\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.$ [i]Proposed by Netherlands.[/i]

2013 NIMO Problems, 2

Tags: ratio

At a certain school, the ratio of boys to girls is $1:3$. Suppose that: $\bullet$ Every boy has most $2013$ distinct girlfriends. $\bullet$ Every girl has at least $n$ boyfriends. $\bullet$ Friendship is mutual. Compute the largest possible value of $n$. [i]Proposed by Evan Chen[/i]

1979 IMO Longlists, 16

Tags: combinatorics

Let $Q$ be a square with side length $6$. Find the smallest integer $n$ such that in $Q$ there exists a set $S$ of $n$ points with the property that any square with side $1$ completely contained in $Q$ contains in its interior at least one point from $S$.

2006 District Olympiad, 4

Tags: function , real analysis

We say that a function $f: \mathbb R \to \mathbb R$ has the property $(P)$ if, for any real numbers $x$, \[ \sup_{t\leq x} f(x) = x. \] a) Give an example of a function with property $(P)$ which has a discontinuity in every real point. b) Prove that if $f$ is continuous and satisfies $(P)$ then $f(x) = x$, for all $x\in \mathbb R$.

2007-2008 SDML (Middle School), 1

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Find $x$ if $\frac{1+\frac{3}{x}}{2-\frac{2}{x}}=7$.

1980 Vietnam National Olympiad, 3

Tags: geometry , analytic geometry

Let $P$ be a point inside a triangle $A_1A_2A_3$. For $i = 1, 2, 3$, line $PA_i$ intersects the side opposite to $A_i$ at $B_i$. Let $C_i$ and $D_i$ be the midpoints of $A_iB_i$ and $PB_i$, respectively. Prove that the areas of the triangles $C_1C_2C_3$ and $D_1D_2D_3$ are equal.

2024 All-Russian Olympiad, 2

Tags: symmetry , algebra

Call a triple $(a,b,c)$ of positive numbers [i]mysterious [/i]if \[\sqrt{a^2+\frac{1}{a^2c^2}+2ab}+\sqrt{b^2+\frac{1}{b^2a^2}+2bc}+\sqrt{c^2+\frac{1}{c^2b^2}+2ca}=2(a+b+c).\] Prove that if the triple $(a,b,c)$ is mysterious, then so is the triple $(c,b,a)$. [i]Proposed by A. Kuznetsov, K. Sukhov[/i]

LMT Team Rounds 2010-20, A16

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Two circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively, and intersect at points $M$ and $N$. The radii of $\omega_1$ and $\omega_2$ are $12$ and $15$, respectively, and $O_1O_2 = 18$. A point $X$ is chosen on segment $MN$. Line $O_1X$ intersects $\omega_2$ at points $A$ and $C$, where $A$ is inside $\omega_1$. Similarly, line $O_2X$ intersects $\omega_1$ at points $B$ and $D$, where $B$ is inside $\omega_2$. The perpendicular bisectors of segments $AB$ and $CD$ intersect at point $P$. Given that $PO_1 = 30$, find $PO_2^2$. [i]Proposed by Andrew Zhao[/i]

2016 Kosovo National Mathematical Olympiad, 5

Tags: geometry

In angle $\angle AOB=60^{\circ}$ are two circle which circumscribed and tangjent to each other . If we write with $r$ and $R$ the radius of smaller and bigger circle respectively and if $r=1$ find $R$ .

1997 Romania National Olympiad, 2

Tags: calculus , integration , function , real analysis

Prove that: $\int_{-1}^1f^2(x)dx\ge \frac 1 2 (\int_{-1}^1f(x)dx)^2 +\frac 3 2(\int_{-1}^1xf(x)dx)^2$ Please give a proof without using even and odd functions. (the oficial proof uses those and seems to be un-natural) :D

2009 All-Russian Olympiad Regional Round, 10.3

Tags: combinatorics , weight

Kostya had two sets of $17$ coins: in one set all the coins were real, and in the other set there were exactly $5$ fakes (all the coins look the same; all real coins weigh the same, all fake coins also weigh the same, but it is unknown lighter or heavier than real ones). Kostya gave away one of the sets friend, and subsequently forgot which of the two sets had stayed. With the help of two weighings, can Kostya on a cup scale without weights, find out which of the two did he give away the sets?

2004 AMC 10, 2

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How many two-digit positive integers have at least one $ 7$ as a digit? $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 18\qquad \textbf{(C)}\ 19\qquad \textbf{(D)}\ 20\qquad \textbf{(E)}\ 30$

2002 District Olympiad, 2

Tags: algebra , system of equations

Solve in $ \mathbb{C}^3 $ the following chain of equalities: $$ x(x-y)(x-z)=y(y-x)(y-z)=z(z-x)(z-y)=3. $$

2023 239 Open Mathematical Olympiad, 8

Tags: inequalities , derivative , algebra

Let $r\geqslant 0$ be a real number and define $f(x)=1/(1+x^2)^r$. Prove that \[|f^{(k)}(x)|\leqslant\frac{2r\cdot(2r+1)\cdots(2r+k-1)}{(1+x^2)^{r+k/2}},\]for every natural number $k{}$. Here, $f^{(k)}(x)$ denotes the $k^{\text{th}}$ derivative of $f$.

2003 Cuba MO, 6

Tags: incenter , rectangle , geometry

Let $P_1, P_2, P_3, P_4$ be four points on a circle, let $I_1$ be incenter of the triangle of vertices $P_2P_3P_4$, $I_2$ the incenter of the triangle $P_1P_3P_4$, $I_3$ the incenter of the triangle $P_1P_2P_4$, $I_4$ the incenter of the triangle $P_2P_3P_1$. Prove that $I_1I_2I_3I_4$ is a rectangle.

2010 Princeton University Math Competition, 5

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$3n$ people take part in a chess tournament: $n$ girls and $2n$ boys. Each participant plays with each of the others exactly once. There were no ties and the number of games won by the girls is $\displaystyle\frac75$ the number of games won by the boys. How many people took part in the tournament?

2021 Israel TST, 1

Tags: game , combinatorics

An ordered quadruple of numbers is called [i]ten-esque[/i] if it is composed of 4 nonnegative integers whose sum is equal to $10$. Ana chooses a ten-esque quadruple $(a_1, a_2, a_3, a_4)$ and Banana tries to guess it. At each stage Banana offers a ten-esque quadtruple $(x_1,x_2,x_3,x_4)$ and Ana tells her the value of \[|a_1-x_1|+|a_2-x_2|+|a_3-x_3|+|a_4-x_4|\] How many guesses are needed for Banana to figure out the quadruple Ana chose?

1996 Nordic, 4

Tags: algebra , functional equation in n

The real-valued function $f$ is defined for positive integers, and the positive integer $a$ satisfies $f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1}$ for all positive integers $n$. (i) Show that $f(n+ 4a) = f(n)$ for all positive integers $n$. (ii) Determine the smallest possible $a$.

1998 Miklós Schweitzer, 1

Tags: set theory , cardinality

Can there be a continuum set of continuum sets such that (i) the intersection of any two is finite, and (ii) every set that intersects all sets intersects any in an infinite set? note: a continuum set is a set that can be put into a 1-to-1 bijection with the reals.

2012 Pan African, 1

Tags: geometry

$AB$ is a chord (not a diameter) of a circle with centre $O$. Let $T$ be a point on segment $OB$. The line through $T$ perpendicular to $OB$ meets $AB$ at $C$ and the circle at $D$ and $E$. Denote by $S$ the orthogonal projection of $T$ onto $AB$ . Prove that $AS \cdot BC = TE \cdot TD$.

2011 Turkey MO (2nd round), 2

Tags: circumcircle , trigonometry , geometric transformation , reflection , perpendicular bisector , cyclic quadrilateral , geometry

Let $ABC$ be a triangle $D\in[BC]$ (different than $A$ and $B$).$E$ is the midpoint of $[CD]$. $F\in[AC]$ such that $\widehat{FEC}=90$ and $|AF|.|BC|=|AC|.|EC|.$ Circumcircle of $ADC$ intersect $[AB]$ at $G$ different than $A$.Prove that tangent to circumcircle of $AGF$ at $F$ is touch circumcircle of $BGE$ too.

2010 CHMMC Fall, 10

Tags: algebra

The $100$th degree polynomial $P(x)$ satisfies $P(2^k) = k$ for $k = 0, 1, . . . 100$. Let $a$ denote the leading coefficient of $P(x)$. Find the unique integer $M$ such that $2^M < |a| < 2^{M+1}$. .

2021 AMC 12/AHSME Spring, 11

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Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$ $\textbf{(A) }\frac{42}5 \qquad \textbf{(B) }6\sqrt2 \qquad \textbf{(C) }\frac{84}5\qquad \textbf{(D) }12\sqrt2 \qquad \textbf{(E) }18$