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Line $ l_2$ intersects line $ l_1$ and line $ l_3$ is parallel to $ l_1$. The three lines are distinct and lie in a plane. Determine the number of points equidistant from all three lines. | 2 | 0.333333 |
What is the probability of having 2 adjacent white balls or 2 adjacent blue balls in a random arrangement of 3 red, 2 white and 2 blue balls? | \frac{10}{21} | 0.25 |
Convex pentagon $ABCDE$ has side lengths $AB=5$ , $BC=CD=DE=6$ , and $EA=7$ . Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$ . | 60 | 0.333333 |
What is the minimum distance between $(2019, 470)$ and $(21a - 19b, 19b + 21a)$ for $a, b \in Z$ ? | \sqrt{101} | 0.166667 |
There are $2024$ cities in a country, every two of which are bidirectionally connected by exactly one of three modes of transportation - rail, air, or road. A tourist has arrived in this country and has the entire transportation scheme. He chooses a travel ticket for one of the modes of transportation and the city from which he starts his trip. He wants to visit as many cities as possible, but using only the ticket for the specified type of transportation. What is the largest $k$ for which the tourist will always be able to visit at least $k$ cities? During the route, he can return to the cities he has already visited.
*Proposed by Bogdan Rublov* | 1012 | 0.166667 |
1. The numbers $1,2, \ldots$ are placed in a triangle as following:
\[
\begin{matrix}
1 & & &
2 & 3 & &
4 & 5 & 6 &
7 & 8 & 9 & 10
\ldots
\end{matrix}
\]
What is the sum of the numbers on the $n$ -th row? | \frac{n(n^2 + 1)}{2} | 0.833333 |
The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation.
Find the smallest possible number of independent deputies after the vacation. | 50 | 0.083333 |
$ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\vartriangle ACE$ and $\vartriangle BDF$ . What is the area of $R$ divided by the area of $ABCDEF$ ? | \frac{1}{3} | 0.083333 |
Consider the recurrence relation $$ a_{n+3}=\frac{a_{n+2}a_{n+1}-2}{a_n} $$ with initial condition $(a_0,a_1,a_2)=(1,2,5)$ . Let $b_n=a_{2n}$ for nonnegative integral $n$ . It turns out that $b_{n+2}+xb_{n+1}+yb_n=0$ for some pair of real numbers $(x,y)$ . Compute $(x,y)$ . | (-4, 1) | 0.583333 |
Let $\mathcal{S}$ be the set $\{1,2,3,\ldots,10\}.$ Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}.$ (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when $n$ is divided by $1000.$ | 501 | 0.083333 |
In right triangle ABC with legs 5 and 12, arcs of circles are drawn, one with center A and radius 12, the other with center B and radius 5. They intersect the hypotenuse at M and N. Find the length of MN. | 4 | 0.083333 |
Magicman and his helper want to do some magic trick. They have special card desk. Back of all cards is common color and face is one of $2017$ colors.
Magic trick: magicman go away from scene. Then viewers should put on the table $n>1$ cards in the row face up. Helper looks at these cards, then he turn all cards face down, except one, without changing order in row. Then magicman returns on the scene, looks at cards, then show on the one card, that lays face down and names it face color.
What is minimal $n$ such that magicman and his helper can has strategy to make magic trick successfully? | n = 2018 | 0.25 |
What is the smallest positive integer $n$ such that $2013^n$ ends in $001$ (i.e. the rightmost three digits of $2013^n$ are $001$ ? | 100 | 0.833333 |
Find the greatest value of $M$ $\in \mathbb{R}$ such that the following inequality is true $\forall$ $x, y, z$ $\in \mathbb{R}$ $x^4+y^4+z^4+xyz(x+y+z)\geq M(xy+yz+zx)^2$ . | \frac{2}{3} | 0.916667 |
$\triangle ABC$ has side lengths $AB=20$ , $BC=15$ , and $CA=7$ . Let the altitudes of $\triangle ABC$ be $AD$ , $BE$ , and $CF$ . What is the distance between the orthocenter (intersection of the altitudes) of $\triangle ABC$ and the incenter of $\triangle DEF$ ? | 15 | 0.083333 |
4. Suppose that $\overline{A2021B}$ is a six-digit integer divisible by $9$ . Find the maximum possible value of $A \cdot B$ .
5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments?
6. Suppose that $a$ and $b$ are positive integers such that $\frac{a}{b-20}$ and $\frac{b+21}{a}$ are positive integers. Find the maximum possible value of $a + b$ . | 143 | 0.083333 |
Peter is chasing after Rob. Rob is running on the line $y=2x+5$ at a speed of $2$ units a second, starting at the point $(0,5)$ . Peter starts running $t$ seconds after Rob, running at $3$ units a second. Peter also starts at $(0,5)$ and catches up to Rob at the point $(17,39)$ . What is the value of t? | \frac{\sqrt{1445}}{6} | 0.083333 |
$K, L, M$ and $N$ are points on sides $AB, BC, CD$ and $DA$ , respectively, of the unit square $ABCD$ such that $KM$ is parallel to $BC$ and $LN$ is parallel to $AB$ . The perimeter of triangle $KLB$ is equal to $1$ . What is the area of triangle $MND$ ? | \frac{1}{4} | 0.333333 |
Let $n$ be a positive integer. Positive numbers $a$ , $b$ , $c$ satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$ . Find the greatest possible value of $$ E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}} $$ | \frac{1}{3^{n+1}} | 0.416667 |
Four disks with disjoint interiors are mutually tangent. Three of them are equal in size and the fourth one is smaller. Find the ratio of the radius of the smaller disk to one of the larger disks. | \frac{2\sqrt{3} - 3}{3} | 0.25 |
$162$ pluses and $144$ minuses are placed in a $30\times 30$ table in such a way that each row and each column contains at most $17$ signs. (No cell contains more than one sign.) For every plus we count the number of minuses in its row and for every minus we count the number of pluses in its column. Find the maximum of the sum of these numbers. | 2592 | 0.083333 |
Let $M$ be a finite subset of the plane such that for any two different points $A,B\in M$ there is a point $C\in M$ such that $ABC$ is equilateral. What is the maximal number of points in $M?$ | 3 | 0.833333 |
Let Akbar and Birbal together have $n$ marbles, where $n > 0$ .
Akbar says to Birbal, “ If I give you some marbles then you will have twice as many marbles as I will have.”
Birbal says to Akbar, “ If I give you some marbles then you will have thrice as many marbles as I will have.”
What is the minimum possible value of $n$ for which the above statements are true? | 12 | 0.833333 |
In parallelogram $ABCD$ , the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$ . If the area of $XYZW$ is $10$ , find the area of $ABCD$ | 40 | 0.25 |
How many of the numbers $2,6,12,20,\ldots,14520$ are divisible by 120? | 8 | 0.083333 |
The fraction $\tfrac1{2015}$ has a unique "(restricted) partial fraction decomposition'' of the form \[\dfrac1{2015}=\dfrac a5+\dfrac b{13}+\dfrac c{31},\] where $a$ , $b$ , and $c$ are integers with $0\leq a<5$ and $0\leq b<13$ . Find $a+b$ . | 14 | 0.083333 |
Consider all possible integers $n \ge 0$ such that $(5 \cdot 3^m) + 4 = n^2$ holds for some corresponding integer $m \ge 0$ . Find the sum of all such $n$ . | 10 | 0.583333 |
In triangle $ABC$ , let $J$ be the center of the excircle tangent to side $BC$ at $A_{1}$ and to the extensions of the sides $AC$ and $AB$ at $B_{1}$ and $C_{1}$ respectively. Suppose that the lines $A_{1}B_{1}$ and $AB$ are perpendicular and intersect at $D$ . Let $E$ be the foot of the perpendicular from $C_{1}$ to line $DJ$ . Determine the angles $\angle{BEA_{1}}$ and $\angle{AEB_{1}}$ .
*Proposed by Dimitris Kontogiannis, Greece* | \angle BEA_1 = \angle AEB_1 = 90^\circ | 0.75 |
$ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$ , different from $C$ . What is the length of the segment $IF$ ? | 10 | 0.5 |
Suppose that $a_0=1$ and that $a_{n+1}=a_n+e^{-a_n}$ for $n=0,1,2,\dots.$ Does $a_n-\log n$ have a finite limit as $n\to\infty?$ (Here $\log n=\log_en=\ln n.$ ) | \lim_{n \to \infty} (a_n - \log n) = 0 | 0.083333 |
Given a square with sides of length $\frac{2}{\pi}$, find the perimeter of the region bounded by the semicircular arcs constructed on the sides of the square. | 4 | 0.5 |
Let $n \ge 3$ be an integer. Find the minimal value of the real number $k_n$ such that for all positive numbers $x_1, x_2, ..., x_n$ with product $1$ , we have $$ \frac{1}{\sqrt{1 + k_nx_1}}+\frac{1}{\sqrt{1 + k_nx_2}}+ ... + \frac{1}{\sqrt{1 + k_nx_n}} \le n - 1. $$ | \frac{2n-1}{(n-1)^2} | 0.666667 |
In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D$, $E$, and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$? | 56 | 0.583333 |
Let $a_n$ be a sequence de fined by some $a_0$ and the recursion $a_{n+1} = a_n + 2 \cdot 3^n$ for $n \ge 0$ .
Determine all rational values of $a_0$ such that $a^j_k / a^k_j$ is an integer for all integers $j$ and $k$ with $0 < j < k$ . | a_0 = 1 | 0.25 |
Let $O$ be the center of the circle $\omega$ circumscribed around the acute-angled triangle $\vartriangle ABC$ , and $W$ be the midpoint of the arc $BC$ of the circle $\omega$ , which does not contain the point $A$ , and $H$ be the point of intersection of the heights of the triangle $\vartriangle ABC$ . Find the angle $\angle BAC$ , if $WO = WH$ .
(O. Clurman) | \angle BAC = 60^\circ | 0.916667 |
Given a positive integer $n,$ let $s(n)$ denote the sum of the digits of $n.$ Compute the largest positive integer $n$ such that $n = s(n)^2 + 2s(n) - 2.$ | 397 | 0.166667 |
Find all pairs of real number $x$ and $y$ which simultaneously satisfy the following 2 relations: $x+y+4=\frac{12x+11y}{x^2+y^2}$ $y-x+3=\frac{11x-12y}{x^2+y^2}$ | (2, 1) | 0.083333 |
Define $ a \circledast b = a + b-2ab $ . Calculate the value of $$ A=\left( ...\left(\left(\frac{1}{2014}\circledast \frac{2}{2014}\right)\circledast\frac{3}{2014}\right)...\right)\circledast\frac{2013}{2014} $$ | \frac{1}{2} | 0.166667 |
The polynomial $ x^3\minus{}ax^2\plus{}bx\minus{}2010$ has three positive integer zeros. Determine the smallest possible value of $ a$. | 78 | 0.916667 |
If $\frac{1}{\sqrt{2011+\sqrt{2011^2-1}}}=\sqrt{m}-\sqrt{n}$ , where $m$ and $n$ are positive integers, what is the value of $m+n$ ? | 2011 | 0.25 |
A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$ . | 324 | 0.083333 |
If facilities for division are not available, it is sometimes convenient in determining the decimal expansion of $1/a$ , $a>0$ , to use the iteration $$ x_{k+1}=x_k(2-ax_k), \quad \quad k=0,1,2,\dots , $$ where $x_0$ is a selected “starting” value. Find the limitations, if any, on the starting values $x_0$ , in order that the above iteration converges to the desired value $1/a$ . | 0 < x_0 < \frac{2}{a} | 0.333333 |
Given that $\sin x \cos x \leq C(\sin^6x+\cos^6x)$ for every real number $x$, determine the least real number $C$. | 2 | 0.5 |
Find the area of quadrilateral $ABCD$ if: two opposite angles are right;two sides which form right angle are of equal length and sum of lengths of other two sides is $10$ | 25 | 0.75 |
Given an integer $n \geq 2$ determine the integral part of the number $ \sum_{k=1}^{n-1} \frac {1} {({1+\frac{1} {n}}) \dots ({1+\frac {k} {n})}}$ - $\sum_{k=1}^{n-1} (1-\frac {1} {n}) \dots(1-\frac{k}{n})$ | 0 | 0.666667 |
For a natural number $k>1$ , define $S_k$ to be the set of all triplets $(n,a,b)$ of natural numbers, with $n$ odd and $\gcd (a,b)=1$ , such that $a+b=k$ and $n$ divides $a^n+b^n$ . Find all values of $k$ for which $S_k$ is finite. | k = 2^m | 0.166667 |
Consider decompositions of an $8\times 8$ chessboard into $p$ non-overlapping rectangles subject to the following conditions:
(i) Each rectangle has as many white squares as black squares.
(ii) If $a_i$ is the number of white squares in the $i$ -th rectangle, then $a_1<a_2<\ldots <a_p$ .
Find the maximum value of $p$ for which such a decomposition is possible. For this value of $p$ , determine all possible sequences $a_1,a_2,\ldots ,a_p$ . | p = 7 | 0.083333 |
$2020$ positive integers are written in one line. Each of them starting with the third is divisible by previous and by the sum of two previous numbers. What is the smallest value the last number can take?
A. Gribalko | 2019! | 0.083333 |
Solve in the interval $ (2,\infty ) $ the following equation: $$ 1=\cos\left( \pi\log_3 (x+6)\right)\cdot\cos\left( \pi\log_3 (x-2)\right) . $$ | x = 3 | 0.333333 |
If $a!+\left(a+2\right)!$ divides $\left(a+4\right)!$ for some nonnegative integer $a$ , what are all possible values of $a$ ?
*2019 CCA Math Bonanza Individual Round #8* | 0, 3 | 0.833333 |
The quadratic polynomial $f(x)$ has the expansion $2x^2 - 3x + r$ . What is the largest real value of $r$ for which the ranges of the functions $f(x)$ and $f(f(x))$ are the same set? | \frac{15}{8} | 0.333333 |
Given that the inequality $1+\sqrt{n^2-9n+20} > \sqrt{n^2-7n+12}$ holds true, determine the number of positive integer $n$ satisfying this inequality. | 4 | 0.083333 |
Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \end{eqnarray*} | x_1 = x_2 = x_3 = x_4 = x_5 | 0.333333 |
$N\geq9$ distinct real numbers are written on a blackboard. All these numbers are nonnegative, and all are less than $1$ . It happens that for very $8$ distinct numbers on the board, the board contains the ninth number distinct from eight such that the sum of all these nine numbers is integer. Find all values $N$ for which this is possible. *(F. Nilov)* | N = 9 | 0.75 |
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy conditions: $f(x) + f(y) \ge xy$ for all real $x, y$ and
for each real $x$ exists real $y$ , such that $f(x) + f(y) = xy$ . | f(x) = \frac{x^2}{2} | 0.666667 |
Find all positive integers $a$ , $b$ , $n$ and prime numbers $p$ that satisfy
\[ a^{2013} + b^{2013} = p^n\text{.}\]
*Proposed by Matija Bucić.* | (a, b, n, p) = (2^k, 2^k, 2013k + 1, 2) | 0.25 |
Let $x$ and $y$ be two real numbers such that $2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7$ . Find $\tan^2 x + 2 \tan^2 y$ . | 9 | 0.166667 |
Define a function $f:\mathbb{N}\rightarrow\mathbb{N}$ , \[f(1)=p+1,\] \[f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,\] where $p$ is a prime number. Find all $p$ such that there exists a natural number $k$ such that $f(k)$ is a perfect square. | p = 3 | 0.833333 |
Alan, Jason, and Shervin are playing a game with MafsCounts questions. They each start with $2$ tokens. In each round, they are given the same MafsCounts question. The first person to solve the MafsCounts question wins the round and steals one token from each of the other players in the game. They all have the same probability of winning any given round. If a player runs out of tokens, they are removed from the game. The last player remaining wins the game.
If Alan wins the first round but does not win the second round, what is the probability that he wins the game?
*2020 CCA Math Bonanza Individual Round #4* | \frac{1}{2} | 0.416667 |
Let $f(x) = x^2 + 6x + c$ for all real number s $x$ , where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly $3$ distinct real roots? | c = \frac{11 - \sqrt{13}}{2} | 0.083333 |
On Binary Island, residents communicate using special paper. Each piece of paper is a $1 \times n$ row of initially uncolored squares. To send a message, each square on the paper must either be colored either red or green. Unfortunately the paper on the island has become damaged, and each sheet of paper has $10$ random consecutive squares each of which is randomly colored red or green.
Malmer and Weven would like to develop a scheme that allows them to send messages of length $2016$ between one another. They would like to be able to send any message of length $2016$ , and they want their scheme to work with perfect accuracy. What is the smallest value of $n$ for which they can develop such a strategy?
*Note that when sending a message, one can see which 10 squares are colored and what colors they are. One also knows on which square the message begins, and on which square the message ends.* | 2026 | 0.166667 |
Determine all positive integers $a,b,c$ such that $ab + ac + bc$ is a prime number and $$ \frac{a+b}{a+c}=\frac{b+c}{b+a}. $$ | (1, 1, 1) | 0.75 |
A permutation of a finite set $S$ is a one-to-one function from $S$ to $S$ . A permutation $P$ of the set $\{ 1, 2, 3, 4, 5 \}$ is called a W-permutation if $P(1) > P(2) < P(3) > P(4) < P(5)$ . A permutation of the set $\{1, 2, 3, 4, 5 \}$ is selected at random. Compute the probability that it is a W-permutation. | \frac{2}{15} | 0.083333 |
A $2.00 \text{L}$ balloon at $20.0^{\circ} \text{C}$ and $745 \text{mmHg}$ floats to an altitude where the temperature is $10.0^{\circ} \text{C}$ and the air pressure is $700 \text{mmHg}$ . Calculate the new volume of the balloon. | 2.06 \text{ L} | 0.25 |
<span style="color:darkred">Find all differentiable functions $f\colon [0,\infty)\to [0,\infty)$ for which $f(0)=0$ and $f^{\prime}(x^2)=f(x)$ for any $x\in [0,\infty)$ .</span> | f(x) = 0 | 0.75 |
The positive integer $ n$ has the following property:
if the three last digits of $n$ are removed, the number $\sqrt[3]{n}$ remains.
Find $n$ . | 32768 | 0.25 |
We are given a convex quadrilateral $ABCD$ in the plane.
(*i*) If there exists a point $P$ in the plane such that the areas of $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$ are equal, what condition must be satisfied by the quadrilateral $ABCD$ ?
(*ii*) Find (with proof) the maximum possible number of such point $P$ which satisfies the condition in (*i*). | 1 | 0.916667 |
Source: 1976 Euclid Part B Problem 2
-----
Given that $x$ , $y$ , and $2$ are in geometric progression, and that $x^{-1}$ , $y^{-1}$ , and $9x^{-2}$ are in are in arithmetic progression, then find the numerical value of $xy$ . | \frac{27}{2} | 0.333333 |
$ P(x)$ is a quadratic trinomial. What maximum number of terms equal to the sum of the two preceding terms can occur in the sequence $ P(1)$ , $ P(2)$ , $ P(3)$ , $ \dots?$
*Proposed by A. Golovanov* | 2 | 0.583333 |
Find all real numbers $a$ for which there exists a non-constant function $f :\Bbb R \to \Bbb R$ satisfying the following two equations for all $x\in \Bbb R:$ i) $f(ax) = a^2f(x)$ and
ii) $f(f(x)) = a f(x).$ | 0, 1 | 0.75 |
Find all triples $(a, b, c)$ of positive integers for which $$ \begin{cases} a + bc=2010 b + ca = 250\end{cases} $$ | (3, 223, 9) | 0.083333 |
Two real numbers $x$ and $y$ are such that $8y^4+4x^2y^2+4xy^2+2x^3+2y^2+2x=x^2+1$ .
Find all possible values of $x+2y^2$ | \frac{1}{2} | 0.416667 |
Find the maximum value of real number $A$ such that $$ 3x^2 + y^2 + 1 \geq A(x^2 + xy + x) $$ for all positive integers $x, y.$ | \frac{5}{3} | 0.833333 |
Compute the determinant of the $n\times n$ matrix $A=(a_{ij})_{ij}$ , $$ a_{ij}=\begin{cases}
(-1)^{|i-j|} & \text{if}\, i\ne j,
2 & \text{if}\, i= j.
\end{cases} $$ | n + 1 | 0.166667 |
Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if $z_1+z_2+z_3=0$ , then $$ \left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda . $$ | \lambda = 1 | 0.333333 |
Given real numbers $a$ and $b$ are chosen with $1<a<b$ such that no triangle with positive area has side lengths $1,a,$ and $b$ or $\tfrac{1}{b}, \tfrac{1}{a},$ and $1$, find the smallest possible value of $b$. | \frac{3 + \sqrt{5}}{2} | 0.583333 |
given a positive integer $n$ .
the set $\{ 1,2,..,2n \}$ is partitioned into $a_1<a_2<...<a_n $ and $b_1>b_2>...>b_n$ .
find the value of : $ \sum_{i=1}^{n}|a_i - b_i| $ | n^2 | 0.666667 |
Given $a,b,c$ are distinct real roots of $x^3-3x+1=0$, calculate the value of $a^8+b^8+c^8$. | 186 | 0.75 |
Let $E(n)$ denote the largest integer $k$ such that $5^k$ divides $1^{1}\cdot 2^{2} \cdot 3^{3} \cdot \ldots \cdot n^{n}.$ Calculate $$ \lim_{n\to \infty} \frac{E(n)}{n^2 }. $$ | \frac{1}{8} | 0.083333 |
Assuming that the roots of $x^3 +px^2 +qx +r=0$ are all real and positive, find the relation between $p,q,r$ which is a necessary and sufficient condition that the roots are the cosines of the angles of a triangle. | p^2 - 2q - 2r - 1 = 0 | 0.166667 |
Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality $$ \left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right) $$ is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx =\alpha.$ When does equality occur?
*(Proposed by Walther Janous)* | 16 | 0.916667 |
The points $P$ and $Q$ are the midpoints of the edges $AE$ and $CG$ on the cube $ABCD.EFGH$ respectively. If the length of the cube edges is $1$ unit, determine the area of the quadrilateral $DPFQ$ . | \frac{\sqrt{6}}{2} | 0.5 |
Find the max. value of $ M$ ,such that for all $ a,b,c>0$ :
$ a^{3}+b^{3}+c^{3}-3abc\geq M(|a-b|^{3}+|a-c|^{3}+|c-b|^{3})$ | \frac{1}{2} | 0.166667 |
The cells of a $2021\times 2021$ table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate $(1, 1)$ , contains the number $0$ . For every other cell $C$ , we consider a route from $(1, 1)$ to $C$ , where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell $C$ . For example, the cell with coordinate $(2, 1)$ contains $1 = 1 + 0$ , the cell with coordinate $(3, 1)$ contains $3 = 2 + 0 + 1$ , and the cell with coordinate $(3, 2)$ contains $7 = 3 + 0 + 1 + 3$ . What is the last digit of the number in the cell $(2021, 2021)$ ? | 5 | 0.083333 |
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 $ABCD$ .
*Proposed by Evan Chen* | 118 | 0.25 |
Given is a chessboard 8x8. We have to place $n$ black queens and $n$ white queens, so that no two queens attack. Find the maximal possible $n$ .
(Two queens attack each other when they have different colors. The queens of the same color don't attack each other) | 8 | 0.666667 |
Given that P be a polynomial with each root is real and each coefficient is either $1$ or $-1$, determine the maximum degree of P. | 3 | 0.416667 |
All the numbers $1,2,...,9$ are written in the cells of a $3\times 3$ table (exactly one number in a cell) . Per move it is allowed to choose an arbitrary $2\times2$ square of the table and either decrease by $1$ or increase by $1$ all four numbers of the square. After some number of such moves all numbers of the table become equal to some number $a$ . Find all possible values of $a$ .
I.Voronovich | 5 | 0.5 |
Determine the number of solutions of the simultaneous equations $ x^2 \plus{} y^3 \equal{} 29$ and $ \log_3 x \cdot \log_2 y \equal{} 1.$ | 2 | 0.416667 |
$201$ positive integers are written on a line, such that both the first one and the last one are equal to $19999$ . Each one of the remaining numbers is less than the average of its neighbouring numbers, and the differences between each one of the remaining numbers and the average of its neighbouring numbers are all equal to a unique integer. Find the second-to-last term on the line. | 19800 | 0.083333 |
Find minimum of $x+y+z$ where $x$ , $y$ and $z$ are real numbers such that $x \geq 4$ , $y \geq 5$ , $z \geq 6$ and $x^2+y^2+z^2 \geq 90$ | 16 | 0.833333 |
Determine all positive real $M$ such that for any positive reals $a,b,c$ , at least one of $a + \dfrac{M}{ab}, b + \dfrac{M}{bc}, c + \dfrac{M}{ca}$ is greater than or equal to $1+M$ . | M = \frac{1}{2} | 0.166667 |
Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6
| x - 2y + 3z| \le 6
| x - 2y - 3z| \le 6
| x + 2y + 3z| \le 6 \end{cases}$ Determine the greatest value of $M = |x| + |y| + |z|$ . | 6 | 0.333333 |
For reals $x\ge3$ , let $f(x)$ denote the function
\[f(x) = \frac {-x + x\sqrt{4x-3} } { 2} .\]Let $a_1, a_2, \ldots$ , be the sequence satisfying $a_1 > 3$ , $a_{2013} = 2013$ , and for $n=1,2,\ldots,2012$ , $a_{n+1} = f(a_n)$ . Determine the value of
\[a_1 + \sum_{i=1}^{2012} \frac{a_{i+1}^3} {a_i^2 + a_ia_{i+1} + a_{i+1}^2} .\]
*Ray Li.* | 4025 | 0.083333 |
For all $x,y,z$ positive real numbers, find the all $c$ positive real numbers that providing $$ \frac{x^3y+y^3z+z^3x}{x+y+z}+\frac{4c}{xyz}\ge2c+2 $$ | c = 1 | 0.666667 |
Find all odd prime numbers $p$ for which there exists a natural number $g$ for which the sets \[A=\left\{ \left( {{k}^{2}}+1 \right)\,\bmod p|\,k=1,2,\ldots ,\frac{p-1}{2} \right\}\] and \[B=\left\{ {{g}^{k}}\bmod \,p|\,k=1,2,...,\frac{p-1}{2} \right\}\] are equal. | p = 3 | 0.083333 |
Find all triples of positive integers $(a,b,c)$ such that the following equations are both true:
I- $a^2+b^2=c^2$ II- $a^3+b^3+1=(c-1)^3$ | (6, 8, 10) | 0.5 |
$N $ different numbers are written on blackboard and one of these numbers is equal to $0$ .One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between $-2016$ and $2016$ were written on blackboard(and some other numbers maybe). Find the smallest possible value of $N $ . | N = 2 | 0.083333 |
For a positive integer $n>1$ , let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n)=n-g(n)$ . For example, $g(10)=5, f(10)=5$ and $g(13)=1,f(13)=12$ . Let $N$ be the smallest positive integer such that $f(f(f(N)))=97$ . Find the largest integer not exceeding $\sqrt{N}$ | 19 | 0.166667 |
Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy \[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\] for every positive integer $n$ . Here, $f^k(n)$ means $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$ .
*Proposed by Bojan Bašić, Serbia* | (f(n) = n, g(n) = 1) | 0.166667 |
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